Graph Wavelets for Spatial Traffic Analysis

19StandardMath ToolsDisplay up to four math function traces (F1-F4). The easy-to-use graphical interface simplifies setup of up to two operations on each function trace;and function traces can be chained together to perform math-on-math.absolute value integralaverage (summed)invert (negate)average (continuous)log (base e)custom (MATLAB) – limited points product (x)derivativeratio (/)deskew (resample)reciprocaldifference (–)rescale (with units)enhanced resolution (to 11 bits vertical)roof envelope (sinx)/x exp (base e)square exp (base 10)square root fft (power spectrum, magnitude, phase,sum (+)up to 50 kpts) trend (datalog) of 1000 events floorzoom (identity)histogram of 1000 eventsMeasure ToolsDisplay any 6 parameters together with statistics, including their average,high, low, and standard deviations. Histicons provide a fast, dynamic view of parameters and wave-shape characteristics.Pass/Fail TestingSimultaneously test multiple parameters against selectable parameter limits or pre-defined masks. Pass or fail conditions can initiate actions including document to local or networked files, e-mail the image of the failure, save waveforms, send a pulse out at the rear panel auxiliary BNC output, or (with the GPIB option) send a GPIB SRQ.Jitter and Timing Analysis Software Package (WRXi-JTA2)(Standard with MXi-A model oscilloscopes)•Jitter and timing parameters, with “Track”graphs of •Edge@lv parameter (counts edges)• Persistence histogram, persistence trace (mean, range, sigma)Software Options –Advanced Math and WaveShape AnalysisStatistics Package (WRXi-STAT)This package provides additional capability to statistically display measurement information and to analyze results:• Histograms expanded with 19 histogram parameters/up to 2 billion events.• Persistence Histogram• Persistence Trace (mean, range, sigma)Master Analysis Software Package (WRXi-XMAP)(Standard with MXi-A model oscilloscopes)This package provides maximum capability and flexibility, and includes all the functionality present in XMATH, XDEV, and JTA2.Advanced Math Software Package (WRXi-XMATH)(Standard with MXi-A model oscilloscopes)This package provides a comprehensive set of WaveShape Analysis tools providing insight into the wave shape of complex signals. Includes:•Parameter math – add, subtract, multiply, or divide two different parameters.Invert a parameter and rescale parameter values.•Histograms expanded with 19 histogram parameters/up to 2 billion events.•Trend (datalog) of up to 1 million events•Track graphs of any measurement parameter•FFT capability includes: power averaging, power density, real and imaginary components, frequency domain parameters, and FFT on up to 24 Mpts.•Narrow-band power measurements •Auto-correlation function •Sparse function• Cubic interpolation functionAdvanced Customization Software Package (WRXi-XDEV)(Standard with MXi-A model oscilloscopes)This package provides a set of tools to modify the scope and customize it to meet your unique needs. Additional capability provided by XDEV includes:•Creation of your own measurement parameter or math function, using third-party software packages, and display of the result in the scope. Supported third-party software packages include:– VBScript – MATLAB – Excel•CustomDSO – create your own user interface in a scope dialog box.• Addition of macro keys to run VBScript files •Support for plug-insValue Analysis Software Package (WRXi-XVAP)(Standard with MXi-A model oscilloscopes)Measurements:•Jitter and Timing parameters (period@level,width@level, edge@level,duty@level, time interval error@level, frequency@level, half period, setup, skew, Δ period@level, Δ width@level).Math:•Persistence histogram •Persistence trace (mean, sigma, range)•1 Mpts FFTs with power spectrum density, power averaging, real, imaginary, and real+imaginary settings)Statistical and Graphical Analysis•1 Mpts Trends and Histograms •19 histogram parameters •Track graphs of any measurement parameterIntermediate Math Software Package (WRXi-XWAV)Math:•1 Mpts FFTs with power spectrum density, power averaging, real, and imaginary componentsStatistical and Graphical Analysis •1 Mpts Trends and Histograms •19 histogram parameters•Track graphs of any measurement parameteramplitude area base cyclescustom (MATLAB,VBScript) –limited points delay Δdelay duration duty cyclefalltime (90–10%, 80–20%, @ level)firstfrequency lastlevel @ x maximum mean median minimumnumber of points +overshoot –overshoot peak-to-peak period phaserisetime (10–90%, 20–80%, @ level)rmsstd. deviation time @ level topΔ time @ levelΔ time @ level from triggerwidth (positive + negative)x@ max.x@ min.– Cycle-Cycle Jitter – N-Cycle– N-Cycle with start selection – Frequency– Period – Half Period – Width– Time Interval Error – Setup– Hold – Skew– Duty Cycle– Duty Cycle Error20WaveRunner WaveRunner WaveRunner WaveRunner WaveRunner 44Xi-A64Xi-A62Xi-A104Xi-A204Xi-AVertical System44MXi-A64MXi-A104MXi-A204MXi-ANominal Analog Bandwidth 400 MHz600 MHz600 MHz 1 GHz 2 GHz@ 50 Ω, 10 mV–1 V/divRise Time (Typical)875 ps500 ps500 ps300 ps180 psInput Channels44244Bandwidth Limiters20 MHz; 200 MHzInput Impedance 1 MΩ||16 pF or 50 Ω 1 MΩ||20 pF or 50 ΩInput Coupling50 Ω: DC, 1 MΩ: AC, DC, GNDMaximum Input Voltage50 Ω: 5 V rms, 1 MΩ: 400 V max.50 Ω: 5 V rms, 1 MΩ: 250 V max.(DC + Peak AC ≤ 5 kHz)(DC + Peak AC ≤ 10 kHz)Vertical Resolution8 bits; up to 11 with enhanced resolution (ERES)Sensitivity50 Ω: 2 mV/div–1 V/div fully variable; 1 MΩ: 2 mV–10 V/div fully variableDC Gain Accuracy±1.0% of full scale (typical); ±1.5% of full scale, ≥ 10 mV/div (warranted)Offset Range50 Ω: ±1 V @ 2–98 mV/div, ±10 V @ 100 mV/div–1 V/div; 50Ω:±400mV@2–4.95mV/div,±1V@5–99mv/div,1 M Ω: ±1 V @ 2–98 mV/div, ±10 V @ 100 mV/div–1 V/div,±10 V @ 100 mV–1 V/div±**********/div–10V/div 1 M Ω: ±400 mV @ 2–4.95 mV/div, ±1 V @5–99 mV/div, ±10 V @ 100 mV–1 V/div,±*********–10V/divInput Connector ProBus/BNCTimebase SystemTimebases Internal timebase common to all input channels; an external clock may be applied at the auxiliary inputTime/Division Range Real time: 200 ps/div–10 s/div, RIS mode: 200 ps/div to 10 ns/div, Roll mode: up to 1,000 s/divClock Accuracy≤ 5 ppm @ 25 °C (typical) (≤ 10 ppm @ 5–40 °C)Sample Rate and Delay Time Accuracy Equal to Clock AccuracyChannel to Channel Deskew Range±9 x time/div setting, 100 ms max., each channelExternal Sample Clock DC to 600 MHz; (DC to 1 GHz for 104Xi-A/104MXi-A and 204Xi-A/204MXi-A) 50 Ω, (limited BW in 1 MΩ),BNC input, limited to 2 Ch operation (1 Ch in 62Xi-A), (minimum rise time and amplitude requirements applyat low frequencies)Roll Mode User selectable at ≥ 500 ms/div and ≤100 kS/s44Xi-A64Xi-A62Xi-A104Xi-A204Xi-A Acquisition System44MXi-A64MXi-A104MXi-A204MXi-ASingle-Shot Sample Rate/Ch 5 GS/sInterleaved Sample Rate (2 Ch) 5 GS/s10 GS/s10 GS/s10 GS/s10 GS/sRandom Interleaved Sampling (RIS)200 GS/sRIS Mode User selectable from 200 ps/div to 10 ns/div User selectable from 100 ps/div to 10 ns/div Trigger Rate (Maximum) 1,250,000 waveforms/secondSequence Time Stamp Resolution 1 nsMinimum Time Between 800 nsSequential SegmentsAcquisition Memory Options Max. Acquisition Points (4 Ch/2 Ch, 2 Ch/1 Ch in 62Xi-A)Segments (Sequence Mode)Standard12.5M/25M10,00044Xi-A64Xi-A62Xi-A104Xi-A204Xi-A Acquisition Processing44MXi-A64MXi-A104MXi-A204MXi-ATime Resolution (min, Single-shot)200 ps (5 GS/s)100 ps (10 GS/s)100 ps (10 GS/s)100 ps (10 GS/s)100 ps (10 GS/s) Averaging Summed and continuous averaging to 1 million sweepsERES From 8.5 to 11 bits vertical resolutionEnvelope (Extrema)Envelope, floor, or roof for up to 1 million sweepsInterpolation Linear or (Sinx)/xTrigger SystemTrigger Modes Normal, Auto, Single, StopSources Any input channel, External, Ext/10, or Line; slope and level unique to each source, except LineTrigger Coupling DC, AC (typically 7.5 Hz), HF Reject, LF RejectPre-trigger Delay 0–100% of memory size (adjustable in 1% increments, or 100 ns)Post-trigger Delay Up to 10,000 divisions in real time mode, limited at slower time/div settings in roll modeHold-off 1 ns to 20 s or 1 to 1,000,000,000 events21WaveRunner WaveRunner WaveRunner WaveRunner WaveRunner 44Xi-A 64Xi-A 62Xi-A104Xi-A 204Xi-A Trigger System (cont’d)44MXi-A64MXi-A104MXi-A204MXi-AInternal Trigger Level Range ±4.1 div from center (typical)Trigger and Interpolator Jitter≤ 3 ps rms (typical)Trigger Sensitivity with Edge Trigger 2 div @ < 400 MHz 2 div @ < 600 MHz 2 div @ < 600 MHz 2 div @ < 1 GHz 2 div @ < 2 GHz (Ch 1–4 + external, DC, AC, and 1 div @ < 200 MHz 1 div @ < 200 MHz 1 div @ < 200 MHz 1 div @ < 200 MHz 1 div @ < 200 MHz LFrej coupling)Max. Trigger Frequency with400 MHz 600 MHz 600 MHz 1 GHz2 GHzSMART Trigger™ (Ch 1–4 + external)@ ≥ 10 mV@ ≥ 10 mV@ ≥ 10 mV@ ≥ 10 mV@ ≥ 10 mVExternal Trigger RangeEXT/10 ±4 V; EXT ±400 mVBasic TriggersEdgeTriggers when signal meets slope (positive, negative, either, or Window) and level conditionTV-Composite VideoT riggers NTSC or PAL with selectable line and field; HDTV (720p, 1080i, 1080p) with selectable frame rate (50 or 60 Hz)and Line; or CUSTOM with selectable Fields (1–8), Lines (up to 2000), Frame Rates (25, 30, 50, or 60 Hz), Interlacing (1:1, 2:1, 4:1, 8:1), or Synch Pulse Slope (Positive or Negative)SMART TriggersState or Edge Qualified Triggers on any input source only if a defined state or edge occurred on another input source.Delay between sources is selectable by time or eventsQualified First In Sequence acquisition mode, triggers repeatedly on event B only if a defined pattern, state, or edge (event A) is satisfied in the first segment of the acquisition. Delay between sources is selectable by time or events Dropout Triggers if signal drops out for longer than selected time between 1 ns and 20 s.PatternLogic combination (AND, NAND, OR, NOR) of 5 inputs (4 channels and external trigger input – 2 Ch+EXT on WaveRunner 62Xi-A). Each source can be high, low, or don’t care. The High and Low level can be selected independently. Triggers at start or end of the patternSMART Triggers with Exclusion TechnologyGlitch and Pulse Width Triggers on positive or negative glitches with widths selectable from 500 ps to 20 s or on intermittent faults (subject to bandwidth limit of oscilloscope)Signal or Pattern IntervalTriggers on intervals selectable between 1 ns and 20 sTimeout (State/Edge Qualified)Triggers on any source if a given state (or transition edge) has occurred on another source.Delay between sources is 1 ns to 20 s, or 1 to 99,999,999 eventsRuntTrigger on positive or negative runts defined by two voltage limits and two time limits. Select between 1 ns and 20 sSlew RateTrigger on edge rates. Select limits for dV, dt, and slope. Select edge limits between 1 ns and 20 s Exclusion TriggeringTrigger on intermittent faults by specifying the normal width or periodLeCroy WaveStream Fast Viewing ModeIntensity256 Intensity Levels, 1–100% adjustable via front panel control Number of Channels up to 4 simultaneouslyMax Sampling Rate5 GS/s (10 GS/s for WR 62Xi-A, 64Xi-A/64MXi-A,104Xi-A/104MXi-A, 204Xi-A/204MXi-A in interleaved mode)Waveforms/second (continuous)Up to 20,000 waveforms/secondOperationFront panel toggle between normal real-time mode and LeCroy WaveStream Fast Viewing modeAutomatic SetupAuto SetupAutomatically sets timebase, trigger, and sensitivity to display a wide range of repetitive signalsVertical Find ScaleAutomatically sets the vertical sensitivity and offset for the selected channels to display a waveform with maximum dynamic range44Xi-A 64Xi-A 62Xi-A104Xi-A 204Xi-A Probes44MXi-A 64MXi-A104MXi-A 204MXi-AProbesOne Passive probe per channel; Optional passive and active probes available Probe System; ProBus Automatically detects and supports a variety of compatible probes Scale FactorsAutomatically or manually selected, depending on probe usedColor Waveform DisplayTypeColor 10.4" flat-panel TFT-LCD with high resolution touch screenResolutionSVGA; 800 x 600 pixels; maximum external monitor output resolution of 2048 x 1536 pixelsNumber of Traces Display a maximum of 8 traces. Simultaneously display channel, zoom, memory, and math traces Grid StylesAuto, Single, Dual, Quad, Octal, XY , Single + XY , Dual + XY Waveform StylesSample dots joined or dots only in real-time mode22Zoom Expansion TracesDisplay up to 4 Zoom/Math traces with 16 bits/data pointInternal Waveform MemoryM1, M2, M3, M4 Internal Waveform Memory (store full-length waveform with 16 bits/data point) or store to any number of files limited only by data storage mediaSetup StorageFront Panel and Instrument StatusStore to the internal hard drive, over the network, or to a USB-connected peripheral deviceInterfaceRemote ControlVia Windows Automation, or via LeCroy Remote Command Set Network Communication Standard VXI-11 or VICP , LXI Class C Compliant GPIB Port (Accessory)Supports IEEE – 488.2Ethernet Port 10/100/1000Base-T Ethernet interface (RJ-45 connector)USB Ports5 USB 2.0 ports (one on front of instrument) supports Windows-compatible devices External Monitor Port Standard 15-pin D-Type SVGA-compatible DB-15; connect a second monitor to use extended desktop display mode with XGA resolution Serial PortDB-9 RS-232 port (not for remote oscilloscope control)44Xi-A 64Xi-A 62Xi-A104Xi-A 204Xi-A Auxiliary Input44MXi-A 64MXi-A104MXi-A 204MXi-ASignal Types Selected from External Trigger or External Clock input on front panel Coupling50 Ω: DC, 1 M Ω: AC, DC, GND Maximum Input Voltage50 Ω: 5 V rms , 1 M Ω: 400 V max.50 Ω: 5 V rms , 1 M Ω: 250 V max. (DC + Peak AC ≤ 5 kHz)(DC + Peak AC ≤ 10 kHz)Auxiliary OutputSignal TypeTrigger Enabled, Trigger Output. Pass/Fail, or Off Output Level TTL, ≈3.3 VConnector TypeBNC, located on rear panelGeneralAuto Calibration Ensures specified DC and timing accuracy is maintained for 1 year minimumCalibratorOutput available on front panel connector provides a variety of signals for probe calibration and compensationPower Requirements90–264 V rms at 50/60 Hz; 115 V rms (±10%) at 400 Hz, Automatic AC Voltage SelectionInstallation Category: 300 V CAT II; Max. Power Consumption: 340 VA/340 W; 290 VA/290 W for WaveRunner 62Xi-AEnvironmentalTemperature: Operating+5 °C to +40 °C Temperature: Non-Operating -20 °C to +60 °CHumidity: Operating Maximum relative humidity 80% for temperatures up to 31 °C decreasing linearly to 50% relative humidity at 40 °CHumidity: Non-Operating 5% to 95% RH (non-condensing) as tested per MIL-PRF-28800F Altitude: OperatingUp to 3,048 m (10,000 ft.) @ ≤ 25 °C Altitude: Non-OperatingUp to 12,190 m (40,000 ft.)PhysicalDimensions (HWD)260 mm x 340 mm x 152 mm Excluding accessories and projections (10.25" x 13.4" x 6")Net Weight7.26kg. (16.0lbs.)CertificationsCE Compliant, UL and cUL listed; Conforms to EN 61326, EN 61010-1, UL 61010-1 2nd Edition, and CSA C22.2 No. 61010-1-04Warranty and Service3-year warranty; calibration recommended annually. Optional service programs include extended warranty, upgrades, calibration, and customization services23Product DescriptionProduct CodeWaveRunner Xi-A Series Oscilloscopes2 GHz, 4 Ch, 5 GS/s, 12.5 Mpts/ChWaveRunner 204Xi-A(10 GS/s, 25 Mpts/Ch in interleaved mode)with 10.4" Color Touch Screen Display 1 GHz, 4 Ch, 5 GS/s, 12.5 Mpts/ChWaveRunner 104Xi-A(10 GS/s, 25 Mpts/Ch in interleaved mode)with 10.4" Color Touch Screen Display 600 MHz, 4 Ch, 5 GS/s, 12.5 Mpts/Ch WaveRunner 64Xi-A(10 GS/s, 25 Mpts/Ch in interleaved mode)with 10.4" Color Touch Screen Display 600 MHz, 2 Ch, 5 GS/s, 12.5 Mpts/Ch WaveRunner 62Xi-A(10 GS/s, 25 Mpts/Ch in interleaved mode)with 10.4" Color Touch Screen Display 400 MHz, 4 Ch, 5 GS/s, 12.5 Mpts/Ch WaveRunner 44Xi-A(25 Mpts/Ch in interleaved mode)with 10.4" Color Touch Screen DisplayWaveRunner MXi-A Series Oscilloscopes2 GHz, 4 Ch, 5 GS/s, 12.5 Mpts/ChWaveRunner 204MXi-A(10 GS/s, 25 Mpts/Ch in Interleaved Mode)with 10.4" Color Touch Screen Display 1 GHz, 4 Ch, 5 GS/s, 12.5 Mpts/ChWaveRunner 104MXi-A(10 GS/s, 25 Mpts/Ch in Interleaved Mode)with 10.4" Color Touch Screen Display 600 MHz, 4 Ch, 5 GS/s, 12.5 Mpts/Ch WaveRunner 64MXi-A(10 GS/s, 25 Mpts/Ch in Interleaved Mode)with 10.4" Color Touch Screen Display 400 MHz, 4 Ch, 5 GS/s, 12.5 Mpts/Ch WaveRunner 44MXi-A(25 Mpts/Ch in Interleaved Mode)with 10.4" Color Touch Screen DisplayIncluded with Standard Configuration÷10, 500 MHz, 10 M Ω Passive Probe (Total of 1 Per Channel)Standard Ports; 10/100/1000Base-T Ethernet, USB 2.0 (5), SVGA Video out, Audio in/out, RS-232Optical 3-button Wheel Mouse – USB 2.0Protective Front Cover Accessory PouchGetting Started Manual Quick Reference GuideAnti-virus Software (Trial Version)Commercial NIST Traceable Calibration with Certificate 3-year WarrantyGeneral Purpose Software OptionsStatistics Software Package WRXi-STAT Master Analysis Software Package WRXi-XMAP (Standard with MXi-A model oscilloscopes)Advanced Math Software Package WRXi-XMATH (Standard with MXi-A model oscilloscopes)Intermediate Math Software Package WRXi-XWAV (Standard with MXi-A model oscilloscopes)Value Analysis Software Package (Includes XWAV and JTA2) WRXi-XVAP (Standard with MXi-A model oscilloscopes)Advanced Customization Software Package WRXi-XDEV (Standard with MXi-A model oscilloscopes)Spectrum Analyzer and Advanced FFT Option WRXi-SPECTRUM Processing Web Editor Software Package WRXi-XWEBProduct Description Product CodeApplication Specific Software OptionsJitter and Timing Analysis Software Package WRXi-JTA2(Standard with MXi-A model oscilloscopes)Digital Filter Software PackageWRXi-DFP2Disk Drive Measurement Software Package WRXi-DDM2PowerMeasure Analysis Software Package WRXi-PMA2Serial Data Mask Software PackageWRXi-SDM QualiPHY Enabled Ethernet Software Option QPHY-ENET*QualiPHY Enabled USB 2.0 Software Option QPHY-USB †EMC Pulse Parameter Software Package WRXi-EMC Electrical Telecom Mask Test PackageET-PMT* TF-ENET-B required. †TF-USB-B required.Serial Data OptionsI 2C Trigger and Decode Option WRXi-I2Cbus TD SPI Trigger and Decode Option WRXi-SPIbus TD UART and RS-232 Trigger and Decode Option WRXi-UART-RS232bus TD LIN Trigger and Decode Option WRXi-LINbus TD CANbus TD Trigger and Decode Option CANbus TD CANbus TDM Trigger, Decode, and Measure/Graph Option CANbus TDM FlexRay Trigger and Decode Option WRXi-FlexRaybus TD FlexRay Trigger and Decode Physical Layer WRXi-FlexRaybus TDP Test OptionAudiobus Trigger and Decode Option WRXi-Audiobus TDfor I 2S , LJ, RJ, and TDMAudiobus Trigger, Decode, and Graph Option WRXi-Audiobus TDGfor I 2S LJ, RJ, and TDMMIL-STD-1553 Trigger and Decode Option WRXi-1553 TDA variety of Vehicle Bus Analyzers based on the WaveRunner Xi-A platform are available.These units are equipped with a Symbolic CAN trigger and decode.Mixed Signal Oscilloscope Options500 MHz, 18 Ch, 2 GS/s, 50 Mpts/Ch MS-500Mixed Signal Oscilloscope Option 250 MHz, 36 Ch, 1 GS/s, 25 Mpts/ChMS-500-36(500 MHz, 18 Ch, 2 GS/s, 50 Mpts/Ch Interleaved) Mixed Signal Oscilloscope Option 250 MHz, 18 Ch, 1 GS/s, 10 Mpts/Ch MS-250Mixed Signal Oscilloscope OptionProbes and Amplifiers*Set of 4 ZS1500, 1.5 GHz, 0.9 pF , 1 M ΩZS1500-QUADPAK High Impedance Active ProbeSet of 4 ZS1000, 1 GHz, 0.9 pF , 1 M ΩZS1000-QUADPAK High Impedance Active Probe 2.5 GHz, 0.7 pF Active Probe HFP25001 GHz Active Differential Probe (÷1, ÷10, ÷20)AP034500 MHz Active Differential Probe (x10, ÷1, ÷10, ÷100)AP03330 A; 100 MHz Current Probe – AC/DC; 30 A rms ; 50 A rms Pulse CP03130 A; 50 MHz Current Probe – AC/DC; 30 A rms ; 50 A rms Pulse CP03030 A; 50 MHz Current Probe – AC/DC; 30 A rms ; 50 A peak Pulse AP015150 A; 10 MHz Current Probe – AC/DC; 150 A rms ; 500 A peak Pulse CP150500 A; 2 MHz Current Probe – AC/DC; 500 A rms ; 700 A peak Pulse CP5001,400 V, 100 MHz High-Voltage Differential Probe ADP3051,400 V, 20 MHz High-Voltage Differential Probe ADP3001 Ch, 100 MHz Differential Amplifier DA1855A*A wide variety of other passive, active, and differential probes are also available.Consult LeCroy for more information.Product Description Product CodeHardware Accessories*10/100/1000Base-T Compliance Test Fixture TF-ENET-B †USB 2.0 Compliance Test Fixture TF-USB-B External GPIB Interface WS-GPIBSoft Carrying Case WRXi-SOFTCASE Hard Transit CaseWRXi-HARDCASE Mounting Stand – Desktop Clamp Style WRXi-MS-CLAMPRackmount Kit WRXi-RACK Mini KeyboardWRXi-KYBD Removable Hard Drive Package (Includes removeable WRXi-A-RHD hard drive kit and two hard drives)Additional Removable Hard DriveWRXi-A-RHD-02* A variety of local language front panel overlays are also available .† Includes ENET-2CAB-SMA018 and ENET-2ADA-BNCSMA.Customer ServiceLeCroy oscilloscopes and probes are designed, built, and tested to ensure high reliability. In the unlikely event you experience difficulties, our digital oscilloscopes are fully warranted for three years, and our probes are warranted for one year.This warranty includes:• No charge for return shipping • Long-term 7-year support• Upgrade to latest software at no chargeLocal sales offices are located throughout the world. Visit our website to find the most convenient location.© 2010 by LeCroy Corporation. All rights reserved. Specifications, prices, availability, and delivery subject to change without notice. Product or brand names are trademarks or requested trademarks of their respective holders.1-800-5-LeCroy WRXi-ADS-14Apr10PDF。

设计应用技术ＤＯＩ：１０．１９３９９／ｊ．ｃｎｋｉ．ｔｐｔ．２０２３．０２．０１９STN网络建设策略分析王文华（广西通信规划设计咨询有限公司，广西南宁530007）摘要：随着5G技术的不断发展，我国5G技术已经从科研阶段正式朝着民用阶段拓展。

在商业化部署过程中，需要重点考虑的问题之一是空间变换网络（Spatial Transformer Network，STN）的建设方法。

5G网络本身具有大带宽、低时延、多连接等特点，支持不同运营商之间的共同建设。

在进行STN回传网络构架建设时，要充分结合5G网络的特点。

重点阐述了骨干层、汇聚层以及接入层的多阶段架构模式，对5G基站流量测算模型和低延时解决方案进行探讨，并对4类不同前传引入段光缆的场景进行了综合造价分析。

关键词：空间变换网络（STN）；共建共享；流量测算；回传网络Analysis of STN Network Construction StrategyWANG Wenhua(Guangxi Communication Planning and Design Consulting Co., Ltd., Nanning 530007, China) Abstract: With the continuous development of 5G technology, China's 5G technology has officially expanded from the scientific research stage to the civilian stage. In the process of commercial deployment, one of the key issues to be considered is the construction method of Spatial Transformer Network (STN). The 5G network itself has the characteristics of large bandwidth, low delay and multi-connection, which supports the joint construction between different operators. When constructing STN backhaul network architecture, we should fully combine the characteristics of 5G network. The multi-stage architecture model of backbone layer, convergence layer layer and access layer is emphasized, the traffic measurement model and low-delay solution of 5G base station are discussed, and the comprehensive cost analysis of four different scenarios of fiber optic cables is made.Keywords: Spatial Transformer Network (STN); building and sharing; flow measurement; return network0 引 言在国家整体规划中，要求积极推进5G通信技术和超宽带技术的发展，5G技术逐渐向商用领域深入渗透。

Graph Wavelets for Spatial Traffic AnalysisBUCS-TR-2002-020Mark Crovella Eric KolaczykDepartment of Computer Science Department of Math and Statistics Boston University Boston UniversityBoston MA02215Boston MA02215July15,2002AbstractA number of problems in network operations and engineering call for new methods of traf-fic analysis.While most existing traffic analysis methods are fundamentally temporal,there isa clear need for the analysis of traffic across multiple network links—that is,for spatial trafficanalysis.In this paper we give examples of problems that can be addressed via spatial trafficanalysis.We then propose a formal approach to spatial traffic analysis based on the wavelettransform.Our approach(graph wavelets)generalizes the traditional wavelet transform so thatit can be applied to data elements connected via an arbitrary graph topology.We explore thenecessary and desirable properties of this approach and consider some of its possible realiza-tions.We then apply graph wavelets to measurements from an operating network.Our resultsshow that graph wavelets are very useful for our motivating problems;for example,they canbe used to form highly summarized views of an entire network’s traffic load,to gain insightinto a network’s global traffic response to a link failure,and to localize the extent of a failureevent within the network.This work was supported in part by NSF awards ANI-9986397and ANI-0095988,and by ONR award N00014-99-1-0219.1IntroductionTo date,the traffic analysis tools developed in the research community and the traffic analysis needs of network engineers and operators have been somewhat disconnected.Most research on traffic analysis has focused on the properties of trafficflowing over individual links,treated as a timeseries[1].However,network engineers and operators are very often more concerned with the properties of traffic over multiple links,or whole networks.In fact,there are many network engineering challenges that could be aided by better tools for traffic analysis.For example,traffic properties play a central role in1)provisioning and capacity planning;2)network configuration and traffic engineering;3)failure detection and diagnosis;and 4)attack detection and prevention.However,traffic analysis tools and methods focused on these kinds of problems are generally not well developed.As a result,many network operators and engineers are forced to address these problems manually or via ad-hoc tools.A common thread running through these problems is the importance of comparison and anal-ysis of traffic patterns across multiple,or all,network links simultaneously.We call this spatial traffic analysis.1Spatial traffic analysis seeks to answer questions about traffic patterns in and between“regions”—topologically localized sets of links—of a network.For example,traffic engineering can be aided by summarizations of average traffic in different network regions;and failure and attack detection can be aided by comparisons of traffic across different network regions.Providing useful,practical tools for spatial traffic analysis is difficult.Two problems arise:first,the large quantity and high dimensionality of the data involved is unmanageable without methods for efficient andflexible data summarization.Second,algorithms must be developed that correctly and intelligently make use of such summaries for the solution of network engineering problems.Given the many degrees of freedom introduced by the wealth of data,such algorithms are not immediately obvious.Good solutions for these two problems are interrelated,because each influences the other.In this paper we present new techniques for spatial traffic analysis.These techniques are based on explicit consideration of network topology;we believe that effective network engineering must consider both the traffic properties on the network’s links and the manner in which the links are connected.Thus our approach is intended to support a whole-network view of data traffic.To enable this view,we develop a new set of formal tools based on wavelet analysis.Our principal insight in this thrust is that traditional wavelet analysis can be generalized for use on data elements connected via an arbitrary graph topology,leading to discrete-space analogues of the well-known wavelet transform.That is,in contrast to the traditional use of wavelets in traffic analysis,we apply wavelets to the spatial domain rather than the temporal domain.In this paper we show one way to accomplish this,and we develop a formal framework for what we call graph wavelets.Graph wavelets are quite general andflexible,and we explore some of the variations that are possible.Using measurements taken from an operating network(Abilene[2])we show that graph wavelets can provide considerable leverage on whole-network traffic analysis.We show how graph wavelets can be used to form highly summarized views of an entire network’s traffic load;how they can be 1More accurately,we might instead write topological to distinguish between this context and that in which the actual spatial geography of the network is incorporated into the analysis.However we use spatial to emphasize the similarity in spirit of our methods to those in the spatial analysis domain.1used to gain insight into a network’s global traffic response to a link failure;and how they can help localize the extent of a failure event.The examples in this paper use link counts available from routers via SNMP.However the methods are general enough to apply to other kinds of measurements associated with the network graph:for example,to study spatial correlation patterns in packet loss.Furthermore,the methods apply equally well to measurements associated with a graph’s edges(links)or vertices(routers).The remainder of the paper is organized as follows.In Section2we review related work.Then in Section3we describe example motivating problems,and we present an informal introduction to graph wavelets as tools for addressing those problems.Section4lays out the formal aspects of graph wavelets:their definition and certain basic properties.Section5then presents detailed examples of how graph wavelets shed light on the nature of measurements taken from the Abilene network.Finally,in Section6we conclude.2Background and Related WorkThe vast majority of research into network traffic analysis has focused on the behavior of traffic on individual links over time.This approach has yielded a number of insights.Most salient among these are observations about the time scaling behavior of traffic:self-similarity and long-range dependence,[3,4];and multifractality and local scaling[5,6].Many of the key results in traffic time scaling analysis have been facilitated by the use of powerful tools,in particular the techniques of wavelet analysis(e.g.,[7]).These temporal traffic analyses have been quite successful in illustrating the presence of sur-prising patterns in the trafficflowing over individual links.However remarkably little research has sought to investigate whether traffic patterns are discernable across multiple links.A similar trend has taken place in the development of network anomaly detection systems.The authors in[8]propose that an anomaly detection system should:1.summarize the nature of typical network conditions in some compact set of metrics,2.continually compare current conditions to the typical metrics,and3.draw operator attention to deviations from typical conditions in as precise and informativemanner as possible.In a style similar to time scaling analysis,work in anomaly detection has generally approached the summarization task in step1from a single-link,temporal analysis standpoint–for example,[9].These timeseries-based approaches to anomaly detection have also made use of wavelet anal-ysis.An example is[10],which explores the diagnostic utility of the traffic energy function;this function is easily obtained using multiresolution analysis.Another approach applying wavelet analysis to anomaly detection is[11],which focuses on analysis of trafficflow measurements.In contrast,the approach we take to anomaly detection—and traffic analysis generally—focuses on the spatial domain:that is,the relationship between traffic on topologically related links.In that sense our work bears a relation to[11],which shows that comparing trafficflows in the incoming and outgoing directions of an access link is useful for identifying anomalies such as denial of service attacks andflash crowd behavior.2Generally speaking,infields ranging from image processing to geography,experience has found that scale is a concept of fundamental importance to the analysis of spatially indexed data. We will argue in Section III(and throughout this paper)that many of the spatial challenges faced by network engineers similarly involve scale in some essential fashion.And therefore our emphasis here is on methods for the multiscale analysis2of spatial network data.The image processing literature arguably has to date the most well known and technically de-veloped body of multiscale analysis techniques for spatial data.Modern representatives from this body perhaps can be said to begin with early work on Laplacian pyramidfiltering[13](which itself formalized still earlier ideas in image processing and computer vision),which was soon fol-lowed and replaced by the current paradigm based on two-dimensional wavelets and their exten-sions.Wavelet-based tools have had a fundamental impact on a variety of standard tasks in image processing,including compression(witness their key role in the JPEG2000standard),denoising, segmentation,and classification.However,wavelet-based methods for images are not immediately applicable to the analysis of spatial network data,for the simple reason that the former are designed for standard topologies and not arbitrary network topologies.On the other hand,there has been recent success in extending the basic wavelet framework to non-standard topologies(e.g.,[14,15,16]),although none of this work so far is relevant to network analysis.What is needed is a notion of wavelets on graphs, which we develop in detail in this paper.3Motivation and ApproachIn this section we provide a more detailed motivation for the notion of spatial traffic analysis,and complement that with an informal illustration of our approach to the problem.3.1Spatial Traffic Analysis:MotivationA number of example problems in network engineering and operations will serve to highlight the need for a whole-network approach to traffic analysis.1.Identifying Spatial Aspects of Typical Operating ConditionsAn important problem facing network operators and engineers is the identification and inter-pretation of typical operating conditions.This is fundamentally a whole-network problem.As an example,spatial analysis plays a role in traffic engineering.The goal of traffic engi-neering is to assign trafficflows to network paths in a way that meets some design criterion.One commonly sought criterion can be load balancing across the network.Engineers may seek to balance load so as to minimize the effects of any single link failure,or to minimize the utilization of the busiest links.2Multiscale analysis,as used in the various literatures,refers simply to the analysis of a given object(s)at multiple scales.While the term“multiresolution analysis”sometimes is used interchangeably with“multiscale analysis,”the former has a specific technical meaning in the mathematics and engineering communities(referring to a sequence of successive approximation spaces,as developed originally by Mallat and Meyer–e.g.,see[12]).The latter more accurately describes the perspective and contributions in this paper and will be used throughout.3A valuable precursor to load balancing is an understanding of which network regions arecarrying the most load,and which regions are relatively less utilized.Armed with this knowledge,network engineers can make high-level,qualitative decisions about the intended outcome of route changes when performing load balancing.Summary information about traffic loads over varying network locations and region sizes provides help in making these kinds of traffic engineering decisions.2.Understanding Shifts in Traffic PatternsA related goal involves understanding the shifts in traffic patterns as a result of traffic engi-neering decisions or network equipment failures.Some networks are engineered with sufficient bandwidth for“protection,”i.e.,so that traffic shifts due to equipment failures can be absorbed without manual intervention in the routing system.In contrast,some networks are provisioned with the expectation that equipment fail-ures will be addressed through explicit traffic engineering actions.In each case,it is essential to have a whole-network view of how traffic patterns shift when equipment fails or traffic is manually re-routed.This whole-network view must provide quantitative information about which regions of the network experienced increased load and which experienced decreased load as a result of the network event.3.Identifying Regions Exhibiting Traffic AnomaliesWhen traffic exhibits unusual characteristics,an immediate and fundamental question con-cerns the size and extent of the region over which the anomaly occurs.For example,if observed traffic load increases to an unusual level,this may be due to a num-ber of factors.Traffic throughout the network may have risen,due to some external driver of increased demand such as a breaking news story.Alternatively,traffic in a localized network region may be increased due to aflash crowd effect(publication of a popular video or report that drives traffic to a single location).Finally,traffic load may be due to a particular pair of hosts engaging in abnormally high traffic.These three scenarios are primarily distinguished by the size of the“neighborhood”over which the anomalously high traffic is observed,and they each demand a different response from network operators.As another example,rapidly detecting denial of service(DoS)attacks is crucial for respon-sive network management.Unfortunately,increased traffic on a single link is not a good indicator of the presence or nature of a DoS attack.Most DoS attacks are distributed,with flooding packets arriving from multiple sources along multiple paths.Accurate identifica-tion of a distributed DoS attack using traffic counts requires the simultaneous assessment of traffic on multiple links of the network.3.2Spatial Traffic Analysis:ApproachThe problems just described all concern questions about one or more“regions”or“neighborhoods”within a given network.To place our discussion of network neighborhoods in a formal setting we consider the graph that is isomorphic to the network as follows:routers or switches in the network correspond to vertices in the graph;and communication links in the network correspond to edges in the graph.We will call the collection of routers and links the network and the corresponding4Figure1:Example Network:Abilenegraph the network graph.Furthermore,we will reserve the terms“links”and“routers”for the network elements and the terms“edges”and“vertices”for the graph elements.3 Furthermore,another graph will be important:the network line graph.For any given graphits corresponding line graph is defined such that and there is an edge in for each pair of edges in that share a common endpoint;i.e.,The network line graph is the line graph of the network graph.The two kinds of graphs are both useful because in a network,certain measurements are as-sociated with the routers,and certain measurements are associated with the links.When we are concerned with comparing measurements associated with routers,then we will be concerned with the adjacency relationships of routers,and so with the network graph.However when we are con-cerned with measurements associated with links(as will be the case in all of the examples in this paper)we will be concerned with the adjacency relationships of links,and so our analyses will involve the network line graph.All of the numerical results that follow in this paper will be based on network line graphs.However,our graph wavelets are defined for arbitrary connected graphs.Our examples in this paper use the network shown in Figure1.This is a map of the Abilene network,which is the network supporting Internet2(this image is from[2]).4This network has11 routers and14links.The corresponding line graph(not shown)has14vertices and23edges.Armed with these tools,we can begin to explore methods for analyzing measurements with respect to network neighborhoods.In the remainder of this section we present an intuitive view of our proposed approach.A formal,rigorous development is deferred to the next section.For purposes of discussion here,let us define the zero-hop neighborhood of a link as the link itself.The one-hop neighborhood is the link,and the set of all links that can be reached in one hop;that is,by traversing a single edge in the network line graph.The two-hop neighborhood of a link is its one-hop neighborhood and all the links that can be reached from any link in that neighborhood in one additional hop,and so on.3We will consider edges in the graph to be undirected.This is a simplification,and we discuss some implications of this simplification below.4We have omitted one link from thisfigure for which we have no data.Internet2is a project developing new network applications and technologies;it has built and uses the Abilene network for testing and deploying these experimental systems.All of the links shown are OC-48,running at2.48Gbps.5Consider the NYC-Cleveland link.Its one-hop neighborhood consists of the three links fromIndianapolis to Washington DC,and its two-hop neighborhood consists of those three links plusthe Indianapolis-Kansas City,Indianapolis-Atlanta,and Washington DC-Atlanta links.The central idea in our approach to spatial network data analysis is the comparison of metricsbetween neighborhoods.For example,for any given link and metric,we might define a comparisonat level(where,for convenience,is an even number)as the average of that metric over all linksin the-hop neighborhood,minus the average over all links that are in the-hop but not the -hop neighborhood.That is,we compare the average measurement in a“disk”around the link to the average measurement in the corresponding“ring”around the disk;if the metric is largeron average closer to the link,the comparison will be positive.So the level2comparison for theNYC-Cleveland link might consist of the average measurements on the Indianapolis-Clevelandand NYC-Washington links,minus the measurements on the NYC-Cleveland link.5A number of considerations motivate this general style of data analysis.First,data traffic onnearby links may often be highly correlated;this will occur for a number of reasons,includingthe fact that each link carries dataflows which themselves likely pass over multiple links.Thus itmay often be reasonable to summarize the traffic on many links in a neighborhood with a singlevalue.Second,differences between neighborhoods are important,as can be seen from the exampleproblems in this section;we wish to draw attention to such differences in our analysis.Third,different effects will be expected to occur at different spatial scales in the network;hence,wedefine comparisons at varying levels so as toflexibly detect a wide range of phenomena.The general notion of summarization and comparison over varying locations and scales is theunderlying principle of wavelet analysis.Indeed,the example problems and approach described inthis section bear strong similarity to problems in signal and image processing,domains in whichwavelet analysis has provided considerable leverage.However,traditional wavelet analysis hasrestricted itself primarily to regular spaces,e.g.,[12].Therefore,in pursuit of a formal basisfor attacking the problems described here,it is necessary and appropriate to extend the notion ofwavelets to certain graph topologies,which we do in the next section.4Graph WaveletsIn this section we approach the topic of whole-network wavelet analysis in a more formal fashion.After reviewing some basic concepts and principles from traditional wavelet analysis we developa framework for a class of graph-based wavelets.4.1Traditional Wavelets and Multiscale AnalysisAt the most basic level,a multiscale analysis(based on wavelets or otherwise)is simply a coor-dinated way of examining local differences in a set of measurements,across a range of scales.Multiscale analyses based on wavelets,of course,are now the most common and well-knownexample of this approach.Although there are currently a host of wavelet functions available,pos-sessed of a variety of different properties and characteristics,at their most basic these functionsshare the two fundamental properties that they(i)are localized(having eitherfinite or essentially 5This description in terms of simple averages over neighborhoods of different levels is a simplification for purposes of illustration.6Figure 2:Haar (dot-dashed)and Mexican Hat (solid)Wavelet Functions on the Real Line.−50555Figure 3:Mexican Hat Wavelet Function on the Plane.The central disk is strongly positive,and the surrounding ring is strongly negative.finite support)and (ii)have zero integral (and hence,excluding the trivial case,they must oscillate positive and negative).By virtue of this locally oscillating behavior,the inner product of a wavelet,say ,with another function,say ,yields coef ficients that are essentially the weighted difference of information in on the regions of positive and negative support of .Any other properties or characterisics of ,such as compact support or smoothness,are the result of using additional “degrees of freedom ”in the overall design process.Figure 2shows two examples of wavelet functions on the real line I R.The first is a symmetric variant of the well-known Haar wavelet,piecewise constant and of finite support.The second is the one-dimensional analogue of the so-called “mexican hat ”wavelet,,which is in finitely differentiable and of in finite support.Both have zero integral and unit norm.When these two functions are rotated about their point of symmetry,the results are radially sym-metric analogues in the plane I R .The analogue of the latter is the mexican hat wavelet,whose relative shape and magnitude are shown in Figure 3in the form of an image.Traditionally,a wavelet analysis of ,for functions de fined on some subset of a Euclidean space (e.g.,I R or I R )is based on the collection of coef ficients ,where is a shifted and dilated version of by and ,respectively.For I R,7the mapping is known as a continuous wavelet transform.If on the other hand and,for Z,this mapping is known as a discrete wavelet transform.And in the latter case,when the function is chosen appropriately,it is possible to create a system of wavelet functions that are orthonormal.Within each of these classes of wavelet transforms(con-tinuous,discrete-redundant,and discrete-orthogonal)there are numerous examples to be found, customized to have various additional properties felt to be useful for a particular application(s). See[12],for example.Regardless of the specifics,the end result of a wavelet transform is an alternative representation of the information in with respect to an indexing in scale and location.Which particular class of wavelet transform is preferable(as well as the choice of wavelet function within class),if any,typically depends on the uses to which one intends to put such a representation.For example, the continuous wavelet transform has been quite popular in astronomy,particularly for the detec-tion of point sources and anomolies in satellite image data(e.g.,[17]).Alternatively,the discrete wavelet transform and its extensions have proven especially useful for the tasks of denoising and compression(e.g.,[18]).Our own contribution in this paper can be said to more closely resemble the traditional continuous wavelet transform in spirit.More recently,there has been much work on so-called“second generation”wavelets(e.g., [19]).Systems of such wavelets are not necessarily composed of either shifts or dilations of some single function.Nevertheless,the members are localized and indexed across a range of scales and locations within scales,have zero integral,and share some common characteristic(s)in their definition.Examples include piecewise constant wavelets defined on general measure spaces[14], wavelets on triangular meshes of arbitrary topology[15],and wavelets on the sphere[20,16].The wavelets we develop in this paper,in extending the traditional framework described above to the context of network graphs,are a new contribution to this second generation of wavelets.4.2A Class of Wavelets for GraphsLet be a connected graph,of degree,corresponding to a network of interest. Without loss of generality,we assume that measurements correspond to vertices,. That is,is either the network graph itself or the network line graph(as defined in Section3.2) depending on whether the actual measurements are taken at routers or on links of the underlying network.The vertex set is a(finite)metric space when equipped with the shortest path distance metric (in units of“hops”)defined with respect to.In fact,it is a measure space when equipped with simple counting measure as well,say.In extending the notion of wavelets to graphs,we seek a collection of functions I R,localized with respect to a range of scale/location indices,such that at a minimum we have.Additional properties are built in by choice.As foreshadowed by our discussion in Section3.2,the construction of our graph wavelets centers on the notion of comparing network measurements within a given region(s)(e.g.,a simple “disk”)to those in a surrounding region(s)(e.g.,a simple“ring”),with both sets of regions centered on a particular vertex and calibrated to a scale.The notion of regions will be made explicit through the concept of hop-neighborhoods.Specifically,we define the-hop neighborhood about,,to be the set of vertices that are less than or equal to hops from. The zero-hop neighborhood of will simply be itself i.e.,.Similarly,we let8Figure4:Schematic Illustration of Graph Wavelet Weighting Scheme:Weights obtained from analogue of mexican hat wavelet.be the set of vertices exactly hops away from.We callthe-hop ring about.In addition to the condition of having zero integral,we will require that each function be constant within hop rings and zero on hop rings outside.These constraints have the effect of imposing a type of symmetry on and a scaling of the underlying support.Figure4 shows an illustration of this effect,based on the construction given below,which may be compared to Figure3for example.Let denote the largest for which the hop ring is non-empty.Given the nature of the graph topology,in contrast to that of Euclidean space,this is a necessary and well-defined parameter in our construction.Within this framework there is still a good deal of freedom in choosing the form of our wavelet functions.Specifically,we note that our symmetry condition implies that definition of can be reduced to that of a set of constants on rings,for .Note then that(1)With the choice of,we reduce the problem further to that offinding an appropriate set of constants that depend only on scale and hop distances.But it can be seen from(1)that,for each location and scale,if and only if.We therefore have the following result regarding wavelets on graphs.Theorem1Let be as above.For each and,define I R as(2)9。

Effective wavelet-based compression method with adaptive quantizationthreshold and zerotree codingArtur Przelaskowski, Marian Kazubek, Tomasz JamrógiewiczInstitute of Radioelectronics, Warsaw University of Technology, Nowowiejska 15/19, 00-665 Warszawa,PolandABSTRACTEfficient image compression technique especially for medical applications is presented. Dyadic wavelet decomposition by use of Antonini and Villasenor bank filters is followed by adaptive space-frequency quantization and zerotree-based entropy coding of wavelet coefficients. Threshold selection and uniform quantization is made on a base of spatial variance estimate built on the lowest frequency subband data set. Threshold value for each coefficient is evaluated as linear function of 9-order binary context. After quantization zerotree construction, pruning and arithmetic coding is applied for efficient lossless data coding. Presented compression method is less complex than the most effective EZW-based techniques but allows to achieve comparable compression efficiency. Specifically our method has similar to SPIHT efficiency in MR image compression, slightly better for CT image and significantly better in US image compression. Thus the compression efficiency of presented method is competitive with the best published algorithms in the literature across diverse classes of medical images. Keywords: wavelet transform, image compression, medical image archiving, adaptive quantization1. INTRODUCTIONLossy image compression techniques allow significantly diminish the length of original image representation at the cost of certain original data changes. At range of lower bit rates these changes are mostly observed as distortion but sometimes improved image quality is visible. Compression of the concrete image with its all important features preserving and the noise and all redundancy of original representation removing is do required. The choice of proper compression method depends on many factors, especially on statistical image characteristics (global and local) and application. Medical applications seem to be challenged because of restricted demands on image quality (in the meaning of diagnostic accuracy) preserving. Perfect reconstruction of very small structures which are often very important for diagnosis even at low bit rates is possible by increasing adaptability of the algorithm. Fitting data processing method to changeable data behaviour within an image and taking into account a priori data knowledge allow to achieve sufficient compression efficiency. Recent achievements clearly show that nowadays wavelet-based techniques can realise these ideas in the best way.Wavelet transform features are useful for better representation of the actual nonstationary signals and allow to use a priori and a posteriori data knowledge for diagnostically important image elements preserving. Wavelets are very efficient for image compression as entire transformation basis function set. This transformation gives similar level of data decorrelation in comparison to very popular discrete cosine transform and has additional very important features. It often provides a more natural basis set than the sinusoids of the Fourier analysis, enables widen set of solution to construct effective adaptive scalar or vector quantization in time-frequency domain and correlated entropy coding techniques, does not create blocking artefacts and is well suited for hardware implementation. Wavelet-based compression is naturally multiresolution and scalable in different applications so that a single decomposition provides reconstruction at a variety of sizes and resolutions (limited by compressed representation) and progressive coding and transmission in multiuser environments.Wavelet decomposition can be implemented in terms of filters and realised as subband coding approach. The fundamental issue in construction of efficient subband coding techniques is to select, design or modify the analysis and synthesis filters.1Wavelets are good tool to create wide class of new filters which occur very effective in compression schemes. The choice of suitable wavelet family, with such criteria as regularity, linearity, symmetry, orthogonality or impulse and step response of corresponding filter bank, can significantly improve compression efficiency. For compactly supported wavelets corresponding filter length is proportional to the degree of smoothness and regularity of the wavelet. Butwhen the wavelets are orthogonal (the greatest data decorrelation) they also have non-linear phase in the associated FIR filters. The symmetry, compact support and linear phase of filters may be achieved by biorthogonal wavelet bases application. Then quadrature mirror and perfect reconstruction subband filters are used to compute the wavelet transform. Biorthogonal wavelet-based filters occurred very efficient in compression algorithms. A construction of wavelet transformation by fitting local defined basis transformation function (or finite length filters) into image data characteristics is possible but very difficult. Because of nonstationary of image data, miscellaneous image futures which could be important for good reconstruction, significant various image quality (signal to noise level, spatial resolution etc.) from different imaging systems it is very difficult to elaborate the construction method of the optimal-for-compression filters. Many issues relating to the choice of the most efficient filter bank for image compression remain still unresolved.2The demands of preserving the diagnostic accuracy in reconstructed medical images are exacting. Important high frequency coefficients which appear at the place of small structure edges in CT and MR images should be saved. Accurate global organ shapes reconstruction in US images and strong noise reduction in MN images is also required. It is rather difficult to imagine that one filter bank can do it in the best way. Rather choosing the best wavelet families for each modality is expected.Our aim is to increase the image compression efficiency, especially for medical applications, by applying suitable wavelet transformation, adaptive quantization scheme and corresponding processed decomposition tree entropy coding. We want to achieve higher acceptable compression ratios for medical images by better preserving the diagnostic accuracy of images. Many bit allocation techniques applied in quantization scheme are based on data distribution assumptions, quantiser distortion function etc. All statistical assumptions built on global data characteristics do not cover exactly local data behaviour and important detail of original image, e.g., different texture small area may be lost. Thus we decided to build quantization scheme on the base of local data characteristics such a direct data context in two dimensions mentioned earlier. We do data variance estimation on the base of real data set as spatial estimate for corresponding coefficient positions in successive subbands. The details of quantization process and correlated coding technique as a part of effective simple wavelet-based compression method which allows to achieve high reconstructed image quality at low bit rates are presented.2. THE COMPRESSION TECHNIQUEScheme of our algorithm is very simple: dyadic, 3 levels decomposition of original image (256×256 images were used) done by selected filters. For symmetrical filters symmetry boundary extension at the image borders was used and for asymmetrical filters - a periodic (or circular) boundary extension.Figure 1. Dyadic wavelet image decomposition scheme. - horizontal relations, - parent - children relations. LL - the lowest frequency subband.Our approach to filters is utilitarian one, making use of the literature to select the proper filters rather than to design them. We conducted an experiment using different kinds of wavelet transformation in presented algorithm. Long list of wavelet families and corresponding filters were tested: Daubechies, Adelson, Brislawn, Odegard, Villasenor, Spline, Antonini, Coiflet, Symmlet, Beylkin, Vaid etc.3 Generally Antonini 4 filters occurred to be the most efficient. Villasenor, Odegard and Brislawn filters allow to achieve similar compression efficiency. Finally: Antonini 7/9 tap filters are used for MR and US image compression and Villasenor 18/10 tap filters for CT image compression.2.1 Adaptive space-frequency quantizationPresented space-frequency quantization technique is realised as entire data pre-selection, threshold selection and scalar uniform quantization with step size conditioned by chosen compression ratio. For adaptive estimation of threshold and quantization step values two extra data structure are build. Entire data pre-selection allows to evaluate zero-quantized data set and predict the spatial context of each coefficient. Next simple quantization of the lowest frequency subband (LL) allows to estimate quantized coefficient variance prediction as a space function across sequential subbands. Next the value of quantization step is slightly modified by a model build on variance estimate. Additionally, a set of coefficients is reduced by threshold selection. The threshold value is increased in the areas with the dominant zero-valued coefficients and the level of growth depends on coefficient spatial position according variance estimation function.Firstly zero-quantized data prediction is performed. The step size w is assumed to be constant for all coefficients at each decomposition level. For such quantization model the threshold value is equal to w /2. Each coefficient whose value is less than threshold is predicted to be zero-valued after quantization (insignificant). In opposite case coefficient is predicted to be not equal to zero (significant). It allows to create predictive zero-quantized coefficients P map for threshold evaluation in the next step. The process of P map creation is as follows:if c w then p else p i i i <==/201, (1)where i m n m n =⋅−12,,...,;, horizontal and vertical image size , c i - wavelet coefficient value. The coefficient variance estimation is made on the base of LL data for coefficients from next subbands in corresponding spatial positions. The quantization with mentioned step size w is performed in LL and the most often occurring coefficient value is estimated. This value is named MHC (mode of histogram coefficient). The areas of MHC appearance are strongly correlated with zero-valued data areas in the successive subbands. The absolute difference of the LL quantized data and MHC is used as variance estimate for next subband coefficients in corresponding spatial positions. We tested many different schemes but this model allows to achieve the best results in the final meaning of compression efficiency. The variance estimation is rather coarse but this simple adaptive model built on real data does not need additional information for reconstruction process and increases the compression efficiency. Let lc i , i =1,2,...,lm , be a set ofLL quantized coefficient values, lm - size of this set . Furthermore let mode of histogram coefficient MHC value be estimated as follows:f MHC f lc MHC Al lc Al i i ()max ()=∈∈ and , (2)where Al - alphabet of data source which describes the values of the coefficient set and f lc n lmi lc i ()=, n lc i - number of lc i -valued coefficients. The normalised values of variance estimate ve si for next subband coefficients in corresponding to i spatial positions (parent - children relations from the top to the bottom of zerotree - see fig. 1) are simply expressed by the following equation: ve lc MHC ve si i =−max . (3)These set of ve si data is treated as top parent estimation and is applied to all corresponding child nodes in wavelet hierarchical decomposition tree.9-th order context model is applied for coarser data reduction in ‘unimportant' areas (usually with low diagnostic importance). The unimportance means that in these areas the majority of the data are equal to zero and significant values are separated. If single significant values appear in these areas it most often suggests that these high frequency coefficients are caused by noise. Thus the coarser data reduction by higher threshold allows to increase signal to noise ratio by removing the noise. At the edges of diagnostically important structures significant values are grouped together and the threshold value is lower at this fields. P map is used for each coefficient context estimation. Noncausal prediction of the coefficient importance is made as linear function of the binary surrounding data excluding considered coefficient significance. The other polynomial, exponential or hyperbolic function were tested but linear function occurred the most efficient. The data context shown on fig. 2 is formed for each coefficient. This context is modified in the previous data points of processing stream by the results of the selection with the actual threshold values at these points instead of w /2 (causal modification). Values of the coefficient importance - cim are evaluated for each c i coefficient from the following equation:cim coeff p i i j j =⋅−=∑1199(),, where i m n =⋅12,,...,. (4)Next the threshold value is evaluated for each c i coefficient: th w cim w ve i i si =⋅+⋅⋅−/(())211, (5)where i m n =⋅12,,...,, si - corresponding to LL parent spatial location in lower decomposition levels.The modified quantization step model uses the LL-based variance estimate to slightly increase the step size for less variance coefficients. Threshold data selection and uniform quantization is made as follows: each coefficient value is firstly compared to its threshold value and then quantized using w step for LL and modified step value mw si for next subbands . Threshold selection and quantization for each c i coefficient can be clearly described by the following equations:LLif c then c c welse if c th then c else c c mw i i i i i i i i si∈=<==//0, (6)where mw w coeff ve si si =⋅+⋅−(())112. (7)The coeff 1 and coeff 2 values are fitted to actual data characteristic by using a priori image knowledge and performingentire tests on groups of similar characteristic images.a) b)Figure 2. a) 9-order coefficient context for evaluating the coefficient importance value in procedure of adaptive threshold P map context of single edge coefficient.2.2 Zerotrees construction and codingSophisticated entropy coding methods which can significantly improve compression efficiency should retain progressive way of data reconstruction. Progressive reconstruction is simple and natural after wavelet-based decomposition. Thus the wavelet coefficient values are coded subband-sequentially and spectral selection is made typically for wavelet methods. The same scale subbands are coded as follows: firstly the lowest frequency subband, then right side coefficient block, down-left and down-right block at the end. After that next larger scale data blocks are coded in the same order. To reduce a redundancy of such data representation zerotree structure is built. Zerotree describes well the correlation between data values in horizontal and vertical directions, especially between large areas with zero-valued data. These correlated fragments of zerotree are removed and final data streams for entropy coding are significantly diminish. Also zerotree structure allows to create different characteristics data streams to increase the coding efficiency. We used simple arithmetic coders for these data streams coding instead of applied in many techniques bit map (from MSB to LSB) coding with necessity of applying the efficient context model construction. Because of refusing the successive approximation we lost full progression. But the simplicity of the algorithm and sometimes even higher coding efficiency was achieved. Two slightly different arithmetic coders for producing ending data stream were used.2.2.1 Construction and pruning of zerotreeThe dyadic hierarchical image data decomposition is presented on fig. 1. Decomposition tree structure reflects this hierarchical data processing and strictly corresponds to created in transformation process data streams. The four lowest frequency subbands which belong to the coarsest scale level are located at the top of the tree. These data have not got parent values but they are the parents for the coefficients in lower tree level of greater scale in corresponding spatial positions. These correspondence is shown on the fig. 1 as parent-children relations. Each parent coefficient has got four direct children and each child is under one direct parent. Additionally, horizontal relations at top tree level are introduced to describe the data correlation in better way.The decomposition tree becomes zerotree when node values of quantized coefficients are signed by symbols of binary alphabet. Each tree node is checked to be significant (not equal to zero) or insignificant (equal to zero) - binary tree is built. For LL nodes way of significance estimation is slightly different. The MHC value is used again because of the LL areas of MHC appearance strong correlation with zero-valued data areas in the next subbands. Node is signed to be significant if its value is not equal to MHC value or insignificant if its value is equal to MHC. The value of MHC must be sent to a decoder for correct tree reconstruction.Next step of algorithm is a pruning of this tree. Only the branches to insignificant nodes can be pruned and the procedure is slightly other at different levels of the zerotree. Procedure of zerotree pruning starts at the bottom of wavelet zerotree. Sequential values of four children data and their parent from higher level are tested. If the parent and the children are insignificant - the tree branch with child nodes is removed and the parent is signed as pruned branch node (PBN). Because of this the tree alphabet is widened to three symbols. At the middle levels the pruning of the tree is performed if the parent value is insignificant and all children are recognised as PBN. From conducted research we found out that adding extra symbols to the tree alphabet is not efficient for decreasing the code bit rate. The zerotree pruning at top level is different. The checking node values is made in horizontal tree directions by exploiting the spatial correlation of the quantized coefficients in the subbands of the coarsest scale - see fig. 1. Sequentially the four coefficients from the same spatial positions and different subbands are compared with one another. The tree is pruned if the LL node is insignificant and three corresponding coefficients are PBN. Thus three branches with nodes are removed and LL node is signed as PBN. It means that all its children across zerotree are insignificant. The spatial horizontal correlation between the data at other tree levels is not strong enough to increase the coding efficiency by its utilisation.2.2.2 Making three data streams and codingPruned zerotree structure is handy to create data streams for ending efficient entropy coding. Instead of PBN zero or MHC values (nodes of LL) additional code value is inserted into data set of coded values. Also bit maps of PBN spatial distribution at different tree levels can be applied. We used optionally only PBN bit map of LL data to slightly increase the coding efficiency. The zerotree coding is performed sequentially from the top to the bottom to support progressive reconstruction. Because of various quantized data characteristics and wider alphabet of data source model after zerotree pruning three separated different data streams and optionally fourth bit map stream are produced for efficient data coding. It is well known from information theory that if we deal with a data set with significant variability of data statistics anddifferent statistics (alphabet and estimate of conditional probabilities) data may be grouped together it is better to separate these data and encode each group independently to increase the coding efficiency. Especially is true when context-based arithmetic coder is used. The data separation is made on the base of zerotree and than the following data are coded independently:- the LL data set which has usually smaller number of insignificant (MHC-valued) coefficients, less PBN and less spatial data correlation than next subband data (word- or charwise arithmetic coder is less efficient then bitwise coder);optionally this data stream is divided on PBN distribution bit map and word or char data set without PBNs,- the rest of top level (three next subbands) and middle level subband data set with a considerable number of zero-valued (insignificant) coefficients and PBN code values; level of data correlation is greater, thus word- or charwise arithmetic coder is efficient enough,- the lowest level data set with usually great number of insignificant coefficients and without PBN code value; data correlation is very high.Urban Koistinen arithmetic coder (DDJ Compression Contest public domain code accessible by internet) with simple bitwise algorithm is used for first data stream coding. For the second and third data stream coding 1-st order arithmetic coder built on the base of code presented in Nelson book 5 is applied. Urban coder occurred up to 10% more efficient than Nelson coder for first data stream coding. Combining a rest of top level data and the similar statistics middle level data allows to increase the coding efficiency approximately up to 3%.The procedure of the zerotree construction, pruning and coding is presented on fig. 3.Construction ofbinary zerotreeBitwise arithmetic codingFinal compressed data representationFigure 3. Quantized wavelet coefficients coding scheme with using zerotree structure. PBN - pruned branch node.3. TESTS, RESULTS AND DISCUSSIONIn our tests many different medical modality images were used. For chosen results presentation we applied three 256×256×8-bit images from various medical imaging systems: CT (computed tomography), MR (magnetic resonance) and US(ultrasound) images. These images are shown on fig. 4. Mean square error - MSE and peak signal to noise ratio - PSNR were assumed to be reconstructed image quality evaluation criteria. Subjective quality appreciation was conducted in very simple way - only by psychovisual impression of the non-professional observer.Application of adaptive quantization scheme based on modified threshold value and quantization step size is more efficient than simple uniform scalar quantization up to 10% in a sense of better compression of all algorithm. Generally applying zerotree structure and its processing improved coding efficiency up to 10% in comparison to direct arithmetic coding of quantized data set.The comparison of the compression efficiency of three methods: DCT-based algorithm,6,7 SPIHT 8 and presented compression technique, called MBWT (modified basic wavelet-based technique) were performed for efficiency evaluation of MBWT. The results of MSE and PSNR-based evaluation are presented in table 1. Two wavelet-based compression techniques are clearly more efficient than DCT-based compression in terms of MSE/PSNR and also in our subjective evaluation for all cases. MBWT overcomes SPIHT method for US images and slightly for CT test image at lower bit rate range.The concept of adaptive threshold and modified quantization step size is effective for strong reduction of noise but it occurs sometimes too coarse at lower bit rate range and very small details of the image structures are put out of shape. US images contain significant noise level and diagnostically important small structures do not appear (image resolution is poor). Thus these images can be efficiently compressed by MBWT with image quality preserved. It is clearly shown on fig.5. An improvement of compression efficiency in relatio to SPIHT is almost constant at wide range of bit rates (0.3 - 0.6 dB of PSNR).a) b)c)Figure 4. Examples of images used in the tests of compression efficiency evaluation. The results presented in table 1 and on fig. 5 were achieved for those images. The images are as follows: a ) echocardiography image, b) CT head image, c) MR head image.Table 1. Comparison of the three techniques compression efficiency: DCT-based, SPIHT and MBWT. The bit rates are chosen in diagnostically interesting range (near the borders of acceptance).Modality - bit rateDCT-based SPIHT MBWTMSE PSNR[dB] MSE PSNR[dB] MSE PSNR[db] MRI - 0.70 bpp8.93 38.62 4.65 41.45 4.75 41.36 MRI - 0.50 bpp13.8 36.72 8.00 39.10 7.96 39.12 CT - 0.50 bpp6.41 40.06 3.17 43.12 3.1843.11 CT - 0.30 bpp18.5 35.46 8.30 38.94 8.0639.07 US - 0.40 bpp54.5 30.08 31.3 33.18 28.3 33.61 US - 0.25 bpp 91.5 28.61 51.5 31.01 46.8 31.43The level of noise in CT and MR images is lower and small structures are often important in image analysis. That is the reason why the benefits of MBWT in this case are smaller. Generally compression efficiency of MBWT is comparable to SPIHT for these images. Presented method lost its effectiveness for higher bit rates (see PSNR of 0.7 bpp MR representation) but for lower bit rates both MR and CT images are compressed significantly better. Maybe the reason is that the coefficients are reduced relatively stronger because of its importance reduction in MBWT threshold selection at lower bits rate range.0,20,30,40,50,60,70,8Rate in bits/pixel PSNR in dBFigure 5. Comparison of SPIHT and presented in this paper technique (MBWT) compression efficiency at range of low bit rates. US test image was compressed.4. CONCLUSIONSAdaptive space-frequency quantization scheme and zerotree-based entropy coding are not time-consuming and allow to achieve significant compression efficiency. Generally our algorithm is simpler than EZW-based algorithms 9 and other algorithms with extended subband classification or space -frequency quantization models 10 but compression efficiency of presented method is competitive with the best published algorithms in the literature across diverse classes of medical images. The MBWT-based compression gives slightly better results than SPIHT for high quality images: CT and MR and significantly better efficiency for US images. Presented compression technique occurred very useful and promising for medical applications. Appropriate reconstructed image quality evaluation is desirable to delimit the acceptable lossy compression ratios for each medical modality. We intend to improve the efficiency of this method by: the design a construction method of adaptive filter banks and correlated more sufficient quantization scheme. It seems to be possible byapplying proper a priori model of image features which determine diagnostic accuracy. Also more efficient context-based arithmetic coders should be applied and more sophisticated zerotree structures should be tested.REFERENCES1.Hui, C. W. Kok, T. Q. Nguyen, …Image Compression Using Shift-Invariant Dydiadic Wavelet Transform”, subbmited toIEEE Trans. Image Proc., April 3nd, 1996.2.J. D. Villasenor, B. Belzer and J. Liao, …Wavelet Filter Evaluation for Image Compression”, IEEE Trans. Image Proc.,August 1995.3. A. Przelaskowski, M.Kazubek, T. Jamrógiewicz, …Optimalization of the Wavelet-Based Algorithm for Increasing theMedical Image Compression Efficiency”, submitted and accepted to TFTS'97 2nd IEEE UK Symposium on Applications of Time-Frequency and Time-Scale Methods, Coventry, UK 27-29 August 1997.4.M. Antonini, M. Barlaud, P. Mathieu and I. Daubechies, …Image coding using wavelet transform”, IEEE Trans. ImageProc., vol. IP-1, pp.205-220, April 1992.5.M. Nelson, The Data Compression Book, chapter 6, M&T Books, 1991.6.M. Kazubek, A. Przelaskowski and T. Jamrógiewicz, …Using A Priori Information for Improving the Compression ofMedical Images”, Analysis of Biomedical Signals and Images, vol. 13,pp. 32-34, 1996.7. A. Przelaskowski, M. Kazubek and T. Jamrógiewicz, …Application of Medical Image Data Characteristics forConstructing DCT-based Compression Algorithm”, Medical & Biological Engineering & Computing,vol. 34, Supplement I, part I, pp.243-244, 1996.8. A. Said and W. A. Pearlman, …A New Fast and Efficient Image Codec Based on Set Partitioning in Hierarchical Trees”,submitted to IEEE Trans. Circ. & Syst. Video Tech., 1996.9.J. M. Shapiro, …Embedded Image Coding Using Zerotrees of Wavelet Coefficients”, IEEE Trans. Signal Proces., vol.41, no.12, pp. 3445-3462, December 1993.10.Z. Xiong, K. Ramchandran and M. T. Orchard, …Space-Frequency Quantization for Wavelet Image Coding”, IEEETrans. Image Proc., to appear in 1997.。

WAVELET TRANSFORM MOMENTS FOR FEATURE EXTRACTIONFROM TEMPORAL SIGNALSIgnacio Rodriguez CarreñoDepartment of Electrical and Electronic Engineering, Public University of Navarre, Arrosadia, Pamplona, SpainEmail: irodriguez@unavarra.esMarko VuskovicDepartment of Computer Science, San Diego State University, San Diego, California, USAEmail: marko@Keywords: Pattern recognition, EMG, feature extraction, wavelets, moments, support vector machinesAbstract: A new feature extraction method based on five moments applied to three wavelet transform sequences has been proposed and used in classification of prehensile surface EMG patterns. The new method has essentially ex-tended the Englehart's discrete wavelet transform and wavelet packet transform by introducing more efficientfeature reduction method that also offered better generalization. The approaches were empirically evaluated onthe same set of signals recorded from two real subjects, and by using the same classifier, which was the Vapnik'ssupport vector machine.1 INTRODUCTIONThe electromyographic signal (EMG), measured at the surface of the skin, provides valuable information about the neuromuscular activity of a muscle and this has been essential to its application in clinical diagno-sis, and as a source for controlling assistive devices, and schemes for functional electrical stimulation. Its application to control prosthetic limbs has also pre-sented a great challenge, due to the complexity of the EMG signals.An important requirement in this area is to accu-rately classify different EMG patterns for controlling a prosthetic device. For this reason, effective feature extraction is a crucial step to improve the accuracy of pattern classification, therefore many signal represen-tations have been suggested.Various temporal and spectral approaches have been applied to extract features from these signals. A comparison of some effective temporal and spectral approaches is given in (Du & Vuskovic 2004), where the authors have applied moments to short time Fou-rier transform (STFT), and short time Thompson transform (STTT) on prehensile EMG patterns.The wavelet transform-based feature extraction techniques have also been successfully applied with promising results in EMG pattern recognition by Englehart and others (1998).The discrete wavelet transform (DWT) and its generalization, the wavelet packet transform (WPT), were elaborated in (Englehart 1989a). These tech-niques have shown better performance than the oth-ers in this area because of its multilevel decomposi-tion with variable trade-off in time and frequency resolution. The WPT generates a full decomposition tree in the transform space in which different wavelet bases can be considered to represent the signal. The techniques were applied to feature extraction from surface EMG signals.However, these techniques produce a large amount of coefficients, since the transform space has very large dimension. This fact suggests the system-atic application of feature selection or projection methods and dimensionality reduction techniques to enable the methodology for real time applications.Englehart applied feature selection and feature pro-jection that yielded better classification results and improved time efficiency. Specifically, the principal component analysis (PCA) was used due to its ability to model linear dependencies and to reject irrelevant information in the feature set (Englehart etal. 1999).This paper continues the work described above by taking a different approach to feature reduction. Extending the idea of spectral moments suggested in (Du & Vuskovic 2004) the sequences of wavelet coefficients are further subjected to the calculation of their temporal moments. The main goal of this work is to propose and empirically compare two different novel feature extraction approaches based on simple two-scale DWT and WPT with the two best Engle-hart’s approaches using the DWT and the WPT in combination with principal component analysis (PCA).In this new approach, the first five raw moments were applied to DWT transformed prehensile EMG sequences, which has proven to be very advanta-geous in the classification stage. The methods em-ployed a simple DWT or WPT with only three trans-form sequences, instead of the full DWT or WPT used by Englehart. This has eliminated the tedious feature reduction procedures and PCA.The evaluation of the three approaches was car-ried out on the same set of data, and with an identical classifier based on Vapnik's support vector machines (SVM) with a linear kernel.2 PREHENSILE EMGSThe research presented here was motivated by the need for classification of prehensile electromiographic signals (EMG) for control of a multifunctional pros-thetic hand (Vuskovic etal. 1995). Since the hand-preshaping phase in an average object grasp takes about 500 ms, it is important to accomplish the feature extraction and classification in less than 400 ms, pref-erably in 200 ms. Such a difficult task requires very strong feature extractor and classifier.The mioelectric control of multifingered hand prostheses was studied in several papers, for example (Nishikava 1991), (Uchida 1992), (Farry 1996), and (Huang 1999). Most of the ideas in these efforts were inspired by Hudgins (Hudgins etal. 1991). In this work the concept of preshaping of multifunctional grasps was based on the recognition of a particular finger joint movement. In an earlier work done at San Diego State University, the approach was rather dif-ferent, based on grasp types, instead of hand configu-rations in joint space. Once a grasp type is recognized from the recorded EMGs, it can be then synergisti-cally mapped into the desired joint configuration (Vuskovic 1995) for any hand, with any number of degrees of freedom. We have considered four basic grasp types according to the Schlesinger classification (Schlesinger 1919): cylindrical grasp (C), spherical grasp (S), lateral grasp (L) and precision grasp (P), see figure 1.3 EXPERIMENTAL SETUPFour-channel surface EMG signals from two healthy subjects were recorded at 1000 Hz sampling fre-quency. The recording was done while the subject has repeatedly performed the four grasp motions. There were 216 grasp recordings evenly distributed across the four grasps types: 60 (subject 1) + 4 (sub-ject 2) for cylindrical grasp, 30+10 for precision grasp, 30+10 for lateral grasp and 60 + 12 for spheri-cal grasp. Three different EMG sequence lengths were used: 200 ms, 300 ms and 400 ms. The 200 and 300 ms sequences were obtained by truncating the recordings of 400 ms sequences. (The sequences of300 ms were not presented in this paper.)Figure 1: Four grasp types.4 DISCRETE WA VELET TRANS-FORMThe DWT is a transformation of the original tem-poral signal into a wavelet basis space. The time-frequency wavelet representation is performed byrepeatedly filtering the signal with a pair of filtersthat cut the frequency domain in the middle.Specifically, the DWT decomposes a signal into an approximation signal and a detail signal. The ap-proximation signal is subsequently divided into new approximation and detail signals. This process is carried out iteratively producing a set of approxima-tion signals at different detail levels (scales) and a final gross approximation of the signal.The detail Dj and the approximation Aj at level j can be obtained by filtering the signal with an L -sample high pass filter g , and an L -sample low pass filter h . Both approximation and detail signals are downsampled by a factor of two. This can be expressed as follows:1110[][][][2],L j j j k A n A n h k A n k −−−===∑Η− (1) 1110[][][][2],L j j j k D n D n g k A n k −−−===∑G − (2) where 0, n = 0,1,…N-1 is the original temporalsequence, while H and G represent the convolu-tion/down sampling operators. Sequences g[n] andh[n] are associated with wavelet function []A n ()ψt andthe scaling function through inner products:()ϕt[]((2),g n t t n =ψ− (3)[]((2).h n t t n =ϕ− (4)Operators H and G can be applied repeatedly in al-teration, for example: 0AA A =H H , DD=0A =G G , AD =0A G H ,DA =0The A and D sequences obtained as the result ofDWT are still massive in terms of the number ofsamples, which contributes to large dimensionality offeature space. Besides, the sequences have a highnoise component inherited from the original EMGsignal.A =H G , etc.A feature extraction approach based on DWTapplied by Englehart (1998, 1998a) consists of fourdifferentiated phases:1. Perform full DWT decomposition of the EMGsignals, until scale j = log2 (N), with the Coifletwavelet of order 4 (C4);2. Square the DWT coefficients;3. Apply PCA for dimensionality reduction tech-niques;4. Determine the optimal number of features perchannel based on the target classifier.An optimization phase is needed before selecting the adequate number of PCA features in order to maxi-mize the performance of the target classifier. The optimum number of features was 100 DWT coeffi-cients per channel of the EMG signal used in this work.5 WA VELET PACKET TRANS-FORMThe WPT is a generalization of DWT. It generates a full wavelet basis decomposition tree. In each scale,not only the approximation signal as in DWT, butalso the detail signals are filtered to obtain another two low and high frequency signals. Many different representations of a signal can be obtained by select-ing different wavelet packet basis. In this regardWPT is superior to DWT, as the chosen basis can be optimized with respect to frequency or time resolu-tion. Englehart (1999) generated a feature extraction method based on the WPT for EMG signals. In this method a previous phase must be applied to the set of training signals. The underlying idea is to select theWPT basis that best classifies all classes of signals. For this purpose, Englehart proposed a modified ver-sion of the local discriminant basis (LDB) algorithm(Englehart 1998a, 2001), to maximize the discrimi-nation ability of the WPT by using a class separabil-ity cost function (Saito & Coifman 1995). Once thebest basis for classification is defined (for differentchannels and different signal lengths), the followingsteps must be performed:1. Perform the full WPT decomposition untilscale j = log 2 (N), with the Symlet wavelet oforder 5 (S5);2. Square the WPT coefficients;3. Average energy maps within each subband;4. Select the WPT coefficients from a basis cho-sen previously for each channel and for differ-ent signal lengths;5. Extract the optimal number of features basedon the target classifier;6. Apply PCA transform to the feature space fordimensionality reduction (removing the eigen-vectors whose eigenvalues are zero);7. Extract the optimum number of features per channel for the target classifier;The optimal number of features for Englehart’s WPT based approach and for the support vector ma-chine as the target classifier (see section 7) was found to be three features per channel, per signal length.6 DWT AND WPT MOMENTS 1The new approach for feature extraction presented here is based on DWT and WPT, and on the calcula-tion of their temporal moments. The approach was first proposed in (Rodriguez & Vuskovic 2005) as an extension of the idea of spectral moments (Du & Vuskovic 2004).Specifically, we used two different wavelets suc-cessfully applied by Englehart on surface EMG sig-nals: C4 and S5In order to reduce the dimensionality and to smooth out the noise, we applied six moments to transformed signals (DWT and WPT):10[],0,1,2,...,5,j mN m n jnM S n m N −=⎛⎞==⎜⎟⎜⎟⎝⎠∑ (5) where represents sequences A, D, AA, DD, AD and DA used in algorithms described below, while N []S n j is number of samples at the corresponding level of decomposition.The new approach based on DWT consists of the following steps:1. Perform two-scale decomposition of the inputsignal;2. Compute moments for three transform se-quences (D, AA, AD);3. Apply logarithm transform to each feature,log(0.1+f);4. Normalize all features using mean value andstandard deviation computed for each feature across all samples.The choice of sequences D, AA and AD was made empirically; it has given the best results in av-1DWT and WPT moments should not be confused with wavelet vanishing moments.erage for the given set of data. Similar choice was made for WPT algorithm.The WPT-based method has the following steps:1. Perform two-scale decomposition of the inputsignal;2. Select basis obtained from previous applicationof the best basis Coifman algorithm; 3. Compute moments for three transform se-quences (A, DA, DD);4. Apply logarithm transform to each feature,log(0.1+f);5. Normalize all features using mean value andstandard deviation computed for each feature across all samples.The optimal basis selection in this method was based on a single channel. The same basis thus ob-tained was subsequently used for single and multiple channels, and for different sequence lengths.Log transformation was applied to moments as it effectively reduces the skewness and the kurtosis of data, consequently resulting in an estimated probabil-ity density that appears more like normal distribution (Vuskovic atal. 1995). The nonlinear transformation of features has significantly improved the classifier performance.7 THE SVM CLASSIFIERThe support vector machines ( Christianini & Shaw-Taylor 2000) are a family of learning algorithms based on the work of Vapnik (1998), which have recently gained a considerable interest in pattern rec-ognition community. The success of SVM comes from their good generalization ability, robustness in high dimensional feature spaces and good computa-tional efficiency.In this work, a standard SVM classifier with lin-ear kernel has been used for dichotomic (binary) clas-sification (Gunn 1997). The multiclass SVM can also be considered, but this is out of the scope of this pa-per.The previous work on the classification of pre-hensile EMG patterns (Vuskovic 1996) has shown that the most difficult is to discriminate cylindrical from spherical grasps (C/S), and then lateral from precision grasps (L/P). Therefore the SVM is applied to these pairs of grasp types and the feature extraction methods were evaluated accordingly.The classification tests were performed with leave-one-out method, where one sample was re-moved from the data set and the rest of the samples were used to train the SVM. The procedure was re-peated for each sample in the data set, and the aver-age hit rate was computed afterwards.8 COMPUTATIONAL COMPLEX-ITYApplication of WPT and calculation of J scales, 2log ≤J N , where N is the length of the original temporal signal, results in JN coefficients. Conse-quently, the computational cost of the full-scale WPT is in the order of 2 (Englehart 2001). Similarly, the computational complexity of full-scale DWT is half the computational complexity of the WPT, i.e.()≤O JN (log )O N N (2log 2O N N ). Since our new ap-decomposition, we can enumerate all the approaches with respect to their computational complexity in the increasing order: DWT(new) < WPT (new) < DWT (Englehart) < WPT (Englehart). The complexities are summarized in table 1.Table 1: Computational complexityNew approachEnglehartDWT WPT DWT WPTO(N)O(2N)O(N logN/2)O(N logN)9 EXPERIMENTAL EV ALUATIONIn this section we discuss the methodology for the experimental evaluation of DWT and WPT ap-proaches.9.1 Cluster VisualizationIn order to compare the effectiveness of a feature extraction method there is needed some method to compare the discrimination of clusters in feature space, either by 2D or 3D scatter plots, or by some distance measure between clusters. Both methods are normally based on the transformation of the feature space through PCA or Fisher-Rao transform, which both use the inverse of the cluster covariance matri-ces. Unfortunately the dimensionality of the feature space is often larger than the number of samples, which makes the methods inapplicable due to the singularity or ill-conditioning of the covariance ma-trices. However, the support vector machines offered new possibilities. SVM maximize the margin be-tween clusters and the separation hyperplane in the original or kernel-induced feature space without a need to use covariance matrices.We use in this work a projection of the original feature space onto the line perpendicular to the maximal-margin separation hyperplane: (6) ,T p X w =where X is N ×d sample (feature) matrix, w is unit, d -dimensional normal to the separation hyperplane, and p is N -vector of projected samples. In order to get a 2D plot of samples another projection vector is needed: (7) .T q X u =The d -dimensional projection vector u doesn't have to be orthonormal to w , but has to be unique in some way. Therefore we used the direction of the minimal variance of both clusters, which is nearly laying in the separation hyperplane. The vector coin-cides with the eigenvector that corresponds to the smallest non-zero eigenvalue of the pooled covari-ance matrix:11221212(1)(1)(,),2N S N S S pool S S N N −+−==+− (8) where N i (N 1+N 2 = N ) and S i are sizes and covari-ance matrices of the two clusters. An example of cluster diagrams, plot(p,q), is shown in figure 3, which will be discussed later.9.2 Hotelling DistanceA useful quantitative measure of cluster discrimina-tion in multidimensional space is Hotelling distance between cluster means (T 2 statistic). The T 2 can be computed for projected clusters:211212121212()()(,),T N N T c c C c N N C pool C C −=−+=,c − (9), where i and C c i are sample means and sample co-variance matrices of projected clusters respectively. In order to establish the significance of the distance under some confidence level, the T 2 distance needs to be compared with the corresponding critical value2c T . The critical value can be obtained if we assume that the quantity21212(1(2N N r T N N r+−−+−)) has F-distribution with degrees of freedom r and 121f N N r =+−−, where r = 2 in case of 2D pro-jections (Seber 1984). The above is true under the assumption that clusters have normal distributions with nearly equal sizes and covariance matrices. If this is not the case, a stronger statistic has to be used. In this work we used statistic suggested in (Yao 1965), where the cluster distance was computed as:()1212112212()//()T T c c C N C N c c −=−+−T . (10)The degrees of freedom for the F-distribution were estimated from the data (Seber 1984) (not pre-sented here due to limited space). The test works for unequal clusters that can have any bell-shaped distri-bution. The T 2 values are shown in tables 2 and 3, and in the scatter diagrams in figure 3. The critical values c were all below 11. The value of cluster distances as a quantitative measure of cluster dis-crimination is that they can be easily and quickly computed.29.3 Number of MomentsOnce the classification pairs are determined, the next step is to determine the optimal number of DWT and WPT moments, which will be used for feature reduction. This was done experimentally by extensive application of feature extractions and classiffications to different EMG signal lengths and different number of channels.Figure 2: Hotelling distances versus number of momentsfor WPT: (a) 200 ms, single channel, (b) 200ms, four channels, (c) 400 ms, single channel, (d) 400 ms, four channels (C/S grasps – lower bars, L/P grasps upper barBased on the bar graphs the selection of five mo-ments (M 0, M 1,…,M 4) was a clear choice.10 THE RESULTSThe comparison of four different approaches: the five-moment DWT and WPT as proposed in this paper, and the DWT and WPT of Englehart (1998a, 1999) have been measured by Hotteling distances and by the classification hit rates applied to two cluster pairs (C/S) and (L/P).The results are presented in tables 2 through 5. The feature extraction was performed for 200 and 400 ms time sequences recorded from a single channel and from four simultaneous channels. Each channel repre-sented one surface EMG electrode attached to the up-per-forehand of the subject. Several different wavelets were used in experiments, but only the two most suc-cessful ones were shown here: the fourth-order Coif-man wavelets (C4) and the fifth-order symlets (S5). The two tables show a roughly good correlation be-tween the Hotelling distances and the classification hit-rates. The small differences can be explained by the fact that the Hotelling distances point the goodness of clustering, while the hit rates stress the generaliza-tion of the trained SVM.An example of four different cluster scatter dia-grams is shown in figure 3.Figure 3: SVM-projected clusters, 200 ms, and four channels, WPT: (a) C/S - new approach, (b) L/P - new approach, (c) C/S - Englehart, (d) L/P - EnglehartThe results suggest clear advantage of our novel method over the Englehart’s approaches mainly due to the moments used for dimensionality reduction, instead of applying PCA. In addition, the application of log transformation on features has helped consid-erably. Our WPT novel method seems to behave bet-ter at classifying the 200 ms sequences. This is due to the WPT basis selection, which better characterizes the frequency structure of the transient signals.Table 2: Hotelling distances (C/S)New approach EnglehartWT WPT Sig. length /chnlsC4 S5 C4 S5 DWT C4WPT S5200/1 75 61 109 97 49 13200/4352 466 424 421 201 73 400/1 92 79 96 79 480 45 400/4 366 570 535 488 295 100Table 3: Hotelling distances (L/P)New approach Englehart DWT WPT Sig.length /chnls C4 S5 C4 S5 DWT C4 WPT S5 200/1 33 65 44 100 362 11 200/4 289 3462 1724 723 756 107 400/1 178 166 233 262 118 60 400/4 560 24680 1472 718 2388 168Table 4: Classification hit rates in % (C/S)New approach EnglehartWT WPT Sig. length /chnlsCO4 SY5 CO4 SY5 DWT C4WPT S5200/1 75.0 76.7 79.2 79.2 60.8 62.5 200/4 90.0 94.2 94.2 95.0 86.7 88.3 400/1 80.8 80.0 80.8 77.5 60.8 59.2 400/4 99.2 96. 7 98.3 97.5 88.3 93.3Table 5: Classification hit rates in % (L/P)New approach Englehart WT WPTSig. length /chnls CO4 SY5 CO4 SY5WT WPT200/1 73.3 81.7 81.7 83.3 56.7 53.3 200/4 91.7 96.7 90.0 98.3 80.0 93.3 400/1 96.7 95.0 93.3 91.7 56.7 63.3 400/4 99.9 99.9 99.9 99.9 88.3 95.011 CONCLUSIONSA new approach of wavelet-based feature extraction from temporal signals has been proposed. The ap-proach extends the Englehart's discrete wavelet trans-form and wavelet packet transform by subjecting the two-scale, three-sequence wavelet coefficients to temporal moment computation. This has helped re-duce significantly the dimensionality of the resulting feature vectors without loosing the essential informa-tion in the original patterns. It was found experimen-tally that first five raw moments represent a good compromise. The new methods are applied to prehen-sile EMG signals of various lengths and various amounts of input signals (surface EMG channels) and compared to the best approaches of Englehart, on the same set of signals. For the comparison are used two quantitative measures: Hotelling statistic and classifi-cation hit rates. The classifier applied to the extracted features was linear support vector machine, which has exceptionally good performance in case of large fea-ture spaces and fewer training samples. The results have shown superior performance of the new ap-proach. A brief complexity analysis also shows that the new approach is more efficient time wise.Although the methodology was demonstrated on EMG signals, we believe the methodology can equally successfully be applied to other temporal sig-nals.REFERENCESChristianini N., Shawe-Taylor, 2000. An Introduction toSupport Vector Machines . Cambridge Univ. Press. Du S. and Vuskovic M., 2004. Temporal vs. Spectral ap-proach to Feature Extraction from Prehensile EMG Sig-nals. In IEEE Int. Conf. on Information Reuse and Inte-gration (IEEE IRI-2004), Las Vegas, Nevada.Englehart K., Hudgins B., Parker P. and Stevenson M., 1998. Time-frequency representation for classification of the transient myoelectric signal. In ICEMBS’98.Proceedings of the 20th Annual International Confer-ence on Engineering in Medicine and Biology Society.ICEMBS Press.Englehart K., 1998a. Signal Representation for Classifica-tion of the Transient Myoelectric Signal. Doctoral The-sis. University of New Brunswick, Fredericton, New Brunswick, Canada.Englehart K., Hudgins B., Parker P. and Stevenson M., 1999. Improving Myoelectric Signal Classification us-ing Wavelet Packets and Principle Component Analy-sis. In ICEMBS’99. Proceedings of the 21st Annual In-ternational Conference on Engineering in Medicine and Biology Society, ICEMBS Press.Englehart K., Hudgins B., Parker P., 2001. A Wavelet –Based Continuous Classification Scheme for Multi-fucntion Myoelectric Control. In IEEE Transactions on Biomedical Engineering, vol. 48, No. 3, pp. 302-311. Farry, K. A., Walker I. D., Baraniuk R. G., 1996. Myoelec-tric Teleoperation of a Complex Robotic Hand. IEEE Trans On Robotic and Automation, Vol. 12, No.5. Gunn S.R., 1997. Support Vector Machines for Classifica-tion and Regression. Technical Report, Image Speech and Intelligent Systems Research Group, University of Southampton.Hannaford B. and Lehman S., 1986. Short Time Fourier Analysis of the Electromyogram: Fast Movements and Constant Contraction. In IEEE Transactions On Bio-medical Engineering. BME-33,Han-Pan Huang H-P, Chen C_Y., 1999. Development of a Myoelectric Discrimination System for Multi-Degree Prosthetic Hand. Proc. of the 1999 International Con-ference on Robotics and Automation, Detroit, May pp.2392-2397.Hudgins, P. Parker and R.N. Scott, 1991. A Neural Net-work Classifier for Multifunctional Myoelectric Con-trol. Annual Int. Conf. Of the EMBS, Vol. 13, No. 3, pp.1454-1455.Hudgins B., Parker P. and Scott R. N., 1993. A New Strat-egy for Multifunctional Myoelectric Control. In IEEE Transactions on Biomedical Engineering, vol. 40, No.1, pp. 82-94.Saito N. and Coifman R. R., 1995. Local Discriminant Basis and their applications. J. Math. Imag. Vis., Vol.5, no 4, pp. 337-358.Nishikawa D, Yu W. Yokoi H, and Kakazu Y, 1991. EMG Prosthetic Hand Controller using Real-Time Learning Method. In Proc. of the IEEE Conf. on SMC, Vol. 1, pp. I 153-158. Carreño I. R., Vuskovic M., 2005. Wavelet-Based Feature Extraction from Prehensile EMG Signals. In 13th Nor-dicBaltic on Biomedical Engineering and Medical Physics (NBC'05 UMEA), Umea, Sweden, 13-17. Schlesinger, D., 1919. Der Mechanische Aufbau der Kunstlishen Glieder. In Ersatzglieder und Arbeitshil-fen, Springer, Berlin.Seber G.A.F., 1984. Multivariate Observations, John Wiley & Sons, pp 102-117.Uchida N. U., Hiraiwa A., Sonehara N., Shimohara K., 1992. EMG Pattern Recognition by Neural Networks for Multi Fingers Control. Proc. of the Annual Int.Conf. of the Engineering in Medicine and Biology So-ciety. Vol 14, Paris, pp.1016-1018.Vapnik V. N., 1998. Statistical Learning Theory. John Wiley & Sons.Vuskovic M., Pozos A. L., Pozos R, 1995. Classification of Grasp Modes Based on Electromyographic Patternsof Preshaping Motions. Proc. of the Internat. Confer-ence on Systems, Man and Cybernetics. Vancouver,B.C., Canada, pp. 89-95, 1995.Vuskovic M., Schmit J., Dundon B. Konopka C., 1996.Hierachical Discrimination of Grasp Modes Using Sur-face EMGs. Proc. of the Internat. IEEE Conference on Robotics and Automation, Minneapolis, Minnesota, April 22-28. 2477-2483.Yao, Y., 1965. An Approximate Degrees of Freedom Solu-tion to the Multivariate Behrens-Fisher Problem, Biometrica, Vol. 52, 139-147.。

《小波分析》课程教学大纲课程名称小波分析Wavelet Analysis授课教师裘国永课程类别专业选修课先修课程高等数学，泛函分析适用学科范围计算机科学与技术开课形式讲解，论文选读开课学期第1学期学时40 学分 2 一课程目的和基本要求小波分析是在20世纪80年代初发展起来的一个应用数学分支，它是传统Fourier分析的改进与发展。

它一方面保留了Fourier分析的优点，更重要的是克服了Fourier分析不能在时域局部化的不足。

它是计算机应用、信号处理、图像分析、非线性科学和工程技术近几年来在方法上的重大突破。

实际上，小波分析在它产生、发展、完善和应用的整个过程中都广泛受惠于计算机科学、信号和图像处理科学、应用数学和纯粹数学、物理科学和地球科学等众多科学研究领域和工程应用技术领域的工作者们的共同努力。

原则上讲，传统上使用Fourier分析的地方，都可以用小波分析取代。

小波分析优于Fourier分析之处是：它在时域和频域同时具有良好的局部化性质，而且对于高频成分采用逐渐精细的时域或空域取样步长，从而可以充分突出研究对象的任何细节。

在学习过程中以教师的专题讲解为主，学生结合自己的研究领域阅读若干小波分析应用的论文，了解和熟悉小波分析方法在本研究领域的应用现状、应用前景和重点。

要求学生最好有高等数学、线性代数和泛函分析的知识。

二课程主要内容本课程介绍离散型小波变换、连续型小波变换的基本理论、正交小波、Mallat分解和重构算法以及小波变换的应用背景。

课程主要内容：小波分析简介、数值泛函基础知识、连续小波变换和离散小波变换、MRA（多分辨率分析、多尺度分析）和小波函数构造、Mallat 算法和小波变换、小波分析应用等。

三课程主要教材[1]冯象初等编著. 数值泛函与小波理论，西安电子科大出版社[2]葛哲学等编著. 小波分析理论与MA TLAB R2007实现，电子工业出版社[3]J. Walker著. A Primer on Wavelets and Their Scientific Applications. 1四主要参考文献[1]Dwight F. Mix, Kraig J. Olejniczak著. 杨志华，杨力华译. 小波基础及应用教程. 机械工业出版社[2]Jaideva C. Goswami, Andrew K.Chan著. 许天周，黄春光译. 小波分析. 国防工业出版社[3]彭玉华著. 小波变换与工程应用. 科学出版社[4]徐长发，李国宽著. 实用小波方法. 华中科技大学出版社[5]杨福生著. 小波变换的工程与应用. 科学出版社[6] A. Boggess, F. J. Narcowich著. 芮国胜，康健译. 小波与傅里叶分析基础. 电子工业出版社[7]崔锦泰著，程正兴译. 小波分析导论. 西安交通大学出版社[8]孙延奎著. 小波分析及其应用. 机械工业出版社[9]陈武凡著. 小波分析及其在图像处理中的应用. 科学出版社[10]胡昌华等著. 基于MA TLAB的系统分析与设计—小波分析. 西安电子科技大学出版社五考核方式考核方式为笔试占50%，论文阅读报告占50%。

美国金融学专业必读教材去美国读金融研究生推荐看的书：1. Real Analysis(实分析)Calculus/ Michael Spivak2. Ordinary Differential Equations(常微分方程)Ordinary Differential Equations With Applications/ Larry C. Andrews本书涵盖了几乎所有基本的常微分知识：一阶、二阶、拉普拉斯变换、方程系统等。

更为可贵的是，有两个章节详细介绍了ODE 在物理中的应用，这对Finance的学习会大有裨益，毕竟Finance和Physics有千丝万缕的联系。

3. Partial Differential Equations(偏微分方程)Partial Differential Equations for Scientists & Engineers/ S. J. Farlow4. Numerical Analysis(数值分析)Numerical Analysis/ Burden, Faires/ 8th Edition5. Intro to Topology(拓扑学引论)Topology：A First Class/ James R. MunkresIntroduction to Topology and Mordern Analysis/ George Simmons6. Applied Linear Algebra(线代)Numerical Linear Algebra/ Lloyd N. Trefethen, David Bau, III.Matrix Computations/Golub and Van Loan.7. Probability & Statistics (概率与统计)Mathematical Statistics with Applications/ Dennis Wackerly, William Mendenhall, Richard L. Scheaffer.Statistical Inference/ George Casella, Roger Berger8. Complex Variables(复变函数)9. Wavelet(小波分析)Introduction to Wavelets and Wavelets Transforms/ Sidney Burrus, Ramesh GopinathAn introduction to Wavelets/ David WalnutAn introcution to Wavelets and other Filtering Method in Finance and Economics/ Ramazan Gencay需要提醒的是，这些原版的书不容易在国内买到，但是，有志向去美国读金融或经济的同学可以选择内容相近的中文书进行学习，毕竟一个扎实的数学功底能够使将来的美国金融专业的学习更有收获。

INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONSInt.J.Circ.Theor.Appl.2006;34:559–582Published online in Wiley InterScience().DOI:10.1002/cta.375A wavelet-based piecewise approach for steady-state analysisof power electronics circuitsK.C.Tam,S.C.Wong∗,†and C.K.TseDepartment of Electronic and Information Engineering,Hong Kong Polytechnic University,Hong KongSUMMARYSimulation of steady-state waveforms is important to the design of power electronics circuits,as it reveals the maximum voltage and current stresses being imposed upon specific devices and components.This paper proposes an improved approach tofinding steady-state waveforms of power electronics circuits based on wavelet approximation.The proposed method exploits the time-domain piecewise property of power electronics circuits in order to improve the accuracy and computational efficiency.Instead of applying one wavelet approximation to the whole period,several wavelet approximations are applied in a piecewise manner tofit the entire waveform.This wavelet-based piecewise approximation approach can provide very accurate and efficient solution,with much less number of wavelet terms,for approximating steady-state waveforms of power electronics circuits.Copyright2006John Wiley&Sons,Ltd.Received26July2005;Revised26February2006KEY WORDS:power electronics;switching circuits;wavelet approximation;steady-state waveform1.INTRODUCTIONIn the design of power electronics systems,knowledge of the detailed steady-state waveforms is often indispensable as it provides important information about the likely maximum voltage and current stresses that are imposed upon certain semiconductor devices and passive compo-nents[1–3],even though such high stresses may occur for only a brief portion of the switching period.Conventional methods,such as brute-force transient simulation,for obtaining the steady-state waveforms are usually time consuming and may suffer from numerical instabilities, especially for power electronics circuits consisting of slow and fast variations in different parts of the same waveform.Recently,wavelets have been shown to be highly suitable for describingCorrespondence to:S.C.Wong,Department of Electronic and Information Engineering,Hong Kong Polytechnic University,Hunghom,Hong Kong.†E-mail:enscwong@.hkContract/sponsor:Hong Kong Research Grants Council;contract/grant number:PolyU5237/04ECopyright2006John Wiley&Sons,Ltd.560K.C.TAM,S.C.WONG AND C.K.TSEwaveforms with fast changing edges embedded in slowly varying backgrounds[4,5].Liu et al.[6] demonstrated a systematic algorithm for approximating steady-state waveforms arising from power electronics circuits using Chebyshev-polynomial wavelets.Moreover,power electronics circuits are piecewise varying in the time domain.Thus,approx-imating a waveform with one wavelet approximation(ing one set of wavelet functions and hence one set of wavelet coefficients)is rather inefficient as it may require an unnecessarily large wavelet set.In this paper,we propose a piecewise approach to solving the problem,using as many wavelet approximations as the number of switch states.The method yields an accurate steady-state waveform descriptions with much less number of wavelet terms.The paper is organized as follows.Section2reviews the systematic(standard)algorithm for approximating steady-state waveforms using polynomial wavelets,which was proposed by Liu et al.[6].Section3describes the procedure and formulation for approximating steady-state waveforms of piecewise switched systems.In Section4,application examples are presented to evaluate and compare the effectiveness of the proposed piecewise wavelet approximation with that of the standard wavelet approximation.Finally,we give the conclusion in Section5.2.REVIEW OF WA VELET APPROXIMATIONIt has been shown that wavelet approximation is effective for approximating steady-state waveforms of power electronics circuits as it takes advantage of the inherent nature of wavelets in describing fast edges which have been embedded in slowly moving backgrounds[6].Typically,power electronics circuits can be represented by a time-varying state-space equation˙x=A(t)x+U(t)(1) where x is the m-dim state vector,A(t)is an m×m time-varying matrix,and U is the inputfunction.Specifically,we writeA(t)=⎡⎢⎢⎢⎣a11(t)a12(t)···a1m(t)............a m1(t)a m2(t)···a mm(t)⎤⎥⎥⎥⎦(2)andU(t)=⎡⎢⎢⎢⎣u1(t)...u m(t)⎤⎥⎥⎥⎦(3)In the steady state,the solution satisfiesx(t)=x(t+T)for0 t T(4) where T is the period.For an appropriate translation and scaling,the boundary condition can be mapped to the closed interval[−1,1]x(+1)=x(−1)(5) Copyright2006John Wiley&Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582A WA VELET-BASED PIECEWISE APPROACH FOR STEADY-STATE ANALYSIS561 Assume that the basic time-invariant approximation equation isx i(t)=K T i W(t)for−1 t 1and i=1,2,...,m(6) where W(t)is any wavelet basis of size2n+1+1(n being the wavelet level),K T i=[k i,0,...,k i,2n+1] is a coefficient vector of dimension2n+1+1,which is to be found.‡The wavelet transformedequation of(1)isKD W=A(t)K W+U(t)(7)whereK=⎡⎢⎢⎢⎢⎢⎢⎢⎣k1,0k1,1···k1,2n+1k2,0k2,1···k2,2n+1............k m,0k m,1···k m,2n+1⎤⎥⎥⎥⎥⎥⎥⎥⎦(8)Thus,(7)can be written generally asF(t)K=−U(t)(9) where F(t)is a m×(2n+1+1)m matrix and K is a(2n+1+1)m-dim vector,given byF(t)=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣a11(t)W T(t)−W T(t)D T···a1i(t)W T(t)···a1m W T(t)...............a i1(t)W T(t)···a ii(t)W T(t)−W T(t)D T···a im W T(t)...............a m1(t)W T(t)···a mi(t)W T(t)···a mm W T(t)−W T(t)D T⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(10)K=[K T1···K T m]T(11)Note that since the unknown K is of dimension(2n+1+1)m,we need(2n+1+1)m equations. Now,the boundary condition(5)provides m equations,i.e.[W(+1)−W(−1)]T K i=0for i=1,...,m(12) This equation can be easily solved by applying an appropriate interpolation technique or via direct numerical convolution[11].Liu et al.[6]suggested that the remaining2n+1m equations‡The construction of wavelet basis has been discussed in detail in Reference[6]and more formally in Reference[7].For more details on polynomial wavelets,see References[8–10].Copyright2006John Wiley&Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582562K.C.TAM,S.C.WONG AND C.K.TSEare obtained by interpolating at2n+1distinct points, i,in the closed interval[−1,1],and the interpolation points can be chosen arbitrarily.Then,the approximation equation can be written as˜FK=˜U(13)where˜F= ˜F1˜F2and˜U=˜U1˜U2(14)with˜F1,˜F2,˜U1and˜U2given by˜F1=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣[W(+1)−W(−1)]T(00···0)···(00···0)(00···0)[W(+1)−W(−1)]T···(00···0)............(00···0)2n+1+1columns(00···0)···[W(+1)−W(−1)]T(2n+1+1)m columns⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭m rows(15)˜F2=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣F( 1)F( 2)...F( 2n+1)(2n+1+1)m columns⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭2n+1m rows(16)˜U1=⎡⎢⎢⎢⎣...⎤⎥⎥⎥⎦⎫⎪⎪⎪⎬⎪⎪⎪⎭m elements(17)˜U2=⎡⎢⎢⎢⎢⎢⎣−U( 1)−U( 2)...−U( 2n+1)⎤⎥⎥⎥⎥⎥⎦⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭2n+1m elements(18)Finally,by solving(13),we obtain all the coefficients necessary for generating an approximate solution for the steady-state system.Copyright2006John Wiley&Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582A WA VELET-BASED PIECEWISE APPROACH FOR STEADY-STATE ANALYSIS5633.WA VELET-BASED PIECEWISE APPROXIMATION METHODAlthough the above standard algorithm,given in Reference[6],provides a well approximated steady-state solution,it does not exploit the piecewise switched nature of power electronics circuits to ease computation and to improve accuracy.Power electronics circuits are defined by a set of linear differential equations governing the dynamics for different intervals of time corresponding to different switch states.In the following,we propose a wavelet approximation algorithm specifically for treating power electronics circuits.For each interval(switch state),we canfind a wavelet representation.Then,a set of wavelet representations for all switch states can be‘glued’together to give a complete steady-state waveform.Formally,consider a p-switch-state converter.We can write the describing differential equation, for switch state j,as˙x j=A j x+U j for j=1,2,...,p(19) where A j is a time invariant matrix at state j.Equation(19)is the piecewise state equation of the system.In the steady state,the solution satisfies the following boundary conditions:x j−1(T j−1)=x j(0)for j=2,3,...,p(20) andx1(0)=x p(T p)(21)where T j is the time duration of state j and pj=1T j=T.Thus,mapping all switch states to the close interval[−1,1]in the wavelet space,the basic approximate equation becomesx j,i(t)=K T j,i W(t)for−1 t 1(22) with j=1,2,...,p and i=1,2,...,m,where K T j,i=[k1,i,0···k1,i,2n+1,k2,i,0···k2,i,2n+1,k j,i,0···k j,i,2n+1]is a coefficient vector of dimension(2n+1+1)×p,which is to be found.Asmentioned previously,the state equation is transformed to the wavelet space and then solved by using interpolation.The approximation equation is˜F(t)K=−˜U(t)(23) where˜F=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣˜F˜F1˜F2...˜Fp⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦and˜U=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣˜U˜U1˜U2...˜Up⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(24)Copyright2006John Wiley&Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582564K.C.TAM,S.C.WONG AND C.K.TSEwith ˜F0,˜F 1,˜F 2,˜F p ,˜U 0,˜U 1,˜U 2and ˜U p given by ˜F 0=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣F a 00···F b F b F a 0···00F b F a ···0...............00···F b F a (2n +1+1)×m ×p columns⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭m ×p rows (F a and F b are given in (33)and (34))(25)˜F 1=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣F ( 1)0 0F ( 2)0 0............F ( 2n +1) (2n +1+1)m columns 0(2n +1+1)m columns···0 (2n +1+1)m columns(2n +1+1)×m ×p columns⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭2n +1m rows(26)˜F 2=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣0F ( 1)···00F ( 2)···0............0(2n +1+1)m columnsF ( 2n +1)(2n +1+1)m columns···(2n +1+1)m columns⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦(27)˜F p =⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣0···0F ( 1)0···0F ( 2)...... 0(2n +1+1)m columns···(2n +1+1)m columnsF ( 2n +1)(2n +1+1)m columns⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦(28)˜U0=⎡⎢⎢⎢⎣0 0⎤⎥⎥⎥⎦⎫⎪⎪⎪⎬⎪⎪⎪⎭m ×p elements(29)Copyright 2006John Wiley &Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582A WA VELET-BASED PIECEWISE APPROACH FOR STEADY-STATE ANALYSIS565˜U1=⎡⎢⎢⎢⎢⎢⎣−U( 1)−U( 2)...−U( 2n+1)⎤⎥⎥⎥⎥⎥⎦⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭2n+1m elements(30)˜U2=⎡⎢⎢⎢⎢⎣−U( 1)−U( 2)...−U( 2n+1)⎤⎥⎥⎥⎥⎦(31)˜Up=⎡⎢⎢⎢⎢⎢⎣−U( 1)−U( 2)...−U( 2n+1)⎤⎥⎥⎥⎥⎥⎦(32)F a=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣[W(−1)]T0 00[W(−1)]T 0............00···[W(−1)]T(2n+1+1)m columns⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭m rows(33)F b=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣[−W(+1)]T0 00[−W(+1)]T 0............00···[−W(+1)]T(2n+1+1)m columns⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭m rows(34)Similar to the standard approach outlined in Section2,all the coefficients necessary for gener-ating approximate solutions for each switch state for the steady-state system can be obtained by solving(23).It should be noted that the wavelet-based piecewise method can be further enhanced for approx-imating steady-state solution using different wavelet levels for different switch states.Essentially, Copyright2006John Wiley&Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582566K.C.TAM,S.C.WONG AND C.K.TSEwavelets of high levels should only be needed to represent waveforms in switch states where high-frequency details are present.By using different choices of wavelet levels for different switch states,solutions can be obtained more quickly.Such an application of varying wavelet levels for different switch intervals can be easily incorporated in the afore-described algorithm.4.APPLICATION EXAMPLESIn this section,we present four examples to demonstrate the effectiveness of our proposed wavelet-based piecewise method for steady-state analysis of switching circuits.The results will be evaluated using the mean relative error (MRE)and mean absolute error (MAE),which are defined byMRE =12 1−1ˆx (t )−x (t )x (t )d t (35)MAE =12 1−1|ˆx (t )−x (t )|d t (36)where ˆx (t )is the wavelet-approximated value and x (t )is the SPICE simulated result.The SPICE result,being generated from exact time-domain simulation of the actual circuit at device level,can be used for comparison and evaluation.In discrete forms,MAE and MRE are simply given byMRE =1N Ni =1ˆx i −x i x i(37)MAE =1N Ni =1|ˆx i −x i |(38)where N is the total number of points sampled along the interval [−1,1]for error calculation.In the following,we use uniform sampling (i.e.equal spacing)with N =1001,including boundary points.4.1.Example 1:a single pulse waveformConsider the single pulse waveform shown in Figure 1.This is an example of a waveform that cannot be efficiently approximated by the standard wavelet algorithm.The waveform consists of five segments corresponding to five switch states (S1–S5),and the corresponding state equations are given by (19),where A j and U j are given specifically asA j =⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩0if 0 t <t 10if t 1 t <t 21if t 2 t <t 30if t 3 t <t 40if t 4 t T(39)Copyright 2006John Wiley &Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582A WA VELET-BASED PIECEWISE APPROACH FOR STEADY-STATE ANALYSIS567S1S2S3S4S50t1t2t3t4THFigure 1.A single pulse waveform consisting of 5switch states.andU j =⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩0if 0 t <t 1H /(t 2−t 1)if t 1 t <t 2−Hif t 2 t <t 3−H /(t 4−t 3)if t 3 t <t 40if t 4 t T(40)where H is the amplitude (see Figure 1).Switch states 2(S2)and 4(S4)correspond to the rising edge and falling edge,respectively.Obviously,when the widths of rising and falling edges are small (relative to the whole switching period),the standard wavelet method cannot provide a satisfactory approximation for this waveform unless very high wavelet levels are used.Theoretically,the entire pulse-like waveform can be very accurately approximated by a very large number of wavelet terms,but the computational efforts required are excessive.As mentioned before,since the piecewise approach describes each switch interval separately,it yields an accurate steady-state waveform description for each switch interval with much less number of wavelet terms.Figures 2(a)and (b)compare the approximated pulse waveforms using the proposed wavelet-based piecewise method and the standard wavelet method for two different choices of wavelet levels with different widths of rising and falling edges.This example clearly shows the benefits of the wavelet-based piecewise approximation using separate sets of wavelet coefficients for the different switch states.Copyright 2006John Wiley &Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582568K.C.TAM,S.C.WONG AND C.K.TSE0−0.2−0.4−0.6−0.8−1−20−15−10−50.20.40.60.81−0.2−0.4−0.6−0.8−10.20.40.60.81(a)051015(b)Figure 2.Approximated pulse waveforms with amplitude 10.Dotted line is the standard wavelet approx-imated waveforms using wavelets of levels from −1to 5.Solid lines are the actual waveforms and the wavelet-based piecewise approximated waveforms using wavelets of levels from −1to 1:(a)switch states 2and 4with rising and falling times both equal to 5per cent of the period;and (b)switch states 2and 4with rising and falling times both equal to 1per cent of the period.4.2.Example 2:simple buck converterThe second example is the simple buck converter shown in Figure 3.Suppose the switch has a resistance of R s when it is turned on,and is practically open-circuit when it is turned off.The diode has a forward voltage drop of V f and an on-resistance of R d .The on-time and off-time equivalent circuits are shown in Figure 4.The basic system equation can be readily found as˙x=A (t )x +U (t )(41)where x =[i L v o ]T ,and A (t )and U (t )are given byA (t )=⎡⎢⎣−R d s (t )L −1L 1C −1RC⎤⎥⎦(42)U (t )=⎡⎣E (1−s (t ))+V f s (t )L⎤⎦(43)Copyright 2006John Wiley &Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582Figure3.Simple buck convertercircuit.Figure4.Equivalent linear circuits of the buck converter:(a)during on time;and(b)during off time.Table ponent and parameter values for simulationof the simple buck converter.Component/parameter ValueMain inductance,L0.5mHCapacitance,C0.1mFLoad resistance,R10Input voltage,E100VDiode forward drop,V f0.8VSwitching period,T100 sOn-time,T D40 sSwitch on-resistance,R s0.001Diode on-resistance,R d0.001with s(t)defined bys(t)=⎧⎪⎨⎪⎩0for0 t T D1for T D t Ts(t−T)for all t>T(44)We have performed waveform approximations using the standard wavelet method and the proposed wavelet-based piecewise method.The circuit parameters are shown in Table I.We also generate waveforms from SPICE simulations which are used as references for comparison. The approximated inductor current is shown in Figure5.Simple visual inspection reveals that the wavelet-based piecewise approach always gives more accurate waveforms than the standard method.Copyright2006John Wiley&Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582−0.5−10.51−0.5−10.51012345670123456712345671234567(a)(b)(c)(d)Figure 5.Inductor current waveforms of the buck converter.Solid line is waveform from piecewise wavelet approximation,dotted line is waveform from SPICE simulation and dot-dashed line is waveform using standard wavelet approximation.Note that the solid lines are nearly overlapping with the dotted lines:(a)using wavelets of levels from −1to 0;(b)using wavelets of levels from −1to 1;(c)using wavelets oflevels from −1to 4;and (d)using wavelets of levels from −1to 5.Table parison of MREs for approximating waveforms for the simple buck converter.Wavelet Number of MRE for i L MRE for v C CPU time (s)MRE for i L MRE for v C CPU time (s)levels wavelets (standard)(standard)(standard)(piecewise)(piecewise)(piecewise)−1to 030.9773300.9802850.0150.0041640.0033580.016−1to 150.2501360.1651870.0160.0030220.0024000.016−1to 290.0266670.0208900.0320.0030220.0024000.046−1to 3170.1281940.1180920.1090.0030220.0024000.110−1to 4330.0593070.0538670.3750.0030220.0024000.407−1to 5650.0280970.025478 1.4380.0030220.002400 1.735−1to 61290.0122120.011025 6.1880.0030220.0024009.344−1to 72570.0043420.00373328.6410.0030220.00240050.453In order to compare the results quantitatively,MREs are computed,as reported in Table II and plotted in Figure 6.Finally we note that the inductor current waveform has been very well approximated by using only 5wavelets of levels up to 1in the piecewise method with extremelyCopyright 2006John Wiley &Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582123456700.10.20.30.40.50.60.70.80.91M R E (m e a n r e l a t i v e e r r o r )Wavelet Levelsinductor current : standard method inductor current : piecewise methodFigure parison of MREs for approximating inductor current for the simple buck converter.small MREs.Furthermore,as shown in Table II,the CPU time required by the standard method to achieve an MRE of about 0.0043for i L is 28.64s,while it is less than 0.016s with the proposed piecewise approach.Thus,we see that the piecewise method is significantly faster than the standard method.4.3.Example 3:boost converter with parasitic ringingsNext,we consider the boost converter shown in Figure 7.The equivalent on-time and off-time circuits are shown in Figure 8.Note that the parasitic capacitance across the switch and the leakage inductance are deliberately included to reveal waveform ringings which are realistic phenomena requiring rather long simulation time if a brute-force time-domain simulation method is used.The state equation of this converter is given by˙x=A (t )x +U (t )(45)where x =[i m i l v s v o ]T ,and A (t )and U (t )are given byA (t )=A 1(1−s (t ))+A 2s (t )(46)U (t )=U 1(1−s (t ))+U 2s (t )(47)Copyright 2006John Wiley &Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582Figure7.Simple boost convertercircuit.Figure8.Equivalent linear circuits of the boost converter including parasitic components:(a)for on time;and(b)for off time.with s(t)defined earlier in(44)andA1=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−R mL mR mL m00R mL l−R l+R mL l−1L l1C s−1R s C s000−1RC⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(48)A2=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−R mR dL mR m R dL m0−R mL m d mR m R dL l−R mR d+R lL l−1L lR mL l d m1C s00R mC(R d+R m)−R mC(R d+R m)0−R+R m+R dC R(R d+R m)⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(49)Copyright2006John Wiley&Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582U1=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣EL m⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(50)U2=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣EL m−R m V fL m d mR m V fL l(R d+R m)−V f R mC(R d m⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(51)Again we compare the approximated waveforms of the leakage inductor current using the proposed piecewise method and the standard wavelet method.The circuit parameters are listed in Table III.Figures9(a)and(b)show the approximated waveforms using the piecewise and standard wavelet methods for two different choices of wavelet levels.As expected,the piecewise method gives more accurate results with wavelets of relatively low levels.Since the waveform contains a substantial portion where the value is near zero,we use the mean absolute error(MAE)forTable ponent and parameter values for simulation ofthe boost converter.Component/parameter ValueMain inductance,L m200 HLeakage inductance,L l1 HParasitic resistance,R m1MOutput capacitance,C200 FLoad resistance,R10Input voltage,E10VDiode forward drop,V f0.8VSwitching period,T100 sOn-time,T D40 sParasitic lead resistance,R l0.5Switch on-resistance,R s0.001Switch capacitance,C s200nFDiode on-resistance,R d0.001Copyright2006John Wiley&Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–5820−0.2−0.4−0.6−0.8−1−50.20.40.60.815100(a)(b)−50.20.40.60.81510Figure 9.Leakage inductor waveforms of the boost converter.Solid line is waveform from wavelet-based piecewise approximation,dotted line is waveform from SPICE simulation and dot-dashed line is waveform using standard wavelet approximation:(a)using wavelets oflevels from −1to 4;and (b)using wavelets of levels from −1to 5.Table IV .Comparison of MAEs for approximating the leakage inductor currentfor the boost converter.Wavelet Number MAE for i l CPU time (s)MAE for i l CPU time (s)levels of wavelets(standard)(standard)(piecewise)(piecewise)−1to 3170.4501710.1250.2401820.156−1to 4330.3263290.4060.1448180.625−1to 5650.269990 1.6410.067127 3.500−1to 61290.2118157.7970.06399521.656−1to 72570.13254340.6250.063175171.563evaluation.From Table IV and Figure 10,the result clearly verifies the advantage of using the proposed wavelet-based piecewise method.Furthermore,inspecting the two switch states of the boost converter,it is obvious that switch state 2(off-time)is richer in high-frequency details,and therefore should be approximated with wavelets of higher levels.A more educated choice of wavelet levels can shorten the simulationCopyright 2006John Wiley &Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582345670.050.10.150.20.250.30.350.40.450.5M A E (m e a n a b s o l u t e e r r o r )Wavelet Levelsleakage inductor current : standard method leakage inductor current : piecewise methodFigure parison of MAEs for approximating the leakage inductor current for the boost converter.time.Figure 11shows the approximated waveforms with different (more appropriate)choices of wavelet levels for switch states 1(on-time)and 2(off-time).Here,we note that smaller MAEs can generally be achieved with a less total number of wavelets,compared to the case where the same wavelet levels are employed for both switch states.Also,from Table IV,we see that the CPU time required for the standard method to achieve an MAE of about 0.13for i l is 40.625s,while it takes only slightly more than 0.6s with the piecewise method.Thus,the gain in computational speed is significant with the piecewise approach.4.4.Example 4:flyback converter with parasitic ringingsThe final example is a flyback converter,which is shown in Figure 12.The equivalent on-time and off-time circuits are shown in Figure 13.The parasitic capacitance across the switch and the transformer leakage inductance are included to reveal realistic waveform ringings.The state equation of this converter is given by˙x=A (t )x +U (t )(52)where x =[i m i l v s v o ]T ,and A (t )and U (t )are given byA (t )=A 1(1−s (t ))+A 2s (t )(53)U (t )=U 1(1−s (t ))+U 2s (t )(54)Copyright 2006John Wiley &Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–5820−0.2−0.4−0.6−0.8−1−6−4−20.20.40.60.81024680−0.2−0.4−0.6−0.8−1−6−4−20.20.40.60.81024680−0.2−0.4−0.6−0.8−1−6−4−20.20.40.60.81024680−0.2−0.4−0.6−0.8−1−6−4−20.20.40.60.8102468il(A)il(A)il(A)il(A)(a)(b)(c)(d)Figure 11.Leakage inductor waveforms of the boost converter with different choice of wavelet levels for the two switch states.Dotted line is waveform from SPICE simulation.Solid line is waveform using wavelet-based piecewise approximation.Two different wavelet levels,shown in brackets,are used for approximating switch states 1and 2,respectively:(a)(3,4)with MAE =0.154674;(b)(3,5)withMAE =0.082159;(c)(4,5)with MAE =0.071915;and (d)(5,6)with MAE =0.066218.Copyright 2006John Wiley &Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582。