DSP讲义

2.6.1 Frequency Response
H ( e jω ) =
n = −∞
∑ h[n]e
∞
− j ωn
Ideal Frequency-Selective Filters: lowpass,highpass,bandstop,bandpass. 2.6.2 The condition for Frequency Response existence (system stability):
n = −∞
∑ h[n] < ∞
∞
2.7 Representation of Sequences by Fourier Transform
X ( e jω ) = 1 x[n] = 2π
n = −∞
∑ x[n]e
jω
∞
− j ωn
π
−
∫π X (e
)e jωn dω
2.8 Symmetry Properties of the Fourier Transform
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Chaoshu Jiang Discrete-Time Signal Processing
R x− < z < R x+
5)
Stability, Casualty, and the ROC of the System transfer function a. Stability: ROC: includes the unit circle. b. Casualty: ROC: R x − < z ≤ ∞ c. Stable and casual: ROC: R x − < z ≤ ∞ ,and R x − < 1 .
H ( z ) = 1 + 2 z −1 + 3z −2 − 4 z −3 ⇒ h[n] = δ [n] + 2δ [n − 1] + 3δ [n − 2] − 4δ [n − 3]
2. Long division method
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Chaoshu Jiang Discrete-Time Signal Processing
2.3 Linear Time-Invariant Systems
y[n] = x[n] * h[n]
2.4 Properties of Linear Time-Invariant Systems
Finite-duration impulse response(FIR) systems Infinite-duration impulse response(IIR) systems
3.3 The Inverse z-Transform
3.3.1 Inspection Method 3.3.2 Partial Fraction Expansion 1. For casual sequences: if N > M and the poles are all first order,
2.5 Linear Constant-Coefficient Difference Equations
∑
k =0
N
a k y[n − k ] =
m =0
∑a
M
m x[ n − m]
2.6 Frequency-Domain Representation of Discrete-Time Signal and Systems
CH4. Sampling of Continous-Time Signals
4.0 Introduction
Contents covered: A way to acquire D-T signals from C-T signals, and what’s happened when a C-T signal is converted to a D-T signal.
+
∑ (1 − d z
m =1 i
s
Cm
−1 m
)
,
s −m ⎧ ⎫ ⎪d s −1 ⎪ ⎨ s − m (1 − d i w) X ( w ) ⎬ ⎪ ⎪ ⎩ dw ⎭ w= d i −1
Ak = (1 − d k z ) X ( z ) z = d k , and C m =
1 ( s − m)! (− d i ) s − m
3. ROCs of different sequences 1) The ROC of a finite-duration sequence: if the none-zero interval is N 1 ≤ n ≤ N 2 , a. N 1 > 0 ,ROC: 0 < z ≤ ∞ b. N 2 < 0 ,ROC: 0 ≤ z < ∞ c. N 2 > 0 and N 1 < 0 ,ROC: 0 < z < ∞ d N 2 = N 1 = 0 ,ROC: 0 ≤ z ≤ ∞ 2) The ROC of a right-sided sequence: if the none-zero interval is N 1 ≤ n , a. N 1 ≥ 0 ,ROC: R x − < z ≤ ∞ b. N 1 < 0 ,ROC: R x − < z < ∞ 3) The ROC of a left-sided sequence: if the none-zero interval is n ≤ N 2 , a. N 2 ≤ 0 ,ROC: 0 ≤ z < R x + b. N 2 > 0 ,ROC: 0 < z < R x + 4) The ROC of a two-sided sequence: ,
3. If N ≤ M and if X ( z ) has a pole of order s at z = d i ,and all other poles are first order
M −N
X ( z) =
∑
r =1
Br Z −r +
−1
k =1, k ≠ i 1 − d k z
∑
N
Ak
−1
[
]
,
Example: For a causal system with system function H ( z ) =
(1 + z −1 ) 2 , determine h[n] . 1 − 1.5 z −1 + 0.5 z −2
3.3.3 Power Series Expansion 1. FIR systems Example:
kz −k
3.2 Properties of the Region of Convergence
1. Definition of ROC The condition that X(z) can be expressed in a closed form. 2. Properties of ROC 1) ROC is a ring or a disk; 2) 3) Fourier transforms convergence condition: ROC of X ( z ) includes the unit circle( z = 1 ). The ROC cannot contain any poles.
Discrete-Time Signal Processing
Lecture Notes
Chaoshu Jiang
Sept. 2007 modified
Chaoshu Jiang Discrete-Time Signal Processing
Discrete-Time Signal Processing
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Chaoshu Jiang Discrete-Time Signal Processing
CH2.Discrete-Time Signals and Systems
2.1 Discrete-Time Signals: Sequences: x[n]
2.1.1 Basic Sequences and Sequence Operations Basic Sequences:
X R (e jω ) = X R (e − jω ), X I (e jω ) = − X I (e − sform Theorems
pp59 table2.2
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Chaoshu Jiang Discrete-Time Signal Processing
CH3.The Z-Transform
δ [n],u[n], Ae j (ω n +φ ) , RN [n]
0
2.2 Discrete-Time Systems
Ideal Delay System, Moving Average System 2.2.1 Memoryless Systems 2.2.2 Linear Systems 2.2.3 Time-Invariant Systems 2.2.4 Causality 2.2.5 Stability
1 1 − d k z −1
n
z > dk ,
so
x[n] =
∑A d
k k =1 −r
N
k
2. If N ≤ M and the poles are all first order
M −N
X ( z) =
∑B Z
r r =1
+
∑ 1− d
k =1
N
Ak
−1 kz
,
Ak = (1 − d k z −1 ) X ( z ) z = d k ,
x C (t )
C/D
Quantizer &Coding