数字信号处理chapter4

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数字信号处理chapter4 Structure of Digital Filter

（1）Direct Form
M
aizi
System function: H ( z )
i0 N
1 bi z i
i1
Difference equation:
N
y( n) ai x( n i ) i0 N bi y( n i ) i1
Direct Form
2N delay units are required.
h(n) is a finite-length sequence, its system function is:
N1
H(z) h(n)zn n0
FIR doesn’t have feedback loop, it is not a recursive form filter,
so there is no problem of stability.
01
11
z
1
11
21
z 1
1M 2M
2020/4/13
0M z 1
1M z 1
（4）Parallel Form
y n
We can see: a) Pole can be adjusted
independently, zero cannot; b) If there are multi-order pole,
)
A
2 i 1
1
1i
z
1
1 1i z 1
2i z2 2i z2
[•]表示取整
x(n)
11 z-1 11 21 z-1 21
y(n)
1M z-1 1M 2M z-1 2M
H1( z )
We can see:
HM( z )

数字信号处理(第四版)第四章ppt

Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Systems Outline Discrete-time system examples Classification of DT systems Impulse and step responses Time-domain characteristics of LTI Simple interconnection schemes
Process a given sequence, called the input system, to generate another sequence, called the output sequence, with more desirable properties or to extract certain information about the input signal. DT system is usually also called the digital filter
12
Digital Signal Processing
© 2013 Jimin Liang
Discrete-Time Systems 4.2 Classification of DT systems Stable system
A system is stable if and only if for every bounded input, the output is also bounded, called BIBO stable.
Discrete-Time Systems 4.1 Discrete-time system examples (4) Linear Interpolator Linear factor-2 interpolator

数字信号处理第四章04

同理：∀n1 , 0 ≤ n ≤ L − 1
详见(4-38) P.142
L
M
例：N=12=4×3, M=4 , L=3 算法流图：图4-20,P.144
这是一个蝶形三点蝶形
这仍然是一个蝶形四点蝶形
x(n)={x(0), x(1), x(2)…x(11)}
x(0) x(1) x(5) x(9)
= ∑ x( n)WNkn = = =
=
n=0 M −1 L −1
n0 = 0 n1 = 0 M −1 L −1
N −1
x(Mn1+n0)
一列一列求DFT
= DFTn1 [ x( n1 , n0 )] 0 ≤ k0 ≤ L − 1, ∀n0
∆
n1 = 0
( Mn1 + n0 )( Lk1 + k0 ) x ( n , n ) W ∑∑ 1 0 N Mn1k0 Lk1n0 k0 n0 MLk1n1 x ( n , n ) W W W W ∑∑ 1 0 N N N N
n1 = 0 L −1
n1 = 0,1,..., L − 1
行号
k0 = 0,1,..., L − 1
′ k 0 n0 (3) X 1 (k0 , n0 ) = X 1 (k0 , n0 )WN
0 ≤ k0 ≤ L − 1
0 ≤ n0 ≤ M − 1 (4) ∀k0 , 0 ≤ k0 ≤ L − 1 （针对每一行） M −1 ′ kn ′ X 2 (k 0 , k1 ) = DFTn0 [ X 1 (k 0 , n0 )] = ∑ X 1 (k0 , n0 )WM , k 0 = 0,1,..., M − 1
0 ≤ n0 ≤ M − 1, ∀k

数字信号处理(丁玉美版)教案第4章

7
4.2.1 直接计算DFT的问题及改进的途径
DFT及IDFT的定义
X (k ) x (n )W
n 0
N 1
kn N
k=0, 1, …, N-1
kn N
1 x(n) N
X (k )W
k 0
N 1
n=0, 1, …, N-1
8
可见，DFT 与 IDFT 的计算成本基本相同。直接计算N点DFT 时：对应一个k需要N次复数乘和（N-1）次复数加；对所有N个k值，则需要 N² 复数乘和N (N-1)次复数加。其中: 一次复数乘需要4次实数乘和2次实数加方能完成。当N较大时，运算量很大。
分成四个1点的序列
24
the butterfly（蝶形运算）
2点DFT 4点DFT
x(0)
X1(0)
X(0)
x(2)
W20
-1
X1(1)
X(1)
x(1)
X2(0)
W40 W41
-1
X(2)
x(3)
W20
X2(1)
-1
-1
X(3)
1点序列的DFT就是序列本身，即不用计算
25
如N>4，则将 x1(r) 再按r的奇偶进一步分解成两个 N/4点长的子序列：
x2(3)
0 2
W
W
0 8 1 8 2 8 3 8
34
求IFFT，也可用DIT-FFT的流程来实现。
x 3(l ) x1(2l )
N l 0,1,..., 1 4 x 4(l ) x1(2l 1)
26
X 1(k )
N / 4 1 l 0
x (2l )W

数字信号处理课件第四章资料

k 0,1,..., N 1 2
5、时间抽取蝶形运算流图符号
X1(k)
X1(k) WNk X 2 (k)
X 2 (k )
WNk
1 X1(k) WNk X 2 (k)
返回DIF 返回例题
设 N 23 8
X1(k)
X 2 (k )
WNk
k 0
W80
1
W81
2
W82
3
W83
X (k)
k 0,1,,7
l0
l 0
X1(k) X 3(k) WNk X 4 (k)
2
X1(k
N 4
)
X 3 (k )
W Nk
2
X
4
(k)
k 0,1,..., N 1 4
x2(r)也进行同样的分解：
x5 (l) x2 (2l)
x6 (l) x2 (2l 1)
l 0,1,..., N 1 4
)
N
/ 21
x1(r)WNrk/ 2
X1(k)
r 0
r 0
X2(k N / 2) X2(k) X (k) X1(k) WNk X 2 (k)
W (kN N
/
2)
WNkWNN
/
2
WNk
N点X(k)可以表示成前 N点和后点N 两部分：
2
2
前半部分X(k)：
X (k) X1(k) WNk X 2 (k)
N 1
X (k) x(n)WNnk k = 0, 1, …, N-1
n0
x(n)
1 N
N 1
X (k )WNnk
k 0
n = 0, 1, …, N-1
二者的差别只在于WN 的指数符号不同，以及差一个常数因子1/N，所以IDFT与DFT具有相同的运算量。

数字信号处理第4章习题解答教材

DFT [x2 (n)]

DFT {Im[ w(n )]}

1 j Wop (k )

1 2j
[W
((k )) N
W
* (( N

k )) N
]RN
(k)
解：由题意 X k DFT xn，Y k DFT y n 构造序列 Z k X k jY k 对Z k 作一次N点IFFT可得序列z n z(n) IDFT Z k
Re[w(n)] j Im[w(n)]
Wep (k) Wop (k)
由x1(n) Re[w(n)]得
X1(k) DFT[x1(n)] DFT{Re[w(n)]} Wep (k)

1 2
[W
((k
))
N
W *((N

k ))N
]RN
(k)
由x2 (n) Im[w(n)]得
X 2 (k )
(2) 按频率抽取的基-2FFT流图
同样共有L = 4级蝶形运算，每级N / 2 = 8个蝶形运算
基本蝶形是DIT 蝶形的转置
X m1(k )
X m1( j)
WNr
-1
X m (k ) Xm( j)
每个蝶形的两节点距离为2Lm ，即从第一级到第四级两节点距离分别为8，4，2，1。
系数WNr的确定：r (k )2 2m1 即k的二进制左移m 1位补零
3. N=16 时，画出基 -2 按时间抽取法及按频率抽取法的 FFT 流图（时间抽取采用输入倒位序，输出自然数顺序，频率抽取采用输入自然顺序，输出倒位序）。
解：自然序
倒位序
0 0000 0000 0 1 0001 1000 8 2 0010 0100 4 3 0011 1100 12 4 0100 0010 2 5 0101 1010 10 6 0110 0110 6 7 0111 1110 14

数字信号处理-第四章(加绪论共八章)精品PPT课件

= 2- (cosω+cos2ω)
21
4.1.2 小结
• 四种FIR数字滤波器的相位特性只取决于 h(n)的对称性，而与h(n)的值无关。 •幅度特性取决于h(n)。 •设计FIR数字滤波器时，在保证h(n)对称的条件下，只要完成幅度特性的逼近即可。
注意：当H(ω)用│H(ω)│表示时，当H(ω)为奇对称时，其相频特性中还应加一个固定相移π。
22
4.1.3 线性相位FIR滤波器的零点特性
h(n) h(N 1 n)
H z zN1H z1
N 1
N 1
H z hnzn hN 1 nzn
N 1 n0
n0
N 1
h(m)z N 1m z N 1 h m z m
m0
m0
zN 1H z1
1）若 z = zi 是H(z)的零点，则 z = zi-1 也是零点
4.1.1 线性相位的条件线性相位意味着一个系统的相频特性是频率的线性函数，即 ()
式中为常数，此时通过这一系统的各频率分
量的时延为一相同的常数，系统的群时延为
g
d () d
5
线性相位FIR滤波器的DTFT为
N1
H e j h n e jn H e j () H ()e j
26
截短并移位的脉冲响应
过渡带带宽＝阻带边缘频率－通带边缘频率设计中用的通带边缘频率=所要求的通带边缘频率+(过渡带带宽/2) 27
20
例1 N=5, h (0) = h (1) = h (3) = h (4) = -1/2, h (2) = 2，求幅度函数H (ω)。
解Ｎ为奇数并且h(n)满足偶
对称关系 a (0) = h (2) = 2 a (1) = 2 h (3) = -1 a (2) = 2 h (4) = -1 H (ω) = 2 - cosω- cos2ω

精品课件-数字信号处理-第4章

(4.1.3)
第4章快速傅里叶变换(FFT)
可见，1次复数乘法相当于4次实数乘法和2次实数加法，1次复数加法相当于2次实数加法。因此，计算一个X(k)需要4N次实数乘法和2N+2(N－1)=4N－2次复数加法，整个DFT运算需要 4N2次实数乘法和N(4N－2)次实数加法。
当N ＞＞1时，N(N－1)≈N2，N(4N－2)≈4N2，因此，不管以复数还是实数进行统计，直接计算N点DFT的乘法或加法次数都与N2成正比，随着N的增加，运算量增加越来越快，特别是N很大时，运算量将非常可观。例如，N=8时，次数为N2=64； N=1024时，次数为N2=1 048 576，超过100万次。对于各种实时性很强的信号处理应用来说，要求计算速度特别快，因此必须改进DFT的计算方法，减少运算量。
第4章快速傅里叶变换(FFT) 第4章快速傅里叶变换(FFT)
4.1 直接计算DFT的运算量及改进途径 4.2 时间抽取法(DIT)基-2 FFT算法 4.3 频域抽取法(DIF)基-2 FFT算法 4.4 快速傅里叶逆变换(IFFT)算法 4.5 FFT算法的工程实现考虑习题
第4章快速傅里叶变换(FFT)
DIF-FFT
第4章快速傅里叶变换(FFT)
4.2 时间抽取法(DIT)基-2 FFT算法 4.2.1 DIT-FFT算法原理
设序列x(n)长度N=2M，N点DIT-FFT算法对应序列奇偶分解，令x1(r)、x2(r)
x1(r) x(2r)， r 0,1, 2,
, N 1 2
x2 (r) x(2r 1)， r 0,1, 2,
4.1 直接计算DFT的运算量及改进途径 4.1.1 直接计算DFT的运算量
设x(n)是长度为N的有限长序列，其N点DFT为

精品文档-数字信号处理(吴瑛)-第4章

第4章快速傅里叶变换(FFT)
由于1点序列的DFT值为其序列本身，因此在最后一次分
解后，流图中已经没有直接计算DFT的环节。第三次分解旋转
因子为
W20 ，WN运0 算流图如图4.3.4
由以上例子我们可以看出，由于每一次分解都是按输入序
列在时域上的次序是偶数还是奇数来抽取的，最终分解成N个1
点DFT，因此称为基2时分FFT。
第4章快速傅里叶变换(FFT)
4.1 引言 4.2 提高DFT运算效率的基本途径 4.3 基2时分FFT算法 4.4 基2频分FFT算法 4.5 IDFT的快速算法 4.6 实序列DFT的有效计算方法 4.7 线性调频Z(Chirp-Z)变换算法习题与上机题
第4章快速傅里叶变换(FFT)
4.1 引言
利用旋转因子的可约性，即
W nkm Nm
WNnk
，
X(k)
X
(k)
N 2
1
x1
(r
)WNrk/
2
WNk
N 2
1
x2
(r
)WNrk/
2
,
0 k N 1
r 0
r 0
第4章快速傅里叶变换(FFT)
由于X1(k)和X2(k)都隐含周期性，周期为 N ，因此上式 2
X (k) X~1(k) WNk X~2 (k), 0 k N 1
N 2
)
X1(k
)
WNk
X
2
(k
)
第4章快速傅里叶变换(FFT)
X (k) X1(k) WNk X 2 (k),
0 k N 1 2
X
(k
N 2
)
X1(k

数字信号处理第4章PPT课件

第24页/共27页
4 快速卷积型
利用圆周卷积定理，采用FFT实现有限长序列x(n)和h(n)的线性卷积，则可得到FIR滤波器的快速卷积结构
x(n)
L 点 X(k)
X(k)·H(k) L 点
y(n)
FFT
IFFT
H(k )
L点 FFT
h(n) FIR的快速卷积型结构
第25页/共27页
THE END
i0
i1
令M=N时，方程对应的信号流图可表示成
第12页/共27页
N
N
y(n) bi x(n 1) ai y(n i)
i0
i 1
直接3;1个乘法器
第13页/共27页
2. 直接型(II型 )---正准型结构
N
N
y(n) bi x(n 1) ai y(n i)
FIR滤波器结构通常采用非递归结构。基本网络结构包括直接型、级联型、频率采样型与快速卷积型
1 直接型 (卷积型、横截型)
FIR数字滤波器的h(n)，传递函数和差分方程分别为
N 1
H (z) h(n)zn
n0 N 1
y(n) h(m)x(n m)
m0
第18页/共27页
▪ FIR的直接型结构
|Hc(e j)|
x(n)
yc(n)
－z－N
o 2 / N
▪ FIR滤波器的频率采样型结构
H ( z)
1 N
(1
z
N
)
N 1 k 0
1
H (k ) WNk z
1
第23页/共27页
▪ 频率采样型结构的优点： • 可直接控制滤波器的响应 • 结构便于标准化、模块化
▪ 结构的缺点： • H(k)和WN-k一般为复数，硬件实现不方便 • 寄存器的有限字长效应会影响系统的稳定性

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Advanved Digital Signal Processing
©山东大学信息科学与工程学院
4.1.1 Linear Nonparametric Signal Models
Consider a stable LTI system with impulse response h(n) and input w(n). The output x(n) is given by the convolution summation which is known as a nonrecursive system representation because the output is computed by linearly weighting samples of the input signal. • Linear random signal model If the input w(n) is a zero-mean white noise process with variance σw2, autocorrelation rw(l)=σw2δ(l) ,and PSD Rw(ejω)=σw2, -π<ω≤π, then from Table 3.2 the autocorrelation, complex PSD, and PSD of the output x(n) are given by, respectively,
2/109 2016/4/13
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©山东大学信息科学与工程学院
Chapter 4 Linear Signal Models
Low-order models are studied in detail. An understanding of the correlation and spectral properties of a signal model is very important for the selection of the appropriate model in practical applications. Finally, we investigate a special case of pole-zero models with one or more unit poles. Pole-zero models are widely used for the modeling of stationary signals with short memory whereas models with unit poles are useful for the modeling of certain nonstationarity processes with trends.
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Advanved Digital Signal Processing
©山东大学信息科学与工程学院arametric signal model When the LTI filter is specified by its impulse response, we have a nonparametric signal model because there is no restriction regarding the form of the model and the number of parameters is infinite. However, if we specify the filter by a finite-order rational system function, we have a parametric signal model described by a finite number of parameters. We focus on parametric models because they are simpler to deal with in practical applications.
3/109 2016/4/13
Advanved Digital Signal Processing
©山东大学信息科学与工程学院
4.1 Introduction
• Motive In practice, we need to generate random signals that possess certain known, second-order characteristics, or we need to describe observed signals in terms of the parameters of known random processes. • Harmonic process model The simplest random signal model is the wide sense stationary white noise sequence w(n) ~ WN(0,σw2) that has uncorrelated samples and a flat PSD. It is also easy to generate in practice by using simple algorithms. If we wish to generate a random signal with a line PSD using filtering approach, we need an oscillator. Unfortunately, such a system is not stable, and its output cannot be stationary.
8/109 2016/4/13
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4.1.1 Linear Nonparametric Signal Models
• Linear random signal model
When the input is a white noise process, the shape of the autocorrelation and the power spectrum (second-order moments) of the output signal are completely characterized by the system.
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4.1 Introduction
• Two major topics The derivation of the second-order moments of AP, AZ, and PZ models, given the coefficients of their system function. The design of an AP, AZ, or PZ system that produces a random signal with a given autocorrelation sequence or PSD function. The second problem is known as signal modeling and theoretically is equivalent to the spectral factorization procedure developed in Section 2.4.4. In this chapter we primarily focus on parametric models that replicate the second-order properties (autocorrelation of PSD) of stationary random sequences.
In this chapter we introduce and analyze the properties of a special class of stationary random sequences that are obtained by driving a LTI system with white noise. We focus on filters having a system function that is rational. We will use the term pole-zero models when we want to emphasize the system viewpoint and the term autoregressive moving-average models to refer to the resulting random sequences. We discuss the impulse response, autocorrelation, power spectrum, partial autocorrelation, and cepstrum of all-pole, all-zero, an pole-zero models. We express all these quantities in terms of the model coefficients and develop procedures to convert from one parameter set to another.
We use the term system-based signal model to refer to the signal generated by a system with a white noise input. If the system is linear, we use the term linear random signal model. In the statistical literature, the resulting model is known as the general linear process model. In some applications it is more appropriate to use a deterministic input with flat spectral envelope or a “white” harmonic process input.

数字信号处理chapter4

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