数字信号处理英文课件Chapter4
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
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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 DSP 英文版课件4.0
w1(n), w2(n): contents of the two registers
At time n:
Example 4.1.5： An IIR filter h(n) 0.75 u (n) . Using convolution, derive
n
closed-form expressions(闭合形式表示) for the output signal y(n) when the input is: （a）A unit step(单位阶跃) x(n)=u(n) （b）An alternating step x(n) (1) u (n) (正负交替的阶跃)
Algorithmic form:
for each input sample x，do：
y：= w1
w1：＝x
The delay register is typically initialized (初始化) to zero： w1(0)=0.
double delay：
I/O equation: y(n)=x(n-2)
State-space representation：
y(n)=w1(n) (output equation) w1 (n+1)=x (n) (state updating equation) w1(n)： content of the register or internal state at time n
4.1.6 Flip-and-Slide Form
Flip-and-Slide Form of convolution:
h(n) is flipped(反转) around and slid (slide平移) over the input sequence x(n). y(n) is obtained by computing the dot product(点积) of flipped h with the input samples aligned(对准) below it.
数字信号处理课件第4章
N −1
2 ( = ∑ x(2r )WN rk + ∑ x(2r + 1)WN2 r +1) k r =0
N 2
−1
N 2
−1
r =0
2 k 2 = ∑ x1 (r )(WN ) rk + WN ∑ x2 (r )(WN ) rk r =0 r =0
−1
N 2
−1
根据可约性，W = e
2 N
N 2
X1(k + N ) = X3 (k) −WNk X4 (k), k = 0,1,L, N −1 4 4
2
(二) N/4点DFT
同样对n为奇数时 ， 点分为两个N/4点的 同样对 为奇数时， N/2点分为两个 为奇数时 点分为两个 点的 序列: 序列 x5 (l) = x2 (2l), l = 0,1,L, N −1 4
3
k 则有：X ( k ) = X 1 (k ) + WN X 2 (k ) k X (k + 4) = X 1 (k ) − WN X 2 (k ), k = 0,1,2,3
(一) N/2点DFT 一 点
整个过程如下图所示: 整个过程如下图所示
x1(0)=x(0) )= ( x1(1)= (2) )=x( x1(2)= (4) )=x( x1(3)= (6) )=x( x2(0)= (1) )=x( x2(1)= (3) )=x( x2(2)= (5) )=x( x2(3)= (7) )=x( X1(0) N/2点 N/2点 X1(1) X1(2) DFT X1(3) X2(0) X2(1) X2(2) X2(3)
2
N 2 N2 ( ) = 2 4 N N ( − 1) 2 2
数字信号处理DigitalSignalProcessingppt课件
存储二维的图像信号或多维的阵列信号，实现二维或多维 的滤波及谱分析等。
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▪ 缺点1：增加了系统的复杂性，它需要模拟
接口以及比较复杂的数字系统；
▪ 缺点2：应用的频范围受到限制，主要是
A/D转换的采样频率的限制；
▪ 缺点3：系统的功率消耗比较大。数字信号
交大出版社,2002.
▪ 陈后金.数字信号处理。高等教育出版社，2004.
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课程考核标准
▪ 作业10％ ▪ 考勤10％ ▪ 实验10％ ▪ 期终考试70％
3
绪论
▪ 基本概念 ▪ 基本组成 ▪ 实现方法 ▪ 数字信号处理的特点 ▪ 应用领域 ▪ 发展历史
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1. 基本概念
▪ 信息科学 ▪ 信号 ▪ 信号分类 ▪ 模拟信号 ▪ 数字信号 ▪ 数字信号处理
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▪ 信息科学
▪ 信息科学是研究信息的获取、传输、处理和利 用的一门科学。
▪ 信号
▪ 是信息的表现形式。（而信息则是信号所含有 的具体内容）
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▪ 信号的分类
▪ 依载体：电信号、磁信号、声信号、光信号、热信 号、机械信号。
▪ 依变量个数：一维、二维、多维（矢量）信号。 ▪ 依周期性：周期信号x(t)=x(t+kT); 非周期信号。 ▪ 依是否为确定函数：确定信号；随机信号。 ▪ 依能量或功率是否有限：能量信号；功率信号。 ▪ 依时间和幅度是否连续：模拟信号；数字信号。
▪ 理论基础，其中最主要的是离散时间信号和离 散时间系统理论以及一些数学理论。
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2. 基本组成
模拟信 号
连续时间信 号
数字信号
采样 保持器
A/D 变换器
数字信号处理Chapter_4(第三版教材)
Digital Processing of ContinuousTime Signals
Complete block-diagram
Antialiasing filter
S/H
A/D
DSP
D/A
Reconstruction filter
• Since both the anti-aliasing filter and the reconstruction filter are analog lowpass filters, we review first the theory behind the design of such filters • Also, the most widely used IIR digitae conversion of an analog lowpass prototype
-<n<
with T being the sampling period • The reciprocal of T is called the sampling frequency FT, i.e., FT =1/T
Sampling of Continuous-time Signals
• Now, the frequency-domain representation of ga(t) is given by its continuos-time Fourier transform (CTFT):
• gp(t) is a continuous-time signal consisting of a train of uniformly spaced impulses with the impulse at t = nT weighted by the sampled value ga(nT) of ga(t) at that instant t=nT
数字信号处理第四章(南理工)
(4.12)
• According to the modulation theorem of CTFT 1 Gp (jΩ) = Ga (jΩ)*∆p (jΩ) 2π
1 +∞ = ∑ Ga (j(Ω+ kΩT )) T k=−∞ (4.16)
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Effect of sampling in the frequency-domain ─ Gp(jΩ) is a periodic function of frequency Ω Ω consisting of a sum of shifted and scaled replicas of Ga(jΩ), shifted by integer of ΩT and Ω scaled by 1/T. ─ Baseband signal: the term on the right-hand side of Eq.(4.16) for k=0 is called baseband portion of Gp(jΩ). ─ Baseband / Nyquist band: frequency range −ΩT /2≤ Ω< ΩT /2
a
+∞
(4.11)
8
CTFT Gp(jΩ) of gp(t) Ω • According to the definition of CTFT +∞ +∞ − jΩt Gp (jΩ) = ∫ ∑ ga (nT)δ (t − nT) e dt −∞ n=−∞ +∞ = ∑ ga (nT)e− jΩnT
• Most signals in the real world are continuous in time; • DT signal processing algorithms are being used increasingly; • Digital processing of a CT signal involves 3 basic steps: ─ Sample a CT signal into a DT signal; (analog-to-digital (A/D) converter) ─ Process the DT signal (binary word); ─ Convert the processed DT signal back into CT signal. (digital-to-analog (D/A) converter)
数字信号处理课件第四章1-精选文档
例 ： 已 知 序 列 x ( nR ) ( n ) , 将 x ( n ) 以 N 8 为 周 期 4 进 行 周 期 延 拓 成 x ( n ) ， 求 x ( nD ) 的 F S 。
解 法 一 ： 数 值 解
n k X (k) x (nW ) N n 0 N 1
x(n)W8nk
1 1N X ( l)X ( k l) 1 2 Nl 0 1 1N X l)X ( k l) 2( 1 Nl 0
N 1
例 ： 已 知 序 列 xn () Rn () ， x () n ( n 1 ) Rn () 1 4 2 5 分 别 将 序 列 以 周 期 为 6 周 期 延 拓 成 周 期 序 列 xn () 和 x () n ， 求 两 个 周 期 序 列 的 周 期 卷 积 和 。 1 2
n0
7
nk W 8 n0
3
1 e e
2 j k 8
2 j 2 k 8
e
2 j 3 k 8
X ( 4 )0 X ( 5 )1 j 1 ( 6 )0 X ( 7 )1 j 1 2 X 2
X ( 0 )4 X ( 1 )1 j2 1X ( 2 )0 X ( 3 )1 j2 1
( n ) I D F S [( Y k ) ] x () m x ( n m ) 则 y 1 2
m 0 N 1
x mx ) 1(nm ) 2(
m 0
N 1
同样，利用对称性
若 则
y () n x () n x () n 1 2
n k Y () k D F S [() y n ] y () n W N n 0
数字信号处理4
1
N=3
band and mono-
0.8
N=8
decrease in stop band.
0.6
(2) The larger N, the
0.4
more ripples.
0.2
(3) Three parameters.
0
0
1
2
3
Ω
Magnitude
Dr. Yibiao YU
Infinite Impulse Response Filter
IIR Filter: Infinite Impulse Response Filter
( 9 hours including exercise )
Study Background
X(ejω) ω
x(n)
H(ejω) 1
ω
ωc
Filter
Y(e jω) ω
ωc
y(n)
H(ejω) 1
ω
ωc
Low pass
Dr. Yibiao YU
Infinite Impulse Response Filter
2005
TOPICS
1. Structure characteristics of IIR filter 2. Butterworth, Chebyshev and Elliptic Filter 3. Impulse Invariant IIR filter design 4. Bilinear Transformation 5. From Low pass to High pass, Band pass
N−1
G=
Ω
ce ;
2 2N
Ha (s) s=0 = 1
《数字信号处理》课件第4章
2
N 1
N2 2
第4章 快速傅里叶变换(FFT)
复数加法次数为
N N 1 2N N 2 2 2 2
由此可见，仅仅经过一次分解，就使运算量减少近一半。 既然这样分解对减少DFT的运算量是有效的，且N=2M, N/2仍然是偶数，故可以对N/2点DFT再作进一步分解。
J 0, 1, 2, 3
这种算法使DFT的运算效率提高了1 ~ 2个数量级， 为数字信号处理技术应用于各种信号的实时处理创造 了条件，大大推动了数字信号处理技术的发展。
第4章 快速傅里叶变换(FFT)
人类的求知欲和科学的发展是永无止境的。多年来， 人们继续寻求更快、更灵活的好算法。1984年，法国的 杜哈梅尔(P. Dohamel)和霍尔曼(H. Hollmann)提出的分裂 基快速算法，使运算效率进一步提高。本章主要讨论基 2FFT
x1
(2l
1)WNk
( /
2l 2
1)
l 0
l 0
N / 41
N / 41
x3 (l)WNkl/ 4 WNk / 2
x4
(l
)WNk
l /
4
l 0
l 0
X 3 (k ) WNk/ 2 X 4 (k )
k 0， 1， ， N 1 2
(4.2.9)
第4章 快速傅里叶变换(FFT)
式中
N / 41
204.8
N lbN 5120
2
这样，就使运算效率提高200多倍。图4.2.5为FFT算法
和直接计算DFT所需复数乘法次数CM与变换点数N的关 系曲线。由此图更加直观地看出FFT算法的优越性，显
然，N
第4章 快速傅里叶变换(FFT)
数字信号处理4
Chapter3
22
C QJ
Digital Signal Processing
DFT
圆周共轭对称分量和圆周共轭反对称分量的 特点。
xep (n) = x*ep ((N − n))N RN (n)
xop (n) = −x*op ((N − n))N RN (n)
Chapter3
23
C QJ
Digital Signal Processing
DFT
� 对偶性
DFS[X~ (n)] = N~x(−k)
� 周期卷积和
∑ DFS[ N−1~x1(m)~x2(n − m)] = X~ 1(k)X~ 2(k)
m=0
∑ DFS[ ~x1 (n)~x2 (n)]
=
1 N
N−1X~ 1(l)X~ 2(k
l=0
−
l)
Chapter3
11
C QJ
Digital Signal Processing
WN = e N
⎧
∑ ⎪⎪
⎨ ⎪
∑ ⎪⎩
x~
(
X~ n)
(k =
) = DFS[ x~(n)] IDFS[ X~ (k)] =
=
N
−1
x~
(
n)W
kn N
n=0
1 N
N
−1
X~
(k
)W
− N
k=0
kn
Chapter3
9
C QJ
Digital Signal Processing
DFT
3.3 离散傅里叶级数的性质
C QJ
Digital Signal Processing
DFT
最新版《数字信号处理(英)》精品课件Chap 4 Digital Processing of CT Signals
Normalized digital angular frequency
Example :
CT signal:
xa (t ) A cos(2 f0t ) A cos(0t )
The sampled DT signal:
x[n] A cos(0nT ) 2 A cos(0 n ) A cos(0 n ) T Normalized digital angular frequency 0:
2
• DT signal processing algorithms are being used increasingly;
Simplified Block diagram of a CT signal processed by DT system
xa (t )
x[n] C/D Converter
Chap 4 Digital Processing of CT Signals
Discrete-Time Signal Processing of CTS; Sampling of CT Signals;
Analog Lowpass Filter Design;
1
4.1 Introduction
3
Other additional circuits
• To prevent aliasing, an analog anti-aliasing filter is employed before the S/H circuit; • To smooth the output signal of the D/A converter, which is a staircase-like waveform, an analog reconstruction filter is used.
数字信号处理(英文版)4-LTI离散时间系统的频域表示
§4.3 Frequency Response
Computation Using MATLAB
• The function freqz(h,w) can be used to determine the values of the frequency response vector h at a set of given frequency points w • From h, the real and imaginary parts can be computed using the functions real and imag, and the magnitude and phase functions using the functions abs and angle
k 0
d k z Y ( z ) pk z X ( z )
k 0
N
k
M
k
where Y(z) and X(z) denote the z-transforms of y[n] and x[n] with associated ROCs, respectively
§4.1 LTI Discrete-Time Systems
§4.2 The Frequency
Response
• The function | H(ej) | is called the magnitude response and the function () is called the phase response of the LTI discrete-time system • Design specifications for the LTI discrete-time system, in many applications, are given in terms of the magnitude response or the phase response or both
数字信号处理英文影印版课件4-1
数字信号处理Digital Signal Processing 电子信息工程系韩建峰KeywordsSections⏹Sampling sinusoids⏹Sampling theorem⏹Discrete-to-Continuous Conversion SummaryLECTURE 1Reading assignments This lecture •Chapter 4•Section 4-1KeywordsPart AContinuous-to-Discrete Conversion Sampling & ReconstructionAliasing & FoldingLECTURE OBJECTIVES•SAMPLING can cause ALIASING •Spectrum for digital signals,x [n]•Normalized Frequencyππωω22ˆ+==ss f f T ALIASINGReviewSignalsSampling Reconstruction•Continuous-time Signal•But the key point is that any computer represent ation is discrete.•So, do sampling!•And, how?()cos()x t A tωϕ=+•Sample a continuous-time signal at equally spaced time instants.Take a “snapshot” every Ts.Speech, audio andso on.•Or, compute the values of a discrete-time signal directly from a formula.2=-+[]53x n n nSAMPLING x(t)•SAMPLING PROCESS•Convert x(t) to numbers x[n]•“n” is an integer; x[n] is a sequence ofvalues•Think of “n” as the storage address inmemory•UNIFORM SAMPLING at t = nTs•IDEAL: x[n] = x(nT)sSAMPLING RATE, f s •SAMPLING RATE (f)s–f=1/T ss•NUMBER of SAMPLES PER SECOND –T= 125 microsec f s= 8000 samples/secs–UNITS ARE HERTZ: 8000 Hz •UNIFORM SAMPLING at t = nT= n/f ss–IDEAL: x[n] = x(nT)=x(n/f s)s•Examples of continuous-time signals exist in the “real-world” outside the computer.•Simple mathematical formula.•More general continuous-time signals can be represented as sum of sinusoids.•So, we will use sinusoidal signal as the basis for our study of sampling.sf s T n A n x ωωωϕω==+=ˆ)ˆcos(][)cos()(][)cos()(ϕωϕω+==+=s s nT A nT x n x t A t x •Change x(t) into x[n] DERIV ATION))cos((][ϕω+=n T A n x s DEFINE DIGITAL FREQUENCYDigital Frequency ωˆ•V ARIES from 0to 2π, as f varies from 0 to the sampling frequency•UNITS are radians, not rad/sec–DIGITAL FREQUENCY is NORMALIZEDss f f T πωω2ˆ==Sample RateHow to select theT sSample TheoremA interesting phenomenon•Exercise 4.1•Is this the only possible answer?Hz 1000at sampled )2400cos()(2==s f t t x π21000[]cos(2400)cos(2.4)cos(0.42)cos(0.4)nx n n n n n πππππ===+=()cos(400)x t t π⇒=Aliasing[]cos(0.4)x n n π=Illustration of aliasingDifferent frequency, but same values at n=0,1,2,3…•2.4πis an alias of 0.4π•Exercise 4.2Aliasing•How does aliasing arise in a mathematical treat ment of discrete-time signal?•The last example:12[]cos(0.4)[]cos(2.4)x n n x n n ππ==2[]cos(0.42)cos(0.4)x n n n n πππ=+=Periodic function with period 2πAliasing Derivation-1and we substitute: t ←n f sIf x (t )=A cos(2π(f + f s )t +ϕ)then: x [n ]=A cos(2π(f + f s )n f s +ϕ)or, x [n ]=A cos(2πf f s n +2π n +ϕ)Aliasing Derivation-22ˆs sfT f πωω==+2π 2()22ˆthen: s s s s sf f f f f f f πππω+==+ˆand we want: []cos()x n A n ωϕ=+If x (t )=A cos(2π(f + f s )t +ϕ)t ←nf sFolded Aliasx (t )=A cos(2π(-f + f s )t -ϕ)SAME DIGITAL SIGNALˆ[]cos()x n A n ωϕ=+x [n ]=A cos((2πf T s )n -2π n +ϕ) x [n ]=A cos((-2πfT s )n +(2π f s T s )n -ϕ)x [n ]=x (nT s )=A cos(2π(-f + f s )nT s -ϕ)Aliasing2ˆs sfT f πωω==+2π 2ˆ2s sfT f πωωπ==-+Folded AliasAlisingPrincipal Aliasingˆˆˆ, 2, 2 integer l l l ωωππω+-=General FormulaSpectrum of a Discrete-Time Signal•PLOT versus NORMALIZED FREQUENCY •INCLUDE ALL SPECTRUM LINES –ALIASES•ADD MULTIPLES of 2π•SUBTRACT MULTIPLES of 2π–FOLDED ALIASES•ALIASES of NEGATIVE FREQS12X*–0.5π12X–1.5π12X0.5π2.5π–2.5πˆω12X12X*12X*1.5π))80/)(100(2cos(][ϕπ+=n A n x 80s f Hz=sf fπω2ˆ=ˆ2sff ωπ=f s =125Hz12X*0.4π12X–0.4π1.6π–1.6πˆω12X12X*))125/)(100(2cos(][ϕπ+=n A n x•DEMO: Strobe Movies 12•What is the meaning of this DEMO?•Can you give us more examples in the real world?f Camera: 30 Frames/s Human Eyessf'fSummary2ˆs sfT f πωω==+2π 2ˆ2s sfT f πωωπ==-+Folded AliasAlisingPrincipal Aliasingˆˆˆ, 2, 2 integer l l l ωωππω+-=General FormulaHomeworkP-4.1Review: Chapter 4, Section 4-1 Preview:Chapter 4, Section 4-2,4-4。