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**Chapter 2 Discrete-time signals and systems**

2.1 Discrete-time signals:sequences 2.2 Discrete-time system 2.3 Frequency-domain representation of discrete-time signal and system

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**2.1 Discrete-time signals:sequences**

2.1.1 Definition Classification of sequence Basic sequences Period of sequence 2.1.5 Symmetry of sequence Energy of sequence 2.1.7 The basic operations of sequences

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**2.1.1 Definition EXAMPLE Enumerative representation**

Function representation

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**Graphical representation**

-2 2 4 6 -3 -1 1 5 10 -0.5 0.5 Graphical representation

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**Generate and plot the sequence in MATLAB**

x=[1,2,1.2,0,-1,-2,-2.5] stem(n,x, '.') n=0:9 y=0.9.^n.*cos(0.2*pi*n+pi/2) stem(n,y,'.')

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**Sampling the analog waveform**

Figure 2.2 EXAMPLE Sampling the analog waveform

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**Display the wav speech signal in ULTRAEDIT**

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**Display the wav speech signal in COOLEDIT**

The whole waveform Display the wav speech signal in local Blowup

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**2.1.2 Classification of sequence**

Right-side Left-side Two-side Finite-length Causal Noncausal

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**2.1.3 Basic sequences 1. Unit sample sequence 2．The unit step sequence**

3．The rectangular sequence

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4. Exponential sequence

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5. Sinusoidal sequence

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**For convenience, sinusoidal signals are usually expressed by exponential sequences.**

The relationship between ω and Ω：

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Period of sequence

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**Three kinds of period of sequence**

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**2.1.5 Symmetry of sequence Conjugate-symmetric sequence**

Conjugate-antisymmetric sequence

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**Real sequences can be decomposed into two symmetrical sequences.**

EXAMPLE n=[-5:5]; x=[0,0,0,0,0,1,2,3,4,5,6]; xe=(x+fliplr(x))/2 ; xo=(x-fliplr(x))/2; subplot(3,1,1) stem(n,x) subplot(3,1,2) stem(n,xe) subplot(3,1,3) stem(n,xo) Real sequences can be decomposed into two symmetrical sequences.

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**Complex sequences can be decomposed into two symmetrical sequences.**

EXAMPLE Complex sequences can be decomposed into two symmetrical sequences. n=[-5:5]; x=zeros(1,11); x((n>=0)&(n<=5))=(1+j).^[0:5] xe=(x+conj(fliplr(x)))/2; xo=(x-conj(fliplr(x)))/2 subplot(3,2,1); stem(n,real(x)) subplot(3,2,2); stem(n,imag(x)) subplot(3,2,3); stem(n,real(xe)) subplot(3,2,4); stem(n,imag(xe)) subplot(3,2,5); stem(n,real(xo)) subplot(3,2,6); stem(n,imag(xo))

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Energy of sequence

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**2.1.7 The basic operations of sequences**

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**Basic operations of sequences**

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**Original speech sequences Original music sequence**

sequences after scalar multiplication sequences after vector addition sequences after vector multiplication echo

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**The matlab codes on the processions**

x=wavread('test1.wav',36000); y=wavread('test2.wav ',36000); z=(x+y)/2.0; wavwrite(z,22050,'test3.wav') y1=y*0.5; wavwrite(y1,22050,'test4.wav') y2=zeros(36000,1); for i=2000:36000 y2(i)=y(i ); end y3=0.6*y+0.4*y2; wavwrite(y3,22050,'test5.wav') w=[0:1/36000:1-1/36000]'; y4=y.*w; wavwrite(y4,22050,'test6.wav') Vector addition realizes composition. scalar multiplication changes the volume. Delay, scalar multiplication and vector addition produce echo. vector multiplication realizes fade-in.

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**The matlab codes on the addition of two sequences**

EXAMPLE

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n=[-4:2] ; x=[1,-2,4,6,-5,8,10] ; %x1[n]=x[n+2] n1=n-2; x1=x; %x2[n]=x[n-4] n2=n+4; x2=x; %y[n] m=[min(min(n1),min(n2)): max(max(n1),max(n2))] ; y1=zeros(1,length(m)) ; y2=y1; y1((m>=min(n1))&(m<=max(n1)))=x1;y2((m>=min(n2))&(m<=max(n2)))=x2; y=3*y1+y2; stem(m,y) Output:y =

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7.convolution sum: steps：turnover, shift, vector multiplication, addition

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**EXAMPLE nx=0:10; x=0.5.^nx; nh=-1:4; h=ones(1,length(nh))**

y=conv(x,h); stem([min(nx)+min(nh):max(nx)+max(nh)],y)

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8.crosscorrelation: aotocorrelation:

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**example：correlation detection in digital audio watermark**

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2.1 summary Definition Classification of sequence Basic sequences Period of sequence Symmetry of sequence Energy of sequence 2.1.7 The basic operations of sequences

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**key： convolution requirements：judge the period of sequence ;**

calculate convolution with graphical and analytical evaluation . key： convolution

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2.2 Discrete-time system Definition：input-output description of systems 2.2.2 Classification of discrete-time system Linear time-invariant system（LTI） 2.2.4 Linear constant-coefficient difference equation Direct implementation of discrete-time system

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**2.2.1 definition：input-output description of systems**

the impulse response

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EXAMPLE

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**2.2.2 classification of discrete-time system**

1．Memoryless (static) system the output depends only on the current input. 2．Linear system 3．Time-invariant system： 4．Causal system： the output does not depend on the latter input. 5．Stable system：

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**2.2.3 linear time-invariant system（LTI）**

How to get h[n] from the input and output：

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**the impulse response in LTI**

EXAMPLE

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Properties of LTI Figure 2.12 h[n]

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**classification of linear time-invariant system**

IIR: h[n]’s length is infinite the latter input the former input FIR must be stable。

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**2.2.4 linear constant-coefficient difference equation**

1.relation with input-output description and convolution EXAMPLE For IIR，the latter two are consistent. input-output description convolution description infinite items，unrealizable difference equation description Finite items, realizable

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**EXAMPLE For FIR，the followings are consistent**

input-output description and difference equation description （non-recursion） Convolution description Another difference equation description，recursion，lower rank For FIR and IIR，difference equations are not exclusive.

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**EXAMPLE 2.Recursive computation of difference equations：**

For IIR, there needs N initial conditions , then ,the solution is unique. For FIR, there needs no initial conditions. With initial-rest conditions (linear, time invariant, and causal), the solution is unique. EXAMPLE

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**3.computation of difference equations with homogeneous**

and particular solution

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**2.2.5. Direct implementation of discrete-time system**

EXAMPLE

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EXAMPLE

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**The matlab codes on the direct realization of LTI**

EXAMPLE The matlab codes on the direct realization of LTI B=1; A=[1,-1] n=[0:100]; x=[n>=0]; y=filter(B,A,x); stem(n,y); axis([0,20,0,20])

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2.2 summary Definition：input-output description of systems 2.2.2 Classification of discrete-time system Linear time-invariant system（LTI） 2.2.4 Linear constant-coefficient difference equation Direct implementation of discrete-time system

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**judge the type of a system（from the relationship between**

keys： judge the type of a system（from the relationship between the input and output, and from h[n] for LTI). the physics meaning of convolution representation for LTI： the output signals are the weighted combination of the input signals， h[n] is the weight。 the similarities and differences between linear constant-coefficient difference equations and convolution representation，recursive computation。 the difference between IIR and FIR： FIR IIR h[n] finite length infinite length y[n]是x[n]的加权 finite items infinite items realization convolution or difference difference , recursion stability stable maybe stable

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**2.3 frequency-domain representation of discrete-time signal and system**

2.3.1 definition of fourier transform frequency response of system 2.3.3 properties of fourier transform

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EXAMPLE The intuitionistic meaning of frequency-domain representation of signals

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**The intuitionistic meaning of frequency-domain representation of systems**

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**The effect of lowpass and highpass filters to image signals**

EXAMPLE

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**Frequency-domain analysis of de-noise process through bandstop filter**

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**Derivation of Fourier transform**

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**2.3.1 definition of fourier transform**

arbitrary phase

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**Matlab codes to draw the frequency chart of signals**

EXAMPLE subplot(2,2,1); fplot('real(1/(1-0.2*exp(-1*j*w)))',[-2*pi,2*pi]); title('实部') subplot(2,2,2); fplot('imag(1/(1-0.2*exp(-1*j*w)))',[-2*pi ,2*pi]); title('虚部') subplot(2,2,3); fplot('abs(1/(1-0.2*exp(-1*j*w)))',[-2*pi,2*pi]); title('幅度') subplot(2,2,4); fplot('angle(1/(1-0.2*exp(-1*j*w)))',[-2*pi,2*pi]); title('相位') Matlab codes to draw the frequency chart of signals

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**Fourier transforms of non-absolutely summable or non-square summable signals**

EXAMPLE EXAMPLE

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**2.3.2 frequency response of system**

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**Ideal filter in frequency and time domain**

EXAMPLE

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**Matlab codes to draw the frequency response of a system**

EXAMPLE h=[1,0,0,0,0,0,0,0,0,0.5] freqz(h,1)

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**Eigenfunction and steady-state response：**

Steady-state response transient response

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**Figure 2.20 causal FIR system acts on causal signal**

Causal and stable IIR system acts on causal signal Figure 2.20

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**example of steady-state response**

Sin(0.1*pi*n) h[n]=[1,1,1,1,1,1,1,1,1,1]/4 B=[1,0,1,0,1];A=[1,0.81,0.81,0.81]

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**2.3.3 properties of fourier transform**

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2.3 summary 2.3.1 definition of fourier transform frequency response of system 2.3.3 properties of fourier transform requirements：calculation of fourier transforms steady-state response linearity time shifting frequency shifting the convolution theorem windowing theorem Parseval’s theorem symmetry properties

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**exercises： 2.35 2.45 2.57 Keys and difficulties：**

the convolution theorem; the frequency spectrum of a real sequence is conjugate symmetric; the frequency spectrum of a conjugate symmetric sequence is a real function. exercises：

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