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

2.1 Discrete-time signals:sequences 2.1.1 Definition 2.1.2 Classification of sequence 2.1.3 Basic sequences 2.1.4 Period of sequence 2.1.5 Symmetry of sequence 2.1.6 Energy of sequence 2.1.7 The basic operations of sequences

2.1.1 Definition EXAMPLE Enumerative representation Function representation

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

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,'.')

Sampling the analog waveform Figure 2.2 EXAMPLE Sampling the analog waveform

Display the wav speech signal in ULTRAEDIT

Display the wav speech signal in COOLEDIT The whole waveform Display the wav speech signal in local Blowup

2.1.2 Classification of sequence Right-side Left-side Two-side Finite-length Causal Noncausal

2.1.3 Basic sequences 1. Unit sample sequence 2．The unit step sequence 3．The rectangular sequence

 4.  Exponential sequence

5. Sinusoidal sequence

For convenience, sinusoidal signals are usually expressed by exponential sequences. The relationship between ω and Ω：

2.1.4 Period of sequence

Three kinds of period of sequence

2.1.5 Symmetry of sequence Conjugate-symmetric sequence Conjugate-antisymmetric sequence

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.

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))

2.1.6 Energy of sequence

2.1.7 The basic operations of sequences

Basic operations of sequences

Original speech sequences Original music sequence sequences after scalar multiplication sequences after vector addition sequences after vector multiplication echo

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-2000+1); 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.

The matlab codes on the addition of two sequences EXAMPLE

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 =3 -6 12 18 -15 24 31 -2 4 6 -5 8 10

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

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)

8.crosscorrelation: aotocorrelation:

example：correlation detection in digital audio watermark

2.1 summary 2.1.1 Definition 2.1.2 Classification of sequence 2.1.3 Basic sequences 2.1.4 Period of sequence 2.1.5 Symmetry of sequence 2.1.6 Energy of sequence 2.1.7 The basic operations of sequences

key： convolution requirements：judge the period of sequence ; calculate convolution with graphical and analytical evaluation . key： convolution

2.2 Discrete-time system 2.2.1 Definition：input-output description of systems 2.2.2 Classification of discrete-time system 2.2.3 Linear time-invariant system（LTI） 2.2.4 Linear constant-coefficient difference equation 2.2.5. Direct implementation of discrete-time system

2.2.1 definition：input-output description of systems the impulse response

EXAMPLE

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：

2.2.3 linear time-invariant system（LTI） How to get h[n] from the input and output：

the impulse response in LTI EXAMPLE

Properties of LTI Figure 2.12 h[n]

classification of linear time-invariant system IIR: h[n]’s length is infinite the latter input the former input FIR must be stable。

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

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.

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

3.computation of difference equations with homogeneous and particular solution

2.2.5. Direct implementation of discrete-time system EXAMPLE

EXAMPLE

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])

2.2 summary 2.2.1 Definition：input-output description of systems 2.2.2 Classification of discrete-time system 2.2.3 Linear time-invariant system（LTI） 2.2.4 Linear constant-coefficient difference equation 2.2.5. Direct implementation of discrete-time system

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

2.3 frequency-domain representation of discrete-time signal and system 2.3.1 definition of fourier transform 2.3.2 frequency response of system 2.3.3 properties of fourier transform

EXAMPLE The intuitionistic meaning of frequency-domain representation of signals

The intuitionistic meaning of frequency-domain representation of systems

The effect of lowpass and highpass filters to image signals EXAMPLE

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

Derivation of Fourier transform

2.3.1 definition of fourier transform arbitrary phase

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

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

2.3.2 frequency response of system

Ideal filter in frequency and time domain EXAMPLE

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)

Eigenfunction and steady-state response： Steady-state response transient response

Figure 2.20 causal FIR system acts on causal signal Causal and stable IIR system acts on causal signal Figure 2.20

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]

2.3.3 properties of fourier transform

2.3 summary 2.3.1 definition of fourier transform 2.3.2 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

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： 2.35 2.45 2.57