19 Real sequences can be decomposed into two symmetrical sequences. EXAMPLEn=[-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.
20 Complex sequences can be decomposed into two symmetrical sequences. EXAMPLEComplex 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)))/2subplot(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))
24 Original speech sequences Original music sequence sequences after scalar multiplicationsequences after vector additionsequences after vector multiplicationecho
25 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=2000:36000y2(i)=y(i );endy3=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.
26 The matlab codes on the addition of two sequences EXAMPLE
35 2.1 summaryDefinitionClassification of sequenceBasic sequencesPeriod of sequenceSymmetry of sequenceEnergy of sequence2.1.7 The basic operations of sequences
36 key： convolution requirements：judge the period of sequence ; calculate convolution with graphicaland analytical evaluation .key： convolution
37 2.2 Discrete-time systemDefinition：input-output description of systems2.2.2 Classification of discrete-time systemLinear time-invariant system（LTI）2.2.4 Linear constant-coefficient difference equationDirect implementation of discrete-time system
38 2.2.1 definition：input-output description of systems the impulse response
40 2.2.2 classification of discrete-time system 1．Memoryless (static) systemthe output depends only on the current input.2．Linear system3．Time-invariant system：4．Causal system：the output does not depend on the latter input.5．Stable system：
41 2.2.3 linear time-invariant system（LTI） How to get h[n] from the input and output：
44 classification of linear time-invariant system IIR: h[n]’s length is infinitethe latter input the former inputFIR must be stable。
45 2.2.4 linear constant-coefficient difference equation 1.relation with input-output description and convolutionEXAMPLEFor IIR，the latter two are consistent.input-output descriptionconvolution descriptioninfinite items，unrealizabledifference equation descriptionFinite items, realizable
46 EXAMPLE For FIR，the followings are consistent input-output description anddifference equation description（non-recursion）Convolution descriptionAnother difference equation description，recursion，lower rankFor FIR and IIR，difference equations are not exclusive.
47 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
48 3.computation of difference equations with homogeneous and particular solution
49 2.2.5. Direct implementation of discrete-time system EXAMPLE
51 The matlab codes on the direct realization of LTI EXAMPLEThe matlab codes on the direct realization of LTIB=1; A=[1,-1]n=[0:100]; x=[n>=0]; y=filter(B,A,x); stem(n,y); axis([0,20,0,20])
52 2.2 summaryDefinition：input-output description of systems2.2.2 Classification of discrete-time systemLinear time-invariant system（LTI）2.2.4 Linear constant-coefficient difference equationDirect implementation of discrete-time system
53 judge the type of a system（from the relationship between keys：judge the type of a system（from the relationship betweenthe 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-coefficientdifference equations and convolution representation，recursive computation。the difference between IIR and FIR：FIR IIRh[n] finite length infinite lengthy[n]是x[n]的加权 finite items infinite itemsrealization convolution or difference difference , recursionstability stable maybe stable
54 2.3 frequency-domain representation of discrete-time signal and system 2.3.1 definition of fourier transformfrequency response of system2.3.3 properties of fourier transform
55 EXAMPLEThe intuitionistic meaning of frequency-domain representation of signals
56 The intuitionistic meaning of frequency-domain representation of systems
57 The effect of lowpass and highpass filters to image signals EXAMPLE
58 Frequency-domain analysis of de-noise process through bandstop filter
60 2.3.1 definition of fourier transform arbitrary phase
61 Matlab codes to draw the frequency chart of signals EXAMPLEsubplot(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
62 Fourier transforms of non-absolutely summable or non-square summable signals EXAMPLEEXAMPLE
73 2.3 summary2.3.1 definition of fourier transformfrequency response of system2.3.3 properties of fourier transformrequirements：calculation of fourier transformssteady-state responselinearitytime shiftingfrequency shiftingthe convolution theoremwindowing theoremParseval’s theoremsymmetry properties
74 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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