Linear Systems: Discrete case & 2DECE 178: Linear Systems Review 2Linear systems-reviewPart 1: Review from G&W (continuous case)Part 2: Discrete case & 2D2D impulse functionLine functionStep functionLinear systems and Shift invarianceImpulse Response of LSI Systems2-D ConvolutionECE 178: Linear Systems Review 32-D SystemsX(n1,n2)n1n2ECE 178: Linear Systems Review 4Impulse Function(Kronecker Delta)δ(n1,n2)=1 if n1=0 and n2=0. =0 otherwise.n1n2ECE 178: Linear Systems Review 5Line ImpulseδT(n1,n2)=1 for all n1=0. =0 otherwise.n1n2ECE 178: Linear Systems Review 6Unit Step Functionu(n1,n2)=1 for all n1,n2>0. =0 otherwise.n1n2ECE 178: Linear Systems Review 7“ System”An input-output relationship is called a system if there is aunique output for any given input.Y(n1,n2) = T[X(n1,n2)]XYTECE 178: Linear Systems Review 8Linear SystemsThe linearity of a system T is defined asLinearity:T[a X1(n1,n2) + b X2(n1,n2)] = a Y1 + b Y2(i.e., principle of superposition holds).Are these linear? (a) y(m,n) = x(m,n) g(m,n) (b) y(m,n) = [x(m,n)]2ECE 178: Linear Systems Review 9Linear Shift Invariant SystemsShift Invariance:T[X(m-k, n-l)] = Y(m-k, n-l) whereY(m,n) = T[X(m,n)].A LSI system is completely characterizedby its response to the impulse functionδ(m,n).ECE 178: Linear Systems Review 10ConvolutionLet h(n1,n2) = T[δ(n1,n2)]; y(n1,n2) = T[x(n1,n2)]; thenh(n1- k1 ,n2 - k2) = T[ δ(n1- k1 ,n2 - k2) ], andECE 178: Linear Systems Review 11Convolution: example1 1 4 10 2 5 3 0 1 2x(m,n)1 1 10 1 -1 0 1h(m,n)mnmn-1 1 1 1h(-m, -n)-1 1 1 1h(1-m, n)y(1,0) = Σ k,l x(k,l)h(1-k, -l) = 0 0 0 00 -2 5 00 0 0 0= 31 5 5 13 10 5 22 3 -2 -3mny(m,n)=verify!ECE 178: Linear Systems Review 12Discrete Convolution in 2Dx(m’,n’)h(m’,n’)m’n’h(m-m’,n-n’)mnoutput=sum of the productof the two in the overlapped region.ECE 178: Linear Systems Review 13Correlation vs ConvolutionECE 178: Linear Systems Review 141-D exampleECE 178: Linear Systems Review 152-D exampleECE 178: Linear Systems Review 16Matlab command:
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