Linear Systems: Discrete case & 2DJanuary 14, 200301/09/03ECE 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 Convolution01/09/03ECE 178: Linear Systems Review 32-D SystemsX(n1,n2)n1n201/09/03ECE 178: Linear Systems Review 4Impulse Function(Kronecker Delta)δ(n1,n2)=1 if n1=0 and n2=0.=0 otherwise.n1n201/09/03ECE 178: Linear Systems Review 5Line ImpulseδT(n1,n2)=1 for all n1=0.=0 otherwise.n1n201/09/03ECE 178: Linear Systems Review 6Unit Step Functionu(n1,n2)=1 for all n1,n2>0.=0 otherwise.n1n201/09/03ECE 178: Linear Systems Review 7“ System”An input-output relationship is called a system if there is a unique output for any given input.Y(n1,n2) = T[X(n1,n2)]XYT01/09/03ECE 178: Linear Systems Review 8Linear SystemsThe linearity of a system T is defined as Linearity: 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)]201/09/03ECE 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 characterized by its response to the impulse function δ(δ(δ(δ(m,n).01/09/03ECE 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) ], andyn n T xk k n k n kxk k T n k n kxk k hn k n kyn n hn n xn nkkkkkk(, ) (, )( , )(, ) ( , )(, )( , )(, ) (, ) ,12 12 1 12 212 1 12 212 1 12 212 12 12212121=--LNMOQP=--=--=*=-••=-••=-••=-••=-••=-••ÂÂÂÂÂÂδδbg01/09/03ECE 178: Linear Systems Review 11Convolution: example1 1 4 10 2 5 3012x(m,n) 1 1 10 1 -101h(m,n)mnmn-1 11 1h(-m, -n)-1 11 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!01/09/03ECE 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
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