Linear Systems: Discrete case & 2D Required Reading: 3.4 ECE 178: Linear Systems Review 2 Linear systems-review Discrete case & 2D 2D impulse function Line function Step function Linear systems and Shift invariance Impulse Response of LSI Systems 2-D ConvolutionECE 178: Linear Systems Review 3 2-D Systems X(n1,n2) n1 n2 ECE 178: Linear Systems Review 4 Impulse Function(Kronecker Delta) δ(n1,n2)=1 if n1=0 and n2=0. =0 otherwise. n1 n2ECE 178: Linear Systems Review 5 Line Impulse δT(n1,n2)=1 for all n1=0. =0 otherwise. n1 n2 ECE 178: Linear Systems Review 6 Unit Step Function u(n1,n2)=1 for all n1,n2>0. =0 otherwise. n1 n2ECE 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)] X Y T ECE 178: Linear Systems Review 8 Linear Systems The 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)]2ECE 178: Linear Systems Review 9 Linear Shift Invariant Systems Shift Invariance: T[X(m-k, n-l)] = Y(m-k, n-l) where Y(m,n) = T[X(m,n)]. A LSI system is completely characterized by its response to the impulse function δ(m,n). ECE 178: Linear Systems Review 10 Convolution Let h(n1,n2) = T[δ(n1,n2)]; y(n1,n2) = T[x(n1,n2)]; then h(n1- k1 ,n2 - k2) = T[ δ(n1- k1 ,n2 - k2) ], andECE 178: Linear Systems Review 11 Convolution: example 1 1 4 1 0 2 5 3 0 1 2 x(m,n) 1 1 1 0 1 -1 0 1 h(m,n) m n m n -1 1 1 1 h(-m, -n) -1 1 1 1 h(1-m, n) y(1,0) = Σ k,l x(k,l)h(1-k, -l) = 0 0 0 0 0 -2 5 0 0 0 0 0 = 3 1 5 5 1 3 10 5 2 2 3 -2 -3 m n y(m,n)= verify! ECE 178: Linear Systems Review 12 Discrete Convolution in 2D x(m’,n’) h(m’,n’) m’ n’ h(m-m’,n-n’) m n output=sum of the product of the two in the overlapped region.ECE 178: Linear Systems Review 13 Correlation vs Convolution ECE 178: Linear Systems Review 14 1-D exampleECE 178: Linear Systems Review 15 2-D example ECE 178: Linear Systems Review 16 Matlab command:
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