Question

The discrete Fourier transform of an mxn matrix X = (Xj.k) is an m x n matrix X = (ĉik) m,n ypj yqk pqSm Sn, p.q=1 where Šm = e270i/m The corresponding inverse Fourier transform is m.n Xj,k = (mn)-1 » počinje-k. p, Sm Sn . peq=1 Let X and Y be mxn matrices with the discrete Fourier transforms X and Y respectively. Define two dimensional circular convolution Z = X * Y to be min Za,b = XXj,kYa–j,b-k j,k=1 where the subscripts are computed modulo m or n as appropriate. Verify X* Y→ Ñ.î. That is, circular convolution in the spatial domain becomes a point-wise product in the frequency domain. Hint: Look at the proof for the 1D case.

Answer 1

Solution Guven data, The discrete fourier transform of an man matrix x = ( xj ,K) is an man matrix 3: () ,x) min eli tak 0221 where 5m= 20 lm The corresponding Inverse fawrier transform xjr = (mm) mang som sa Pal het 'X' discrete & 'y be man matrices with the Fourier transforms 'î'" & "ů respectively given, x(K) = DFT {x{m}, YCK) : DFT { 4 (n)} 30, XK9 (K) = OF {{4cm)} Ⓡ {41033}

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