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3): Convolution in original space corresponds to ordinary (elementwise) multiplication in z-space. ) Note that the special case z = e±2 π i/n is the discrete Fourier transform. 2 (chirp z-transform) Algorithm for the chirp z-transform: 1. Multiply f elementwise with z x 2 /2 . 2. Convolve (acyclically) the resulting sequence with the sequence z −x is required here. 3. Multiply elementwise with the sequence z k 2 /2 2 /2 , zero padding of the sequences . The above algorithm constitutes a ‘fast’ (∼ n log(n)) algorithm for the ZT because fast convolution is possible via FFT.

1 (cyclic convolution by definition) Compute the cyclic convolution of a[] with b[] using the definition, result is returned in c[] procedure convolution(a[],b[],c[],n) { for tau:=0 to n-1 { s := 0 for x:=0 to n-1 { tx := tau - x if tx<0 then tx := tx + n s := s + a[x] * b[tx] } c[tau] := s } } This procedure uses (for length-n sequences a, b) proportional n2 operations, therefore it is slow for large values of n. The Fourier transform provides us with a more efficient way to compute convolutions that only uses proportional n log(n) operations.

Memory access is without skips) 6. Multiply each matrix element (index r, c) by exp(∓2 π i r c/n). 7. Apply a (length R) FFT on each column (of the transposed matrix). (memory access with C-skips) Note that steps 3, 4 and 5 constitute a length-C convolution. e. convolutions of data sets that do not fit into the RAM workspace. An important consideration is the Minimization of the number of disk seeks The number of disk seeks has to be kept minimal because these are slow operations which, if occur too often, degrade performance unacceptably.

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