By Arndt J.

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**Example text**

11) x>τ x≤τ where the rhs. sums are silently understood as restricted to 0 ≤ x < n. For 0 ≤ τ < n the sum Sτ is always zero because b2n+τ −x is zero (n ≤ 2n + τ − x < 2n for 0 ≤ τ − x < n); (0) the sum Rτ is already equal to hτ . For n ≤ τ < 2n the sum Sτ is again zero, this time because it (1) extends over nothing (simultaneous conditions x < n and x > τ ≥ n); Rτ can be identified with hτ (0 ≤ τ < n) by setting τ = n + τ . e. the right half of the linear convolution element-wise added to the left half.

11) x>τ x≤τ where the rhs. sums are silently understood as restricted to 0 ≤ x < n. For 0 ≤ τ < n the sum Sτ is always zero because b2n+τ −x is zero (n ≤ 2n + τ − x < 2n for 0 ≤ τ − x < n); (0) the sum Rτ is already equal to hτ . For n ≤ τ < 2n the sum Sτ is again zero, this time because it (1) extends over nothing (simultaneous conditions x < n and x > τ ≥ n); Rτ can be identified with hτ (0 ≤ τ < n) by setting τ = n + τ . e. the right half of the linear convolution element-wise added to the left half.

9 as CHAPTER 1. 9 (matrix Fourier algorithm) The matrix Fourier algorithm (MFA) for the FFT: 1. Apply a (length R) FFT on each column. 2. Multiply each matrix element (index r, c) by exp(±2 π i r c/n) (sign is that of the transform). 3. Apply a (length C) FFT on each row. 4. Transpose the matrix. Note the elegance! 10 (transposed matrix Fourier algorithm) The (TMFA) for the FFT: transposed matrix Fourier algorithm 1. Transpose the matrix. 2. Apply a (length C) FFT on each column (transposed row).