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Apply a FFT on each column (of the transposed matrix). 21) For the acyclic (or linear) convolution of sequences one can use the cyclic convolution of the zero padded sequences zx := {x0 , x1 , . . , nn−1 , 0, 0, . . e. x with n zeros appended). ω) B {ω} B is the cc. of C {ω2 } C and therefore every B {} B-term is the cc. of the C {} C-term in the same line. Is there a nice and general scheme for real valued convolutions based on the MFA? Read on for the positive answer. 6 s, d lower half plus/minus higher half of x CHAPTER 2.

Division by n is multiplication by the inverse of n. 2) Cf. [1], [3], [14] or [2] and books on number theory. 1 Prime modulus: Z/pZ = Fp If the modulus is a prime p then Z/pZ is the field Fp : All elements except 0 have inverses and ‘division is possible’ in Z/pZ. Thereby the second condition is trivially fulfilled for all FFT lengthes n < p: a prime p is coprime to all integers n < p. 1n coprime to m ⇐⇒ gcd(n, m) = 1 63 CHAPTER 4. NUMBERTHEORETIC TRANSFORMS (NTTS) 64 Roots of unity are available for the maximal order R = p−1 and its divisors: Therefore the first condition on n for a length-n mod p FFT being possible is that n divides p − 1.

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. g. L is a power of 2).

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