By Arndt J
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Extra info for Algorithms for Programmers
8 The z-transform (ZT) In this section we will learn a technique to compute the FT by a (linear) convolution. In fact, the transform computed is the z-transform, a more general transform that in a special case is identical to the FT. 25) x=0 The z-transform is a linear transformation, its most important property is the convolution property CHAPTER 2. 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.
G. L is a power of 2). Second remember that the FT is the special case z = e±2 π i/n of the ZT: With the chirp ZT algorithm one also has an (arbitrary length) FFT algorithm The transform takes a few times more than an optimal transform (by direct FFT) would take. The worst case (if only FFTs for n a power of 2 are available) is n = 2p + 1: One must perform 3 FFTs of length 2p+2 ≈ 4 n for the computation of the convolution. So the total work amounts to about 12 times the work a FFT of length n = 2p would cost.
17) x>τ Final division of this element (by V τ ) gives h(0) + V n h(1) as stated. 18) This gives a nice possibility to directly use complex FFTs for the computation of a linear (acycclic) convolution of two real sequences: for length-n sequences the elements of the linear convolution with indices 0, 1, . . , n − 1 are then found in the real part of the result, the elements n, n + 1, . . , 2 n − 1 are the imaginary part. 19) Cyclic, negacyclic and right-angle convolution can be understood as a polynomial product modulo z n − 1, z n + 1 and z n ± i, respectively (cf.
Algorithms for Programmers by Arndt J