Fix any nonzero complex number z. Engineers working in audio signal processing invented the chirp z-transform,...

 

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Fix any nonzero complex number z. Engineers working in audio signal processing invented the chirp z-transform, in which an input vector (xo,...,X-1) is transformed into the vector y = (yo,..,Yn-1) defined by n-1 Y; = E x2 (3) i=0 (In the special case that z is a primitive nth root of unity, this is a Fourier transform over Z/n, but here we consider the more general case.) Show how to compute the chirp transform in time O(n log n). You may assume that arithmetic operations on complex numbers take unit time. Hint: use the identity ij -f/2 = /2-(-/2 to rewrite Eq. 3 to yield an expression for the function yjz-/2 as a convolution of two functions. Fix any nonzero complex number z. Engineers working in audio signal processing invented the chirp z-transform, in which an input vector (xo,...,X-1) is transformed into the vector y = (yo,..,Yn-1) defined by n-1 Y; = E x2 (3) i=0 (In the special case that z is a primitive nth root of unity, this is a Fourier transform over Z/n, but here we consider the more general case.) Show how to compute the chirp transform in time O(n log n). You may assume that arithmetic operations on complex numbers take unit time. Hint: use the identity ij -f/2 = /2-(-/2 to rewrite Eq. 3 to yield an expression for the function yjz-/2 as a convolution of two functions.

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