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2. Discrete Fourier Transform.(/25) 1. N-th roots of unity are defined as solutions to the equation:...

2. Discrete Fourier Transform.(/25) 1. N-th roots of unity are defined as solutions to the equation: w = 1. There are exactly

2. Discrete Fourier Transform.(/25) 1. N-th roots of unity are defined as solutions to the equation: w = 1. There are exactly N distinct N-th roots of unity. Let w be a primitive root of unity, for example w = exp(2 i/N). Show the following: N, if N divides m k=0 10, otherwise N -1 N wmk 2. Fix and integer N > 2. Let f = (f(0), ..., f(N − 1)) a vector (func- tion) f : [N] → C. The Discrete Fourier Transform of f is another complex vector (function) F: [N] → C, F = (F(0),...,F(N − 1)) of the same dimension N, F(k) = wkn f(n) . Thus Fourier trsansofrm is a linear operator represented by the NxN matrix A= (akn), akn = win • Write explicitly the Fourier matrix od order 4 using w = exp(2mi/4) = • Find the Fourier Transform of the vector (2,1, -2, 1) using the matrix above. • Using item (1) from this problem, verify that the inverse matrix is A-1 = $w-kn). In other words, a vector f can be recovered from its Fourier Transform F by the Fourier Inversion Formula: f(n) = 1 Çw=nk F(x)
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e given w m Suppose wm- o ou 1ON W IM? - 2 [Summation formula for. Q.P] work e di Co k me NE] Now, assume, N/m NA) worx N = N

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