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Unformatted text preview: X(exp(j*w)) = (1-exp(-j*w*(M+1))/(1-exp(-j*w)) for w taking values other than multiples of 2*pi. (Otherwise, X(exp(j*W)) = M+1.) The same sum was encountered in Section 3.8.2, as the inner product <v,s> where v = exp(j*w*n) , n = 0:M(i.e., L=M+1) s = exp(j*0*n) = 1, n = 0:M(i.e., w0 = 1) The final expression for that sum was exp(-j*M*w/2) * F(w) Sheet1 Page 2 where F(w) = sin((M+1)*w/2)/sin(w/2), a symmetric function of w. (ii) x = ones(24,1) would be the simplest such definition. fft(x,500) would then compute the DFT of a vector of 24 ones padded with 477 zeros. This is the same as the DTFT of the given (finite-duration) sequence, computed at 500 equally spaced frequencies ranging from 0 to 2*(0.998)*pi....
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- Spring '08