Evaluation of the DTFT by the DFT If the DTFT of a finite sequence is taken, the result is a continuous function of ω . If the DFT of the same sequence is taken, the results are N evenly spaced samples of the DTFT. In other words, the DTFT of a finite signal can be evaluated at N points with the DFT. X ( ω ) = D T F T { x ( n ) } = ∑ n = − ∞ ∞ x ( n ) e − j ω n and because of the finite length X ( ω ) = ∑ n = 0 N − 1 x ( n ) e − j ω n . If we evaluate ω at N equally space points, this becomes X ( 2 π N k ) = ∑ n = 0 N − 1 x ( n ) e − j 2 π N k n which is the DFT of x ( n ) . By adding zeros to the end of x ( n ) and taking a longer DFT, any density of points can be evaluated. This is useful in interpolation and in plotting the spectrum of a finite length signal. This is discussed further in Chapter . There is an interesting variation of the Parseval's theorem for the DTFT of a finite length- N signal. If x ( n ) ≠ 0 for 0 ≥ n ≥ N − 1 , and if L ≥ N , then ∑ n = 0 N − 1 | x ( n ) | 2 = 1 L ∑ k = 0 L − 1 | X ( 2 π k / L ) | 2 = 1 π ∫ 0 π | X ( ω ) | 2 ⅆ ω . The second term in () says the Riemann sum is equal to its limit in this case.