We stated in Section 8.4 that a direct relationship between X(e j ) and X[k] can be

We stated in Section 8.4 that a direct relationship between X(e^jω) and X[k] can be derived, where X[k] is the DFS coefficients of a periodic sequence and X(e^jω) is the “Fourier transform of one period. Since X[k] corresponds to samples of X(e^jω), the relationship then corresponds to an interpolation formula. 

One approach to obtaining the desired relationship is to rely on the discussion of Section 8.4, the relationship of Eq (8.54).and the modulation property of Section2.9.7. The procedure is as follows:  

1. With X[k] denoting the DFS coefficients of x[n], express the Fourier transform X(e^jω) of x[n] as an impulse train,

2. From Eq. (8.57) x[n] can be expressed as x[n] = x[n]w[n], where w[n] is an appropriate finite-length window.

3. Since x[n] = x[n]w[n], from Section 2.9.7, X(e^jω) can be expressed as the (periodic) convolution of X(e^jω) and W(e^jω).

By carrying out the details in this procedure, show that X(e^jω) can be expressed as

specify explicitly the limits on the summation. 

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