We stated in Section 8.4 that a direct relationship between X(ejω) and ˜X [k] can be der...

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 Section 2.9.7. The procedure is as follows:

1. With as an impulse train; i.e., scaled and shifted impulse functions S(ω).

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

3. Since 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.