In deriving the DFS analysis Eq. (8.11), we used the identity of Eq. (8.7). To verify th...

In deriving the DFS analysis Eq. (8.11), we used the identity of Eq. (8.7). To verify this identity, we will consider the two conditions k – r =mN and k –r ≠ mN separately.

(a) For k − r = mN, show that e^j (2π/N)(k−r)n = 1 and, from this, that

(b) Since k and r are both integers in Eq. (8.7), we can make the substitution k − r = and consider the summation

Because this is the sum of a finite number of terms in a geometric series, it can be expressed in closed form as

For what values of is the right-hand side of Eq. (P8.54-3) equation indeterminate? That is, are the numerator and denominator both zero?

(c) From the result in part (b), show that if k –r ≠ mN, then