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 ej (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
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