(Solution) - In Section 2 8 the conjugate symmetric and conjugate antisymmet -(2025 Original AI-Free Solution)

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In section 8.2, we stated the property that if
x1[n] = x[n ? m],
then
X1[k] = WkmN X[k].
Where X[k] and X1[k] are the DFS coefficients of x[n] and x1[n], respectively. In this problem, we consider the proof of that property.
(a) Using Eq (8.11) together with an appropriate substitution of variables, show that X1[k] can be expressed as
(b) The summation in Eq. (p8.52-1) can be rewritten as, using the fact that x[r] and WkrN are both periodic, show that
(c) From your results in parts (a) and (b), show that