In section 8 2, we stated the property that if x 1 n x n m , then X 1 k W km N X k Where X k and X 1 k are the DFS coefficients of x n and x 1 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 X 1 k can be expressed as (b) The summation in Eq (p8 52 1) can be rewritten as, using the fact that x r and W kr N are both periodic, show that (c) From your results in parts (a) and (b), show that N 1 n W (8 11) n 0 Part a N 1 m ( ( W j W (P8 52 1) Part b N I m N 1 m wy jw (P8 52 2) rel) N 1 w w r W (P8 52 3) r N m Part c (k w WW WW'X1k 3D 0

Question: In section 8.2, we stated the property that if? x 1 [n] = x[n ? m], ?then? X 1 [k] = W km N X[k].

In section 8.2, we stated the property that if?

x₁[n] = x[n ? m],

?then?

X₁[k] = W^km_N 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 W^kr_N are both periodic, show that

N-1 -n] W". (8.11) n=0 Part a N-1-m ( (]- W" j

N-1 -n] W". (8.11) n=0 Part a N-1-m ( (]- W" j W. (P8.52-1) - Part b N-I-m N-1-m wy + jw. (P8.52-2) rel) N-1 w. w r]W. (P8.52-3) r=N-m [! Part c . (k-w]WW = WW'X1k %3D -0

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