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

(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