Question

3.11 [3] If the N-point DCT of a length-N sequence x[] is denoted by C(k) (see Eq. (3.21)),s n-N sequence x[n] is den that the DCT is an energy-preserving transform, i.e., N-1 N-1 rIn n=0 k-0

Need help on question 3.11, thanks a lot.

Answer 1

C(K) = squareroot 2/N alpha (k) sigma^N - 1_n = 0 x(n) cos[(2n + 1) k pi/2N] x(n) = inverse DCT x(n) = squareroot 2/N sigma^N - 1_k = 0 alpha(k) middot C(k) middot cos[(2n + 1)k pi/2N] sigma^N - 1_n = 0 x(n) middot x(n) = sigma^N - 1_n = 0 x(n) middot {squareroot 2/N sigma^N - 1_k = 0 alpha (k) C(k) cos(2n + 1)k pi/2N} interchanging the summation = sigma^N - 1_k = 0 c(k) {sigma^N - 1_n = 0 squareroot 2/N middot alpha(k) x(n) cos(2n + 1)k pi/2N} = sigma^N - 1_k = 0 C(k) {alpha (k) squareroot 2/N sigma^N - 1_n = 0 x(n) cos(2n + 1)k pi/2N} sigma^N - 1_n = 0 x^2(n) = sigma^N - 1_k = 0 C(k) middot C(k) sigma^N - 1_n = 0 x^2(n) = sigma^N - 1_K = 0 C^2(k)