Repeat Exercise 3.9 for the case of two complex antisymmetric sequences. Exercise 3.9 Show how to compute

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Repeat Exercise 3.9 for the case of two complex antisymmetric sequences.

Exercise 3.9

Show how to compute the DFT of two even complex length- \(N\) sequences performing only one length \(N\) transform calculation. Follow the steps below:

(i) Build the auxiliary sequence \(y(n)=W_{N}^{n} x_{1}(n)+x_{2}(n)\).

(ii) Show that \(Y(k)=X_{1}(k+1)+X_{2}(k)\).

(iii) Using properties of symmetric sequences, show that \(Y(-k-1)=X_{1}(k)+X_{2}(k+1)\).

(iv) Use the results of (ii) and (iii) to create a recursion to compute \(X_{1}(k)\) and \(X_{2}(k)\). Note that \(X(0)=\sum_{n=0}^{N-1} x(n)\).

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Digital Signal Processing System Analysis And Design

ISBN: 9780521887755

2nd Edition

Authors: Paulo S. R. Diniz, Eduardo A. B. Da Silva , Sergio L. Netto

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