Page 362 - Schaum's Outline of Theory and Problems of Signals and Systems
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CHAP. 61 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS 349
Noting that the term in brackets in the last term is the DFT of X*[k], we get
which shows that the same algorithm used to evaluate the DFT can be used to evaluate the
IDFT.
6.54. The DFT definition in Eq. (6.92) can be expressed in a matrix operation form as
X=WNx (6.206)
where
x=
The N x N matrix WN is known as the DFT matrix. Note that WN is symmetric; that is,
Wz = WN, where W: is the transpose of WN.
(a) Show that
where W; ' is the inverse of WN and W,* is the complex conjugate of WN.
(6) Find W, and W;' explicitly.
(a) If we assume that the inverse of W, exists, then multiplying both sides of Eq. (6.206) by
Wi ', we obtain
which is just an expression for the IDFT. The IDFT as given by Eq. (6.94) can be
expressed in matrix form as
Comparing Eq. (6.210) with Eq. (6.2091, we conclude that
