Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 34 exercise 2.3
Author: Panagiotis
27
Sep
In exercise
[1, p. 34 exercise 2.3] we are asked to prove that the normalized DFT matrix given in
[1, p. 21, (2.22)] is unitary.
Solution:
The normalized
DFT matrix is given by:
The above multiplication results in:
The elements of the matrix
for
are sums of geometric series which are given by
[2, p. 16] .
Thus e.g. for
the following result is derived:
Similar the other terms
of the matrix are also equal to zero and therefore
, where
is the identity matrix. Thus the normalized DFT matrix is unitary. QED.
[1] Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”, Prentice Hall, ISBN: 0-13-598582-X. [2] Bronstein and Semdjajew and Musiol and Muehlig: “Taschenbuch der Mathematik”, Verlag Harri Deutsch Thun und Frankfurt am Main, ISBN: 3-8171-2003-6.
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