Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 34 exercise 2.11
Author: Panagiotis
24
Jan
In
[1, p. 34 exercise 2.11] we are asked to find the eigenvalues of the circulant matrix given in
[1, (2.27),p.22].
Solution:
The matrix given in
[1, (2.27),p.22] is
According to
[1, (2.42),p.27] we can write the matrix as
with

and where

are the elements of the first row of the matrix

.
From the eigenvectors

and eigenvalues

of

we can derive the eigenvalues of

by:
Thus the eigenvalues of the matrix

are given by

and are related to the eigenvectors
![\mathbf{v_i}=\frac{1}{\sqrt{n}} \left[ {\begin{array}{*{20}c}
1 & {e^{ - j\frac{{2\pi }}{n}(i - 1)} } & {e^{ - j\frac{{2\pi }}{n}2(i - 1)} } & \cdots & {e^{ - j\frac{{2\pi }}{n}(n - 1)(i - 1)} } \\
\end{array}} \right]^T](https://lysario.de/wp-content/cache/tex_1f405cf76d1d0320892ae84d0cfb4691.png)
where

with

. The elements of the first row of the matrix

are

,

,

,

, as can be seen in the solution of
[1, p. 34 exercise 2.5] given in
[2, solution of exercise 2.5].
[1] Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”, Prentice Hall, ISBN: 0-13-598582-X. [2] Panagiotis Chatzichrisafis: “Solution of exercise 2.5 from Kay’s Modern Spectral Estimation -Theory
and Applications”, lysario.de.
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