Newsgroups: comp.speech
Path: lyra.csx.cam.ac.uk!warwick!uknet!EU.net!howland.reston.ans.net!cs.utexas.edu!natinst.com!news.dell.com!tadpole.com!uunet!news.iij.ad.jp!wnoc-tyo-news!news.u-tokyo.ac.jp!news.tisn.ad.jp!s.u-tokyo!kuis-news!wnoc-kyo-news!atrwide!atr-la!lucke
From: lucke@itl.atr.co.jp (Helmut Lucke)
Subject: question on discrete cosine transform
Message-ID: <LUCKE.94Jun20104111@atrq28.itl.atr.co.jp>
Sender: news@itl.atr.co.jp (USENET News System)
Nntp-Posting-Host: atrq28
Organization: ATR Telecommunications Research Labs., Kyoto, JAPAN
Date: Mon, 20 Jun 1994 01:41:11 GMT
Lines: 40


I recently came across the following excerpt in a proof and got stuck on it.
I thought that the answer may seem obvious to someone with more
of a signal processing background so I post it here. I'll be greatful
for any hints that may lead to a solution. Here it goes:

"Let
        [ 1-a   -a    0    .    .     0 ]
        [  -a    1   -a    0    .     . ]
D(a) =  [  0    -a    1   -a    0     . ]
        [  .     .    .    .    .     . ]
        [  .     .    0   -a    1    -a ]
        [  0     .    .    0    -a  1-a ]

This is the transform matrix of the Discrete Cosine Transform. Its eigenvalues
are given by

                       (k-1)pi
\lambda_k = 1 - 2a cos -------            (K = dimension of D(a))
                          K

" (end of quote)


My questions are:

(1) in what sense is D(a) the transfer matrix of the DCT and what is the
    role of a ?
(2) Why are lambda_k the eigenvalues of D(a) ?


Helmut
--------------------------------------------------------------------
Helmut Lucke                                <lucke@itl.atr.co.jp>
ATR Interpreting Telecommunications Research Laboratories
2-2 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02, JAPAN
Tel: +81-7749-5-1382 (direct)               Fax:   +81-7749-5-1308
     +81-7749-5-1301 (switchboard)
--------------------------------------------------------------------

