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From: karsh@geowhiz.UUCP (Bruce Karsh)
Newsgroups: net.math
Subject: Normalization for DFT
Message-ID: <175@geowhiz.UUCP>
Date: Mon, 8-Apr-85 00:09:46 EST
Article-I.D.: geowhiz.175
Posted: Mon Apr  8 00:09:46 1985
Date-Received: Tue, 9-Apr-85 03:36:37 EST
Distribution: net
Organization: UW Madison, Geology Dept.
Lines: 45


  The DFT (Discrete Fourier Transform) is usually defined as:
                              N
                            ------          -2 PI i m n
                            \               -----------
FORWARD TRANSFORM:  F(n) =   >      f(m)  e      N
                            /
                            ------
                             m=0
                                 N
                               ------          2 PI i m n
                            1  \               ----------
REVERSE TRANSFORM: f(m) =   -   >      F(n)  e      N
                            N  /
                               ------
                                n=0

  Why is that?  Wouldn't it make more sense to normalize both
the forward and inverse transforms with (1/sqrt(N))?  Like this:

                                     N
                                   ------          -2 PI i m n
                             1     \               -----------
FORWARD TRANSFORM:  F(n) = -----    >      f(m)  e      N
                          sqrt(N)  /
                                   ------
                                    m=0
                                    N
                                  ------          2 PI i m n
                             1    \               ----------
REVERSE TRANSFORM: f(m) =  -----   >      F(n)  e      N
                          sqrt(N) /
                                  ------
                                   n=0
  This way looks more symetrical.  Besides, when you do it this way, the
sum of the sqares of all the transformed coefficients equals the sum
of the squares of all the un-transformed coeffiecients.

  So why do people define it the other way?
-- 
Bruce Karsh                           |
U. Wisc. Dept. Geology and Geophysics |
1215 W Dayton, Madison, WI 53706      | This space for rent.
(608) 262-1697                        |
{ihnp4,seismo}!uwvax!geowhiz!karsh    |