DCT by convolution and convolution theorem ?

From: Bernie (a.buisson_at_nextamp.com)
Date: 08/03/04

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    Date: Tue, 3 Aug 2004 17:42:28 +0200

    Cp(u,S,p) u-eme coefficient of a n point DCT on the signal S with pixel
    origin at p as (1):

                                            __n-1
                                            \
    Cp(u,S,p) = alpha(u)* | S(x+p)cos(pi*u*(2*x+1)/(2*n)
                                            /__x=0

    with S signal of m real data and p in [0..n-1] (n=8 and m =16)

    to write Cp as a convolution product (2):
                        __m-1
                        \
    Cp(u,S,p) = | S(x)*G(u-x,u,p)
                        /__x=0

    we introduce G(x,u,p) define by :
    if p<=u-x<p+n then alpha(u)cos(pi*u*(2*(u-x-p)+1))
    else 0

    we can also write the following relation (3):

    __m-1 __m-1 __ m-1
    \ (-2*Pi*I*u*t/m) \ \
    (-2*Pi*I*u*t/m)
    |Cp(u,S,p) *e = | [ |
    S(x)*G(u-x,u,p) ] * e
    /__u=0 /__u=0 /__x=0

    with :
    (-2*Pi*I*u*t/m) (-2*Pi*I*x*t/m) (-2*Pi*I*(u-x)*t/m)
    e =e *e

    (3) can be rewrite as (4) :
    __m-1 __m-1 __
    m-1
    \ (-2*Pi*I*u*t/m) \ (-2*Pi*I*x*t/m) \
    (-2*Pi*I*(u-x)*t/m)
    |Cp(u,S,p) *e = | S(x)*e [
    |G(u-x,u,p)*e ]
    /__u=0 /__x=0
    /__u=0

    we can see appear the DFT of S on m points in 0..m-1

    but my question is how transform (4) to find in the second part of the
    equation a DFT of G and obtain a relation as describe in lots of book by
    "Convolution Theorem" ?

                    -1
    g*h = F [F(g).F(h)]

    with * is a convolution product and . a multiplication

    i know the conclusion, but if i cannot understand why and proof that this is
    ok with my G function...
    so if someone could help me, any advice would be welcome.
    thanks

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    • Re: DCT by convolution and convolution theorem ?
      ... > Cpu-eme coefficient of a n point DCT on the signal S with pixel ... > to write Cp as a convolution product: ... > equation a DFT of G and obtain a relation as describe in lots of book by ...
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    • Re: DCT by convolution and convolution theorem ?
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