Digital Signal Processing Reference
In-Depth Information
Naturally, inverting the multiplication
M
1
= M
2
• M
3
is not
M
2
= M
1
/M
3
-1
but is defined by
M
2
= M
1
• M
3
-1
;
i.e. by the multiplication by the transposed matrix.
In principle, a matrix multiplication is defined as follows:
⎡
⎤
n
∑
=
A
⋅
B
=
a
⋅
b
;
⎢
⎣
⎥
⎦
ij
jk
j
1
a
,
a
b
,
b
a
b
+
a
b
,
a
b
+
a
b
⎡
⎤
⎡
⎤
⎡
⎤
11
12
11
12
11
11
12
21
11
21
12
22
⋅
=
;
⎣
⎦
⎣
⎦
⎣
⎦
a
,
a
b
,
b
a
b
+
a
b
,
a
b
+
a
b
21
22
21
22
21
11
22
21
21
12
22
22
Fig. 7.33.
Definition of a matrix multiplication
Apart from the Discrete Cosine Transform (DCT), other transformation
processes are also conceivable for compressing frames and can be repre-
sented as matrix multiplications, these being the
•
Karhunen Loeve Transform (1948/1960)
•
Haar's Transform (1910)
•
Walsh-Hadamard Transform (1923)
•
Slant Transform (Enomoto, Shibata, 1971)
•
Discrete Cosine Transform (DCT, Ahmet, Natarajan, Rao,
1974)
•
Short Wavelet Transform
A great advantage of the DCT is the great energy concentration (Fig.
7.34.) to a very few values in the spectral domain, and the avoidance of
Gibbs' phenomenon which would lead to overshoots in the inverse trans-
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