Digital Signal Processing Reference
In-Depth Information
Fig. P4.3. Legendre
polynomials with order 0-3.
1
P
0
(
x
)
P
1
(
x
)
P
2
(
x
)
P
3
(
x
)
0.5
0
−0.5
−1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Fig. P4.4. First few orders of
Chebyshev polynomials of the
first kind.
1
T
0
(
x
)
T
1
(
x
)
T
5
(
x
)
0.5
T
2
(
x
)
0
T
4
(
x
)
T
3
(
x
)
−0.5
−1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
follows:
⌊
n
/
2
⌋
T
n
(
x
)
=
n
2
(
−
1)
k
(
n
−
k
−
1)!
k
!(
n
−
2
k
)!
(2
x
)
n
−
2
k
,
n
=
0
,
1
,
2
,
3
,...
k
=
0
The first few Chebyshev polynomials are given by
T
3
(
x
)
=
4
x
3
−
3
x
;
T
0
(
x
)
=
1;
T
4
(
x
)
=
8
x
4
−
8
x
2
+
1;
T
1
(
x
)
=
x
;
T
2
(
x
)
=
2
x
2
−
1;
T
5
(
x
)
=
16
x
5
−
20
x
3
+
5
x
;
which satisfy the following relationship:
T
n
+
1
(
x
)
=
2
xT
n
(
x
)
−
T
n
−
1
(
x
)
and are shown in Fig. P4.4.
The Chebyshev polynomials
{
T
n
(
x
)
,
n
=
0
,
1
,
2
,...}
form an orthog-
onal set on the interval [
−
1, 1] with respect to the weighting function by
satisfying the following:
1
π
m
=
n
=
0
1
√
1
−
x
2
T
m
(
x
)
T
n
(
x
)d
x
=
π/
2
m
=
n
=
1
,
2
,
3
0
m
=
n
.
−
1
Verify the above orthogonality condition for
m
,
n
=
0, 1, 2, 3, 4.
4.5
The Haar functions are very popular in signal processing and wavelet appli-
cations. These functions are generated using a scale parameter (
m
) and a
translation parameter (
n
). Let the mother Haar function (
m
=
n
=
0) be
defined as follows:
10
≤
t
<
0
.
5
H
0
,
0
(
t
)
=
−
10
.
5
≤
t
≤
1
otherwise
.
0
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