Digital Signal Processing Reference
In-Depth Information
where
β
(
p
,
N
)
is a normalization factor. The simplest choice of this factor is
N
p
. The recur‐
rence relation of Tchebichef polynomials with respect to the chosen order is:
(
p
+1
)
t
p
+1
(
n
)
-
(
2
p
+1
)(
2
n
-
N
+1
)
t
p
(
n
)
+
p
(
N
2
-
p
2
)
t
p
-1
(
n
)
=0
(23)
where
p
≥1 and the first two polynomials are
t
0
(
n
)
=1 and
t
1
(
n
)
=2
n
-
N
+1 . The orthogonality
property satisfies the following squared-norm:
ρ
(
p
,
N
)
=
(
2
p
)
!
(
N
+
p
2
p
+ 1
)
(24)
2.4.2. Krawtchouk moments
The definition of the
n
th
order classical Krawtchouk polynomial is defined as
a
k
,
n
,
p
x
k
=
F
1
(
-
n
, -
x
; -
N
;
p
)
n
K
n
(
x
;
p
,
N
)
=
k
=0
(25)
2
where
x
,
n
=0,
1,
2,
…,
N
,
N
>0,
p
∈(0,1) and
F
1
(∙)
2
is the generalized hypergeometric
function
∞
(
a
)
k
(
b
)
k
(
c
)
k
z
k
k
!
F
1
(
a
,
b
;
c
;
z
)
=
k
=0
(26)
2
and
(
x
)
k
is the Pochhammer symbol given by
(
x
)
k
=
x
(
x
+1
)(
x
+2
)
⇯
(
x
+
k
-1
)
k
≥1 and
(
x
)
0
=1
(27)
The normalized and weighted Krawtchouk polynomials
{
¯
n
(
x
;
p
,
N
)
}
are defined as [10]
¯
n
(
x
;
p
,
N
)
=
K
n
(
x
;
p
,
N
)
w
(
x
;
p
,
N
)
(28)
ρ
(
n
;
p
,
N
)
where the weight function,
w
(
∙
)
and the square norm,
ρ
(
∙
)
are given as
w
(
x
;
p
,
N
)
=
(
x
)
p
x
(
1-
p
)
N
-
x
(29)
and
ρ
(
n
;
p
,
N
)
=
(
p
-
p
)
n n
!
(30)
(
-
N
)
n
