Digital Signal Processing Reference
In-Depth Information
Appendix
The Sinc Product Expansion Formula
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The goal is to prove the product expansion
1
n
2
∞
(
π
)
t
2
sin
t
=
−
(9.44)
π
t
n
=
1
We present two proofs; the first was proposed by Euler in 1748 and, while
it certainly lacks rigor by modern standards, it has the irresistible charm of
elegance and simplicity in that it relies only on basic algebra. The second
proof is more rigorous, and is based on the theory of Fourier series for pe-
riodic functions; relying on Fourier theory, however, hides most of the con-
vergence issues.
Euler's Proof.
Consider the
N
roots of unity for
N
odd. They comprise
z
e
±j
ω
N
k
for
k
=
1plus
N
−
1 complex conjugate roots of the form
z
=
=
1,...,
(
N
. If we group the complex conjugate roots pairwise
we can factor the polynomial
z
N
N
−
1
)
/
2and
ω
=
2
π/
N
−
1as
(
N
−
1
)
/
2
z
2
1
z
N
−
1
=(
z
−
1
)
−
2
z
cos
(
ω
N
k
)+
k
=
1
The above expression can immediately be generalized to
(
N
−
1
)
/
2
z
2
a
2
z
N
a
N
−
=(
z
−
a
)
−
2
az
cos
(
ω
N
k
)+
k
=
1
Now replace
z
and
a
in the above formula by
z
=(
1
+
x
/
N
)
and
a
=(
1
−
x
/
N
)
;
we obtain the following:
1
N
1
N
x
N
x
N
+
−
−
=
1
)
(
N
−
1
)
/
2
N
2
1
x
2
4
x
N
=
−
cos
(
ω
N
k
)+
+
cos
(
ω
N
k
k
=
1
)
1
(
N
−
1
)
/
2
x
2
N
2
·
4
x
N
1
1
+
cos
(
ω
N
k
)
=
−
cos
(
ω
N
k
+
1
−
cos
(
ω
N
k
)
k
=
1
x
2
1
)
1
Ax
(
N−
1
)
/
2
+
cos
(
ω
N
k
=
+
N
2
1
)
−
cos
(
ω
N
k
k
=
1
)
(
N−
1
)
/
2
k
)
.Thevalue
A
is also the coefficient for the degree-one term
x
in the right-hand side and
1
where
A
is just the finite product
(
4
/
N
−
cos
(
ω
N
k
=
1
