Digital Signal Processing Reference
In-Depth Information
Similarly, for two periodic series
x
(
k
)
and
y
(
h
)
with the same period
N
(or if they
are both of finite support {0,...,
N -
1}), their circular convolution sum
()
x
yk
is
⊗
periodic of period
N,
and defined for all values of
n
by:
N
−
1
1
∑
(
)
()
() ( )
x
yk
=
xmy k m
−
mod
N
[2.42]
⊗
N
m
=
0
The convolution sum and the circular convolution sum verify the commutative
and associative properties and the identity element is:
()
()
()
()
⎧
⎪
⎪
⎨
⎪
⎪
⎩
δ
δ
t
k
Tt
for the functions
for the series
[2.43]
Ξ
for the functions of period
T
T
N
Nk
Ξ
for the series of period
N
(
)
() (
)
In addition, by noting
x
t
0
(
t
)
the delayed function
x
(
t
)
by
tx t xtt
−
00
t
0
we can easily show:
()
()
⎧
⎪
xyt
=
xyt
for the functions
for the series
⊗
⊗
t
t
0
0
()
()
x
y k
=
x
y
k
⎪
⎪
⎨
⊗
⊗
k
k
0
0
[2.44]
()
()
x
y
A
=
x
y
A
T
for the functions of period
for the series of period
⊗
⊗
⎪
⎪
t
t
0
0
()
()
x
y
A
=
x
y
A
N
⊗
⊗
⎪
⎩
k
k
0
0
In particular, the convolution of a function or a series with the delayed identity
element delays it by the same amount.
We can easily verify that the Fourier transform of a convolution sum or circular
convolution sum is the product of transforms:
n
() ()()
n
() ()()
n
() ()()
n
()
()
()
⎧
xyf
=
xfyf
ˆ
ˆ
Fourier transform (continuous time)
⊗
⎪
⎪
ˆ
ˆ
xyv xvyv
xy
=
Fourier transform (discrete time)
⊗
⎪
⎨
[2.45]
A
=
x y
ˆ
A A
ˆ
Fourier series (period
T
)
⎪
⎪
⊗
x
y
A
=
x
ˆ
y
ˆ
A
Discrete Fourier transform (period
N
)
⎪
⎩
⊗
A
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