Digital Signal Processing Reference
In-Depth Information
The distributive property states that convolution is a linear operation with respect
to addition. The periodic convolution also satisfies the distributive property
provided that the three sequences have the same fundamental period
K
0
.
Associative property
x
1
[
k
]
∗
x
2
[
k
]
∗
x
3
[
k
]
=
x
1
[
k
]
∗
x
2
[
k
]
∗
x
3
[
k
]
.
(10.23)
This property states that changing the order of the linear convolution operands
does not affect the result of the linear convolution. The periodic convolution
also satisfies the associative property provided that the three sequences have
the same fundamental period
K
0
.
Shift property
If
x
1
[
k
]
∗
x
2
[
k
]
=
g
[
k
], then
x
1
[
k
−
k
1
]
∗
x
2
[
k
−
k
2
]
=
g
[
k
−
k
1
−
k
2
]
(10.24)
for any arbitrary integer constants
k
1
and
k
2
. In other words, if the two operands
of the linear convolution sum are shifted then the result of the convolution sum
is shifted in time by a duration that is the sum of the individual time shifts
introduced in the operands. The periodic convolution satisfies the shift property
with respect to the circular shift operation.
Length of convolution
Let the non-zero lengths of the convolution operands
x
1
[
k
] and
x
2
[
k
] be denoted by
K
1
and
K
2
time units, respectively. It can be shown
that the non-zero length of the linear convolution (
x
1
[
k
]
∗
x
2
[
k
]) is
K
1
+
K
2
−
1
time units. The periodic convolution does not satisfy the length property. The
circular convolution of two periodic sequences with fundamental period
K
0
is
also of length
K
0
.
Convolution with impulse function
x
1
[
k
]
∗ δ
[
k
−
k
0
]
=
x
1
[
k
−
k
0
]
.
(10.25)
In other words, convolving a DT sequence with a unit impulse function whose
origin is located at
k
=
k
0
shifts the DT sequence by
k
0
time units. Since periodic
convolution is defined in terms of periodic sequences and the impulse function
is not a periodic sequence, Eq. (10.25) is not valid for the periodic convolution.
Convolution with unit step function
∞
k
x
1
[
k
]
∗
u
[
k
]
=
x
[
m
]
u
[
k
−
m
]
=
x
[
m
]
.
(10.26)
m
=−∞
m
=−∞
Equation (10.26) states that convolving a DT sequence
x
[
k
] with a unit step
function produces the running sum of the original sequence
x
[
k
] as a function
of time
k
. Since periodic convolution is defined in terms of periodic sequences
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