Digital Signal Processing Reference
In-Depth Information
or circular convolution. The periodic convolution is discussed in Section 10.6,
where we mentioned that the linear convolution may be efficiently calculated
through periodic convolution. The convolution sum satisfies the commutative,
distributive, associative, and time-shifting properties.
(1) The commutative property states that the order of the convolution operands
does not affect the result of the convolution.
(2) The distributive property states that convolution is a linear operation with
respect to addition.
(3) The associative property is an extension of the commutative property to
more than two convolution operands. It states that changing the order of
the convolution operands does not affect the result of the convolution sum.
(4) The time-shifting property states that if the two operands of the convolution
sum are shifted in time then the result of the convolution sum is shifted
by a duration that is the sum of the individual time shifts introduced in the
convolution operands.
(5) If the lengths of the two functions are
K
1
and
K
2
samples, the convolution
sum of these two functions will have a length of
K
1
−
1 samples.
(6) Convolving a sequence with a unit DT impulse function with the origin at
k
=
k
0
shifts the sequence by
k
0
time units.
(7) Convolving a sequence with a unit DT step function produces the running
sum of the original sequence as a function of time
k
.
+
K
2
Finally, in Section 10.8, we expressed the memoryless, causality, stability, and
invertibility properties of an LTID system in terms of its impulse response.
(1) An LTID system will be memoryless if and only if its impulse response
h
[
k
]
=
0 for
k
=
0.
(2) An LTID system will be causal if and only if its impulse response
h
[
k
]
=
0
for
k
<
0.
(3) The impulse response
h
[
k
] of a (BIBO) stable LTID system is absolutely
summable, i.e.
∞
h
[
k
]
< ∞.
k
=−∞
(4) An LTID system will be invertible if there exists another LTID system with
impulse response
h
i
[
k
] such that
h
[
k
]
∗
h
i
[
k
]
= δ
[
k
]
.
The system with the
impulse response
h
i
[
k
] is the inverse system.
In the next chapter, we consider the frequency representations of DT sequences
and systems.
Problems
10.1
Consider the input sequence
x
[
k
]
=
2
u
[
k
] applied to a DT system modeled
with the following input-output relationship:
y
[
k
+
1]
−
2
y
[
k
]
=
x
[
k
]
,
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