Digital Signal Processing Reference
In-Depth Information
satisfies the commutative, distributive, associative, time-shifting, and scaling
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
integral.
(4) The time-shifting property states that if the two operands of the convolution
integral are shifted in time, then the result of the convolution integral is
shifted by a duration that is the sum of the individual time shifts introduced
in the convolution operands.
(5) The duration of the waveform produced by the convolution integral is the
sum of the durations of the convolved signals.
(6) Convolving a signal with a unit impulse function with origin at
t
=
t
0
shifts
the signal to the origin of the unit impulse function.
(7) Convolving a signal with a unit step function produces the running integral
of the original signal as a function of time
t
.
(8) If the two convolution operands are scaled by a factor
α
, then the result of
the convolution of the two operands is scaled by
α
and amplified by
α
.
In Section 3.7, we expressed the memoryless, causality, inverse, and stability
properties of an LTIC system in terms of its impulse response.
(1) An LTIC system will be memoryless if and only if its impulse response
h
(
t
)
=
0 for
t
=
0.
(2) An LTIC system will be causal if and only if its impulse response
h
(
t
)
=
0
for
t
<
0.
(3) The impulse response of the inverse of an LTIC system satisfies the property
h
i
(
t
)*
h
(
t
)
= δ
(
t
).
(4) The impulse response
h
(
t
) of a (BIBO) stable LTIC system is absolutely
integrable, i.e.
∞
h
(
t
)
d
t
< ∞.
−∞
Finally, in Section 3.8 we presented a few M
ATLAB
examples for solving
constant-coefficient differential equations with initial conditions.
In Chapters 4 and 5, we will introduce the frequency representations for CT
signals and systems. Such representations provide additional tools that simplify
the analysis of LTIC systems.
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