IEEE Signal Processing - July 2018 - 126

lecture notes
Behnam Shahrrava

Closed-Form Impulse Responses of Linear
Time-Invariant Systems: A Unifying Approach

I

n many signal processing applications,
filtering is accomplished through
linear time-invariant (LTI) systems
described by linear constant-coefficient
differential and difference equations
since they are conveniently implemented
using either analog or digital hardware
[1]. An LTI system can be completely
characterized in the time domain by its
impulse response or in the frequency
domain by its frequency response, which
is the Fourier transform of the system's
impulse response. Equivalently, using
the Laplace transform [or the z-transform
in the case of discrete-time (DT) systems] as a generalization of the Fourier
transform, any continuous-time (CT) or
DT LTI system can be characterized by
its transfer function (or system function)
in the s-domain or the z-domain, respectively. In this article, we explain how to
find impulse responses of LTI systems
described by differential and difference
equations directly in the time domain
without resorting to any transform methods or recursive procedures.

Relavance
In the literature, the topic of finding impulse responses of LTI systems
directly in the time domain has not
received proper attention for the following reasons. First, there are approaches
based on transform methods. Second,
Digital Object Identifier 10.1109/MSP.2018.2810300
Date of publication: 27 June 2018

126

in the case of DT systems, even the
impulse response of an LTI system can
be obtained recursively.
For these reasons, in the signal processing community, there are various
groups of researchers that have different
opinions regarding this topic. There is
one group that has no interest in finding
the impulse response of an LTI system
directly in the time domain since it can
be obtained based on transform methods
[7]-[10]. This kind of approach implies
that the topic of time-domain analysis of
LTI systems is handicapped and incomplete since it is mostly relied on the frequency-domain analysis. However, the
following logical question arises: "Why
do we need to resort to a transformation
to find the impulse response of a system
that is one of the characteristics of the
system in the time domain?"
There is another group of researchers that thinks it may not be possible to
determine a closed-form solution for the
impulse response of a DT LTI system
described by a difference equation [5],
[6]. However, there are some researchers
who have been trying to obtain closedform expressions for impulse responses
of LTI systems without resorting to
transform methods or recursive procedures. In the literature [1]-[4], it has
been shown how a closed-form expression for the impulse response of a CT
LTI system described by a differential
equation can be obtained in the time
domain. Even in [1]-[4], the same probIEEE Signal Processing Magazine

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July 2018

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lem has been tackled for DT LTI systems described by difference equations
but with a completely different method
from the one for CT systems. Because
of two different methods and not treating CT and DT systems in parallel, none
of those approaches can be considered
as a unifying approach. In other words,
the disadvantage of all those approaches
is because of a lack of sharing insight
and intuition between CT and DT systems in the time domain.

Prerequisites
To understand this article, the prerequisites consist of basic calculus and the fundamentals of signal and system analysis.
Some familiarity with homogeneous differential and difference equations is necessary. Also, in particular, a background
in the topic of time-domain analysis of
LTI systems is required.

Problem statement and solution
Problem statement
In this article, the problem of finding a
closed-form expression for the impulse
response of a CT LTI system is revisited with the same operational notations
used in [2]. Using the fact that LTI systems can be represented as the cascade
combination of two LTI systems [1],
we provide a new mathematical proof
that not only verifies the results in [2]
but also shows the same path toward
deriving impulse responses of DT LTI
1053-5888/18©2018IEEE



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