Time-domain analysis of discrete-time systems - Continuous and Discrete Time Signals and Systems - page 422

CHAPTER

10

Time-domain analysis of
discrete-time systems

An important subset of discrete-time (DT) systems satisfies the linearity and
time-invariance properties, discussed in Chapter 2. Such DT systems are
referred to as linear, time-invariant, discrete-time (LTID) systems. In this chap-
ter, we will develop techniques for analyzing LTID systems. As was the case
for the LTIC systems discussed in Part II, we are primarily interested in cal-
culating the output response y [ k ] of an LTID system to a DT sequence x [ k ]
applied at the input of the system.
In the time domain, an LTID system is modeled either with a linear, constant-
coefficient difference equation or with its impulse response h [ k ]. Section 10.1
covers linear, constant-coefficient difference equations and develops numer-
ical techniques for solving such equations. Section 10.2 defines the impulse
response h [ k ] as the output of an LTID system to an unit impulse function δ [ k ]
applied at the input of the system and shows how the impulse response can
be derived from a linear, constant-coefficient difference equation. Section 10.3
proves that any arbitrary DT sequence can be represented as a linear combina-
tion of time-shifted DT impulse functions. This development leads to a second
approach for calculating the output y [ k ] based on convolving the applied input
sequence x [ k ] with the impulse response h [ k ] in the DT domain. The resulting
operation is referred to as the convolution sum and is defined in Section 10.4.
Section 10.5 introduces two graphical methods for calculating the convolution
sum, and Section 10.6 lists several important properties of the convolution sum.
A special case of convolution sum, referred to as the periodic or circular con-
volution, occurs when the two operands are periodic sequences. Section 10.7
develops techniques for computing the periodic convolution and shows how
it may be used to compute the linear convolution. In Section 10.8, we revisit
the causality, stability, and invertibility properties of LTID systems and express
these properties in terms of the impulse response h [ k ]. M ATLAB instructions
for computing the convolution sum are listed in Section 10.9. The chapter is
concluded in Section 10.10 with a summary of the important concepts covered
in the chapter.

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