Page 85 - DSP Integrated Circuits

70 Chapter 3 Digital Signal Processing

3.8.1 Linear Systems
Most subsystems used in signal processing belong to the class of linear systems. A
linear system is denned:
If two proper input signals jci(n) and X2(n) yield the output signals yi(ri)
and y2(n), i.e., x±(n) —> y\(ri) and X2(n) —»yzfo), then the system is linear if
and only if a x\(n) + b x%(n) -> a yi(ri) + b y%(n) for all a, b so that the com-
bined input and output signals are within their proper domains.

Figure 3.8 Linear system

A consequence of this property, as illustrated in Figure 3.8, is that the input sig-
nal can be expanded into an orthogonal set of basic functions, e.g., Fourier series for
a periodic input signal. The effect of each Fourier component can then be determined
and the output obtained by summing the contributions from these components.

3.8.2 SI (Shift-Invariant) Systems
Systems can also be classified with respect to their behavior over time. The behav-
ior of a shift-invariant system (SI}, sometimes also called a time-invariant system
(TT), does not vary with time.

Thus, an SI system, as illustrated ir
Figure 3.9, has the same response for £
given input, independent of when th(
input is applied to the system. The out
put is only shifted the same amount as
the input. Figure 3.9 Original and shifted input
Examples of shift-invariant opera and output signals for a
tions and systems are addition, multipli shift-invariant system
cation, normal digital filters, and cosine
transforms. An example of a system that is not shift-invariant is a decimator that
reduces the sample frequency.

3.8.3 LSI (Linear Shift-Invariant) Systems
A system that is both linear and shift-invariant is called an LSI system.^ The out-
put signal of an LSI system is determined by convolution of the input signal and
the impulse response.

L
This abbreviation should not be confused with LSI (large scale integration).