Chapter-08-Frequency-Response - Chapter 8 Frequency...

Chapter 8
Frequency Response
A class of systems that yield to sophisticated analysis techniques is the class of linear time-
invariant systems ( LTI system ), discussed in chapter 5 . LTI systems have a key property: given
a sinusoidal input, the output is a sinusoidal signal with the same frequency, but possibly different
amplitude and phase. Given an input that is a sum of sinusoids, the output will be a sum of the same
sinusoids, each with its amplitude and phase (possibly) modified.
We can justify describing audio signals as sums of sinusoids on purely psychoacoustic grounds.
However, because of this property of LTI systems, it is often convenient to describe any signal as a
sum of sinusoids, regardless of whether there is a psychoacoustic justification. The real value in this
mathematical device is that by using the theory of LTI systems, we can design systems that operate
more-or-less independently on the sinusoidal components of a signal. For example, abrupt changes
in the signal value require higher frequency components. Thus, we can enhance or suppress these
abrupt changes by enhancing or suppressing the higher frequency components. Such an operation
is called filtering because it filters frequency components. We design systems by crafting their
frequency response , their response to sinusoidal inputs. An audio equalizer, for example, is a filter
that enhances or suppresses certain frequency components. Images can also be filtered. Enhancing
the high frequency components will sharpen the image, whereas suppressing the high frequency
components will blur the image.
State-space models described in previous chapters are precise and concise, but in a sense, not as
powerful as a frequency response. For an LTI system, given a frequency response, you can assert a
great deal about the relationship between an input signal and an output signal. Fewer assertions are
practical in general with state-space models.
LTI systems, in fact, can also be described with state-space models, using difference equations and
differential equations , as explored in chapter 5 . But state-space models can also describe systems
that are not LTI. Thus, state-space models are more general. It should come as no surprise that the
price we pay for this increased generality is fewer analysis and design techniques. In this chapter,
we explore the (very powerful) analysis and design techniques that apply to the special case of LTI
systems.
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248 CHAPTER 8. FREQUENCY RESPONSE
t
x
0 1 -1
D
+1.0
( x ) D
-1.4
( x )
Figure 8.1: Illustration of the delay system D
τ
. D
- 1 . 4
( x ) is the signal x to the
left by 1.4, and D
+ 1 . 0
( x ) is x moved to the right by 1.0.
8.1 LTI systems
LTI systems have received a great deal of intellectual attention for two reasons. First, they are
relatively easy to understand. Their behavior is predictable, and can be fully characterized in fairly
simple terms, based on the frequency domain representation of signals that we introduced in the
previous chapter. Second, many physical systems can be reasonably approximated by them. Few
physical systems perfectly fit the model, but many fit very well within a certain regime of operation.