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

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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. 247
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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.
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