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Unformatted text preview: Chapter 7 Frequency Domain We are interested in manipulating signals. We may wish to synthesize signals, as modems need to do in order to transmit a voice-like signal through the telephone channel. We may instead wish to analyze signals, as modems need to do in order to extract digital information from a received voice-like signal. In general, the field of communications is all about synthesizing signals with characteristics that match a channel, and then analyzing signals that have often been corrupted by the channel in order to extract the original information. We may also wish to synthesize natural signals such as images or speech. The field of computer graphics puts much of its energy into synthesizing natural-looking images. Image processing includes image understanding , which involves analyzing images to determine their content. The field of signal processing includes analysis and synthesis of speech and music signals. We may wish to control a physical process. The physical process is sensed (using temperature, pressure, position and speed sensors). The sensed signals are processed in order to estimate the internal state of the physical process. The physical process is controlled on the basis of the state estimate. Control system design includes the design of state estimators and controllers . In order to analyze or synthesize signals, we need models of those signals. Since a signal is a func- tion, a model of the signal is a description or a definition of the function. We use two approaches. The first is a declarative (what is) approach. The second is an imperative (how to) approach. These two approaches are complementary. Depending on the situation, one approach is better than the other. Signals are functions. This chapter in particular deals with signals where the domain is time (dis- crete or continuous). It introduces the concept of frequency-domain representation of these signals. The idea is that arbitrary signals can be described as sums of sinusoidal signals. This concept is first motivated by referring to psychoacoustics, how humans hear sounds. Sinusoidal signals have par- ticular psychoacoustic significance. But the real justification for the frequency domain approach is much broader. It turns out to be particularly easy to understand the effect that LTI systems ( linear time invariant systems ), discussed in chapter 5 , have on sinusoidal signals. A powerful set of analysis and design techniques then follow for arbitrary signals and the LTI systems that operate on them. 217 218 CHAPTER 7. FREQUENCY DOMAIN Although we know that few (if any) real-world systems are truly LTI, we can easily construe models where the approximation is valid over some regime of operation. The previous chapter showed how modal models can be constructed to build realistic models over a broader range of operating conditions. Frequency domain methods are amenable to such hybrid system treatment....
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- Spring '08