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Unformatted text preview: Chapter 10 The Four Fourier Transforms In chapter 7 we saw that the Fourier series describes a periodic signal as a sum of complex expo- nentials. In chapter 8 we saw that if the input to an LTI system is a sum of complex exponentials, then the frequency response of the LTI system describes its response to each of the component ex- ponentials. Thus we can calculate the system response to any periodic input signal by combining the responses to the individual components. In chapter 9 we saw that the response of the LTI system to any input signal can also be obtained as the convolution of the input signal and the impulse response. The impulse response and the frequency response give us the same information about the system, but in different forms. The impulse response and the frequency response are related by the Fourier transform, where in chapter 9 we saw both discrete time and continuous time versions. In this chapter, we will see that for discrete-time systems, the frequency response can be described as a sum of weighted complex exponentials (the DTFT), where the weights turn out to be the impulse response samples. We will see that the impulse response is, in fact, a Fourier series representation of the frequency response, with the roles of time and frequency reversed from the uses of the Fourier series that we have seen so far. This reappearance of the Fourier series is not a coincidence. In this chapter, we explore this pattern by showing that the Fourier series is a special case of a family of representations of signals that are collectively called Fourier transforms . The Fourier series applies specifically to continuous-time, periodic signals. The discrete Fourier series applies to discrete-time, periodic signals. We complete the story with the continuous-time Fourier transform (CTFT), which applies to continuous-time signals that are not periodic, and the discrete-time Fourier transform (DTFT), which applies to discrete-time signals that are not periodic. 327 328 CHAPTER 10. THE FOUR FOURIER TRANSFORMS 10.1 Notation We define the following four sets of signals: • ContSignals = [ Reals → Complex ] . Since Reals is included in Complex , ContSignals includes continuous-time signals whose range is Reals , and so we won’t need to consider these sepa- rately. ContSignals includes continuous-time signals, but we are not insisting that the domain be interpreted as time. Indeed, sometimes the domain could be interpreted as space, if we are dealing with images. In this chapter we will see that it is useful sometimes to interpret the domain as frequency. • DiscSignals = [ Integers → Complex ] . This includes discrete-time signals whose domain is time or sample number, but again we are not insisting that the domain be interpreted as time....
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This note was uploaded on 09/29/2010 for the course EE 20N taught by Professor Ayazifar during the Spring '08 term at University of California, Berkeley.
- Spring '08