CH10(1)

CH10(1) - CHAPTER 10 Fourier Transform Properties The time...

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185 CHAPTER 10 Fourier Transform Properties The time and frequency domains are alternative ways of representing signals. The Fourier transform is the mathematical relationship between these two representations. If a signal is modified in one domain, it will also be changed in the other domain, although usually not in the same way. For example, it was shown in the last chapter that convolving time domain signals results in their frequency spectra being multiplied. Other mathematical operations, such as addition, scaling and shifting, also have a matching operation in the opposite domain. These relationships are called properties of the Fourier Transform, how a mathematical change in one domain results in a mathematical change in the other domain. Linearity of the Fourier Transform The Fourier Transform is linear, that is, it possesses the properties of homogeneity and additivity . This is true for all four members of the Fourier transform family (Fourier transform, Fourier Series, DFT, and DTFT). Figure 10-1 provides an example of how homogeneity is a property of the Fourier transform. Figure (a) shows an arbitrary time domain signal, with the corresponding frequency spectrum shown in (b). We will call these two signals: and , respectively. Homogeneity means that a change in x [ ] X [ ] amplitude in one domain produces an identical change in amplitude in the other domain. This should make intuitive sense: when the amplitude of a time domain waveform is changed, the amplitude of the sine and cosine waves making up that waveform must also change by an equal amount. In mathematical form, if and are a Fourier Transform pair, then x [ ] X [ ] kx [ ] and are also a Fourier Transform pair, for any constant k . If the kX [ ] frequency domain is represented in rectangular notation, means that both [ ] the real part and the imaginary part are multiplied by k . If the frequency domain is represented in polar notation, means that the magnitude is [ ] multiplied by k , while the phase remains unchanged.
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The Scientist and Engineer's Guide to Digital Signal Processing 186 Sample number 0 64 128 192 -3 -2 -1 0 1 2 3 255 c. k x[ ] Sample number 0 64 128 192 -3 -2 -1 0 1 2 3 255 a. x[ ] Frequency 0 0.1 0.2 0.3 0.4 0.5 0 10 20 30 40 50 b. X[ ] Frequency 0 0.1 0.2 0.3 0.4 0.5 0 10 20 30 40 50 d. k X[ ] Amplitude Frequency Domain Time Domain FIGURE 10-1 Homogeneity of the Fourier transform. If the amplitude is changed in one domain, it is changed by the same amount in the other domain. In other words, scaling in one domain corresponds to in the other domain. F.T. F.T. Additivity of the Fourier transform means that addition in one domain corresponds to in the other domain. An example of this is shown in Fig. 10-2. In this illustration, (a) and (b) are signals in the time domain called and , respectively. Adding these signals produces a third x 1 [ ] x 2 [ ] time domain signal called , shown in (c). Each of these three signals x 3 [ ] has a frequency spectrum consisting of a real and an imaginary part, shown in (d) through (i). Since the two time domain signals add to produce the third time domain signal, the two corresponding spectra third spectrum.
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This note was uploaded on 02/10/2012 for the course ECE 3551 taught by Professor Staff during the Spring '11 term at FIT.

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CH10(1) - CHAPTER 10 Fourier Transform Properties The time...

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