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ch11_The Scientist and Engineers Guide to Digital Signal Pro

# ch11_The Scientist and Engineers Guide to Digital Signal...

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209 CHAPTER 11 Fourier Transform Pairs For every time domain waveform there is a corresponding frequency domain waveform, and vice versa. For example, a rectangular pulse in the time domain coincides with a sinc function [i.e., sin(x)/x] in the frequency domain. Duality provides that the reverse is also true; a rectangular pulse in the frequency domain matches a sinc function in the time domain. Waveforms that correspond to each other in this manner are called Fourier transform pairs . Several common pairs are presented in this chapter. Delta Function Pairs For discrete signals, the delta function is a simple waveform, and has an equally simple Fourier transform pair. Figure 11-1a shows a delta function in the time domain, with its frequency spectrum in (b) and (c). The magnitude is a constant value, while the phase is entirely zero. As discussed in the last chapter, this can be understood by using the expansion/compression property. When the time domain is compressed until it becomes an impulse, the frequency domain is expanded until it becomes a constant value. In (d) and (g), the time domain waveform is shifted four and eight samples to the right, respectively. As expected from the properties in the last chapter, shifting the time domain waveform does not affect the magnitude, but adds a linear component to the phase. The phase signals in this figure have not been unwrapped , and thus extend only from - B to B . Also notice that the horizontal axes in the frequency domain run from -0.5 to 0.5. That is, they show the negative frequencies in the spectrum, as well as the positive ones. The negative frequencies are redundant information, but they are often included in DSP graphs and you should become accustomed to seeing them. Figure 11-2 presents the same information as Fig. 11-1, but with the frequency domain in rectangular form . There are two lessons to be learned here. First, compare the polar and rectangular representations of the

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The Scientist and Engineer's Guide to Digital Signal Processing 210 Sample number 0 16 32 48 -1 0 1 2 63 d. Impulse at x[4] Sample number 0 16 32 48 -1 0 1 2 63 a. Impulse at x[0] Frequency -0.5 0 0.5 -2 -1 0 1 2 e. Magnitude Frequency -0.5 0 0.5 -6 -4 -2 0 2 4 6 f. Phase Frequency -0.5 0 0.5 -2 -1 0 1 2 h. Magnitude Frequency -0.5 0 0.5 -6 -4 -2 0 2 4 6 i. Phase Frequency -0.5 0 0.5 -2 -1 0 1 2 b. Magnitude Frequency -0.5 0 0.5 -6 -4 -2 0 2 4 6 c. Phase Sample number 0 16 32 48 -1 0 1 2 63 g. Impulse at x[8] Frequency Domain Time Domain Amplitude Phase (radians) FIGURE 11-1 Delta function pairs in polar form . An impulse in the time domain corresponds to a constant magnitude and a linear phase in the frequency domain. frequency domains. As is usually the case, the polar form is much easier to understand; the magnitude is nothing more than a constant, while the phase is a straight line. In comparison, the real and imaginary parts are sinusoidal oscillations that are difficult to attach a meaning to.
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