TAILIEUCHUNG - DISCRETE-SIGNAL ANALYSIS AND DESIGN- P18

DISCRETE-SIGNAL ANALYSIS AND DESIGN- P18:Electronic circuit analysis and design projects often involve time-domain and frequency-domain characteristics that are difÞcult to work with using the traditional and laborious mathematical pencil-and-paper methods of former eras. This is especially true of certain nonlinear circuits and sys- tems that engineering students and experimenters may not yet be com- fortable with. | SMOOTHING AND WINDOWING 71 for the improved attenuation of the side lobes. For the Hann the value of the peak response is -38 dB at k for the Hammimg it is -53 dB at k and for the rectangular it is only 18 dB at k . Comparing the main lobe widths in the vicinity of X k 1 the Hann is at 20 dB and the Hamming is at 20 dB which can perhaps be a worthwhile improvement. Comparing the Hamming and the Hann the Hamming provides deeper attenuation of the first side-lobe which is one of its main goals and limits in the neighborhood of 45 to 60 dB at the higher-frequency lobe peaks another goal . For many applications this is quite satisfactory. On the other hand the Hann is not quite as good up close but is much better at higher frequencies and this is often preferred. In many introductory references these two windows seem to meet the majority of practical requirements for non-integer frequency k values. In the equations for the Hamming and Hann windows we see the sum of a constant term and a cosine term. There are other window types such as the Kaiser window and its variations that have additional cosine terms. These may be found in the references at the end of the chapter and are not pursued further in this book. These other window types are useful in certain applications as discussed in the references. We noted that the time-domain window sequence multiplies a timedomain signal sequence. In the frequency domain the spectrum of the window convolves with the spectrum of the signal. These interesting subjects will be explored in Chapter 5. Figure 4-4 is a modification of Fig. 4-3 that illustrates the use of convolution in the frequency domain. Equation 4-3 contains formulas for the spectra of the windows including frequency translation to . Rectangle X1 k T7 N N-1 I- E n __ 1 i exp Í j2nN - k n 0 L Hamming 1 -1 Í X2 k N X I 0-54 - 0-56 n 0 X cos n j2nN 38-0 - k 4-3 72 DISCRETE-SIGNAL ANALYSIS AND DESIGN N-1 Hann X3 k n 0 cos 2n n N - n exp i .

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