TAILIEUCHUNG - DISCRETE-SIGNAL ANALYSIS AND DESIGN- P21

DISCRETE-SIGNAL ANALYSIS AND DESIGN- P21: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. | 86 DISCRETE-SIGNAL ANALYSIS AND DESIGN and h m must provide an initial separation of at least one m . If we need to reduce L in order to reduce the sequence length the method of circular convolution can be used as illustrated in Fig. 5-5. Note that the amplitude scales and the number of samples for x m and h m can be different. The following steps are executed. a x m is plotted. b h m is plotted. c h m is flipped about the m 0 point on the m-axis. The h m line at m 2 length equals zero now appears at m 2 which is the same as m 2 16 14. The line that was at m 11 length 9 now appears at m 11 which is the same as m 11 16 5. d We do not start the sum of products convolution of overlapping x m and h m at this time. e The h m sequence advances 2 places to h 2 m . The line at m 14 length zero moves to m 16 which is identical to m 0. The h n m and x m sequences are now starting to intersect at m 0. f The process of multiply add and shift begins. The result for 11 m is shown for which the convolution value is 224. For values greater than 18 m the convolution value is 0 because x and h no longer intersect. As we can see this procedure is confusing when done manually we do not automate it in this book and it is suggested that the side-by-side method of Figs. 5-3 and 5-4 be used instead. Adjust the length L x m h m 1 as needed by increasing N 2M and let the computer perform Eq. 5-4 the simple sum of products of the overlapping x m and h m amplitudes. The lines of zero amplitude are taken care of automatically. Design the problem to simplify the process within a reasonable length L. Time and Phase Shift In Fig. 5-6 a sequence x m is shown. A single h 3 is also shown and h 3 is the flip from 3 to 3. The amplitude of h m is . If we now suddenly move h 3 forward six places to become h 6 3 for an n 6 we get y 3 which is a copy of x 6 3 shifted to the right three MULTIPLICATION AND CONVOLUTION 87 16 8 0 12 i x m 10 8 . 6 4 i 2 . 0 _ 0 1 2 3 4 5 6 7 8

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