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DISCRETE-SIGNAL ANALYSIS AND DESIGN- P20

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DISCRETE-SIGNAL ANALYSIS AND DESIGN- P20: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. | MULTIPLICATION AND CONVOLUTION 81 Figure 5-2 Example of polynomial multiplication using double summation to find the area of a figure. CONVOLUTION Convolution is a valuable tool for the analysis and design of communications systems and in many other engineering and scientific activities. Equation 5-4 is the basic equation for discrete-time convolution. œ y n x m h m x m h n m 5-4 m œ where is the convolution operator and y n x m and h m can all be the complex-valued discrete-time sequences I and jQ that we considered carefully in Chapter 1. Note that x m h m and y n are in the time domain but they can also be complex Y k X k and H k in the frequency domain with magnitude and phase attributes. Also all three can have different amplitude scale factors on the same graph or on separate graphs. We focus initially on the time domain. Equation 5-4 appears to be simple enough but actually needs some careful study and practice to develop insight and to assure correct answers. 82 DISCRETE-SIGNAL ANALYSIS AND DESIGN Sequence x m is a signal input time-domain sequence that extends in time from m to to m to. In practical problems this sequence is assumed to have useful amplitude only between two specific limits m min and m max . Sequence h m refers to a system function also a time-domain sequence that is assumed to have useful limits from m min to m max which may not be the same limits as the limits for x m . Sequence h n m is h m that has had two operations imposed 1 h m has been flipped in time and becomes h m and 2 h m has been shifted to the right n places starting from an initial value of n determined by the nature of the problem whose value is now h n m . The expression fold and slide is widely used to describe this two-part operation. One reason for the fold and slide of h m to h n m is that we want the leading edge in time of h m the response to encounter the leading edge in time of x m the excitation as we start the sliding operation increasing n from starting n to .