TAILIEUCHUNG - DISCRETE-SIGNAL ANALYSIS AND DESIGN- P22

DISCRETE-SIGNAL ANALYSIS AND DESIGN- P22: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 91 c The convolution of a and b using the basic convolution equation 5-4 y n x m y m . We see the familiar smoothing and stretching operation that the convolution performs on x m and h m . Convolution needs the additional time region for correct results as noted in the equation in part c and in Eq. 5-6 . d This step gets the convolution of x m and h m and also the spectrum DFT of the convolution in one step using the double summation of Eq. 5-6 . e See step f . f Steps e and f . These steps get the DFT spectrum X k of x m and spectrum H k of h m using the DFT in Eq. 1-2 . g The product X k H k is the spectrum Z k of the output . This product is the sequence multiplication described in Eq. 5-1 . The additional factor N will be explained below. Note that the spectrum of the output in part g is identical to the spectrum of the output found in part d . That is X k H k DFT of x m h m and the IDFT of X k H k x m h m not shown here. h The IDFT in Eq. 1-8 produces the sequence z n which is identical to the convolution x m h m that we found in part c . In part g we introduced the factor N. If we look at the equation in part d we see a single factor 1 N. But the product of X k H k in part g produces the factor 1 N 2. A review of parts e and f verify this. This produces an incorrect scale factor in part h so we correct part g to fix this problem. There are different conventions used for the scale factors for the various forms of the DFT and IDFT. The correction used here takes this problem into account for the Bracewell see Chapter 1 conventions that we are using. This action does not produce an error. it makes the bookkeeping correct and the Mathcad worksheet takes the correct action as needed. These mild discrepancies can show up and need not create concern. In Fig. 5-7 the convolution in parts c and h has a sharp peak at location n 8. When the signal is mildly contaminated with noise this is a good location to place a detector circuit that .

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