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DISCRETE-SIGNAL ANALYSIS AND DESIGN- P17: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. | 66 DISCRETE-SIGNAL ANALYSIS AND DESIGN the 0 to N 1 region. Some additional programming steps can accomplish this but for this example because of the two almost zero-valued end regions we assign the value 0.0 which is very nearly correct. Math-cad assigns 0 to unused locations for example at i 1 and i N. If necessary the two guardbands can be lengthened a little. We see also that the sequence can be a modified time sequence in which case the smoothing is filtering certain regions of the time domain or it can be a modified frequency sequence in which case certain frequency ranges can be modified. For example the sharp edges of a band filter are softened and rounded to obtain benefits such as improved group delay near the band edges. This method is also known as transition sampling Oppenheim and Schafer 1975 pp. 250-254 . In this case the IDFT is the time domain of the modified frequency response that can be used to improve an analog filter or digital filter. Analog mechanical and crystal filters often use the transitional design which is the Bessel response to the 6-dB level for improved phase linearity and Chebyshev beyond that for a good shape factor. All-pass networks further improve group delay variations at the cost of additional time delay. When networks with too much time delay are used inside an automatic gain control AGC loop transient response and stability become much more difficult. We almost always avoid putting these types of filters within a fast-responding gain-control feedback loop. The smoothing should focus on undesired rapid changes without degrading excessively the desired slower-changing signal. A good way to implement this is to use many closely spaced samples and use the three-point method no more times than necessary for adequate results. A further consideration is the rounding off at the corners where smoothing has reduced the 3-dB width of the time or frequency response. This can often be compensated by modifying the time and frequency .
