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DISCRETE-SIGNAL ANALYSIS AND DESIGN- P10: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. | SINE COSINE AND 0 31 Real Imaginary cos --------1-------------------------------1----- a -cos -------1-------------------------------1----- b sin --------1------------------------------- ----- -sin d j cos ------ ------------------------------- ----- e -j cos f jsin -------1-------------------------------1----- g -j sin -------1-------------------------------1----- h Figure 2-2 Phasor polarity and type for all possible sine-cosine waves. Solid lines real dashed lines imaginary. amplitude 8. The spectrum is positive-sided at 2 and 6. x n 5 cos 2n n 2 j N j 8 cos 2n n 6 N 2-1 32 DISCRETE-SIGNAL ANALYSIS AND DESIGN In Fig. 2-3 the n step size 0.1 from 0 to N 1 provides smooth curves in the real and imaginary x n graphs. Figure 2-3 shows the following The real cosine spectrum at k 2 and 14 amplitude 2.5 2.5 5.0 see Fig. 2-2a . The j cosine spectrum at k 6 and 10 amplitude 4 4 8 see Fig. 2-2f . The phase at k 2 and 14 0 . The phase at k 6 and 10 90 . Observe that Mathcad provides the correct two-sided phasors Fig. 2-2 that the subsequent two-sided IDFT restores to the input x n . If x n is viewed from 0 to N 1 the positive frequencies 2 and 6 number of cycles per record length are visible. In more complicated situations it is a good idea to avoid possible confusion by making sure that all of the Re X k and Im X k phasor pairs are combined into the correct one-sided real and imaginary sine and cosine positive-frequency constituents as defined in Fig. 2-2. For the example in Fig. 2-3 the one-sided output is 5 cosine at f 2 at 0 and j 8 cosine at f 6 at 90 . Note the use of the Mathcad function atan2 Re v Im v -180 n which covers the range 180 as compared with atan v 180 n which only covers 90 . There are some possibilities for de-cluttering the results IfIm X k 0.0001 set fy k 0 to avoid clutter in the phase data. If Re X k 0.0001 set Re X k 0.0001. If j Im X k j 1000 set jIm X k j 1000. If j Im X k j 1000 set jIm X k j 1000. Scale the problem to avoid values of X k 0.001
