DISCRETE-SIGNAL ANALYSIS AND DESIGN- P23: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. | 96 DISCRETE-SIGNAL ANALYSIS AND DESIGN adjust scaling factors Chapter 1 as needed to assure adequate coverage and resolution of time and frequency. The smoothing and windowing methods in Chapter 4 are helpful in reducing time and frequency requirements without deleting important data. This is where the art of approximation and practical engineering get involved and we delay the more advanced mathematical considerations for consideration down the road. Also small quantities of noise can very quickly take us into the complexities of statistical analysis. We are for the most part going to avoid this arena because it is does not suit the limited purposes of this book. For this reason we are going to be somewhat less than rigorous in some of the embedded-noise topics to be covered. However we will have brief contacts that will give us a feel for these topics. PROPERTIES OF A DISCRETE SEQUENCE Expected Value of x n In Fig. 6-1a a discrete time sequence 256 points is shown from n 0 to N 1. The first positive half is from 0 to N 2-1 and the second negative half is from N 2 to N 1 as we saw in Figs. 1-1d and 1-2b. For an infinite x n sequence the quantity E x n the statistical expected value also known as the first moment Meyer 1970 Chap. 7 is E x n x n p x n 6-1 n tt where p x n is the probability from to of occurrence of a particular x n of an infinite sequence. In deterministic sequences from 0 to N 1 such as Fig. 6-1a each x n is assumed to have an equal probability 1 N from 0 to N 1 of occurrence the amplitudes of x n can all be different and Eq. 6-1 reverts to the time-average value x n from 0 to N 1 which we have been calling the time-domain dc value and which is also identical to the zero frequency X 0 value for the X k frequency-domain sequence. Note that the angular brackets in x n refer to a time average . We have reasonably assumed that expected value and PROBABILITY AND CORRELATION 97 a b Figure 6-1 Signal without noise a with noise and envelope .
