DISCRETE-SIGNAL ANALYSIS AND DESIGN- P24: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. | PROBABILITY AND CORRELATION 101 fact that linear systems have superposition of average or expected power values that are independent uncorrelated see later in this chapter and Chapter 7 . This Pav is a random variable 0 that has an average value for a large number of repetitions or possibly for one very long sequence. Numerous repeats of Eq. 6-5 converge to values close to W. In dB the ratio of desired signal power to undesired noise power is 10 log - dB 6-7 N We are often interested in the ratio N N 1 S N dB in this example not much different. This exercise illustrates the importance of averaging many calculation results when random noise or other random effects are involved. A single calculation over a single very long sequence may be too time consuming. Advanced texts consider these random effects in more excruciating detail. Variance Signals often have a dc component and we want to identify separately the power in the dc component and the power in the ac component. We have looked at this in previous chapters. Variance is another way to do it in the time domain especially when x n includes an additive random noise term e n and is defined as V x n 02 E x n x n 2 E xr n 2 E x n 2 6-8 x n 2 x n 2 where x x e V x n is the expected or average value of the square of the entire waveform minus the square of the dc component and the result is the average ac power in x n . The distinction between the average-of-the-square and the square-of-the-average should be noted. The positive root VV x n is known as ox the standard deviation of x and has an ac rms volts value which we look at more closely in the next topic. 102 DISCRETE-SIGNAL ANALYSIS AND DESIGN A dozen records of the noise-contaminated signal using Eq. 6-8 followed by averaging of the results produces an ensemble average that is a more accurate estimate of the signal power and the noise power. An example of variance as derived from Fig. 6-1b using Eq. 6-8 is shown in Eq. 6-9 . T r 2i average
