DISCRETE-SIGNAL ANALYSIS AND DESIGN- P26

DISCRETE-SIGNAL ANALYSIS AND DESIGN- P26:Electronic circuit analysis and design projects often involve time-domainand frequency-domain characteristics that are difÞcult to work with usingthe traditional and laborious mathematical pencil-and-paper methods offormer eras. This is especially true of certain nonlinear circuits and sys-tems that engineering students and experimenters may not yet be com-fortable with.
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DISCRETE-SIGNAL ANALYSIS AND DESIGN- P26 PROBABILITY AND CORRELATION 111for it is Eq. (6-16), where we use E (expectation) to mean the same as“averaging”, assuming that many repetitions have been performed: E{[x(n) − E(x(n))][y(n) − E(y(n))]} ρxy = √ (6-16) V (x(n))V (y(n)) E{[x(n) − E(x(n))][y(n) − E(y(n))]} = σX σY After many repetitions and averaging of ρxy , the numerator is theexpected value of the cross-covariance of x (n) and y(n) [Eq. (6-15)],and the denominator is the square root of the product of the variances ofx (n) and y(n), or more simply, just σx σy . V (x (n)) and V (y(n)) or (σx andσy ) must both be greater than 0.0. This equation can be simpliÞed as E[x(n)y(n)] − E[x(n)]E[y(n)] ρxy = √ (6-17) V (x(n))V (y(n)) E[x(n)y(n)] − E[x(n)]E[y(n)] = σX σYIf x (n) and y(n) are independent then the numerator of Eq. (6-17) is zero: E[x(n)y(n)] = E[x(n)]E[y(n)] (6-18)and ρxy = 0.0. However, there are some cases, not to be explored here,where x (n) and y(n) are not independent, yet ρxy is nevertheless equal tozero. So “uncorrelated” and “independent” do not always coincide. Look-ing at Eq. (6-18), we can guess that this might happen. For further insightabout the correlation coefÞcient, see [Meyer, 1970, Chap. 7]. As an example we will calculate ρxy of Fig. 6-5 using Eq. (6-17) andthe time-averaged values instead of expected values because Eq. (6-17)is assumed to be noise-free: (xy) − x y ρXY = (6-19) σX σY 0.096 − (0.31 · 0.277) = = 0.099 0.344 · 0.286The same calculation on Fig. 6-4 produces a value of 1.00.112 DISCRETE-SIGNAL ANALYSIS AND DESIGN This brief introduction to correlation and variance is no more than a“get acquainted” starting point for these topics and is not intended as asubstitute for more advanced study and experience with probability andstatistical methods. We are limited to signal sequences that are discrete inboth time and frequency domains from 0 to N − 1, which makes thingsa little easier. Mathcad calculates very easily all of the equations in thischapter.REFERENCESCarlson, A. B., 1986, Communication Systems, 3rd ed., McGraw-Hill, New York.Meyer, P. L., 1970, Introductory Probability and Statistical Methods, Addison- Wesley, Reading, MA.Oppenheim, A. V., and R. W. Schafer, 1999, Discrete-Time Signal Processing, 2nd ed., Prentice Hall, Upper Saddle River, NJ.Schwartz, M., 1980, Information Transmission, Modulation and Noise, 3rd ed., McGraw-Hill, New York.Zwillinger, D., Ed., 1996, CRC Standard Mathematical Tables and Formulae, CRC Press, Boca Raton, FL. 7The Power SpectrumWe have learned that a time-domain discrete sequence x (n) that extendsfrom 0 ≤ n ≤ N − 1 can be considered as two-sided, positive-time forthe Þrst half and negative-time for the second half. Each sample x (n),considered by itself, is just a magnitude (see Chapter 1). It also hasa time-position attribute but none other, such as frequency or phase orproperties such as real or imaginary. In other words, x (n) is not a phasor.It is what we see on an ordinary oscilloscope. On the other hand, the x (n) sequence (the entire scope screen dis-play) can consist of a set of complex-valued voltage or current waveformsapplied to a complex load network of some kind. However, time-domainanalysis of complex signals combined with complex loads requires mathmethods that we will not explore in this book [Oppenheim and Schafer,1999; Carlson, 1986; Schwartz, 1980; Dorf and Bishop, 2005; Sheareret al., 1971], so we prefer to convert the time sequence x (n) to the fre-quency X (k ) domain using the DFT. After processing the signal in thefrequency domain we can, if we wish, use the IDFT to get the timedomain x (n) sequence representation of the processed discrete signal.Discrete-Signal Analysis and Design, By William E. SabinCopyright  2008 John Wiley & Sons, Inc. 113114 DISCRETE-SIGNAL ANALYSIS AND DESIGNThis is a simple and very useful approach that is widely used, especiallyin computer-aided design. In this chapter we are interested in power. We are also interested inphasors. The problem is that any phasor that has constant amplitude haszero average power, so it makes no sense to talk about average phasorpower. Therefore, we will combine the positive- and negative-frequencyphasors coherently, using the methods described in Fig. 2-2 and employedelsewhere, to get a positive-frequency sine wave or cosine wave at fre-quency (k ) and phase θ(k ) from 1 ≤ k ≤ N/2 − 1. We then have a truesignal that has average power at frequency (k ), and we can look at itspower spectrum. There is another approach available. The real or imaginary part of thephasor Mej ωt is a sinusoidal wave that has a peak value M . The rms valueof this sinusoidal wave, considered by itself, is M · 0.7071. In our Mathcadexamples the method of the previous paragraph, where we combine bothsides of the phasor spectrum coherently, is an excellent and very simpleapproach that takes into account the two complex-conjugate phasors thatare the constituents of the true sine or cosine signal.FINDING THE POWER SPECTRU ...