DISCRETE-SIGNAL ANALYSIS AND DESIGN- P26

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DISCRETE-SIGNAL ANALYSIS AND DESIGN- P26: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.

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PROBABILITY AND CORRELATION 111 for 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 the expected 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 of x (n) and y(n), or more simply, just σx σy . V (x (n)) and V (y(n)) or (σx and σy ) 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 σY If 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 to zero. So “uncorrelated” and “independent” do not always coincide. Look- ing at Eq. (6-18), we can guess that this might happen. For further insight about the correlation coefÞcient, see [Meyer, 1970, Chap. 7]. As an example we will calculate ρxy of Fig. 6-5 using Eq. (6-17) and the 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 · 0.286 The 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 a substitute for more advanced study and experience with probability and statistical methods. We are limited to signal sequences that are discrete in both time and frequency domains from 0 to N − 1, which makes things a little easier. Mathcad calculates very easily all of the equations in this chapter. REFERENCES Carlson, 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.
7 The Power Spectrum We have learned that a time-domain discrete sequence x (n) that extends from 0 ≤ n ≤ N − 1 can be considered as two-sided, positive-time for the 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 has a time-position attribute but none other, such as frequency or phase or properties 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 waveforms applied to a complex load network of some kind. However, time-domain analysis of complex signals combined with complex loads requires math methods that we will not explore in this book [Oppenheim and Schafer, 1999; Carlson, 1986; Schwartz, 1980; Dorf and Bishop, 2005; Shearer et 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 the frequency domain we can, if we wish, use the IDFT to get the time domain x (n) sequence representation of the processed discrete signal. Discrete-Signal Analysis and Design, By William E. Sabin Copyright  2008 John Wiley & Sons, Inc. 113
114 DISCRETE-SIGNAL ANALYSIS AND DESIGN This is a simple and very useful approach that is widely used, especially in computer-aided design. In this chapter we are interested in power. We are also interested in phasors. The problem is that any phasor that has constant amplitude has zero average power, so it makes no sense to talk about average phasor power. Therefore, we will combine the positive- and negative-frequency phasors coherently, using the methods described in Fig. 2-2 and employed elsewhere, 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 true signal that has average power at frequency (k ), and we can look at its power spectrum. There is another approach available. The real or imaginary part of the phasor Mej ωt is a sinusoidal wave that has a peak value M . The rms value of this sinusoidal wave, considered by itself, is M · 0.7071. In our Mathcad examples the method of the previous paragraph, where we combine both sides of the phasor spectrum coherently, is an excellent and very simple approach that takes into account the two complex-conjugate phasors that are the constituents of the true sine or cosine signal. FINDING THE POWER SPECTRUM We will use voltage values, but current values apply equally well, using the Norton source transformation [Shearer et al., 1971]. The discrete Fourier transform DFT [Eq. (1-2)] of an x (n) signal sequence leads to a dis- crete two-sided X (k ) steady-state spectrum of complex voltage phasors oscillating at frequency (k ) from 1 to N − 1 with amplitude X (k ) and relative phase θ(k ). At each (k ) and (N − k ) we will combine a pair of complex-conjugate phasors to get a positive-side sine or cosine V (k ) from k = 1 to k = N /2 − 1. In order to keep the analysis consistent with circuit realities, assume that V (k ) is an open-circuit voltage generator whose internal impedance is for now a constant resistance R g , but a complex Z g (k ) can very easily be used. V (k ) is then the steady-state open-circuit voltage at frequency (k ). The voltage VL(k ) across the load Y (k ) (see Fig. 7-1) is found independently
THE POWER SPECTRUM 115 Y(k) RG P(k) G(k) V(k) ± jB(k) VL(k) GB(k) (XE(k))2 + (XO(k))2 PL(k) := ·Y(k) (1 + R·Y(k))2 V(k) VL(k) := 1 + RG·Y(k) Figure 7-1 Equivalent circuit of frequency-domain power spectrum at frequency position k . at each frequency (k ) as V (k) V (k) V L(k) = = (7-1) 1 + RG Y (k) 1 + ZG (k)Y (k) For a given V (k ) the steady-state value of VL(k ) depends only on the present value of Y (k ) and Z G (k ) or R G , and not on previous or future val- ues of (k ). The complex load admittance Y (k) = [G(k) + GB (k)] ± j B(k) siemens which we have pre-determined by calculation or measurement at each frequency (k ), is driven by the complex signal voltage V L(k) = Re[V L(k)] ± j Im[V L(k)], and the complex power to the load is PL(k) = V L(k)2 Y (k) watts and vars (7-2) Figure 7-1 shows the equivalent circuit for Eq. (7-1). PL(k ) is the power spectrum with real part (watts), imaginary part (vars), and phase angle θ(k ), that is delivered to the complex load admittance Y (k) = G(k) + GB (k) ± j B(k). If B (k ) is zero, the power Pl(k ) is in phase with