96 DISCRETE-SIGNAL ANALYSIS AND DESIGN adjust scaling factors (Chapter 1) as needed to assure “adequate” cover- age and resolution of time and frequency. The smoothing and windowing methods in Chapter 4 are helpful in reducing time and frequency require- ments without deleting important data. This is where the “art” of approx- imation 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 com- plexities 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 Þrst (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. ForaninÞnite x (n) sequence, the quantity E [x (n)], the statistical expected value, also known as the Þrst moment [Meyer, 1970, Chap. 7], is E[x(n)] = ∞ n=−∞ x(n)p(x(n)) (6-1) where p(x (n)istheprobability, from 0.0 to +1.0, of occurrence of a particular x (n)ofaninÞnite sequence. In deterministic sequences from 0toN −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 −2 0 0 (a) (b) (c) 2 255 −0.4 −0.2 0 0.2 0.4 1 10 100 Noise count (vertical) vs. noise amplitude (horizontal) for 256 total counts 7 13 45 55 63 46 17 7 3 0 50 100 150 200 250 0 1 2 Figure 6-1 Signal without noise (a), with noise and envelope detected (b), and a sample histogram of the envelope-detected noise alone (c). time-average value are the same for a repetitive deterministic sequence with little or no aliasing: X(0) = E[x(n)] =x(n)= 1 N N−1 n =0 x(n) (6-2) all of which are zero in Fig. 6-1a. Figure 6-1b introduces a small amount of additive noise voltage ε(n) to each x(n), so called because each x(n) is the same as in Fig. 6-1a but with noise ε(n) that is added to x(n), not multiplied by x(n) as it would be in various nonlinear applications. The noise voltage itself, as found 98 DISCRETE-SIGNAL ANALYSIS AND DESIGN dB dB Linear Plot 0.4 0.3 0.2 0.1 0 s = 1 −5 −4 −3 −2 −1 012345 −10 −8 −6 −4 −20246810 0 −20 −40 −60 −80 −100 −120 0 −20 −40 −60 −80 −100 −120 −140 −4 −3 −2 −1 01234 s = 2 s = 1 s = 2 Figure 6-2 Normal distribution for σ = 1 and 2. in communications systems, is assumed to have the familiar bell-shaped Gaussian (also called) amplitude density function shown in the histogram in Fig. 6-1c and in more detail in Figs. 6-2 and 6-3 [Schwartz, 1980, Chap. 5]. See also Chapter 7 of this book for a more detailed discussion of noise and its multiplication. PROBABILITY AND CORRELATION 99 The noise has random amplitude. For this discussion, the signal plus noise [x (n) +ε(n)] is measured by an ideal (linear) full-wave envelope detector that delivers the value |x (n) +ε(n)| at each (n), where the vertical bars imply the positive magnitude. In Fig. 6-1b the ε(n) noise voltage is a small fraction of the signal voltage. In each calculation of the sequence x(n) a new sequence ε x (n) is also generated. Equation (6-3) Þnds the positive magnitude of [x(n) +ε(n)]. E[|(x +ε)n|] ≈|(x +ε)n| ≈ 1 N N−1 n=0 [|(x +ε)n|] (6-3) This is an example of the wide-sense stationary process, in which the expected value and the time average of a very long single record or the average of many medium-length records are the same and the autocorre- lation value (discussed later in the chapter) depends only on the time shift τ [Carlson, 1986, Chap. 5.1]. The contribution of the sequence ε(n) to the detector output is a ran- dom variable that ßuctuates with each solution of Eq. (6-3). The histogram approximation in Fig. 6-1c for one of the many solutions shows a set of amplitude values of ε(n), arranged in a set of discrete values on the hor- izontal scale, with the number of times that value occurs on the vertical scale. The total count is always 256, one noise sample for each value of n from 0 to 255. The count within each rectangle, divided by 256, is the fraction of the total occurring in that rectangle. This is a sim- ple normalization procedure that we will use [see also Schwartz, 1980, Fig. A-9]. The averaging of a large number of these data records leads to an ensemble average. A single very long record such as N =2 16 =65,536 using Mathcad is also a very good approximation to the expected value. Average Power The average power, also known as the expected value of x(n) 2 and also as the time-average value x(n) 2 of the deterministic (no noise) sequence in Fig. 6-1a is 1.00 W into 1.0 ohm, found in Eq. (6-4) P av = E[x(n)] 2 = x(n) 2 = 1 N N−1 n=0 x(n) 2 = 1.00 W (6-4) 100 DISCRETE-SIGNAL ANALYSIS AND DESIGN This includes any power due to a dc component in x (n), which in Fig. 6-1a is zero. The power in the envelope-detected signal is also 1.00 W. The total power in both cases must be identical because this ideal full-wave envelope detector is assumed to be 100% efÞcient. In Fig. 6-1b with added noise, the power is P av ≈ 1 N N−1 n=0 [ x(n) + ε(n) ] 2 ≈ 1 N N−1 n=0 ( x(n) ) 2 + 2x(n)ε(n) + ( ε(n) ) 2 (6-5) ≈ 1 N N−1 n=0 ( x(n) ) 2 + ( ε(n) ) 2 ≈1.024 W (on average) and the additional 0.024 W is due to noise alone. In most situations, the expected value of the product [2x(n)ε(n)] is assumed to be zero because these two are, on average, statistically inde- pendent of each other. In this example with additive noise and full-wave ideal envelope detection we will be slightly more careful. Comparing noise and signal within the linear envelope detector, and assuming that the signal is usually much greater than the noise, the noise and signal are, as a simpliÞcation, in phase half of the time and in opposite phase half of the time. P(n) can then be approximated as P(n) ≈ [ x(n) + ε(n) ] 2 = x(n) 2 + 2x(n)ε(n) + ε(n) 2 ≈ [ x(n) − ε(n) ] 2 = x(n) 2 − 2x(n)ε(n) + ε(n) 2 (6-6) average value of P(n)≈ x(n) 2 + ε(n) 2 which is the same as Eq. (6-5). Individual calculations of this product in Eq. (6-5) vary from about +20 mW to about −20 mW, and the average approaches zero over many repetitions. A single long record N =2 16 =65,536 produces about ±1.0 mW, and P av ≈1.024 W. The P av result is then the expected value of signal power plus the expected value of noise power. This illustrates the interesting and useful . 96 DISCRETE-SIGNAL ANALYSIS AND DESIGN adjust scaling factors (Chapter 1) as needed to assure “adequate” cover- age and resolution of time and frequency. The smoothing and windowing methods. Figs. 6-2 and 6-3 [Schwartz, 1980, Chap. 5]. See also Chapter 7 of this book for a more detailed discussion of noise and its multiplication. PROBABILITY AND CORRELATION 99 The noise has random amplitude it would be in various nonlinear applications. The noise voltage itself, as found 98 DISCRETE-SIGNAL ANALYSIS AND DESIGN dB dB Linear Plot 0.4 0.3 0.2 0.1 0 s = 1 −5 −4 −3 −2 −1 012345 −10 −8 −6