new Section 34

This is the Title PhL 3.26.05 34. Statistical Amplitude Modulated Pulse Trains Part I (a) Energy and Power Spectra for a General Pulse Train In the previous Section we dealt with simple pulse trains. Here we consider the more general amplitude modulated pulse train. In all equations, one can replace !Syntax Error, I by !Syntax Error, Ito adapt the equation to a finite pulse train instead of an infinite one. x(t) = !Syntax Error, I yn xpulse(t -tn) (34.1) X(ω) = (1/T1)Xpulse(ω) X'ω) = Xpulse(ω) X"(z) (34.2) X"(z) = X'ω)/T1 = !Syntax Error, Iyn e-iωnT. (34.3) To find the frequency-domain power spectrum of a signal x(t), our first task is to compute | X(ω) |2. From (34.2) we get |X(ω)|2 = |Xpulse(ω)|2 (1/T1)2 |X'ω)|2 = |Xpulse(ω)|2 | X"(z) |2 z = eiωT (34.4) We therefore must deal with the following object, using (33.4), | X"(z) |2 = (1/T1)2 |X'ω)|2 = | !Syntax Error, Iyn e-iωnT | 2 = !Syntax Error, Iyn e-iωnT !Syntax Error, Iym* e+iωmT = !Syntax Error, I !Syntax Error, I ym* yn eiω(m-n)T (34.5) so that | X(ω) |2 = | Xpulse(ω) |2 !Syntax Error, I !Syntax Error, I ym* yn eiω(m-n)T (34.6) From box (33.36) we then have E(ω) ≡ |X(ω)|2/2π = (1/2π) | Xpulse(ω) |2 !Syntax Error, I !Syntax Error, I ym* yn eiω(m-n)T (34.7) P(ω) ≡ = (1/2πT) | Xpulse(ω) |2!Syntax Error, I !Syntax Error, I ym* yn eiω(m-n)T (34.8) which we can write as E(ω) = T1 Ppulse(ω) !Syntax Error, I !Syntax Error, I ym* yn eiω(m-n)T (34.9) P(ω) = T1 Ppulse(ω) (1/T) !Syntax Error, I !Syntax Error, I ym* yn eiω(m-n)T (34.10) P = !Syntax Error, Idω P(ω) . (33.31) (34.11) Not knowing details of the yn there is not much else we can do in these expressions. (b) Energy and Power Spectra for a Statistical Pulse Train We imagine now a large ensemble of I pulse trains in which a particular pulse train is labeled by index i. This pulse train has coefficients yn(i) . We are interested in the ensemble averages of E(ω) and P(ω) and P. We indicate the ensemble average of some quantity Q as <Q>. By averaging the last three equations above we obtain <E(ω)> = T1 Ppulse(ω) !Syntax Error, I !Syntax Error, I <ym* yn> eiω(m-n)T (34.12) <P(ω)> = T1 Ppulse(ω) (1/T) !Syntax Error, I !Syntax Error, I <ym* yn> eiω(m-n)T (34.13) <P> = !Syntax Error, Idω <P(ω)> . (34.14) where <ym* yn> = (1/I) < ym(i)* yn(i)> (34.15) Up to this point we have tried to be general, allowing the ym(i) to be complex coefficients, but from now on we consider them to be real, so we can delete the asterisks in the above equations. A very simple case to consider is this, where A and B are real, ym = A probability p ym = B probability 1-p (34.16) Remember that index m labels pulse location m in the pulse train. If probabilities of a pulse value at locations m and at n are uncorrelated (they are independent of each other), then we can write the averaged coefficients as follows, first for two different locations m ≠ n, then for the same location m = n : n ≠ m α ≡ <ymyn> = [pp] AA + [p(1-p)] AB + [(1-p)p]BA + [(1-p)(1-p)]BB n = m β ≡ <ym2> = [p]AA + [(1-p)] BB (34.17) In the square brackets [..] we indicate the probability of some case occurring, and this is multiplied by the value that the quantity in question takes in that case. The reader must now stop reading and stare at the above equations until they make complete sense. Notice that the second line is quite distinct from the first line. In the double summation in (34.13), there are both "diagonal" terms where m = n, and off diagonal terms where m≠n. These groupings must be treated separately according to the above. We have defined new symbols α and β to emphasize that, for the situations shown, there is no longer any dependence on the n and m indices for these quantities. In a later section, we shall encounter a situation where the above equations do not apply because there is correlation between positions m and n (the AMI line code). Inserting (34.17) into (34.13) gives <P(ω)> = T1 Ppulse(ω) (1/T) { α!Syntax Error, I !Syntax Error, I [ eiω(m-n)T] +β!Syntax Error, I !Syntax Error, I [1] } (34.18) In Section 35 we shall evaluate this important result for infinite and finite pulse trains, but we pause momentarily for a digression. (c) Digression on expectation values, correlation and covariance In the language of statistics, one can imagine a "random variable" Ym associated with location m in our pulse train, and this variable can take certain values ym (perhaps 0 and 1). We imagine a statistical ensemble of pulse trains indexed by i, and at location m the ith pulse train has value ym(i) of Ym. If all pulse trains in our ensemble of pulse trains count the same in our average (they do), then one can write <ymyn> in (35.1) as E(YmYn) = (1/I) !Syntax Error, I ym(i)yn(i) in the sense of E(XY) = (1/I) !Syntax Error, Ix(i)y(i) where E(q) means the expectation value of quantity q. This particular expectation value has a special name: it is called the correlation of X and Y. So in fact we have <ymyn> = E(YmYn) = (1/I) !Syntax Error, I ym(i)yn(i) ≡ corr(YmYn) // correlation of Ym and Yn If corr(YmYn) is a significant positive or negative number, then the random variables Ym and Yn are strongly correlated in some way. This means that the probabilities of values of ym(i) and yn(i) are related in some way. One might wonder how there could be a correlation between locations m and n in a pulse train. We shall see this happen below in the discussion of the AMI lines code. If Ym and Yn are independent random variables, we expect corr(YmYn) ≈ 0. Two other expectation values of interest are these, the mean and the covariance : E(Ym) = (1/I) !Syntax Error, Iym(i) = <ym> // mean of Ym E( [Ym - <ym>][ [Yn - <yn>]) = (1/I) !Syntax Error, I[ ym(i) - <ym>] [yn(i)- <yn>] = cov(YmYn) // covariance of Ym and Yn If it happens that <ym> = 0 for a certain pulse train, then covariance and correlation are the same.