Section 32

This is the Title PhL 3.26.05 Chapter 6: Power in Pulse Trains 32. The Autocorrelation Function Here we deal with some preliminary matters before studying the power spectra of pulse trains. Start with a reasonable function x(t). Define the autocorrelation function of x(t) as follows: rx(t) ≡ !Syntax Error, I dt' x(t')* x(t' + t) . (32.1) The integrand is the function evaluated at time t' times the same function evaluated at later time t'+t. Some sources define rx(t) with an extra overall "normalizing" constant factor. For example, if one were to define ax(t) ≡ rx(t)/ T where T is the duration of a pulse train, then ax(t) and rx(t) have different dimensions. Below we show that our rx(t) has dimensions of energy, so ax(t) would have dimensions of power. We prefer defining autocorrelation as shown in (32.1) with no normalizing constant. A simple reflection property follows from the above definition (use t" = t' + t ): rx(-t) = rx(t) * . (32.2) For real x(t) then rx(t) is an even function of t, in (32.1) we could put either +t or -t in the last parentheses. Although we have not yet mentioned statistics and randomness, one could easily imagine the following situation. Suppose x(t') is some sort of random function ("noise") that takes values in the range -1 to 1. It seems likely that for a value of the separation t that is larger than some small value, one might get rx(t) = 0. The vague argument would be that there is no "correlation" between x(t') and x(t'+t), so the product of these two functions ought to be pretty random, and a sum of random numbers in the range -1 to 1 ought to be zero. Even in this case, we can see that the result is not zero if t = 0, since we are then summing a positive quantity. In fact, rx(0) is the area under x(t)2. (a) Autocorrelation function for a Square Pulse Before going any further, let us compute the autocorrelation function for some simple case we are familiar with. A good candidate is x(t) = a square pulse of width τ and amplitude A. As in (9.1), xpulse(t) = A [ θ(t + τ/2) - θ(t - τ/2) ] . (9.1) One can easily do the above integral (32.1) to get the answer, but it is very obvious what the answer is. We are multiplying a box times a box shifted by t. Where they overlap, the integrand is A2. The boxes only overlap if the absolute value of shift tis less than the width τ of the pulse. If this is so, the size of the overlap is τ - |t|. If |t| is larger than τ, there is no overlap, so the integral is 0. Thus, rpulse(t) A2(τ - |t|) θ(τ - |t|) = (A2τ) [ 1 - |t|/τ ] θ(τ - |t|) (32.3) and we find that the autocorrelation function is a triangle whose base is twice the pulse width, Fig 32.1 In general, if x(t) has some finite width τ, rx(t) will have width 2τ. (b) Energy, power and spectral energy density for a finite signal x(t) The total energy in a finite duration signal x(t) can be computed in either the t-domain or the ω-domain using Parseval's formula (10.5), to which we add 1/R to each side, E = !Syntax Error, Idt |x(t)|2/R = !Syntax Error, Idω . (32.4) If we think of x(t) as the voltage across a resistor R, then dt x2(t)/R is the energy delivered to the resistor in time dt, and the integral on the left is the total energy in signal x(t). The dimensions on the right are, looking at (1.1), dω = sec-1 (volt-sec)2/ohms = (volt2/ohms)sec = watts-sec = joules = energy . (32.5) Setting R = 1Ω, we can write this as E = !Syntax Error, Idt p(t) = !Syntax Error, Idω E(ω) (32.6) p(t) ≡ |x(t)|2 = energy density in the t-domain (joule/sec = watt) ( = instantaneous power) p(t)dt = energy in dt (joules) E(ω) ≡ |X(ω)|2/2π = energy density in the ω-domain (joule-sec) E(ω)dω = energy in dω (joules) We shall refer to E(ω) as the spectral energy density of signal x(t) whose Fourier Transform is X(ω). The isolated single pulse xpulse(t) has a corresponding Epulse(ω). The function p(t) is the "temporal energy density". (c) The Wiener-Khintchine theorem Changing to t" = -t', we can trivially rewrite the definition (32.1) as follows: rx(t)  !Syntax Error, Idt" x(t - t") x(-t")* . // energy units (32.7) From now on, we assume x(t) is a real valued function. Recall now the convolution theorem (3.6), a(t) = !Syntax Error, I dt" b(t-t") c(t") A(ω) = B(ω) C(ω) . (3.6) We see that (32.7) has the standard convolution equation form where we select b(t) = x(t) ↔ B(ω) = X(ω) c(t) = x(-t)* ↔ C(ω) = X(ω)* // from (7.2) Thus, the diagonalized frequency domain form A(ω) = B(ω) C(ω) is Rx(ω) = |X(ω)|2 . (32.8) Dividing by 2π we find that E(ω) = (1/2π) Rx(ω) . (32.9) This says that the spectral energy density E(ω) of signal x(t) is 1/2π times the Fourier Integral Transform Rx(ω) of the autocorrelation function rx(t) of the signal x(t). This result is sometimes called the Wiener-Khintchine [Khintchin] theorem. Note also from (32.1) that rx(t) evaluated at t = 0 gives the total energy in signal x(t), rx(0) = !Syntax Error, Idt |x(t)|2 ≡ E = total energy in signal x(t) (32.10) For us, the significance of (32.9) is that we can "inject statistics" into a computation of the autocorrelation function, and then we will know the power spectrum of our statistical signal from (32.9). All we have to do is Fourier transform the autocorrelation function rx(t). Examples will follow. (d) Verification of Wiener-Khintchine for a Square Pulse Equation (32.3) gives the autocorrelation function rx(t) for a square pulse. One can insert this into the Fourier transform (1.1) to compute Rx(ω), Rx(ω) = !Syntax Error, I dt (A2τ) [ 1 - | t | / τ ] e-iωt = 2(A2τ) !Syntax Error, Idt (1-t/τ) cos(ωt) = 2(A2τ) (1-cos(ωτ))/(ω2τ) = 4(A2τ) sin2(ωτ/2)/(ω2τ) = (A2τ2) sin2(ωτ/2)/(ωτ/2)2 = (Aτ)2 [ sinc(ωτ/2) ]2 , (32.11) and this is recognized from (9.2) to be |X(ω)|2 for the square pulse, in agreement with (32.8) . (e) Cross-correlation, convolution, and autocorrelation Notation: a* means complex conjugation, b ∗ c means convolution, b ⋆ c means cross-correlation . The cross-correlation of two functions b and c is defined this way (b⋆c)(t) ≡ !Syntax Error, I dt' b*(t')c(t+t') = (c⋆b)*(-t) (32.12) where the right equality is easy to show setting t+t' = t". So in general, b⋆c ≠ c⋆b . According to our autocorrelation definition (32.1), rx(t) ≡ !Syntax Error, I dt' x(t')* x(t' + t) , (32.1) the autocorrelation of function b is the cross-correlation of b with itself, so rb = b⋆b. In Section 7 we noted that a function is Hermitian if f*(-t) = f(t). Fact: If b and c are both Hermitian, then b⋆c = c⋆b. (32.13) Proof: b⋆c = !Syntax Error, I dt' b*(t')c(t+t') = !Syntax Error, I dt' b(-t')c*(-t-t') = !Syntax Error, I dt' b(t"+t)c*(t") = c⋆b If b and c are both real and both even, they are both Hermitian so again, b⋆c = c⋆b . The convolution of two functions b and c we saw from the (3.1) and (3.2) was this (b∗c)(t) = !Syntax Error, I dt' b(t')c(t-t') = (c∗b)(t) . In order to relate these two operations, we need to show more detail in the notation. Thus, using t" = -t', [b(t)⋆c(t)](t) = !Syntax Error, I dt' b*(t')c(t+t') = !Syntax Error, I dt" b*(-t")c(t-t") = [b*(-t) ∗ c(t)](t) . If b(t) is a Hermitian function so b*(-t) = b(t), then we have shown that : Fact: If b is Hermitian, then b⋆c = b∗c. (32.14) If b = c = real, then we have from (32.1) and (32.2), rb(t) = !Syntax Error, I dt' b(t')b(t'+t) = !Syntax Error, I dt' b(t')b(t'-t) so we have just proven part (a) of this fact, while part (b) was shown above (32.13), Fact: (a) If b is real, then rb = b⋆b = b∗b (b) For any b, rb = b⋆b (32.15) (f) Z Transform Wiener-Khintchine theorem for a Pulse Train The Wiener-Khintchine theorem was stated above in section (c) as Rx(ω) = |X(ω)|2 (32.8) where Rx(ω) is the Fourier Integral transform of the autocorrelation function rx(t) rx(t) ≡ !Syntax Error, I dt' x(t')* x(t' + t) . (32.1) in the case that x(t) is real. Once we know about the convolution theorem (3.6) for the Fourier Integral Transform, we see that the Wiener-Khintchine theorem is just the application of this theorem to the particular convolution equation (32.1). From the amplitudes yn of an infinite pulse train one can define an autocorrelation sequence in analogy with the autocorrelation function, rs ≡ limN→∞ [!Syntax Error, I yn* yn+s ] ≡ <yn* yn+s> . (32.16) Although rx(t) (our particular (32.1) definition) has no normalization factor, we have added to the definition of rs in order to obtain the finite result rs = <yn* yn+s>. It is convenient to use this shorthand notation for rs, rs = !Syntax Error, I yn* yn+s = !Syntax Error, I yn* yn+s (32.17) where T = (2N+1)T1 is the duration of the pulse train. We know that this infinite T is going to cancel another T in any "application" so we allow it to exist temporarily, as in (33.22) where T = [2πδ(0)]T1 . The Z transform of rs is given by R"(z) ≡ !Syntax Error, I rs z-s = !Syntax Error, I{ !Syntax Error, I yn* yn+s } z-s = !Syntax Error, I !Syntax Error, I [yn* zn ] [yn+s z-(n+s)] = !Syntax Error, I !Syntax Error, I [yn* zn ] [ym z-m] m ≡ n+s = [ !Syntax Error, I yn* zn] [!Syntax Error, Iym z-m ] = Y"(z)* Y"(z) = | Y"(z) |2 so we have obtained a Z Transform version of the Wiener-Khintchine theorem. We may regard this simple result as an application of the Z Transform convolution theorem (24.5) to the particular convolution sum (32.17) with Δt → . Here is comparison of the two cases: R"(z) = | Y"(z) |2 Z Transform Wiener-Khintchine (32.18) Rx(ω) = |X(ω)|2 . regular Wiener-Khintchine (32.8) Here X(ω) is the Fourier Integral Transform of the pulse train x(t), x(t) = !Syntax Error, I yn xpulse(t -tn). (25.1) while Y"(z) is the Z transform of the sequence of pulse train amplitudes. The results of this section apply to finite as well as infinite pulse trains. In a way, this fact is obvious if we just pad out the finite pulse train with 0's to make it infinite, but it is still worth showing explicitly. The appropriate autocorrelation sequence for a finite pulse train is given by, rs ≡ !Syntax Error, I yn* yn+s ≡ <yn* yn+s> . // finite pulse train (32.16)' If we assume yn vanishes outside the range -N to N, then rs vanishes outside the range -2N to 2N, in accord with the analog comment above that rx(t) has twice the width of any finite x(t). We shall nevertheless show Σs below as having an infinite range, though we know the range will really be finite when the summand is examined. We then compute R"(z) as above R"(z) ≡ !Syntax Error, I rs z-s = !Syntax Error, I{ !Syntax Error, I yn* yn+sz-s } = !Syntax Error, I !Syntax Error, I yn* yn+s z-s = !Syntax Error, I !Syntax Error, I yn* yn+s z-s Now let m = n+s and replace the s sum with an m sum, = !Syntax Error, I !Syntax Error, I yn* ym z-(m-n) But now the range restrictions from ym on m gives us this final result = !Syntax Error, I !Syntax Error, I yn* ym z-(m-n) = (!Syntax Error, I yn* zn) (!Syntax Error, I ym z-m) so then R"(z) = | Y"(z) |2 = | Y"(z) |2 where now T = (2N+1)T1. (32.18)' which has the exact same form as (32.18).