Connection between Ng notes and Spectral
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A brief note dated 3.22.13 from the temp files of the Spectral Theory book, signed PhL. It compares the pulse-train average <a_m a_n> used in the Spectral text with the covariance and correlation of random variables, treating the index i as the statistical sum index. It then begins on autocorrelation r_x(t). Some equations were lost in extraction.
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Connection: between Ng notes and Spectral PhL 3.22.13
A. In Spectral we deal with
<aman> ≡ (1/I) !Syntax Error, Iaimain
where m and n mark positions in time. so am is a digital thing. You could write
<a(tm)a(tn)> = (1/I) !Syntax Error, Iai(tm)ai(tn)
I know that if random variables xa and xb each have zero mean, we can say
cov(Xa,Xb) = (1/m)Σi=1m xa(i) xb(i)
cov(Xm,Xn) = (1/M)Σi=1m xm(i) xn(i)
cov(Am,An) = (1/M)Σi=1m am(i) an(i)
and this looks a lot like <aman> . If they don't have zero mean, I think you can still say
corr(Am,An) = (1/M)Σi=1m am(i) an(i)
So Am is the random variable associated with the value of the pulse at position m in the pulse train. The index i is the statistical sum index.
B. Now what about autocorrelation.
rx(t) ≡ !Syntax Error, I dt' x(t') x(t' + t) (32.1)