# the demonstration that states that the FT of the ACF function is the square of the DTFT of the signal

I am following the book The Intuitive Guide to Fourier Analysis & Spectral Estimation with MATLAB. I am trying to selflearn the fourier analysis in matlab. I got lost in one passage in the demonstration that states that the FT of the ACF function is the square of the DTFT of the signal. I have attached it here As you can see in the passage that I named 1. a delta_tau is missed in my opinion. Can you confirm that? More important in the point 2. in my opinion there should be a tau and not t. So I don't know how it is demonstrated this formula, because if in the passage 2. there is t the demonstration can be ended. I hope you can help me

• I am the author of this book. Thanks for pointing out the typo. Will fix in the kindle version. Thanks to Matt. Math books are hard to write! Charan Langton – charan langton Oct 22 '19 at 17:51

You are right, the derivation is full of typos. The first equation below Eq. $$(8.39)$$ should read

$$\int_{-\infty}^{\infty}x(t+\tau)e^{\color{red}{-}j\omega\tau}d\tau=X(\omega)e^{j\omega \color{red}{t}}\tag{1}$$

Substituting into $$(8.39)$$ gives

\begin{align}\mathcal{F}\big\{R(\tau)\big\}&=\int_{-\infty}^{\infty}x(t)X(\omega)e^{j\omega t}dt\\&=X(\omega)X(-\omega)=|X(\omega)|^2\end{align}\tag{2}

where the last equality is only true for real-valued $$x(t)$$. However, the overall result is also true for complex-valued $$x(t)$$ because in that case the ACF is defined differently:

$$\mathcal{F}\big\{R(\tau)\big\}=\int_{-\infty}^{\infty}x^*(t)x(t+\tau)dt\tag{3}$$

• Thank you. I am not an expert so I thought I was missing something – Ashish Bhigah Jul 9 '19 at 21:10
• @AshishBhigah: You're welcome. Many books have typos, especially in their first editions, so be aware ... – Matt L. Jul 9 '19 at 21:11
• In other book I have seen that the definition of psd has a multiplication (1/2*pi). How can be handled in this demonstration? Does it desappear somehow? – Ashish Bhigah Jul 10 '19 at 13:49
• @AshishBhigah: It's probably best to formulate a new question with the corresponding formula. This makes sure that we understand the problem, and we know which formula you're referring to. – Matt L. Jul 10 '19 at 14:15