# Power spectral density and auto correlation function: Frequency vs time representation

I have a time correlated periodic signal $$a(t)$$ with $$t=0,1,...,T$$ and $$a(t+T) = a(t)$$ that has been sampled at rate $$\Delta t$$ and I analyse the signal using the PSD via FFT and the ACF. As far as I understand, the standard FFT treats the functions by default as periodic.

For the ACF $$\langle a(t) a(0) \rangle$$, the largest possible time difference is $$T/2$$, since the signal is periodic (everything larger would only approach the starting point from the other side), i.e., $$\langle a(T/2 + t) a(0) \rangle = \langle a(T/2 - t) a(0) \rangle$$, there is no new information after $$t=T/2$$.

However, for the PSD $$\langle a_f a_{-f}\rangle$$ the frequency inherited from the FFT is $$f_i = [0,1,...,T]\cdot \frac{1}{T\Delta t}$$, where $$f_i < f_\mathrm{Ny} = \frac{1}{2\Delta t}$$ (according to Nyquist). It contains information about correlations of the signal at a frequency of $$f_1 = 1/(T\Delta t)$$, which should in my understanding be somewhat equivalent to the longest possible time correlation $$t = T\Delta t$$. But that in the end is equivalent to $$t=0$$ and thus should be equivalent to the shortest of the frequencies. Intuitively, I would think that only frequencies larger then $$f=\frac{1}{T/2}$$ contain useful information.

How is that possible? What am I missing? What is the relation between frequencies in the PSD and times in the ACF?

there is no new information after $$t=T/2$$.

Correct. If the sequence is periodic, so is it's autocorrelation.

It contains information about correlations of the signal at a frequency of $$f_1=1/(TΔt)$$

I don't think that's a useful interpretation. What is "correlation at a specific frequency" supposed to mean?

What is the relation between frequencies in the PSD and times in the ACF?

The PSD is the Fourier Transform of the ACF.

The autocorrelation at the origin is simply the total mean power of a DT signal $$x[n]$$ i.e., $$R_{x}[0]=\mu_{x}^{2}+\sigma_{x}^{2}$$ so its like a DC quantity + AC quantity. Now take the DTFT of $$R_{x}[0]$$ what you will get is the PSD at $$k=0$$ so $$R_{x}[0]$$ can be represented as some inverse DTFT : $$R_{x}[0]=\frac{1}{2\pi}\int_{-\pi}^{\pi}S_{x}(e^{j\Omega})\;\text{d}\Omega$$ So what this is telling me is that I have a DT signal $$x[n]$$ that is contributing a portion of a power at a certain frequency $$\Omega$$ in the frequency domain to the total power of the signal i.e., $$R_{x}[0]$$.