Autocorrelation and Power Spectral Density (PSD) Examples in Digital Communications

Two fundamental examples in digital communication systems are used to explain Autocorrelation and Power Spectral Density (PSD).

* Note that there is a small "typo" in the magnitude of the sinc function PSD that's drawn under the triangle function. The magnitude at f=0 should be (A^2)T. The triangle autocorrelation function can be viewed as the convolution of two identical rectangular functions of base T and height A/sqrt(T). The Fourier transform of that rectangular function is a sinc function with a height A sqrt(T). And since convolution in the time domain is equivalent to multiplication in the frequency domain, the Fourier transform of the triangle function will be a sinc^2 function with height (A^2)T.

** And here's an additional description that might also help with understanding the content in this video (based on a question I received in the comments under the video):
It's important to think about which random variable the expectation (ie. E[.]) is over. In this case the random variable is t_o. For some values of t_o the times t and t+tau will be within the same digital symbol period (and the value will be A*A). For other values, the times t and t+tau will be in different symbol periods (ie. the digital clock transition occurs between those two times). In this case the value will be either A*A, or (-A)*A, or A*(-A), or (-A)*(-A), with equal probability - and the average of these will be 0. So overall, the average of the first case (A^2) and the second case (0) will depend on the percentage of the realisations of t_o that are in each case. The bigger the value of tau, the smaller the chance of t and t+tau being in the same digital period. Hence the Autocorrelation function goes down as tau goes up.

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