EE484: Probability and Introduction to Random Processes Autocorrelation and the Power Spectrum By: Jason Cho

Overview Autocorrelation Power Spectrum Summary 2

Abbreviations WSS: Wide Sense Stationary – is a stochastic process X(t) which has a constant mean. Also, the autocorrelation of the process must be dependent on the time difference, and not on each time individually. 3

Autocorrelation 4 Autocorrelation is a procedure which compares a process or a signal X(t) with a phase shifted copy of itself. This procedure is useful for determining if a pattern exists in the signal (periodic signal?), and also can be used to determine how fast the process varies in time. The autocorrelation function of a WSS random process, X (t), is defined as The autocorrelation of a random process can also be seen as the cross- correlation with itself. By the cross-correlation definition, we have Note: ○ operator is used to denote the “pentagram”, ★, operator which is similar to the convolution operator, without the time-reversal one of the functions.

Autocorrelation 5 So the autocorrelation function can be re-written as Going back to our original definition of the autocorrelation function, τ = time difference If do some manipulation to the above formula, we will arrive at a property of the autocorrelation, which shows that the autocorrelation function is an even function.

Autocorrelation 6 Consider a the autocorrelation function when the time difference is zero, We see that the autocorrelation at zero time difference results in the average power of the random process X (t). Because the expected value for | X (t)| 2 is non-negative, this results in which is another property of the autocorrelation function.

Autocorrelation 7 We can also show that R X (0) is the maximum value of autocorrelation function. To prove this, consider the Cauchy-Schwarz Inequality.

Power Spectrum 8 The power spectrum of a random process generates a plot of a random process’ power is distributed over frequencies. The power spectrum is sometimes referred to as the power spectral density. The power spectrum of a random process X (t) can be defined as the Fourier Transform of the autocorrelation function of the random process. S(ω) = power spectral density R(τ) = autocorrelation function Therefore, the other half of the transform pair is This relationship between the autocorrelation function and the power spectral density is known as the Wiener-Khinchin Theorem.

Power Spectrum 9 We determined previously that R X (0) was the average power of a random process X(t). This way, the average power can be found from This theorem is important because a power signal with non-zero average power is not square-integrable, and therefore the Fourier Transform does not exist. It is important to note that the power spectral density exists only for stationary processes, otherwise, the function is of an additional variable so the spectral density function does not exist.

Summary Autocorrelation Power Spectrum Relationship between autocorrelation and power spectrum 10

Thank You! 11