Generation of random sequences: While deterministic signals such as square pulses, sinc waveforms, sinuses and cosines are used in specific applications, almost all other real-life signals from econometric series to radar returns, from genetic codes to multimedia signals in consumer electronics are information-bearing random signals.
- Time sequences: Generate a sequence of 1000 equally spaced samples of a Gauss-Markov process using the recursive relation: . Here assume that , and is a sequence of independent identically distributed Gaussian random variables. You can use the randn function in MATLAB to generate zero-mean, unit-variance random variables. Below is given a low-pass filter (a>0) excited by a white noise sequence, and the filter has a real pole at z= a. Plot the output waveform for the following values of a: a= 0.5, a = 0.95, a = 0.995. Comment on the effect of the selection of a on the resulting time sequence.
- Autocorrelation function: In order to estimate the autocorrelation function of this discrete process, one can apply the formula: . However, since we have a finite number of samples an approximation would be . Notice we shorten the data and consider only the overlapping samples in the window starting at n=1 and the window starting at n=m+1. The theoretical autocorrelation for such a signal model is also known as: , where is the input process variance. Plot the autocorrelation sequence in three separate graphs for the values of a= 0.5, a = 0.95, a = 0.995, but superpose in each graph the analytical and estimated and correlation functions.
- Power Spectrum: Using the Wiener-Khinchine theorem, we can find the power spectrum in two different ways. Then do the following:
- Estimate , where , for a=0.95 and N=512
- Estimate again as above, but this time averaging ten estimates, each obtained from a different time sequence. In other words, you must use the recursion , ten times, and average the results, i.e., . The estimates are noisy and random, and to improve and smooth the estimates, we must calculate the quantities several times (let us say 10 times) and average them.
- We also know the analytical value of , where is the power spectral density of the input white noise, hence, , and
Plot the power spectra in these three cases and comment on each of them.