Electrical -- 2
Budget: $10 – $30 USD
The purpose of this problem is to study the variance of the standard periodogram and to observe its inconsistency as an estimator. We will also see how averaging periodograms affects the variance of the estimate in the Welch method. Use MATLAB throughout, but do NOT use any built-in functions for the spectral estimates.
a) Generate a 100-point Gaussian random sequence with zero-mean and unit- variance. Compute and plot the periodogram using dB scaling. How does the periodogram compare to the true PSD of the Gaussian white noise process? Comment on the variability of the PSD estimate.
b) Repeat part (a) using a 200-sample sequence from the process. Does the increase in the number of samples result in a decrease in the estimator variance?
c) Repeat part (a) using a 1000-sample sequence. Does this improve the PSD estimate?
d) The variance itself is a random process. Using your code form parts (a) through (c) run the experiment 1000 times. Compute the variance directly on the resulting periodogram and record it for the 100-sample, 200-sample, and 1000-sample sequences for each of the 1000 trials. Compute and plot histograms of the variances you computed. What observations can you make regarding the distribution of the variance itself as the length of the sequence is increased?
e) Repeat part (d) using 10,000 trials. What can you observe about the distribution of the variance? Try 100,000 trials and observe the effect on the histograms. What observations can you make?
f) Estimate the PSD from the 1000-point sequence of part (c) using the Welch method with a segment length of 100 and no overlapping between segments. Use a rectangular window on each segment. Repeat the experiment 1000 times again. Compare the variability of the PSD estimate to that found in part (d).
g) Repeat part (f) using a triangular (Bartlett) window instead of the rectangular window. How does this affect the PSD estimate?
a) Generate a 100-point Gaussian random sequence with zero-mean and unit- variance. Compute and plot the periodogram using dB scaling. How does the periodogram compare to the true PSD of the Gaussian white noise process? Comment on the variability of the PSD estimate.
b) Repeat part (a) using a 200-sample sequence from the process. Does the increase in the number of samples result in a decrease in the estimator variance?
c) Repeat part (a) using a 1000-sample sequence. Does this improve the PSD estimate?
d) The variance itself is a random process. Using your code form parts (a) through (c) run the experiment 1000 times. Compute the variance directly on the resulting periodogram and record it for the 100-sample, 200-sample, and 1000-sample sequences for each of the 1000 trials. Compute and plot histograms of the variances you computed. What observations can you make regarding the distribution of the variance itself as the length of the sequence is increased?
e) Repeat part (d) using 10,000 trials. What can you observe about the distribution of the variance? Try 100,000 trials and observe the effect on the histograms. What observations can you make?
f) Estimate the PSD from the 1000-point sequence of part (c) using the Welch method with a segment length of 100 and no overlapping between segments. Use a rectangular window on each segment. Repeat the experiment 1000 times again. Compare the variability of the PSD estimate to that found in part (d).
g) Repeat part (f) using a triangular (Bartlett) window instead of the rectangular window. How does this affect the PSD estimate?