Problem
Today we look over autocorrelation and power spectral density (PSD) again. Unlike in our last analysis, we now will compare and contrast the autocorrelation and PSD of several different signals.
The first signal we will observe is a Gaussian noise signal that contains two hundred samples. For the autocorrelation plot we will generate one with 20 lags in it. Since our signal is a noise signal we expected that its power across all frequencies to be constant.
The second signal is a single impulse with one hundred samples in it. The lags in our autocorrelation will range up to 20. In similar vein to the Gaussian we expected a flat frequency response since our time domain signal is short in time. Short in time means broad in frequency.
The third signal is also an impulse signal that repeats every 20 samples. Since we have a train of impulses we increase the number of samples to 200 and the number of lags to 60. The PSD should be similar to the single impulse.
The fourth signal generated is a sinewave whose periods occurs every 20 samples. The frequency of said sine wave is dependent on the sample frequency. Up to know we have used a sample frequency of 8KHz therefore our sinewave has as frequency of fs/T=8kHz/20=400Hz. We repeat our analysis of the autocorrelation and PSD as explained before. Also for this sine wave case we analyze the behavior if we were to change the number of samples between 14, 17, 20, 23, and 26.
The sixth signal is a sine-wave with a noise component. Given a SNR of 10db we perform the same analyses.
Approach and Results
The Gaussian white noise is expected to have energy across all frequencies. We can find the autocorrelation function of our white noise and observe how just by chance the signal looks to be correlated a bit. In truth we know the signal is entirely random.
In our second signal we have an impulse. Recall from Signal and systems that short in the time domain correlates to wide in the frequency domain. If we find the autocorrelation of our signal we will note that the signal is never correlated to itself since it is not periodic. Our power spectral density thus is flat like our Gaussian white noise signal.