Digital Signal Processing

Consider the analog signal xt 10cos2 890t 5cos2 385t 4cos2 1450tt where t is random zero mean Gaussian noise. The signal is digitized at a sampling rate of 6000 Hz, and the length of the resulting sequence is 18,000 samples. Determine and display the frequency content of the analog signal, as a function of frequency in Hertz, using both the periodogram and the MUSIC algorithm for signal-to-noise ratios (SNR) ranging from -20 to +15 decibels (dB). For the MUSIC algorithm, use only the first 2000 samples of the sequence. Given a desired SNR in dB, the required noise power can be found by solving the relationship          noise signal P P SNR 10 10log for Pnoise . Psignal is, by definition, the power in the largest sinusoid, i.e.,    2 1 2 max , , , Psignal  A A  AM , where A A AM , , 1 2 are the amplitudes of the M sinusoids present. Having determined Pnoise for a given SNR, generate random zero mean Gaussian noise with variance equal to Pnoise . This can be accomplished in Matlab using the randn command and noting that the variance is the square of the standard deviation,  . Add this random noise to the digitized signal. It is recommended that you seed the random number generator before the initial run so that you can reproduce your results. Compare and contrast the ability of the two techniques to accurately determine the frequency components present under the following conditions: 1. Limit the length of the sequence to the first 2000 samples and vary SNR from -20 to +15 dB. Implement the code using 10 different realizations of noise for each value of SNR and determine the number of times in these 10 realizations that the periodogram and MUSIC successfully locate the three frequencies present within 1 Hz. For each of the 10 realizations, use an identical realization of noise for both the periodogram and MUSIC. Locate the three highest peaks in the spectrum and determine the frequencies at which they occur. Record how many, if any of the three known signal frequencies are located. For example, figures 1, 2, and 3 display the periodogram results for three different realizations of noise when SNR = -10 dB. In these three trials, the 890 and 1425 Hz components were correctly identified within 1 Hz in all three trials; the 1450 Hz component was successfully located within 1 Hz in only one of the three trials. You do not need to display results for all 10 noise realizations—just show results of a representative trial for each SNR and summarize the outcomes of all trials in tabular form. 2. For the MUSIC algorithm only, compare results when using the autocorrelation and the covariance methods of estimating the correlation matrix for SNR values ranging from -20 to +15 dB. 3. For the periodogram only, allow the length of the sequence to be the original length of 18,000 samples. Assess its performance in locating the frequencies when SNR ranges from -20 to +15 dB.