Problem
Chapter 10 of Fundamentals of Applied Probability and Random Processes by Oliver-C-Ibe introduces applied concepts for stochastic processes. A stochastic process is one where the output contains a component of randomness. In this assignment we combine knowledge from Signals and Systems with Stochastic Systems and observe what trends we can conclude. To begin we will create a function that will provide us a sine wave with a component of noise. In the previous assignment we demonstrated how we could create a function using the Box-Muller technique to create an array of normally distributed variables. We now use that function to create a Gaussian noise aspect and combine it with our signal. The amount of noise we combine with our sine wave is dependent on the signal to noise ratio. We can define the SNR as follows:
In our function we will be able to take in the desired output frequency in hertz, the length (in seconds), the sample frequency, and SNR.
Now that we are capable of creating a signal with noise we observe the relationship of the autocorrelation to the SNR. Autocorrelation tells us how correlated a signal is to itself at different shift values (τ). In our case we will take the first 16 shifts of a 500Hz sine wave sampled at 8000Hz. Inspecting the autocorrelation at different shifts will tells us at SNR does our noise impact our sine wave considerably. Also at these various SNRs we inspect the Fourier transform of the signal. We are curious to observe what noise looks like in the frequency domain.
Lastly, we explore the idea of the power spectral density function. The PSD is the Fourier transform of the autocorrelation function and tells us the power at certain frequencies of our signal. To demonstrate the concept we create a signal with 30db SNR. Passing the signal through a digital filter shown below provides us with a clearer output. We find the PSD and plot the Fourier transform of our filtered signal squared. We expect the resulting plots should be equal.
y[n]=0.5y[n-1]+x[n]
Approach and Results
To begin we create a function that produces a sine wave with noise given a certain SNR. Using the Box-Muller method allows us to quickly create an array of normal variables, but we still need to find the variance of the noise. Recall that the variance also equals the power of a signal. Using equation 1 we can find the power of the noise. Then we can feed it into our normal number generator to create a noise signal. We create a table of the power in the entire signal given different SNR.