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AWGN, WGN, Autocorrelation and PSD Explained using Matlab

Using matlab simulation with code in description of this video, we address what is wgn ,i.e., white Gaussian noise, how to generate awgn ,i.e., additive white Gaussian noise, how to find autocorrelation function of white Gaussian noise and how auto-correlation function relates with the psd , i.e., power spectral density. We look into some of the key concepts which are directly related to the topic of Gaussian distribution such and probability density function of Gaussian random variable, the outcomes within a random process, difficulty of finding psd from the auto correlation function and how does the white Gaussian noise sound. %The matlab code used in the simulation setup presented in the video is given below. clc; clear all; close all fs = 1e4; % Sampling Fre t_span = 10; % max time in sec t = 0:1/fs:t_span-1/fs; % time interval L = size(t,2); % Number of Samples mu=0; % mean value var = .1; % variance aka power in W sigma=sqrt(var); % st. deviation var_dB = 10*log(var); % Power in dBW %Generating the random Process X=sqrt(var)*randn(L,1)+mu; %Stem Time Series Plot of White Gaussian Noise. figure(1); stem(X); title('Stem Time Series Plot of White Gaussian Noise'); xlabel('Samples') ylabel('Sample Values') axis([0 25 -.5 .5]); grid on; grid on; keyboard %Sound of White Gaussian Noise. sound(X) keyboard %Plot the PDF of the Gaussian random variable. figure(2); [f,xi]=ksdensity(X); plot(xi,f,'-o') grid on; title('PDF of White Gaussian Noise'); xlabel('x'); ylabel('PDF f_x(x)'); % Auto-correlation function; The argument 'biased' is used for proper scaling by 1/L %Normalize auto-correlation with sample length for proper scaling [acf,lags] = xcorr(X,'biased'); figure(3) plot(lags,acf); axis([-1000 1000 0 0.02]); grid on; title('Auto-correlation Function of White Noise'); xlabel('Lags (\tau)') ylabel('Auto-correlation = \sigma^2 \delta(\tau)') grid on; keyboard %The PSD part is adopted from %https://www.gaussianwaves.com/2013/11... %A very useful resoruce that is. %Since X is iid, we partition it to approximate PSD figure(4) X1= reshape(X,[1000,100]); w = 1/sqrt(size(X1,1))*fft(X1); %Normalizing by sqrt(size(X1,1)); Pzavg = mean(w.*conj(w)); %Computing the mean power from fft N=size(X1,2); normFreq=[-N/2:N/2-1]/N; Pzavg=fftshift(Pzavg); %Shift zero-frequency component to center of spectrum plot(normFreq,10*log10(Pzavg),'r'); axis([-0.5 0.5 -30 5]); grid on; ylabel('PSD (dB/Hz)'); xlabel('Normalized Frequency'); title('Power Spectral Density (PSD) of White Noise'); keyboard %AWGN on Sawtooth wave figure(5) new_sig = 2*sin(2*t)'; new_sig_noise = new_sig+X; plot(t,new_sig_noise, '-.g') hold on plot(t,new_sig, 'r') legend('Signal with AWGN','Original Signal') xlabel('time') ylabel('Values') grid on;