How do I get the probability density function of a sine wave?

Hi, I want to do something very simple in MATLAB which is just to get the probability density function of a sine wave and plot it. I know that the pdf plot has a U-shape, but I am not able to get it by using the pdf function in MATLAB no matter which "name" I use. For example:

x=-4*pi:.01:4*pi; y=sin(x);

So I just want to plot pdf for x between -1 and 1. Any help is appreciated. Thanks.

[Wayne King]

I think you need to tell us what is random here and what the distribution of that random variable is. There is nothing inherently random about sin(x) unless x is a random variable. For example, x may be uniform (0,2*pi). Then, you will need to consider something like sin^2(x) because a PDF cannot be negative.

[Zach]

Hi Wayne,

Thanks for your response. The random variable is just y (not x) between -1 and 1. So my plot should look like: http://imageshack.us/photo/my-images/692/sinepdf.jpg/. I would like to apply this to other signals, but I would first like to understand how to use the pdf function for this simple case. Thanks.

[Star Strider]

I'm not going to enter this as an ‘Answer’ because I can only take credit for Googling until I found a good discussion of it in EXAMPLES for Matlab Signal Processing (curiously without any associated MATLAB scripts). See ‘Probability Distribution of a Sine Wave’ pages 232-3, eqns 8.47-8.55 and figures 8.11 and 8.12.

Answer 1

[Image Analyst]

So why don't you generate a long sine wave signal and then pass it in to hist()? It's really really easy:

clc;    % Clear the command window.
close all;  % Close all figures (except those of imtool.)
workspace;  % Make sure the workspace panel is showing.
format longg;
format compact;
fontSize = 20;

omega = 1:80000;
period = 8000;
y = sin(2*pi*omega / period);
subplot(2, 1, 1);
plot(omega, y, 'b-');
title('Signal', 'FontSize', fontSize);
grid on;
% Enlarge figure to full screen.
set(gcf, 'units','normalized','outerposition',[0 0 1 1]);
[counts binValues] = hist(y, 64);
subplot(2, 1, 2);
bar(binValues, counts);
title('Histogram', 'FontSize', fontSize);
grid on;

Run my code above and you'll see your image.

[Zach]

Hi Image Analyst,

Thank you for the response, this is very close to what I want. However, I am really looking to plot the pdf, is there a way to simply convert this hist() to a pdf? Thanks.

[Image Analyst]

Simply divide the counts by the sum of the counts so that the total "area" equals 1.