9.4: Random Number Applications
- Page ID
- 87911
In engineering applications, such as radar or satellite signals, the data has random noise. A process called filtering can be used to reduce the noise. There are multiple types of filters. The following is one example.
A first-order, low-pass filter smooths noisy data by using a for-loop with the following equation, which sets the output at each time step to be a weighted average of the previous output and the current input:
yk = gainy*yk-1 + gainx*xk
where gainy + gainx = 1.
The bandwidth chosen for the filter sets these gain parameters.
The algorithm steps are:
- Define a time vector
- Create a pure step function input
- Define the parameters for a 1st Order Low-Pass Filter.
- Apply the 1st Order Low-Pass Filter to the pure step input using a for-loop.
- Create a 2nd input = step function + noise
- Apply the 1st Order Low-Pass Filter to the noisy input using a for-loop.
Write a for loop to implement the filter equation.
%% Low_Pass_Filter_Step_2023.m
clear all; format compact; format short; clc; close all;
%% a. Define the time and signal vectors
% Create the time vector
dt = 0.002;
t = 0 : dt : 0.4; %(s)
% Create the step function
s0 = ones(size(t)); % Initialize the step function
s0(1:5) = 0; % set data to zero before the step occurs.
% Plot the pure signal
figure;
plot(t,s0,'k');
grid on;
hold on; % Plot the other graphs on the same figure
%% b. Add noise to the signal
noise_std = 0.2; % Standard deviation of the noise
s1 = s0 + noise_std*randn(size(s0)); % signal + noise
plot(t,s1,'bs'); % Plot as blue squares
%% c. Define the parameters for a 1st Order Low-Pass Filter
% A 1st Order Low-Pass Filter smooths the data
% Each iteration's output is a weighted combination
% of the current measurement + the previous filter output.
bw = 10 %(Hz) Bandwidth
tau = 1/(2*pi*bw) %(s) Time constant
% The output rises to a factor of (1-(1/e)) of the step in tau seconds
gain_y = exp(-2*pi*bw*dt)
gain_s = 1-gain_y
%% d. Apply the 1st Order Low-Pass Filter
% to the noisy signal using a for-loop.
y1 = zeros(size(s1)); % initial the output to zeros
t_len = length(t)
for k = 2:t_len
y1(k) = gain_y*y1(k-1) + gain_s*s1(k);
end
hold on;
plot(t,y1,'go'); % Plot as green circles
title({'First-Order Filter Step Function',...
['Bandwidth= ',num2str(bw),', Time Constant= ',num2str(tau)]});
% The { } allows a 2-line title
xlabel('Time');
legend('Pure Signal', 'Noisy Signal', 'Filtered Signal')
Solution
An example of the result for one realization of random noise is shown in this figure.
.
Modify Low_Pass_Filter_Step.m as follows:
a. Change the input to a sine wave with this code:
dt = 0.002; % s
t = 0 : dt : 0.4; % s
f1 = 10 % Hz
s0 = sin(2*pi*f1*t);
% Don't set any data to zero
b. Because the sine wave has both positive values, delete this line: ylim([0,1.5]);
c. Change the filter bandwidth from 10 to 20 Hz:
bw = 20 %(Hz) Bandwidth
d. Change the legend to be this:
legend('Pure Since', 'Filtered Pure Sine', 'Noisy Sine','Filtered Noisy Sine')
- Answer
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One noise realization looks like this:
.