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9.4: Random Number Applications

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      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.

      Example \(\PageIndex{1}\) Low-Pass Filter with a Step Input

      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:

      1. Define a time vector
      2. Create a pure step function input
      3. Define the parameters for a 1st Order Low-Pass Filter.
      4. Apply the 1st Order Low-Pass Filter to the pure step input using a for-loop.
      5. Create a 2nd input = step function + noise
      6. 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
      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);
      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
      legend('Pure Signal', 'Noisy Signal', 'Filtered Signal')



      An example of the result for one realization of random noise is shown in this figure.



      Exercise \(\PageIndex{1}\) Low-Pass Filter with a Sine Wave Input

      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')


      One noise realization looks like this:



      This page titled 9.4: Random Number Applications is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Carey Smith.

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