MATLAB:Plotting Discrete Responses

Introduction

The following is an HTML export of a MATLAB Live Editor file that looks at generating functions to calculate the impulse and step responses for:

\( \begin{align*} \frac{3}{2}y[n]-\frac{1}{2}y[n-1]&=x[n]\\ y[n]&=\frac{1}{3}y[n-1]+\frac{2}{3}x[n] \end{align*} \)

The impulse response $$h[n]$$ and step resonse $$s_r[n]$$ are:

\( \begin{align*} h[n]&=\frac{2}{3}\left(\frac{1}{3}\right)^nu[n]\\ s_r[n]&=\left(1-\left(\frac{1}{3}\right)^{n+1}\right)u[n] \end{align*} \)

Code

If you have MATLAB, the .m file is available in the collapsed section below. If you want to reconstitute it as a Live Code file:

• Copy the text into a new .m file and save it
• Click the down arrow on the Save button and select Save As
• Change the Save as type to Live Code and then save
• MATLAB .m File

  %% Calculate and Plot Impulse and Step Responses
  % This code will look at solving and plotting the impulse and step responses 
  % for:
  % 
  % $$\frac{3}{2}y[n]-\frac{1}{2}y[n-1]=x[n]$$ 
  % 
  % which can be re-written as:
  % 
  % $$y[n]=\frac{1}{3}y[n-1]+\frac{2}{3}x[n]$$
  %% Initialize
  
  clear
  format short e
  %% Plot impulse and step function
  
  n = -2:5;
  
  % Impulse responses
  figure(1)
  clf
  stem(n, arrayfun(@(n) hr(n), n), 'bs')
  hold on
  stem(n, hf(n), 'r+')
  hold off
  title('Impulse Responses')
  xlabel('n')
  
  % Step responses
  figure(2)
  clf
  stem(n, arrayfun(@(n) srr(n), n), 'bs')
  hold on
  stem(n, srf(n), 'r+')
  hold off
  title('Step Responses')
  xlabel('n')
  %% Function definitions
  % Define unit impulse and step functions
  
  function out=delta(n)
      out = (n==0).*1;
  end
  
  function out=u(n)
      out = (n>=0).*1;
  end
  % Define impulse and step responses using recursion
  
  function out=hr(n)
  % Note: only works for one value of n at a time
  % Use arrayfun(@(n) hr(n), ARRAY) for multiple responses
      if n < 0
          out = 0;
      else
          out = 1/3*hr(n-1) + 2/3*delta(n);
      end
  end
  
  function out=srr(n)
  % Note: only works for one value of n at a time
  % Use arrayfun(@(n) srr(n), ARRAY) for multiple responses
      if n < 0
          out = 0;
      else
          out = 1/3*srr(n-1) + 2/3*u(n);
      end
  end
  % Define impulse and step responses using formula
  
  function out=hf(n)
      out = 2/3*(1/3).^n.*u(n);
  end
  
  function out=srf(n)
      out = (1-(1/3).^(n+1)).*u(n);
  end
  

Live Code Results

Calculate and Plot Impulse and Step Responses

This code will look at solving and plotting the impulse and step responses for:

which can be re-written as:

Initialize

clear

format short e

Plot impulse and step function

n = -2:5;

% Impulse responses

figure(1)

clf

stem(n, arrayfun(@(n) hr(n), n), 'bs')

hold on

stem(n, hf(n), 'r+')

hold off

title('Impulse Responses')

xlabel('n')

% Step responses

figure(2)

clf

stem(n, arrayfun(@(n) srr(n), n), 'bs')

hold on

stem(n, srf(n), 'r+')

hold off

title('Step Responses')

xlabel('n')

Function definitions

Define unit impulse and step functions

function out=delta(n)

out = (n==0).*1;

end

function out=u(n)

out = (n>=0).*1;

end

Define impulse and step responses using recursion

function out=hr(n)

% Note: only works for one value of n at a time

% Use arrayfun(@(n) hr(n), ARRAY) for multiple responses

if n < 0

out = 0;

else

out = 1/3*hr(n-1) + 2/3*delta(n);

end

end

function out=srr(n)

% Note: only works for one value of n at a time

% Use arrayfun(@(n) srr(n), ARRAY) for multiple responses

if n < 0

out = 0;

else

out = 1/3*srr(n-1) + 2/3*u(n);

end

end

Define impulse and step responses using formula

function out=hf(n)

out = 2/3*(1/3).^n.*u(n);

end

function out=srf(n)

out = (1-(1/3).^(n+1)).*u(n);

end