MATLAB:Plotting Discrete Responses
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\(
\begin{align*}
\frac{3}{2}y[n]-\frac{1}{2}y[n-1]&=x[n]\\
y[n]&=\frac{1}{3}y[n-1]+\frac{2}{3}x[n]
\end{align*}
\) \(
\begin{align*}
h[n]&=\frac{2}{3}\left(\frac{1}{3}\right)^nu[n]\\
s_r[n]&=\left(1-\left(\frac{1}{3}\right)^{n+1}\right)u[n]
\end{align*}
\)
Calculate and Plot Impulse and Step 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:
The impulse response $$h[n]$$ and step resonse $$s_r[n]$$ are:
Code
If you have MATLAB, the code is available in the collapsed sections below:
- 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 Editor 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