Impulse Response h[n] of an ARMA System
Step-by-Step Solution
1. Start with the Transfer Function:
Given: H(z) = (1 + 0.3z⁻¹) / (1 - 0.75z⁻¹ + 0.5z⁻²)
This is an ARMA(2,1) system where:
- MA (numerator) coefficients: [1, 0.3]
- AR (denominator) coefficients: [1, -0.75, 0.5]
2. Find the Impulse Response h[n]:
We want the inverse Z-transform of H(z). Instead of doing partial fraction decomposition, we use the system's difference equation.
3. Recursive Computation Using the Difference Equation:
From the system equation:
y[n] + 0.75y[n−1] − 0.5y[n−2] = x[n] + 0.3x[n−1]
Assume x[n] = δ[n] (unit impulse): x[0] = 1, others are 0. Then y[n] = h[n]
4. Compute h[n] values:
- n = 0: y[0] + 0 = 1 → h[0] = 1
- n = 1: y[1] + 0.75*1 = 0.3 → h[1] = -0.45
- n = 2: y[2] - 0.3375 - 0.5 = 0 → h[2] = 0.8375
- n = 3: y[3] + 0.628125 + 0.225 = 0 → h[3] = -0.853125
5. Table of First Few Values:
| n | h[n] |
|---|---|
| 0 | 1 |
| 1 | -0.45 |
| 2 | 0.8375 |
| 3 | -0.853125 |
6. Conclusion:
The impulse response is obtained by applying δ[n] and using the recursive equation. It reveals how past outputs and current/previous inputs shape the system's behavior.
MATLAB Code
clc;
clear;
close all;
%Impulse Response of an ARMA System%
% Define ARMA coefficients
ar_coeffs = [1, -0.75, 0.5]; % AR coefficients (denominator)
ma_coeffs = [1, 0.3]; % MA coefficients (numerator)
% Generate impulse signal
impulse = zeros(1, 100);
impulse(1) = 1; % Delta function (unit impulse)
% Filter the impulse signal through ARMA system
h = filter(ma_coeffs, ar_coeffs, impulse);
% Plot the impulse response
figure;
stem(0:length(h)-1, h, 'filled'); % Use stem for discrete-time signal
xlabel('n');
ylabel('h[n]');
title('Impulse Response of ARMA System');
grid on;
Output
Copy the aforementioned MATLAB code from here
Why Finding h[n] Is Useful in ARMA and WSS Contexts
1. Understand System Behavior:
The impulse response h[n] completely characterizes a linear time-invariant (LTI) system. Knowing h[n] allows us to determine the output y[n] for any input x[n] using convolution:
y[n] = x[n] * h[n]
In ARMA systems, this tells us how the system (or channel) modifies the input signal.
2. Analyze the Effect on Spectral Properties:
For wide-sense stationary (WSS) input signals, the power spectral density (PSD) of the output is given by:
Sy(f) = |H(f)|² · Sx(f)
Here, H(f) is the Fourier transform of h[n]. So knowing h[n] helps us understand how the system alters the frequency content of the input signal.
Further Reading