IIR Filter Simulation

IIR Filter & Feedback Lab

Exploring Recursive Systems: Where Output depends on Output

1 Input Signal $x[n]$

Signal Type Unit Impulse δ[n] Sine Wave Unit Step u[n]

Frequency (for Sine)

Notice: With an Impulse input, the output of an IIR filter can ring forever (Infinite response).

2 Feedback Coefficients

Numerator $b$ (Feed-forward)

Denominator $a$ (Feedback - $a_0$ is always 1)

System Stable

Try setting $a$ to 1.0, -1.1 to see an unstable system!

3 The Difference Equation (Recursive)

$$y[n] = \frac{1}{a_0} \left( \sum_{k=0}^{M} b_k x[n-k] - \sum_{j=1}^{N} a_j y[n-j] \right)$$

Time Index $n$: n = 5

4 Time Domain

5 Magnitude Response

FIR vs IIR: The Key Difference

Finite Impulse Response (FIR)

• No Feedback.
• Always Stable.
• Output becomes zero once the impulse leaves the window.
• Equation: $y[n] = b_0x[n] + b_1x[n-1] \dots$

Infinite Impulse Response (IIR)

• Uses Feedback (Past $y$ values).
• Can become Unstable (Infinite growth).
• A single pulse can cause an output that lasts forever.
• Equation: $y[n] = b_0x[n] \dots - a_1y[n-1] - a_2y[n-2] \dots$

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