An Infinite Impulse Response (IIR) filter is a recursive digital signal processing system that uses feedback loops to process signals, creating an output that theoretically continues indefinitely in response to an impulse.

IIR Filter

An Infinite Impulse Response (IIR) filter is a fundamental component in digital signal processing that processes signals using both current and previous inputs and outputs through feedback mechanisms.

Core Characteristics

Recursive Structure: Unlike FIR filter, IIR filters use feedback paths, making them recursive systems
Memory Effect: The filter "remembers" previous outputs through feedback loops
Infinite Response: Theoretically continues producing output indefinitely after an impulse input
Efficiency: Generally requires fewer coefficients than FIR filters for similar frequency responses

Mathematical Foundation

The general difference equation for an IIR filter is:

y[n] = Σ(b[k]x[n-k]) - Σ(a[k]y[n-k])

Where:

y[n] is the current output
x[n] is the current input
b[k] are the feedforward coefficients
a[k] are the feedback coefficients

Common Types

Butterworth filter: Maximally flat frequency response
Chebyshev filter: Steeper rolloff but with ripples
Elliptic filter: Sharpest rolloff but with ripples in both passband and stopband

Applications

IIR filters find extensive use in:

Advantages and Limitations

Advantages

Lower computational requirements
Smaller memory footprint
Can model analog filter responses directly

Limitations

Potential stability issues
Phase response characteristics
More complex implementation due to feedback

Implementation Considerations

When implementing IIR filters, designers must consider:

Real-world Usage

IIR filters are commonly found in:

The design of IIR filters requires careful consideration of tradeoffs between performance, complexity, and stability requirements. Modern digital filter design tools help engineers optimize these parameters for specific applications.