Overview
An IIR filter uses a recursive difference equation, allowing compact filters with long impulse responses but requiring attention to stability and coefficient precision.
CoreSource checked
IIR Filter
Filters a signal with feedback from previous outputs as well as current and previous inputs.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Signal Processing
- Difficulty
- Intermediate
- Family
- Digital Filter
- Tier
- Core
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 1
- Relationships
- 2
- Code example
- Ready
- Visualization
- Ready
- Aliases
- Infinite Impulse Response Filter
Core Concepts
feedbackdifference equationrecursive filterstability
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(k)
Average
O(n k)
Worst
O(n k)
Space
O(k)
When to Use
- Use when a sharp response is needed with relatively few coefficients.
- Use when nonlinear phase is acceptable or can be compensated.
When Not to Use
- Avoid when exact linear phase is required without extra processing.
- Avoid when coefficient quantization could make the recursive filter unstable.
What It Does
- Filters a signal using feedback from previous output samples
- Uses feedback and difference equation to guide the work.
- Produces filtered signal.
Use Cases
- Filtering noisy signals
- Spectral analysis
- Audio and sensor processing
Inputs
- Sampled signal
- Filter or transform parameters
Outputs
- Filtered signal
- Spectrum or feature representation
Key Formulas (1)
IIR recurrence
y[n] = sum bk x[n−k] − sum ak y[n−k]
y[n] = sum b_k x[n-k] - sum a_k y[n-k]Computes output from current inputs and previous outputs.
Pseudocode
- 1.Load feedforward and feedback coefficients
- 2.Initialize input and output history
- 3.For each sample
- 4.Compute feedforward contribution from recent inputs
- 5.Subtract or add feedback contribution from previous outputs by convention
- 6.Store and return the output sample
Canonical Example
- Input
- y[n] = 0.5*x[n] + 0.5*y[n-1], x = [1,0,0]
- Output
- y starts [0.5, 0.25, 0.125]
- Explanation
- The feedback term keeps producing a decaying response after the input stops.
Implementation Notes
- Use stable design methods and inspect pole locations.
- Consider biquad sections instead of one high-order direct-form filter.
- Test impulse, step, and steady-state responses.
Common Pitfalls
- Feedback coefficients can place poles outside the stable region.
- Direct-form implementations may suffer from numerical precision issues.
- Filter state must be initialized and reset deliberately.
Interactive Mechanism Walkthrough
Iterative Convergence & State TrajectoryShow the input signal beside the filtered, transformed, or reconstructed output.
Step 1 of 6: Load feedforward and feedback coefficients
signal-processing-mechanismNotes
IIR filters are efficient, but stability and numeric representation are part of the implementation.
Sources (1)
What each reference supports for this entry.
Discrete-Time Signal Processing
BookSignal-processing reference for digital filters, convolution, spectral analysis, windows, resampling, and transform-based methods.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Relationships
View full relationship pageAlternative toFIR Filter
IIR filters use feedback, while FIR filters use finite convolution.
Related Algorithms
Tags
Signal ProcessingFilter DesignFilteringRecursive Filter