Overview
A Finite Impulse Response filter is stable by construction and computes each output from a fixed number of input samples.
CoreSource checked
FIR Filter
Filters a signal using a finite weighted sum of current and previous input samples.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Signal Processing
- Difficulty
- Beginner
- Family
- Digital Filter
- Tier
- Core
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 1
- Relationships
- 5
- Code example
- Ready
- Visualization
- Ready
Core Concepts
filter tapsfinite impulse responseweighted sumsample buffer
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(taps)
Average
O(n * taps)
Worst
O(n * taps)
Space
O(taps)
When to Use
- Use when stability and predictable phase behavior matter.
- Use when linear-phase filtering is a central requirement.
When Not to Use
- Avoid when a very sharp response is needed with very few coefficients.
- Avoid when the sampling rate, window size, or stationarity assumptions do not hold.
What It Does
- Computes digital filter within Digital Filter workloads (O(n * taps) time, O(taps) space).
- Works by construction and computes each output from a fixed number of input samples.
- Maintains key invariants across filter taps, finite impulse response, weighted sum.
Use Cases
- Linear-phase filtering
- Audio equalization
- Digital signal conditioning
Inputs
- Signal samples
- FIR coefficients
Outputs
- Filtered signal
Key Formulas (1)
FIR output
y[n] = ∑(b[k] · x[n − k])
y[n] = sum(b[k] * x[n - k])Computes output from a finite set of input samples and coefficients.
Pseudocode
- 1.Keep a buffer of recent input samples
- 2.For each output sample, multiply samples by coefficients
- 3.Sum the products
- 4.Shift in the next input sample
Canonical Example
- Input
- samples = [1, 2, 3], taps = [0.5, 0.5]
- Output
- output sample = average of two recent samples
- Explanation
- The filter computes a weighted sum of finite input history.
Implementation Notes
- Use convolution tests with impulse input to verify tap ordering.
- Plot or inspect both the original and processed signal for small test cases.
- Keep a tiny example with known output beside the implementation.
- Keep sampling rate, window size, latency, and boundary conventions explicit in tests.
Common Pitfalls
- Forgetting coefficient order can reverse the intended impulse response.
- Boundary handling and phase response can change the result in visible or audible ways.
- Not defining boundary, empty, or degenerate inputs before implementation.
- Ignoring sampling rate, boundary handling, or phase behavior can make the output misleading.
Interactive Mechanism Walkthrough
Iterative Convergence & State TrajectoryShow a sliding window of samples multiplied by filter taps.
Step 1 of 4: Keep a buffer of recent input samples
signal-transformNotes
FIR filters can have exact linear phase when coefficients are symmetric.
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
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Has variantsLMS Adaptive Filter
Signal Processing · Core
LMS commonly adapts FIR filter coefficients from an error signal.
Compare with thisAlternativesIIR Filter
Signal Processing · Core
IIR filters use feedback, while FIR filters use finite convolution.
Compare with thisRelationships
View full relationship pageUsesConvolution
FIR filtering is finite impulse response convolution.
Related Algorithms
Tags
Signal ProcessingFilteringFilter Design