Overview
A Butterworth Filter is designed to be maximally flat in the passband, making it a common general-purpose low-pass, high-pass, or band-pass filter.
ExtendedSource checked
Butterworth Filter
Filters signals with a smooth passband and monotonic frequency response.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Signal Processing
- Difficulty
- Intermediate
- Family
- Digital Filter
- Tier
- Extended
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 1
- Relationships
- 2
- Code example
- Ready
- Visualization
- Ready
Core Concepts
cutoff frequencyfilter orderfrequency responseIIR filter
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(n)
Average
O(n * order)
Worst
O(n * order)
Space
O(order)
When to Use
- Use when a smooth passband is preferred over sharp ripple behavior.
- Use when audio filtering is a central requirement.
When Not to Use
- Avoid when a very sharp transition band is required with low order.
- 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 * order) time, O(order) space).
- Works by Filters signals with a smooth passband and monotonic frequency response.
- Maintains key invariants across cutoff frequency, filter order, frequency response.
Use Cases
- Audio filtering
- Sensor smoothing
- Signal conditioning
Inputs
- Signal samples
- Cutoff frequency
- Filter order
Outputs
- Filtered signal
Key Formulas (1)
Magnitude response
|H(f)| = 1 / √1 + (f / cutoff)(2 · order)
|H(f)| = 1 / sqrt(1 + (f / cutoff)^(2 * order))Describes the smooth Butterworth low-pass rolloff.
Pseudocode
- 1.Choose filter type, cutoff, and order
- 2.Compute or load filter coefficients
- 3.Process samples through the filter recurrence
- 4.Return filtered output
Canonical Example
- Input
- signal = noisy sensor samples, cutoff = 5 Hz
- Output
- output = smoothed samples
- Explanation
- Frequencies above the cutoff are attenuated smoothly.
Implementation Notes
- Use stable second-order sections for higher-order IIR filters.
- Plot or inspect both the original and processed signal for small test cases.
- Keep a tiny example with expected output beside the implementation.
Common Pitfalls
- Designing with normalized frequency incorrectly shifts the cutoff.
- Boundary handling and phase response can change the result in visible or audible ways.
- Not defining boundary, empty, or degenerate inputs before implementation.
Interactive Mechanism Walkthrough
Iterative Convergence & State TrajectoryShow a smooth frequency response that rolls off after the cutoff.
Step 1 of 4: Choose filter type, cutoff, and order
signal-transformNotes
Higher order gives steeper rolloff but can add numerical sensitivity.
Sources (1)
What each reference supports for this entry.
Discrete-Time Signal Processing
BookReference text for digital filters, Fourier analysis, and discrete-time signal-processing methods.
Open sourceExplore Next
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Relationships
View full relationship pageVariant ofLow-Pass Filter
Butterworth filters are often designed as maximally flat low-pass filters.
Related Algorithms
Tags
Signal ProcessingFilteringFilter DesignLow Pass