Overview
The Savitzky-Golay Filter preserves local shape better than a simple moving average by replacing each point with a polynomial fit estimate.
ExtendedSource checked
Savitzky-Golay Filter
Smooths data by fitting a low-degree polynomial inside a moving window.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Signal Processing
- Difficulty
- Intermediate
- Family
- Polynomial Smoothing
- Tier
- Extended
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 1
- Relationships
- 1
- Code example
- Ready
- Visualization
- Ready
Core Concepts
local polynomialmoving windowleast squaresshape preservation
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(n)
Average
O(n * window)
Worst
O(n * window)
Space
O(window)
When to Use
- Use when smoothing should preserve peak height or local curvature.
- Use when smoothing spectra is a central requirement.
When Not to Use
- Avoid when impulse noise dominates and a median filter is more robust.
- Avoid when the sampling rate, window size, or stationarity assumptions do not hold.
What It Does
- Slides a local window
- Fits a polynomial to nearby samples
- Outputs the fitted center value
Use Cases
- Smoothing spectra
- Preserving peaks
- Derivative estimation
Inputs
- Signal samples
- Window size
- Polynomial degree
Outputs
- Smoothed signal
Key Formulas (1)
Local polynomial
y(t) ≈ a0 + a1·t + a2·t2 + … + ad·td
y(t) ~= a0 + a1*t + a2*t^2 + ... + ad*t^dFits nearby samples with a low-degree polynomial.
Pseudocode
- 1.Choose an odd window size and polynomial degree
- 2.For each sample, collect neighboring samples
- 3.Fit or apply precomputed polynomial coefficients
- 4.Write the center estimate
Canonical Example
- Input
- samples = [1, 2, 10, 2, 1], window = 5
- Output
- center estimate is smoothed but peak-aware
- Explanation
- The local polynomial fit reduces noise while preserving shape better than plain averaging.
Implementation Notes
- Precompute convolution coefficients for fixed window and degree settings.
- 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
- A window that is too wide can smooth away meaningful narrow features.
- 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 small polynomial curve fitted through each moving window.
Step 1 of 4: Choose an odd window size and polynomial degree
signal-transformNotes
The polynomial degree must be lower than the window size.
Sources (1)
What each reference supports for this entry.
Savitzky and Golay 1964
PaperOriginal paper on smoothing and differentiation by local least-squares polynomial fitting.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Relationships
View full relationship pageAlternative toMoving Average
Savitzky-Golay smoothing preserves local polynomial shape better than a plain average.
Related Algorithms
Tags
Signal ProcessingFilteringSmoothingInterpolation