Overview
The Goertzel Algorithm uses a recurrence to compute a single DFT bin, making it useful when only a few frequencies matter.
SpecializedSource checked
Goertzel Algorithm
Efficiently measures one selected frequency bin from a signal.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Signal Processing
- Difficulty
- Intermediate
- Family
- Frequency Analysis
- Tier
- Specialized
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 1
- Relationships
- 1
- Code example
- Ready
- Visualization
- Ready
Core Concepts
single DFT binsecond-order recurrencefrequency powersample rate
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(n)
Average
O(n)
Worst
O(n)
Space
O(1)
When to Use
- Use when only one or a few frequency bins need to be measured.
- Use when tone detection is a central requirement.
When Not to Use
- Avoid when a full spectrum is needed for each signal window.
- Avoid when the sampling rate, window size, or stationarity assumptions do not hold.
What It Does
- Targets one frequency bin
- Runs a second-order recurrence
- Computes bin power or coefficient
Use Cases
- Tone detection
- DTMF decoding
- Embedded frequency checks
Inputs
- Signal samples
- Target frequency bin
Outputs
- Selected frequency magnitude or coefficient
Key Formulas (1)
Goertzel recurrence
s[n] = x[n] + 2·cos(2·π·k/N)·s[n−1] − s[n−2]
s[n] = x[n] + 2*cos(2*pi*k/N)*s[n-1] - s[n-2]Accumulates the selected frequency bin through a recurrence.
Pseudocode
- 1.Choose the target bin
- 2.Compute the recurrence coefficient
- 3.Run the recurrence across samples
- 4.Convert final state to magnitude or complex bin value
Canonical Example
- Input
- samples contain a 697 Hz tone
- Output
- power at 697 Hz bin is high
- Explanation
- The recurrence measures the target tone without computing every DFT bin.
Implementation Notes
- Keep sample rate, window length, and target bin calculation together.
- 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
- Choosing a bin that does not match the sample rate and target frequency causes leakage.
- 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 one frequency bin accumulating energy while other bins are ignored.
Step 1 of 4: Choose the target bin
signal-transformNotes
FFT is usually better when many bins are needed.
Sources (1)
What each reference supports for this entry.
Goertzel 1958
PaperOriginal reference for evaluating finite trigonometric series efficiently.
Open sourceExplore Next
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Relationships
View full relationship pageSpecializesDiscrete Fourier Transform
Goertzel efficiently evaluates selected DFT bins.
Related Algorithms
Tags
Signal ProcessingFrequency DomainSpectral AnalysisAudio Processing