Overview
The Continuous Wavelet Transform compares a signal with shifted and scaled wavelets, producing a detailed time-scale representation.
ExtendedSource checked
Continuous Wavelet Transform
Analyzes a signal across continuously varying scales and shifts.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Signal Processing
- Difficulty
- Advanced
- Family
- Time-Frequency Transform
- Tier
- Extended
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 1
- Relationships
- 2
- Code example
- Ready
- Visualization
- Ready
- Aliases
- CWT
Core Concepts
scale parametertranslation parametermother wavelettime-scale map
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(n s)
Average
O(n s)
Worst
O(n s)
Space
O(n s)
When to Use
- Use when filtering noisy signals needs scale parameter.
- Use when the input matches the assumptions behind time-Frequency Transform.
When Not to Use
- Avoid when sampling rate, phase, or boundary assumptions are unknown.
- Avoid when a simpler algorithm gives the required result with less setup.
What It Does
- Analyzes a signal across continuously varying scales and shifts
- Uses scale parameter and translation parameter 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)
Wavelet coefficient
W(a, b) = integral x(t) · ψ((t − b) / a) dt
W(a, b) = integral x(t) * psi((t - b) / a) dtMeasures signal content at a scale and position.
Pseudocode
- 1.Prepare the inputs and state needed for Continuous Wavelet Transform.
- 2.Apply the scale parameter step while the stopping condition has not been met.
- 3.Update intermediate state and discard invalid candidates.
- 4.Return filtered signal.
Canonical Example
- Input
- Continuous Wavelet Transform input with a small representative case
- Output
- Filtered signal
- Explanation
- Continuous Wavelet Transform applies scale parameter to produce filtered signal.
Implementation Notes
- Document sample rate, windowing, and normalization conventions.
- Keep the scale parameter step separate enough to test independently.
Common Pitfalls
- Boundary handling and normalization choices can change the interpretation of the result.
- Forgetting to check edge cases can make Continuous Wavelet Transform look correct on only ideal inputs.
Interactive Mechanism Walkthrough
Iterative Convergence & State TrajectoryShow the input signal beside the filtered, transformed, or reconstructed output.
Step 1 of 4: Prepare the inputs and state needed for Continuous Wavelet Transform.
signal-processing-mechanismNotes
Continuous Wavelet Transform should be used with the assumptions and tradeoffs of time-Frequency Transform in mind.
Sources (1)
What each reference supports for this entry.
Daubechies 1992
BookWavelet reference covering continuous and discrete wavelet transforms.
Open sourceExplore Next
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Relationships
View full relationship pageVariant ofWavelet Transform
CWT analyzes a signal across continuous scales and shifts.
Related Algorithms
Tags
Signal ProcessingWaveletTime FrequencySpectral Analysis