Discrete Wavelet Transform Calculator
Haar Discrete Wavelet Transform overview
This calculator computes a one-level Haar discrete wavelet transform (DWT) for a real-valued sequence whose length is a power of two (2, 4, 8, 16, …). It returns two sequences:
- Approximation coefficients (low-frequency, coarse trend)
- Detail coefficients (high-frequency, local changes)
The Haar wavelet is the simplest wavelet basis. It uses piecewise-constant functions to summarize each pair of samples with their average and their difference.
Formulas for the one-level Haar DWT
Suppose you have an input sequence with N even. For a one-level Haar transform we group samples in pairs:
.
For each pair we compute an approximation coefficient and a detail coefficient :
Here . The calculator uses this averages and differences form, which is easy to interpret numerically.
Reconstruction (inverse transform)
From the approximation and detail coefficients, you can recover the original samples pairwise as
and .
In terms of explicit formulas:
This shows that no information is lost at this level: the transform is invertible as long as all and are kept.
Worked example
Consider the simple sequence of length 4:
.
-
Pair (2, 4)
-
Pair (6, 10)
The calculator would report:
- Approximation coefficients: [3, 8]
- Detail coefficients: [-1, -2]
You can check reconstruction:
How to interpret the results
- Approximation coefficients (a) track the local average behavior of the signal. Large smooth trends appear mainly in these values.
- Detail coefficients (d) capture local differences between adjacent samples. Values near zero indicate smooth regions; large magnitudes indicate sharp changes or edges.
For signal denoising or compression, it is common to keep the approximation coefficients and selectively shrink or discard small detail coefficients.
Comparison: Haar DWT vs. direct samples vs. Fourier
| Representation | What it emphasizes | Localization in time | Typical use |
|---|---|---|---|
| Raw samples | Exact values at each index | Perfect in time, no frequency separation | Storage, direct plotting, simple statistics |
| Haar DWT coefficients | Local averages and differences | Good time localization, coarse frequency separation | Edge detection, compression, denoising, multiresolution analysis |
| Fourier coefficients | Global sinusoidal components | Poor time localization, fine frequency resolution | Spectral analysis, filtering of stationary signals |
Limitations and assumptions of this calculator
- One-level only: the tool computes a single Haar DWT level. For multi-level or multiresolution analysis, you would repeatedly apply the transform to the approximation coefficients.
- Length must be a power of two: the input sequence length should be 2, 4, 8, 16, and so on. If the length is not a power of two, the transform is not performed.
- Numeric sequences only: the calculator expects real numbers separated by commas. It does not directly process images, audio files, or complex-valued data.
- Educational focus: the implementation is designed for clarity and small to medium sequences, not for high-performance batch processing or very large data sets.
- Specific normalization: the formulas use averages and differences divided by 2 (not by ). If you are comparing with a textbook that uses factors, coefficients will differ by a constant scaling.
Within these limits, the calculator is well suited for learning how the Haar DWT works, checking hand calculations, and exploring basic signal features.