Wavelets and Time-Frequency Analysis

The CWT of a signal $x(t)$ with respect to a mother wavelet $\psi(t)$ is:

$$W(a, b) = \frac{1}{\sqrt{|a|}} \int_{-\infty}^{\infty} x(t)\, \psi^*\!\left(\frac{t-b}{a}\right) dt$$

where $a$ is the scale (inversely related to frequency) and $b$ is the translation (time shift). The factor $1/\sqrt{|a|}$ ensures energy normalization across scales.

import pywt
scales = np.arange(1, 128)
coeffs, freqs = pywt.cwt(x, scales, 'morl', sampling_period=1/fs)

The DWT decomposes a signal into approximation coefficients (low frequency) and detail coefficients (high frequency) at each level using a pair of complementary filters:

$$c_A[n] = \sum_k x[k]\, \tilde{\phi}[2n - k] \quad \text{(lowpass)}$$ $$c_D[n] = \sum_k x[k]\, \tilde{\psi}[2n - k] \quad \text{(highpass)}$$

followed by downsampling by 2.

import pywt
cA, cD = pywt.dwt(x, 'db4')     # single-level DWT
coeffs = pywt.wavedec(x, 'db4', level=5)  # multi-level DWT
x_rec = pywt.waverec(coeffs, 'db4')       # reconstruction

Different wavelet families have different properties:

wavelet = pywt.Wavelet('db4')
print(f"Filter length: {wavelet.dec_len}")
print(f"Vanishing moments: {wavelet.dec_len // 2}")

Multi-resolution analysis decomposes a signal into components at different resolution levels. At level $j$, the signal is represented as:

$$x(t) = \sum_k c_{J,k}\, \phi_{J,k}(t) + \sum_{j=1}^{J} \sum_k d_{j,k}\, \psi_{j,k}(t)$$

where $\phi_{j,k}$ and $\psi_{j,k}$ are scaled and translated versions of the scaling and wavelet functions. Each level captures a specific frequency band:

• Level 1 details: $[f_s/4, f_s/2]$
• Level 2 details: $[f_s/8, f_s/4]$
• Level $j$ details: $[f_s/2^{j+1}, f_s/2^j]$

Wavelet denoising exploits the fact that signal energy concentrates in a few large wavelet coefficients, while noise spreads across all coefficients. The procedure is:

1. Compute the DWT: $\text{coeffs} = \text{wavedec}(x, \text{wavelet}, \text{level})$
2. Threshold the detail coefficients
3. Reconstruct: $\hat{x} = \text{waverec}(\text{coeffs}, \text{wavelet})$

Common thresholding rules:

• Hard: $\hat{d} = d \cdot \mathbf{1}_{|d| > \lambda}$
• Soft: $\hat{d} = \text{sign}(d) \cdot \max(|d| - \lambda, 0)$

Universal threshold: $\lambda = \sigma \sqrt{2 \ln N}$ where $\sigma = \text{MAD}(d_1) / 0.6745$ (median absolute deviation of level-1 detail coefficients).