10.2 Mother Wavelets and Their Characteristics

Mother wavelets are the core functions behind the Continuous Wavelet Transform (CWT). By scaling and shifting a single mother wavelet, you generate a whole family of wavelets that can probe a signal at different scales (frequencies) and positions (times). Choosing the right mother wavelet directly affects how well you can resolve features in your signal, so understanding their properties is essential.

Mother Wavelets in CWT

Definition and Role

A mother wavelet is a finite-energy function with zero mean that serves as the template for CWT analysis. "Zero mean" here means the wavelet integrates to zero over all time, so it naturally acts as a bandpass filter rather than responding to DC offsets.

The CWT works by comparing your signal against scaled and translated copies of this mother wavelet. Each comparison (an inner product) tells you how much the signal resembles the wavelet at that particular scale and time location. The choice of mother wavelet depends on what you're looking for in the signal and what time-frequency tradeoff you need.

Generating Wavelet Families

A single mother wavelet produces an entire family through two operations: scaling and translation.

• Scaling dilates or compresses the wavelet by a factor $a$:
  • $a > 1$ stretches the wavelet, giving it a longer time duration and lower center frequency (useful for capturing slow, low-frequency behavior)
  • $0 < a < 1$ compresses the wavelet, shortening its duration and raising its center frequency (useful for capturing fast, high-frequency transients)
• Translation shifts the wavelet along the time axis by $b$, letting it "visit" different parts of the signal

Together, these operations produce wavelets at every combination of scale and position. The CWT then computes the inner product (correlation) between the signal and each member of this family, yielding wavelet coefficients that encode the signal's time-frequency content.

Common Mother Wavelets

Morlet Wavelet

The Morlet wavelet is a complex-valued wavelet formed by a complex sinusoid multiplied by a Gaussian envelope. Because it's essentially a windowed oscillation, it excels at detecting and characterizing oscillatory or sinusoidal patterns in signals.

• Provides strong frequency localization, making it a go-to choice for time-frequency analysis (e.g., seismic data, EEG analysis)
• Its complex nature lets you extract both amplitude and phase information from the signal, not just energy
• Time localization is decent but not as sharp as compact-support wavelets

Mexican Hat Wavelet

The Mexican Hat wavelet (also called the Ricker wavelet) is the negative normalized second derivative of a Gaussian function. Its shape looks like a sombrero: a positive central peak flanked by two negative lobes.

• Symmetric and real-valued, so it introduces no phase distortion
• Has a single vanishing moment, which makes it effective at detecting sharp transitions, edges, and singularities
• Offers good time localization but relatively poor frequency localization
• Commonly used in image processing, pattern recognition, and any task where you need to pinpoint abrupt changes

Daubechies Wavelets

Daubechies wavelets are a family of orthogonal, compactly supported wavelets. Each member is labeled by its number of vanishing moments (e.g., db4 has 4 vanishing moments).

• Compact support means the wavelet is nonzero only over a finite interval, which gives excellent time localization
• More vanishing moments produce a smoother wavelet that better suppresses polynomial trends, but the support length grows in return
• Asymmetric in general (only db1, the Haar wavelet, is symmetric), so they can introduce phase shifts in the coefficients
• Widely used in practical applications like compression (JPEG 2000) and denoising, where orthogonality and compact support are valuable

Mother Wavelet Characteristics

Shape and Symmetry

The wavelet's shape determines what signal features it's sensitive to. A wavelet that looks like a sharp spike will respond to transients; one that looks like a smooth oscillation will respond to periodic components.

• Symmetric wavelets (e.g., Mexican Hat, Morlet) preserve phase information and avoid phase distortion in the coefficients. This matters in applications like audio processing where phase relationships are meaningful.
• Asymmetric wavelets (e.g., Daubechies) can shift the apparent timing of features in the coefficients, but they often provide better time localization or other desirable mathematical properties like orthogonality.

Regularity and Vanishing Moments

Regularity describes how smooth the wavelet function is. Smoother wavelets approximate smooth signals well, while rougher wavelets (like the Haar wavelet) are better at catching abrupt discontinuities.

Vanishing moments control the wavelet's ability to ignore polynomial trends. A wavelet with $N$ vanishing moments produces zero coefficients when applied to any polynomial of degree $N - 1$ or lower. In practice:

• More vanishing moments mean the wavelet "sees through" smooth, slowly varying parts of the signal and responds only to genuine detail
• This leads to sparser representations and better compression for signals with smooth regions (e.g., natural images)
• The tradeoff is a longer wavelet, which reduces time localization

Time-Frequency Localization

Every wavelet faces a fundamental constraint: the Heisenberg uncertainty principle prevents perfect localization in both time and frequency simultaneously. The product of time resolution and frequency resolution has a lower bound that no wavelet can beat.

• Compactly supported wavelets (Daubechies) tend toward better time localization
• Wavelets with smooth, extended frequency responses (Morlet) tend toward better frequency localization
• Your application dictates the right balance. If you need to pinpoint when a transient occurs, favor time localization. If you need to distinguish closely spaced frequency components, favor frequency localization.

Wavelet Families from a Mother Wavelet

Scaling and Translation

Both operations are captured in a single formula. Given a mother wavelet $\psi(t)$, the scaled and translated version is:

$\psi_{a,b}(t) = \frac{1}{\sqrt{|a|}} \psi\left(\frac{t - b}{a}\right)$

where:

• $a$ is the scale parameter (controls dilation/compression)
• $b$ is the translation parameter (controls position along the time axis)
• The $\frac{1}{\sqrt{|a|}}$ factor normalizes the wavelet's energy so that every scaled version has the same energy as the original

Larger $a$ values stretch the wavelet for coarse-scale (low-frequency) analysis. Smaller $a$ values compress it for fine-scale (high-frequency) analysis. Varying $b$ slides the wavelet across the signal to examine local behavior at each time instant.

Wavelet Coefficients

The CWT coefficient at scale $a$ and position $b$ is defined as:

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

where $x(t)$ is the input signal and $\psi^*(t)$ is the complex conjugate of the mother wavelet.

Each coefficient measures the similarity between the signal and the wavelet at that specific scale and position:

• A large magnitude means the signal strongly resembles the wavelet shape at that scale and location
• A small magnitude means little resemblance
• For complex wavelets like the Morlet, the coefficient's magnitude gives local amplitude and its angle gives local phase

Plotting these coefficients over all $(a, b)$ pairs produces a scalogram, which is the CWT's time-frequency representation of the signal. This map lets you see how the signal's frequency content evolves over time.