10.1 Definition and Properties of CWT

Continuous Wavelet Transform

Definition and Mathematical Representation

The Continuous Wavelet Transform (CWT) analyzes signals in both time and frequency simultaneously. It does this by decomposing a signal into wavelets: oscillatory functions that are localized in both time and frequency. Where the Fourier Transform tells you what frequencies exist in a signal, the CWT tells you what frequencies exist and when they occur.

The CWT is defined by:

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

where:

• $f(t)$ is the input signal
• $\psi(t)$ is the mother wavelet, and the asterisk $*$ denotes its complex conjugate
• $a > 0$ is the scale parameter, controlling dilation (stretching) or compression of the wavelet
• $b$ is the translation parameter, sliding the wavelet along the time axis

The normalization factor $\frac{1}{\sqrt{a}}$ ensures that the wavelet's energy stays constant across all scales. Without it, a stretched wavelet would carry more energy than a compressed one, and comparisons across scales would be meaningless.

Think of the CWT as a template-matching operation: at every position $b$ and every scale $a$, you're computing how well the signal matches a scaled and shifted copy of the mother wavelet.

Applications and Advantages

CWT is especially useful for non-stationary signals, where frequency content changes over time:

• Speech signals have rapidly shifting frequency content as phonemes and intonation change
• Biomedical signals (EEG, ECG) exhibit time-varying patterns tied to physiological states or abnormalities
• Seismic data contain different wave arrivals at different times, each with distinct frequency characteristics

Because wavelets are localized in time, the CWT can detect transient or short-lived events that traditional Fourier analysis tends to smear out. It also provides multi-resolution analysis: large scales capture broad, low-frequency trends, while small scales capture fine, high-frequency detail. And because you choose the mother wavelet, you can tailor the analysis to the signal at hand.

Properties of CWT

Linearity, Translation, and Scaling

Linearity. The CWT of a sum of signals equals the sum of their individual CWTs:

$CWT\{c_1 f(t) + c_2 g(t)\} = c_1 \, CWT\{f(t)\} + c_2 \, CWT\{g(t)\}$

where $c_1$ and $c_2$ are constants. This means you can analyze complex signals by breaking them into simpler components and transforming each one separately.

Translation invariance. Shifting the input signal in time by $t_0$ shifts the CWT coefficients by the same amount in the translation variable:

$f(t) \rightarrow f(t - t_0) \implies CWT(a, b) \rightarrow CWT(a, b - t_0)$

The shape of the time-frequency representation doesn't change; it just slides along the time axis.

Scaling covariance. Scaling the input signal in time by a factor $s > 0$ rescales the CWT coefficients:

$f(t) \rightarrow f(st) \implies CWT(a, b) \rightarrow \frac{1}{\sqrt{s}} \, CWT\!\left(\frac{a}{s}, \frac{b}{s}\right)$

Note the $\frac{1}{\sqrt{s}}$ amplitude factor that comes along with the rescaling. Together, translation invariance and scaling covariance mean the CWT faithfully tracks features regardless of where they appear or what time scale they occupy.

Invertibility and Signal Reconstruction

The CWT is invertible: you can recover the original signal from its CWT coefficients, provided the mother wavelet satisfies the admissibility condition (discussed below). The inverse CWT is:

$f(t) = \frac{1}{C_\psi} \int_{0}^{\infty} \int_{-\infty}^{\infty} CWT(a, b) \, \psi_{a,b}(t) \, \frac{db \, da}{a^2}$

where $C_\psi$ is the admissibility constant and $\psi_{a,b}(t) = \frac{1}{\sqrt{a}} \psi\!\left(\frac{t-b}{a}\right)$ is the scaled and translated wavelet. Note that the scale integral runs over $a > 0$ (positive scales only), and the $\frac{1}{a^2}$ factor provides the correct weighting in the time-scale plane.

Invertibility is what makes the CWT practical for tasks like signal denoising (transform, modify coefficients, reconstruct) and compression (keep only the significant coefficients).

Admissibility Condition in CWT

Definition and Implications

For a function $\psi(t)$ to serve as a valid mother wavelet, it must satisfy the admissibility condition:

$C_\psi = \int_{0}^{\infty} \frac{|\Psi(\omega)|^2}{\omega} \, d\omega < \infty$

where $\Psi(\omega)$ is the Fourier transform of $\psi(t)$. This condition has a direct physical consequence: it requires $\Psi(0) = 0$, which means:

$\int_{-\infty}^{\infty} \psi(t) \, dt = 0$

In other words, the mother wavelet must have zero mean. It has to oscillate above and below zero, giving it a bandpass-like spectrum. A function that doesn't average to zero (like a Gaussian bump by itself) can't be a mother wavelet.

Why does this matter? The admissibility constant $C_\psi$ appears in the denominator of the inverse CWT formula. If $C_\psi$ is infinite, reconstruction is impossible. So the admissibility condition is what guarantees you can get your signal back.

Examples of Admissible Wavelets

• Mexican Hat wavelet: $\psi(t) = (1 - t^2) \, e^{-t^2/2}$. This is the negative normalized second derivative of the Gaussian. It's real-valued, symmetric, and good at detecting sharp transitions and peaks. Its simple shape makes it a common choice for introductory CWT analysis.
• Morlet wavelet: $\psi(t) = e^{-t^2/2} \, e^{j\omega_0 t}$, typically with $\omega_0 = 5$ or $6$. This is a complex-valued wavelet (a Gaussian-windowed complex exponential), providing excellent time-frequency localization. The complex form gives you both amplitude and phase information. Strictly speaking, the Morlet wavelet only approximately satisfies the admissibility condition, but for $\omega_0 \geq 5$ the error is negligible in practice.
• Daubechies wavelets: A family of orthogonal wavelets with compact support, constructed via a recursive filter-bank algorithm. These are primarily used in the discrete wavelet transform for applications like image compression and denoising, though they can be used in continuous settings as well.

The choice of mother wavelet depends on what you're looking for in the signal. If you need phase information, use a complex wavelet like the Morlet. If you need to detect sharp features, the Mexican Hat is a natural choice.

CWT vs Fourier Transform

Differences in Approach and Information Provided

The Fourier Transform and the CWT both decompose signals, but they answer different questions:

The Fourier Transform tells you the overall frequency content of a signal but can't tell you when those frequencies occurred. If a signal contains a brief burst at 100 Hz followed by a steady tone at 50 Hz, the Fourier Transform will show both frequencies but won't reveal their timing. The CWT will show exactly when each frequency component appears.

Suitability for Different Signal Types

CWT is the better choice for non-stationary signals:

• Speech, where frequency content shifts rapidly with each phoneme
• EEG/ECG, where transient events like spikes or arrhythmias carry diagnostic information
• Seismic recordings, where P-waves and S-waves arrive at different times with different frequency signatures

The Fourier Transform is well-suited for stationary signals:

• Periodic signals (sine waves, square waves) with constant frequency content
• Steady-state vibration from rotating machinery, where frequencies are determined by rotational speed and mechanical components

The CWT also offers more flexibility because you select the mother wavelet to match the signal's characteristics. The Fourier Transform uses a fixed sinusoidal basis. That said, the Fourier Transform is computationally cheaper and perfectly adequate when the signal's frequency content doesn't change over time. The CWT's added power comes at the cost of redundancy (it's an overcomplete representation) and higher computational demand.