10.3 Time-Scale Representation

Time-scale Representation in CWT

2D Representation of Signal

The Continuous Wavelet Transform (CWT) produces a two-dimensional map of a signal, with time on the x-axis and scale on the y-axis. This is the core idea behind time-scale representation: instead of looking at a signal as just amplitude vs. time, you get to see which frequency components are active at each moment.

• Different scales correspond to different frequency ranges, so you can study how the signal behaves across various frequencies
• Each point in the 2D map captures the signal's frequency content at a specific time, revealing how that content evolves

Wavelet Analysis Approach

CWT works by comparing the signal against scaled and shifted copies of a mother wavelet, a single prototype waveform.

• Scaling (dilating or compressing) the mother wavelet changes its frequency sensitivity. A compressed wavelet targets higher frequencies; a dilated one targets lower frequencies.
• Shifting slides the wavelet along the time axis so you can probe different moments in the signal.

At each combination of scale and time shift, the CWT computes a coefficient that measures the similarity between the signal and that particular wavelet copy.

• A large coefficient means the signal strongly resembles the wavelet at that time and scale, indicating significant energy there.
• A small coefficient means little matching content exists at that location.

Multi-resolution Analysis with CWT

Scale-dependent Resolution

One of the most important properties of CWT is that resolution depends on scale. This is sometimes called the Heisenberg trade-off in time-frequency analysis: you can't have perfect resolution in both time and frequency simultaneously.

• Small scales (high frequencies): The wavelet is compressed, giving you high temporal resolution but low frequency resolution. You can pinpoint when a fast event happens, but you get less precision about its exact frequency.
• Large scales (low frequencies): The wavelet is stretched out, giving you high frequency resolution but low temporal resolution. You can precisely identify slow oscillations, but their exact timing becomes blurrier.

This behavior is what makes CWT a multi-resolution tool. It automatically adapts its resolution depending on the frequency range you're examining.

Feature Identification and Signal Behavior

Multi-resolution analysis is valuable because real signals contain features at many different scales.

• Transient events (like a sudden click or spike) show up best at small scales, where temporal resolution is sharpest
• Slow trends or periodic oscillations are better captured at large scales, where frequency resolution is highest
• Scale-dependent patterns are features that stand out only at certain scales. For example, a heartbeat's QRS complex is prominent at one scale range, while the slower T-wave appears at another.

By examining CWT output across all scales, you can extract the frequency bands that carry the most meaningful information about your signal.

Scale vs. Frequency in CWT

Inverse Relationship

Scale and frequency are inversely related in CWT. The relationship is given by:

$f = \frac{f_c}{a}$

where $f$ is the frequency captured, $f_c$ is the center frequency of the mother wavelet (a fixed property of whichever wavelet you choose), and $a$ is the scale parameter.

• Smaller scale → higher frequency (compressed wavelet, shorter duration)
• Larger scale → lower frequency (dilated wavelet, longer duration)

This is why scalograms often label the y-axis with scale increasing downward, so that higher frequencies appear at the top, matching the convention of spectrograms.

Wavelet Dilation and Compression

As scale increases, the wavelet stretches in time. A dilated wavelet covers a longer time window and responds to slower variations in the signal, such as overall trends or low-frequency oscillations.

As scale decreases, the wavelet compresses. A compressed wavelet covers a shorter time window and picks up rapid changes, such as sharp transients, edges, or high-frequency bursts.

This dilation/compression mechanism is what gives CWT its adaptive resolution: the analyzing window naturally widens for low frequencies and narrows for high frequencies.

Interpreting CWT Representations

Scalogram Visualization

The standard way to visualize CWT output is a scalogram: a 2D plot with time on the x-axis, scale (or equivalent frequency) on the y-axis, and color or grayscale intensity representing the magnitude of the CWT coefficients.

• Bright or intense regions indicate strong similarity between the signal and the wavelet at that time-scale location. These are the dominant signal components.
• Dark regions indicate little energy at that combination of time and scale.

The scalogram gives you an immediate visual summary of where (in time) and at what frequency the signal's energy is concentrated.

Analyzing Time-Frequency Characteristics

Several features in a scalogram carry specific meaning:

• Ridges are continuous bands of high-magnitude coefficients that trace out the dominant frequency components over time. The position of a ridge along the scale axis tells you the frequency; its extent along the time axis tells you how long that component persists.
• Isolated bright spots often correspond to transient events, short bursts of energy localized in both time and scale.
• Broadband vertical streaks (spanning many scales at one time instant) typically indicate impulsive events like clicks or discontinuities.

The scalogram is especially powerful for non-stationary signals, where frequency content changes over time. A standard Fourier transform would average over the entire signal and miss these changes, but the CWT's time-scale map reveals exactly when each frequency component appears, strengthens, weakens, or disappears.