6.7 Scalograms and time-scale representations

Scalograms and time-scale representations offer powerful tools for analyzing signals with changing frequency content over time. These techniques use wavelet transforms to break down signals into different scales, providing a detailed view of how frequency components evolve.

Unlike traditional methods, scalograms adapt their analysis to the signal's characteristics. This flexibility allows for better detection of transient events and subtle frequency shifts, making scalograms invaluable in fields like audio processing, biomedical analysis, and pattern recognition.

Scalograms

• Scalograms are a visual representation of a signal's time-scale distribution, providing insights into how the signal's frequency content evolves over time
• They are particularly useful for analyzing non-stationary signals, where the frequency components change over time
• Scalograms are generated using wavelet transforms, which decompose the signal into a set of wavelets at different scales and time positions

Definition of scalograms

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• A scalogram is a two-dimensional plot that displays the absolute values of the wavelet coefficients as a function of time (x-axis) and scale (y-axis)
• The color or intensity of each point in the scalogram represents the magnitude of the wavelet coefficient at a specific time and scale
• Scalograms provide a localized time-scale representation of the signal, allowing for the identification of transient events and changes in frequency content

Scalogram vs spectrogram

• While both scalograms and spectrograms are used to analyze time-frequency information, they differ in their underlying transforms and representations
• Spectrograms are based on the Short-Time Fourier Transform (STFT), which uses a fixed window size to analyze the signal's frequency content over time
• Scalograms, on the other hand, use wavelet transforms, which provide a multi-resolution analysis by using wavelets with varying scales (window sizes) to capture both low and high-frequency information

Scalogram analysis

• Scalogram analysis involves interpreting the patterns and features present in the scalogram to gain insights into the signal's time-scale characteristics
• Key aspects of scalogram analysis include identifying regions of high energy (bright spots), which indicate the presence of significant signal components at specific times and scales
• The scalogram's vertical axis represents the scale, with lower scales corresponding to higher frequencies and vice versa

Scalogram interpretation

• In a scalogram, the horizontal axis represents time, while the vertical axis represents scale (inversely related to frequency)
• Bright regions in the scalogram indicate the presence of significant signal components at specific times and scales
• The scalogram's energy distribution across scales and time provides information about the signal's time-varying frequency content and the presence of transient events or discontinuities
• Interpreting scalograms requires an understanding of the relationship between scale and frequency, as well as the characteristics of the chosen wavelet function

Wavelet transforms for scalograms

• Wavelet transforms are the mathematical tools used to generate scalograms by decomposing a signal into a set of wavelets at different scales and time positions
• Unlike the Fourier transform, which provides a global frequency representation, wavelet transforms offer a localized time-scale analysis
• The choice of wavelet function plays a crucial role in the scalogram generation process, as different wavelets have different time-frequency localization properties

Continuous wavelet transform (CWT)

• The continuous wavelet transform (CWT) is a mathematical operation that computes the inner product between a signal and a set of scaled and translated versions of a chosen wavelet function
• The CWT is defined as: $CWT(a,b) = \frac{1}{\sqrt{|a|}} \int_{-\infty}^{\infty} x(t) \psi^* (\frac{t-b}{a}) dt$, where $a$ is the scale factor, $b$ is the translation factor, and $\psi(t)$ is the wavelet function
• The resulting CWT coefficients represent the similarity between the signal and the wavelet at different scales and time positions

CWT vs Fourier transform

• The Fourier transform decomposes a signal into a sum of sinusoidal basis functions, providing a global frequency representation without localized time information
• The CWT, in contrast, uses wavelets as basis functions, which are localized in both time and frequency domains
• The CWT's ability to adapt the window size based on the scale allows for a multi-resolution analysis, capturing both low and high-frequency information with appropriate time resolution

Wavelet functions for CWT

• Various wavelet functions can be used for the CWT, each with different properties and suitability for specific applications
• Some common wavelet functions include the Morlet wavelet, Mexican Hat wavelet, and Daubechies wavelets
• The choice of wavelet function depends on factors such as the signal's characteristics, the desired time-frequency resolution, and the computational efficiency

CWT scalogram generation

• To generate a CWT scalogram, the CWT coefficients are computed for a range of scales and time positions
• The absolute values of the CWT coefficients are then plotted as a two-dimensional image, with time on the x-axis, scale on the y-axis, and the coefficient magnitude represented by color or intensity
• The resulting scalogram provides a visual representation of the signal's time-scale distribution, allowing for the identification of transient events and changes in frequency content

Time-scale representations

• Time-scale representations, such as scalograms, provide a way to analyze and visualize the time-varying frequency content of signals
• Unlike time-frequency representations, which use a fixed frequency resolution, time-scale representations adapt the frequency resolution based on the scale, providing a multi-resolution analysis

Time-scale vs time-frequency

• Time-frequency representations, such as spectrograms, use a fixed window size (frequency resolution) for all frequencies, resulting in a trade-off between time and frequency resolution
• Time-scale representations, like scalograms, use a variable window size that depends on the scale, allowing for a more flexible and adaptive analysis of the signal's time-varying frequency content
• In time-scale representations, lower scales correspond to higher frequencies with better time resolution, while higher scales correspond to lower frequencies with better frequency resolution

Scale concept in wavelets

• In the context of wavelet transforms, scale refers to the degree of compression or dilation applied to the wavelet function
• Lower scales correspond to more compressed wavelets, which are better suited for capturing high-frequency information with good time resolution
• Higher scales correspond to more dilated wavelets, which are better suited for capturing low-frequency information with good frequency resolution
• The scale concept allows for a multi-resolution analysis of the signal, adapting the time-frequency resolution based on the signal's characteristics

Scale-to-frequency relationship

• There is an inverse relationship between scale and frequency in wavelet analysis
• Lower scales correspond to higher frequencies, while higher scales correspond to lower frequencies
• The exact relationship between scale and frequency depends on the specific wavelet function and its center frequency
• The scale-to-frequency relationship is important for interpreting scalograms and understanding the time-frequency content of the signal

Time-scale resolution

• Time-scale representations, like scalograms, provide a variable time-frequency resolution that depends on the scale
• At lower scales (higher frequencies), the time resolution is better, allowing for the accurate localization of transient events and rapid changes in the signal
• At higher scales (lower frequencies), the frequency resolution is better, allowing for the accurate identification of slow-varying components and trends in the signal
• The multi-resolution nature of time-scale representations allows for a more comprehensive analysis of the signal's time-varying frequency content

Multiresolution analysis

• Multiresolution analysis (MRA) is a mathematical framework that provides a structured way to decompose a signal into a hierarchy of approximations and details at different scales
• MRA forms the basis for the discrete wavelet transform (DWT) and allows for efficient computation and analysis of the signal's time-scale representation

Multiresolution concept

• The multiresolution concept involves representing a signal as a sum of approximations and details at different scales
• At each scale, the signal is decomposed into a low-frequency approximation and a high-frequency detail component
• The approximation at a given scale is further decomposed into an approximation and detail at the next scale, creating a hierarchical representation of the signal

Scaling function

• The scaling function, denoted as $\phi(t)$, is a low-pass filter used in MRA to generate the approximation coefficients at each scale
• The scaling function satisfies a two-scale equation: $\phi(t) = \sum_{k} h[k] \sqrt{2} \phi(2t-k)$, where $h[k]$ are the scaling function coefficients
• The scaling function is designed to capture the low-frequency information of the signal and provide a smooth approximation at each scale

Wavelet function

• The wavelet function, denoted as $\psi(t)$, is a high-pass filter used in MRA to generate the detail coefficients at each scale
• The wavelet function is related to the scaling function by a two-scale equation: $\psi(t) = \sum_{k} g[k] \sqrt{2} \phi(2t-k)$, where $g[k]$ are the wavelet function coefficients
• The wavelet function is designed to capture the high-frequency information of the signal and provide the details that complement the approximation at each scale

Multiresolution decomposition

• Multiresolution decomposition involves iteratively applying the scaling and wavelet functions to the signal to obtain the approximation and detail coefficients at each scale
• At each level of decomposition, the approximation coefficients from the previous level are passed through the scaling function to obtain the approximation coefficients at the current level, while the detail coefficients are obtained by applying the wavelet function
• This process is repeated until the desired level of decomposition is reached, resulting in a hierarchical representation of the signal in terms of approximations and details at different scales

Discrete wavelet transform (DWT)

• The discrete wavelet transform (DWT) is a computational algorithm that implements the multiresolution analysis framework to efficiently decompose a signal into its wavelet coefficients
• DWT is widely used in various signal processing applications, including compression, denoising, and feature extraction

DWT definition

• The DWT is defined as the inner product of a signal $x[n]$ with a set of discretized and scaled versions of the wavelet function $\psi_{j,k}[n]$
• The DWT coefficients are given by: $W_{j,k} = \sum_{n} x[n] \psi_{j,k}[n]$, where $j$ and $k$ are the scale and translation indices, respectively
• The DWT coefficients represent the similarity between the signal and the wavelet at different scales and time positions

DWT implementation

• The DWT is typically implemented using a filter bank structure, consisting of a low-pass filter (scaling function) and a high-pass filter (wavelet function)
• The signal is passed through the filter bank, and the outputs are downsampled by a factor of 2 to obtain the approximation and detail coefficients at each level
• This process is repeated on the approximation coefficients to obtain the coefficients at the next level, creating a dyadic decomposition of the signal

Subband coding with DWT

• Subband coding is a technique that uses the DWT to decompose a signal into a set of frequency subbands, each representing a specific range of frequencies
• The DWT-based subband coding allows for efficient compression and processing of the signal by exploiting the energy compaction property of wavelets
• The signal is reconstructed by upsampling and filtering the subband coefficients, followed by a summation of the reconstructed subbands

DWT scalogram interpretation

• The DWT scalogram is a visual representation of the DWT coefficients, displaying the signal's time-scale distribution
• In the DWT scalogram, the horizontal axis represents time, while the vertical axis represents the scale (or level) of decomposition
• The magnitude of the DWT coefficients is represented by the color or intensity of each pixel in the scalogram
• The DWT scalogram allows for the identification of transient events, discontinuities, and changes in the signal's frequency content at different scales

Wavelet packet transform (WPT)

• The wavelet packet transform (WPT) is an extension of the discrete wavelet transform (DWT) that provides a more flexible and adaptive decomposition of the signal
• Unlike the DWT, which only decomposes the approximation coefficients at each level, the WPT decomposes both the approximation and detail coefficients, resulting in a complete binary tree of subband coefficients

WPT vs DWT

• The DWT decomposes the signal into a set of approximation and detail coefficients at each level, focusing on the low-frequency content of the signal
• The WPT, on the other hand, decomposes both the approximation and detail coefficients at each level, resulting in a more balanced and adaptive representation of the signal
• The WPT allows for a finer frequency resolution and a more flexible analysis of the signal's time-frequency content compared to the DWT

WPT decomposition tree

• The WPT decomposition results in a complete binary tree of subband coefficients, where each node represents a specific frequency subband
• At each level of decomposition, the WPT splits both the approximation and detail coefficients into two new sets of coefficients, creating a more balanced and uniform decomposition
• The depth of the WPT decomposition tree determines the frequency resolution and the number of subbands in the analysis

WPT scalogram generation

• The WPT scalogram is a visual representation of the WPT coefficients, displaying the signal's time-scale distribution across the complete binary tree of subbands
• To generate the WPT scalogram, the absolute values of the WPT coefficients are computed for each subband and plotted as a two-dimensional image
• The horizontal axis represents time, while the vertical axis represents the subband index or the corresponding frequency range
• The magnitude of the WPT coefficients is represented by the color or intensity of each pixel in the scalogram

WPT for signal analysis

• The WPT provides a more flexible and adaptive framework for analyzing the time-frequency content of signals compared to the DWT
• By decomposing both the approximation and detail coefficients at each level, the WPT allows for a more balanced and uniform analysis of the signal's frequency components
• The WPT is particularly useful for signals with complex time-frequency structures, as it provides a finer frequency resolution and a more adaptive representation of the signal
• WPT-based analysis has been applied in various fields, including audio and speech processing, biomedical signal analysis, and pattern recognition

Applications of scalograms

• Scalograms, generated using wavelet transforms such as the CWT, DWT, or WPT, have numerous applications in signal processing and data analysis
• The time-scale representation provided by scalograms allows for the identification of transient events, changes in frequency content, and hidden patterns in signals

Scalograms in signal denoising

• Scalograms can be used for signal denoising by exploiting the localization properties of wavelets
• By applying thresholding techniques to the wavelet coefficients in the scalogram domain, noise can be effectively suppressed while preserving the important signal features
• The denoised signal is then reconstructed from the modified scalogram, resulting in an improved signal-to-noise ratio

Scalograms for feature extraction

• Scalograms provide a rich set of features that can be extracted for various signal processing tasks, such as classification, segmentation, and pattern recognition
• The time-scale distribution captured by the scalogram allows for the identification of discriminative features that characterize different signal classes or events
• Statistical measures, such as energy, entropy, or moments, can be computed from the scalogram to obtain a compact representation of the signal's time-scale properties

Scalograms in pattern recognition

• Scalograms have been successfully applied in pattern recognition tasks, such as speech recognition, image classification, and fault diagnosis
• The multi-resolution analysis provided by scalograms allows for the detection of patterns and features at different scales, enhancing the discriminative power of the recognition system
• Machine learning algorithms, such as support vector machines or neural networks, can be trained on scalogram-based features to classify or recognize patterns in signals

Scalograms for time-varying signals

• Scalograms are particularly useful for analyzing time-varying signals, where the frequency content changes over time
• The time-scale representation provided by scalograms allows for the identification of transient events, such as discontinuities, spikes, or bursts, which may be difficult to detect in the time or frequency domain alone
• Scalograms can be used to track the evolution of the signal's frequency content over time, providing insights into the underlying dynamics and behavior of the system