Digital Signal Processing -EC504- Module4 ( MAKAUT-Syllabus)

 Today, our communication elements are: - The origin of waves, Classification of waves: CWT and DWT, Filter bank in signal processing.

 The origin of waves: -

Wavelets are one of the most influential mathematical devices that have appeared in recent decades, bringing revolution in areas such as signal processing, image compression, and data analysis. The term 'waves' arises from the need to analyse signs of many parameters or resolutions, which could not fully achieve the classic Fourier analysis. While the Fourier transformation provides frequency information, it does not locate signs in time, making it less effective for non-stationary signals. The wavelet theory appeared to cross these boundaries, both time and frequency location.

The roots of waves can be detected in the early 1900s. One of the early contributors was Alfred Haar in 1909, who introduced the Haar feature, which would later be called a wavelength, the first example of this. The Haar wavelet was simple but still powerful, which provides a way to represent signals due to the base functions of the size of the square. Although everyone's work did not gain instant popularity, it laid the foundation for modern wavelet theory.

The next important development came from the late 1970s and early 1980s from a French geophysicist, Jean Morlet. Morlet worked to analyse seismic characters, which are naturally non-stable, which means that their frequency material changes over time. He introduced the concept of a "wavelet" as a small fluctuation function, located in both time and frequency. To study seismic waves more efficiently, Morlet replaced the ideas of the furrowers with the window, which we are now aware of for wave conversion. This allowed researchers to zoom in on indications of different parameters and detect hidden functions in traditional Fourier analysis.

Parallel to mathematicians such as Alex Grosmans, with Morlets to provide a stiff mathematical structure for waves. Their work formalised the constant waves (CWT), which decomposes a signal in the waves of different scales and translations. This was a success because it gave a bridge between geophysical and pure mathematics, making waves a versatile tool in the discipline.

Another important step came with the contribution from Ingrid Debtkis in the late 1980s and early 1990s. He developed a family of compact orthogonal waves, known as a submerged wavelet, which became necessary for practical applications. These waves were smooth, mathematically elegant, and suitable for numerical calculation, making them very effective for digital signals and image processing. Work on Daubechies made wavelets practical for applications such as image compression (e.g., JPEG2000), noise reduction, and biomedical signal analysis.

With this development, waves developed in an area that combines mathematics, physics, engineering, and computer science. The idea of the Multisction Analysis (MRA) introduced by Stephen Mallt further strengthened the frame. The MRA provided a systematic method of analysing signs of different parameters and led to an algorithm that was calculated to be effective and widely useful.

In summary, the origin of the wave is not attributed to a person or moment but is responsible for a number of contributions for almost a century. As the leading task of Haar, which continues through the seismic applications of the peacock, the mathematical stiffness of the Grossman, and the practical waves of the Daubechies, the region has developed as the foundation stone of modern signal processing.

Classification of waves: CWT and DWT: -

The wavelet transform is a powerful mathematical device used to analyse signals and images on several parameters. Unlike furrier analysis, which only provides global frequency information, it can capture both time and frequency properties at the same time. This makes it especially effective for non-stable signals, where frequency material changes over time. Mostly, wavelet transformations are classified into two categories: Continuous Wavelet Transformation (CWT) and Discrete Wavelet Transformation (DWT). Both methods share the basic idea of breaking a signal into scale and changed versions of a basic function called "Mother Wavelet", but they are different in implementation, resolution, and applications.

1. Continuous Wave Transform (CWT)

Continuous Wavel transformation provides a very detailed analysis of signs by continuously separating the scale and translation parameters from the mother's wavelength. In CWT, the wavelength (compressed or stretched) is scaled to the signal to any possible value, causing a two-dimensional representation of time and frequency.

Mathematically, the CWT of a signal $x(t)$ is defined as:

Here, $a$ represents the scale (frequency information), b represents translation (time information), and $\psi(t)$ is the mother wavelet.

The benefits of CWT:

• Provides a complete and detailed representation of the signal.
• Useful for analysing fine functions in non-stable signals.
• High resolution in both time and frequency.

CWT limit:

• It is complex and expensive because of its continuous nature.
• Produces large amounts of fruitless information, which is not always convenient for digital systems.

CWT application:

CWT is mainly used in applications that require accuracy and high resolution, such as biomedical signal analysis (EEG, ECG), geophysics, and vibration analysis.

2. Discrete Wavelet Transform (DWT):

The discussed vowel transform scale and translation parameters simplify the process, usually at a diadic interval (POWERS OF TWO). It eliminates profits and makes the calculation effective, which is essential for digital signal processing.

Instead of analysing all possible scales and positions, DWT uses filters to divide the signal into connection (low existing) and expansion (high existing) components. This process is repeated in a technique called multisavulsion analysis (MRA).

The benefits of DWT:

• Calculation is efficient and suitable for digital implementation.
• Low memory requirements and provides a compact representation.
• Well-adapted real-time applications.

DWT limit:

• Less resolution compared to CWT.
• Some exact details may be lost due to discretion.

DWT application:

DWT is widely used in practical applications such as image compression (JPEG2000), denoising, speech recognition, and multimedia transfer. The efficiency makes it a preferred option in most engineering applications.

Comparison of CWT and DWT: -

Aspect	CWT	DWT
Scale & Translation	Continuous	Discrete (dyadic intervals)
Computation	High, redundant	Efficient, compact
Resolution	High time-frequency resolution	Moderate resolution
Applications	Biomedical, geophysics, research	Compression, denoising, real-time

Filter bank in signal processing: -

A filter bank is a basic concept in digital signal processing (DSP) that includes a set of tap-delay filters designed to divide an input signal into multiple components, each with a specific frequency below the tap. Essentially, a filter bank separates a broadband signal for a small, more manageable frequency range, which can then be treated, analysed, or compressed independently. This concept is widely used in areas such as audio and speech treatment, image compression, and communication.

Basic Concept of Filter Banks:

The main idea of a filter bank is to use multiple filters arranged in parallel to decompose a signal. Each filter removes part of the spectrum of the signal. For example, in speech or sound analysis, a filter bank may separate frequencies in the leash equivalent to low, middle, and high tones.

When filtered, underband signals can be treated in different ways, such as compression, denoising, or functional extraction. Later, if necessary, the indications from the underband can be reconciled using a synthesis filter bank to restore the original signal.

Types of filter banks

Uniform filter bank

• All underbands belong to the same bandwidth.
• Example: Divide an audio signal into a fixed frequency range.

Non-human filter bank

• The width of the underband varies, which is often designed to mimic human hearing perception.
• Examples: postal frequency filter banks in speech recognition.

Analysis and synthesis filter bank

• Analysis filter bank: Divide the signal into an underband.
• Synthesis Filter Bank: Rebuilding the original signal (or an estimate) from these underbands.

Filter Bank's applications

• Audio and voice treatment: Filter banks are used in voice recognition systems, hearing equipment, and sound compression (e.g., MP3). By dividing the sound into frequency bands, important functions can be taken effectively or compressed.
• Image compression: In JPEG2000, the vowel-based filter banks decomposed images to individual resolution levels, making compression more efficient.
• Communication system: Multicarrier modulation techniques, such as OFDM, allow effective transfer and error control to distinguish indications in subcapsules to rely on filter banks.
• Biomedical signal processing: EEG or ECG signals can be decomposed into the underband to analyse frequency-specific patterns associated with medical conditions.

The Advantages/benefits of filter banks

• At the same time, leave analysis of several frequency bands.
• Support effective compression and denoising.
• Provide flexibility for applications in different fields.

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