Transforms in Digital Signal Processing

9 min readMay 1, 2023

Introduction to DSP :

Digital Signal Processing (DSP) is a field of study concerned with the processing, analysis, and manipulation of signals that have been converted from analog to digital form. These signals can be of various types, such as audio, video, images, or any other type of information that can be represented as a sequence of numbers. DSP has revolutionized the way we acquire, analyze, and manipulate signals in a wide range of applications, including telecommunications, audio and video processing, medical imaging, radar systems, and many others.

The main aim of DSP is to extract useful information from signals, remove unwanted noise or artifacts, and enhance or transform the signals in a way that is meaningful for the intended application. DSP algorithms and techniques can be applied to both real-time and offline processing of signals, and can range from simple filtering and smoothing operations to complex signal analysis and machine learning algorithms.

Transforms :

The term “transform” refers to the application of mathematical transformations to change the representation of the signal from one form to another. Decomposition of the signal into orthogonal basis functions. In signal processing, a transform is a mathematical operation that converts a signal from one domain to another.

Transforms can be either continuous or discrete. Continuous transforms are used for continuous-time signals, while discrete transforms are used for discrete-time signals. Transforms are also used in data compression, where they can be used to represent data in a more compact form.

Transforms are a powerful tool in signal processing, enabling the analysis and processing of signals in different domains. Transforms are widely used to analyze, process and compress signals in various applications such as audio and image processing, communications, and control systems.

The most commonly used transforms in signal processing are the Fourier Transform, the Laplace Transform, and the Z-Transform. The Fourier Transform is used to analyze signals in the frequency domain, which is useful for tasks such as filtering, compression, and modulation. The Laplace Transform is used to analyze signals in the complex frequency domain, which is useful for analyzing stability and performance of control systems. The Z-Transform is used to analyze discrete-time signals in the frequency domain.

Types of Transforms :

1. Continuous-Time Fourier Transform(CTFT)

2. Discrete-Time Fourier Transform(DTFT)

3. Discrete Fourier Transform(DFT)

4. Z-Transform

5. Bilinear Transform

6. Discrete Cosine Transform(DCT)

7. HAAR transform

Analysis of Different Transforms:

1. Continuous Time Fourier Transform (CTFT) :

The Continuous-Time Fourier Transform (CTFT) is a mathematical tool used in signal processing to represent a continuous-time, aperiodic signal as a continuous spectrum of complex exponential functions. It is defined as the Fourier transform of a continuous-time signal, and it can be used to analyze the frequency content of a signal in the continuous-time domain.

The Fourier transform of a continuous-time function x(t) is defined as,

The inverse Fourier transform of a continuous-time function is defined as,

Properties of Fourier Transform:

Several significant characteristics of the continuous-time Fourier transform (CTFT) are present. Both driving Fourier transform pairs and determining broad frequency domain relationships can benefit from these properties. These characteristics also aid in determining how different time domain procedures affect the frequency domain. The table below lists some of the crucial characteristics of the continuous time Fourier transform.

2. Discrete-Time Fourier Transform (DTFT):

The discrete-time Fourier transform can be used to represent a discrete-time signal in the frequency domain. Consequently, the discrete-time Fourier transform is the name given to the Fourier transform of a discrete-time sequence (DTFT).

If x(n) is a discrete-time sequence mathematically speaking, then its discrete-time Fourier transform is defined as

Condition for Existence of Discrete-Time Fourier Transform

The Fourier transform of a discrete-time sequence x(n) exists if and only if the sequence x(n) is absolutely summable, i.e.,

Because they are not completely summable, exponentially expanding sequences do not have a discrete-time Fourier transform (DTFT).

Additionally, the DTFT method of system analysis can only be used to analyze asymptotically stable systems and cannot be used to analyze unstable systems; in other words, the DTFT method can only be used to analyze systems whose transfer function has poles inside the unit circle.

3. Discrete Fourier Transform(DFT):

A complex-valued function of frequency, the discrete Fourier transform (DFT) in mathematics transforms a finite sequence of evenly spaced samples of a function into a same-length sequence of evenly spaced samples of the discrete-time Fourier transform (DTFT).

The DFT is the most significant discrete transform and is utilized in many real-world situations to do Fourier analysis. Any quantity or signal that fluctuates over time and is sampled over a finite time period (typically determined by a window function) is referred to as a function in digital signal processing. Examples include the pressure of a sound wave, a radio signal, or daily temperature readings. The samples in image processing can be the pixel values along a row or column of an image. The DFT is also used to efficiently carry out other operations, including convolutions or big integer multiplication, and to solve partial differential equation.

And equation for the DFT is

The DFT has the benefit of being very simple to program, making it simple to implement in any coding language. The required computational time is a drawback, though this is getting less of a problem with the advancement of contemporary computing.

4. Z-Transform:

A mathematical technique called the Z-transform (ZT) is used to translate differential equations in the temporal domain into algebraic equations in the z-domain.

Use of Z-Transform

An extremely helpful tool in the analysis of a linear shift invariant (LSI) system is the Z-transform. Difference equations are used to model an LSI discrete time system. These time-domain difference equations are solved by first converting them into algebraic equations in the z-domain using the Z-transform, manipulating the algebraic equations in the z-domain, and then converting the results back into the time domain using the inverse Z-transform.

Types of Z-Transform

Unilateral Z-Transform
Bilateral Z-Transform

Mathematically, if x(n) is a discrete-time signal or sequence, then its bilateral or two-sided Z-transform is defined as –

Also, the unilateral or one-sided z-transform is defined as –

Advantages and Disadvantages of Z-Transform:

Advantages:

The Z-transform can be used to find the transfer function of a discrete-time system. This makes it a valuable tool in designing digital filters and analyzing their frequency response.
The Z-transform provides a powerful way to analyze discrete-time signals in the frequency domain. By applying the Z-transform to a signal, its frequency content can be analyzed and manipulated using algebraic techniques.
The Z-transform has many useful properties that can be used to simplify calculations and derive closed-form solutions for system analysis and design.

Disadvantages:

The Z-transform is only defined for causal, stable signals. This means that signals that are not causal or unstable cannot be analyzed using the Z-transform.
The Z-transform can be difficult to compute and interpret, especially for complex signals. It requires knowledge of complex analysis and other advanced mathematical concepts.
The Z-transform can be computationally expensive, especially for large signals. This can make it impractical for real-time applications or applications with limited computational resources.

5. Bilinear Transform:

In digital signal processing and discrete-time control theory, the bilinear transform is used to convert continuous-time system representations to discrete-time and vice versa. This technique is also known as Tustin’s approach after Arnold Tustin.

The bilinear transform is the result of a numerical integration of the analog transfer function into the digital domain. We can define the bilinear transform as:

The bilinear transform can be used to produce a piecewise constant magnitude response that approximates the magnitude response of an equivalent analog filter.

Advantages of bilinear transformation method

a. It is a one-to-one mapping.

b. No aliasing impact is present.

c. The steady digital filter is created by converting the stable analog filter.

d. The ability to alter any sort of filter is unrestricted.

e. From the s-domain to the Z-domain, there is a one-to-one transformation.

Disadvantages of bi-linear transformation method

a. The mapping is nonlinear in this method because of this frequency warping effect takes place.

6. Discrete Cosine Transform(DCT):

A finite sequence of data points is expressed using a discrete cosine transform (DCT) as the sum of cosine functions vibrating at various frequencies.

Use of DCT

It is used in the majority of digital media, including speech coding, digital radio, digital television, digital pictures (such as JPEG and HEIF), digital video (such as MPEG and H.26x), digital audio (such as Dolby Digital, MP3 and AAC), and digital audio (such as MP3 and AAC) (such as AAC-LD, Siren and Opus). DCTs are also crucial for a wide range of other scientific and engineering applications, including digital signal processing, telecommunications equipment, lowering network bandwidth consumption, and spectral approaches for partial differential equations’ numerical solution.

The DCT is the most widely used transformation technique in signal processing, and by far the most widely used linear transform in data compression.

The DCT has a strong “energy compaction” property, capable of achieving high quality at high data compression ratios. Discrete Cosine Transform (DCT) has been a crucial role for video and image compression since Chen and Pratt proposed image compression application based on DCT. The energy compaction property of DCT is highly efficient for the compression.

Advantages and Disadvantages of DCT

Therefore, the DCT has a number of characteristics that make it a useful tool for picture compression. Fast algorithms can be employed for computing, the transformation is orthogonal (the inverse is transposed and energy is retained), and the result for (near) constant matrices typically consists of a sizable number of (near) zero values.

The DCT has one significant drawback. While the output values are frequently real-valued, the input values from preprocessed 8 x 8 blocks are integer-valued.

Thus, in order to decide on the values in each DCT block and provide output that is integer-valued, we require a quantization step. For more information, please refer to the Quantization subsection.

7. HAAR Transform:

The Haar wavelet in mathematics is a collection of downscaled “square-shaped” functions that collectively make up a wavelet family or basis. In that it enables the representation of a target function over an interval in terms of an orthonormal basis, wavelet analysis is analogous to Fourier analysis. The Haar sequence is frequently used as a teaching example and is widely acknowledged as the first wavelet basis.

Uses

It is beneficial in electrical and computer engineering applications like signal and image compression because it offers a straightforward and computationally effective method for analysing a signal’s local characteristics. The Haar matrix is the source of the Haar transform.