Scope

1. What is a transform?
2. Why do we use transform?
3. Why are there different types of transform?
4. What kind of transform exists?

Transformation in Signal Processing

1. Intro

   Signal processing involves analyzing, modifying, and synthesizing signals to extract meaningful information or achieve specific goals.
   Signals can be in various forms such as audio, video, images, or data streams.
   

2. Laplace Transform

   The Laplace transform is a mathematical tool used to transform signals from the time domain to the complex frequency domain.
   It helps in analyzing linear time-invariant systems, solving differential equations, and understanding system behaviour in the frequency domain.
   

3. Fourier Transform

   The Fourier transform decomposes a signal into its constituent frequencies.
   It represents a signal in the frequency domain, showing how much of each frequency is present in the signal.
   This transformation is widely used in fields like audio processing, image processing, and communication systems.
   

4. Short-Time Fourier Transform (STFT) and Gabor-Transform

   The STFT is a technique used to analyze non-stationary signals by computing the Fourier transform over short, overlapping time segments.
   It provides information about the frequency content of a signal as it evolves over time.
   The Gabor transform is a specific implementation of the STFT using a Gaussian window function.
   

5. z-Transform

   The z-transform is used to analyze discrete-time systems and signals in the complex z-plane.
   It is analogous to the Laplace transform for continuous-time systems, providing insights into system dynamics, stability, and frequency response.
   

6. Discrete-time Fourier Transform (DTFT) and Discrete Fourier Transform (DFT)

   The DTFT and DFT are used to analyze discrete-time signals in the frequency domain.
   The DTFT represents a signal as a continuous function of frequency, while the DFT computes the spectrum of a signal using a finite number of discrete frequency samples.
   The DFT is widely used in practical applications due to its computational efficiency.
   

7. Linear Block Transform

   The Linear Block Transform is a general term used to describe various linear transformations applied to blocks of data.
   It encompasses techniques like wavelet transforms, cosine transforms, and other orthogonal transforms used for data compression, denoising, and feature extraction.