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Fast Fourier Transform

Fast Fourier Transform

The Fast Fourier Transform (FFT) is a computational algorithm designed to compute the Discrete Fourier Transform (DFT) efficiently. By breaking down the calculations into smaller, manageable sections, the FFT significantly reduces the computational complexity involved. Direct computation of an N-point DFT requires N2 complex multiplications, whereas the FFT algorithm needs only (N/2)log⁡2N multiplications, offering a much faster performance.
The computational efficiency of the FFT becomes...

Discrete-Time Fourier Series

Discrete-Time Fourier Series

The Discrete-Time Fourier Series (DTFS) is a fundamental concept in signal processing, serving as the discrete-time counterpart to the continuous-time Fourier series. It allows for the representation and analysis of discrete-time periodic signals in terms of their frequency components. Unlike its continuous counterpart, which utilizes integrals, the calculation of DTFS expansion coefficients involves summations due to the discrete nature of the signal.
For a discrete-time periodic signal x[n]...

Discrete-time Fourier transform

Discrete-time Fourier transform

The Discrete-Time Fourier Transform (DTFT) is an essential mathematical tool for analyzing discrete-time signals, converting them from the time domain to the frequency domain. This transformation allows for examining the frequency components of discrete signals, providing insights into their spectral characteristics. In the DTFT, the continuous integral used in the continuous-time Fourier transform is replaced by a summation to accommodate the discrete nature of the signal.
One of the notable...

Discrete Fourier Transform

Discrete Fourier Transform

The Discrete Fourier Transform (DFT) is a fundamental tool in signal processing, extending the discrete-time Fourier transform by evaluating discrete signals at uniformly spaced frequency intervals. This transformation converts a finite sequence of time-domain samples into frequency components, each representing complex sinusoids ordered by frequency. The DFT translates these sequences into the frequency domain, effectively indicating the magnitude and phase of each frequency component present...

Basic Discrete Time Signals

Basic Discrete Time Signals

The unit step sequence is defined as 1 for zero and positive values of the integer n. This sequence can be graphically displayed using a set of eight sample points, showing a step function starting from n=0 and remaining constant thereafter.
The unit impulse or sample sequence is mathematically expressed as zero for all n values except at n=0, where it is one. The unit impulse sequence, denoted by δ(n), is the first difference of the unit step sequence, while the unit step sequence u(n) is the...

Basic signals of Fourier Transform

Basic signals of Fourier Transform

The Fourier Transform is a pivotal mathematical tool in signal processing, enabling the transformation of time-domain signals into their frequency-domain representations. Among the numerous elements within this domain, certain functions like the sinc function, delta function, and exponential signals hold significant importance due to their unique properties and implications.
The sinc function, defined as sinc(x) = sin(πx)/(πx), is particularly notable for its symmetry and behavior at zero. It...

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Related Experiment Video

Updated: Jun 26, 2026

Blood Flow Imaging with Ultrafast Doppler

Blood Flow Imaging with Ultrafast Doppler

Published on: October 14, 2020

A fast discrete S-transform for biomedical signal processing.

Robert A Brown¹, Richard Frayne

¹Department of Electrical and Computer Engineering, University of Calgary and the Seaman Family MR Research Centre, Hotchkiss Brain Institute, Foothills Medical Centre, AB, Canada. brownr@ucalgary.ca

Annual International Conference of the IEEE Engineering in Medicine and Biology Society. IEEE Engineering in Medicine and Biology Society. Annual International Conference

|January 24, 2009

Summary

This summary is machine-generated.

The S-transform analyzes signal frequency and phase. A new fast S-transform reduces computational needs, making it more applicable for biomedical signal processing tasks.

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Area of Science:

Signal Processing
Biomedical Engineering
Image Analysis

Background:

The S-transform combines Fourier and wavelet transform benefits, offering precise frequency and local spectral analysis.
Its application in biomedical signal and image processing is promising but hindered by high computational cost.

Purpose of the Study:

Introduce a computationally efficient discrete implementation of the S-transform.
Overcome the limitations of the classic S-transform in medical applications.

Main Methods:

Development of a fast S-transform algorithm.
Discrete implementation focusing on reduced computational complexity.

Main Results:

The fast S-transform significantly decreases computational requirements compared to the classic S-transform.
Enables more practical application of S-transform analysis in medicine.

Conclusions:

The fast S-transform is a viable and efficient alternative for biomedical signal and image analysis.
Reduced computational load expands the potential use of S-transform in clinical settings.