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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-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]...

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 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...

Continuous -time Fourier Transform

Continuous -time Fourier Transform

The Fourier series is instrumental in representing periodic functions, offering a powerful method to decompose such functions into a sum of sinusoids. This technique, however, necessitates modification when applied to nonperiodic functions. Consider a pulse-train waveform consisting of a series of rectangular pulses. When these pulses have a finite period, they can be accurately represented by a Fourier series. Yet, as the period approaches infinity, resulting in a single, isolated pulse, the...

Properties of DTFT II

Properties of DTFT II

In the study of discrete-time signal processing, understanding the properties of the Discrete-Time Fourier Transform (DTFT) is crucial for analyzing and manipulating signals in the frequency domain. Several properties, including frequency differentiation, convolution, accumulation, and Parseval's relation, offer powerful tools for signal analysis.
The frequency differentiation property is illustrated by considering a DTFT pair and differentiating both sides with respect to ω. Multiplying by j...

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Fractional discrete Fourier transforms.

Z T Deng, H J Caulfield, M Schamschula

|November 3, 2009

Summary

Calculating fractional Fourier transforms optically is complex. This study introduces an efficient algorithm extending the discrete Fourier transform, reducing computational complexity to O(N(2)) for faster analysis.

Area of Science:

Optics and Photonics
Signal Processing
Computational Mathematics

Background:

Direct calculation of fractional Fourier transforms (FrFTs) from optical implementation expressions is computationally intensive.
Existing methods for FrFTs can be laborious and time-consuming, hindering practical applications.

Purpose of the Study:

To develop a computationally efficient method for calculating fractional Fourier transforms.
To extend the discrete Fourier transform (DFT) for handling fractional orders.
To provide a general algorithm for computing the fractional discrete Fourier transform (FDFT) matrix.

Main Methods:

Definition of a system for computing the fractional discrete Fourier transform.
Development of a general method to compute the fractional discrete Fourier transform matrix.

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Numerical validation of the proposed algorithm.

Main Results:

The proposed extension of the discrete Fourier transform achieves a computational complexity of O(N(2)).
A general algorithm for computing the fractional discrete Fourier transform matrix has been successfully derived.
Numerical simulations confirmed the accuracy and efficiency of the developed algorithm.

Conclusions:

The developed algorithm offers a significant improvement in computational efficiency for fractional Fourier transforms.
This method provides a practical approach for computing FrFTs, overcoming the limitations of direct optical implementation calculations.
The fractional discrete Fourier transform algorithm is validated and ready for application in relevant scientific and engineering fields.