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Related Concept Videos

Discrete-time Fourier transform01:26

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 Series01:20

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 Fourier Transform01:15

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...
Fast Fourier Transform01:10

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...
Properties of DTFT I01:24

Properties of DTFT I

In signal processing, Discrete-Time Fourier Transforms (DTFTs) play a critical role in analyzing discrete-time signals in the frequency domain. Various properties of the DTFTs such as linearity, time-shifting, frequency-shifting, time reversal, conjugation, and time scaling help understand and manipulate these signals for different applications.
The linearity property of DTFTs is fundamental. If two discrete-time signals are multiplied by constants a and b respectively, and then combined to...
Properties of DTFT II01:24

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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A Multimodal Wide-Field Fourier-Transform Raman Microscope
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Improved discrete fractional Fourier transform.

S C Pei, M H Yeh

    Optics Letters
    |July 15, 1997
    PubMed
    Summary

    A new discrete fractional Fourier transform (DFRFT) is proposed, closely matching the continuous version. This improved DFRFT maintains essential rotation properties, offering a more accurate digital representation of this mathematical operation.

    Area of Science:

    • Mathematics
    • Signal Processing
    • Applied Physics

    Background:

    • The continuous Fourier transform is a fundamental tool in signal processing.
    • Existing discrete fractional Fourier transforms (DFRFTs) do not accurately replicate the continuous transform's behavior.
    • A need exists for a DFRFT that preserves the properties of the continuous fractional Fourier transform.

    Purpose of the Study:

    • To introduce a novel discrete fractional Fourier transform (DFRFT).
    • To develop a DFRFT that accurately approximates the continuous fractional Fourier transform.
    • To ensure the proposed DFRFT retains the important rotation properties.

    Main Methods:

    • Development of a new algorithm for the discrete fractional Fourier transform.
    • Mathematical formulation and derivation of the proposed DFRFT.

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  • Comparative analysis with existing DFRFT methods and the continuous fractional Fourier transform.

    • Main Results:

    • The proposed DFRFT yields results highly similar to the continuous fractional Fourier transform.
    • The new DFRFT successfully preserves the inherent rotation properties.
    • Demonstrated superiority over previous DFRFT implementations in accuracy and property retention.

    Conclusions:

    • The novel DFRFT offers a more faithful digital implementation of the fractional Fourier transform.
    • This improved DFRFT is valuable for applications requiring accurate signal analysis and manipulation.
    • The retained rotation properties enhance its utility in various scientific and engineering fields.