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Related Concept Videos

Properties of Fourier Transform I01:21

Properties of Fourier Transform I

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The application of Fourier Transform properties in radio broadcasting is multifaceted, enabling significant advancements in the way signals are transmitted and received. Key areas where these properties are utilized include simultaneous multi-channel transmission, audio clip speed adjustments, live broadcast delays for different time zones, audio frequency adjustments, and signal demodulation.
In radio broadcasting, multiple audio signals often need to be transmitted simultaneously. The Fourier...
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Properties of Fourier Transform II01:24

Properties of Fourier Transform II

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The Fourier Transform (FT) is an essential mathematical tool in signal processing, transforming a time-domain signal into its frequency-domain representation. This transformation elucidates the relationship between time and frequency domains through several properties, each revealing unique aspects of signal behavior.
The Frequency Shifting property of Fourier Transforms highlights that a shift in the frequency domain corresponds to a phase shift in the time domain. Mathematically, if x(t) has...
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Discrete Fourier Transform01:15

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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...
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Fast Fourier Transform01:10

Fast Fourier Transform

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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.
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Continuous -time Fourier Transform01:11

Continuous -time Fourier Transform

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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...
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Parseval's Theorem for Fourier transform01:15

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Parseval's theorem is a fundamental principle in signal processing that enables the calculation of a signal's energy in either the time domain or the frequency domain. This theorem is pivotal in demonstrating energy conservation between these two domains, ensuring that the computed energy value remains consistent regardless of the domain of analysis.
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A Multimodal Wide-Field Fourier-Transform Raman Microscope
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On-chip interrogator based on Fourier transform spectroscopy.

Fellipe Grillo Peternella, Thomas Esselink, Bas Dorsman

    Optics Express
    |June 6, 2019
    PubMed
    Summary
    This summary is machine-generated.

    This study introduces a new integrated Fourier transform (FT) spectrometer for interrogating photonic sensors. It achieves high precision by compensating for thermal drift and achieving a minimum modulation amplitude of 400 fm.

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    Area of Science:

    • Photonics
    • Spectroscopy
    • Optical Engineering

    Background:

    • Photonic sensors are crucial for various applications.
    • Accurate interrogation of photonic sensors requires high-resolution spectrometers.
    • Existing methods face challenges with thermal drift and resolution.

    Purpose of the Study:

    • To present the design and characterization of a novel integrated Fourier transform (FT) spectrometer.
    • To demonstrate its application in interrogating photonic sensors.
    • To achieve high precision and compensate for environmental factors.

    Main Methods:

    • Implementation of a planar spatial heterodyne spectrometer using Mach-Zehnder interferometers (MZIs).
    • Utilizing 3x3 multi-mode interferometers for complex Fourier coefficient retrieval.
    • Numerical solution of non-linear equations via Newton's method for data analysis.

    Main Results:

    • Demonstration of the first integrated FT spectrometer for photonic sensor interrogation.
    • Compensation of approximately 92% of thermal-induced phase drift using a reference sensor.
    • Achieved a minimum modulation amplitude of 400 fm, exceeding FT spectrometer resolution by two orders of magnitude.

    Conclusions:

    • The novel integrated FT spectrometer offers unprecedented precision for photonic sensor interrogation.
    • The developed method effectively compensates for thermal drift, enhancing measurement stability.
    • This technology paves the way for more sensitive and reliable photonic sensor systems.