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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...

Doppler Effect - II

Doppler Effect - II

The Doppler effect has several practical, real-world applications. For instance, meteorologists use Doppler radars to interpret weather events based on the Doppler effect. Typically, a transmitter emits radio waves at a specific frequency toward the sky from a weather station. The radio waves bounce off the clouds and precipitation and travel back to the weather station. The radio frequency of the waves reflected back to the station appears to decrease if the clouds or precipitation are moving...

Doppler Effect - I

Doppler Effect - I

The Doppler effect and Doppler shift were named after the Austrian physicist and mathematician Christian Johann Doppler in 1842, who conducted experiments with both moving sources and moving observers. Consider an observer standing on a street corner, observing an ambulance with a siren sound passing by at a constant speed. The observer experiences two characteristic changes in the sound of the siren. Initially, the sound increases in loudness as the ambulance approaches and decreases in...

Aliasing

Aliasing

Accurate signal sampling and reconstruction are crucial in various signal-processing applications. A time-domain signal's spectrum can be revealed using its Fourier transform. When this signal is sampled at a specific frequency, it results in multiple scaled replicas of the original spectrum in the frequency domain. The spacing of these replicas is determined by the sampling frequency.
If the sampling frequency is below the Nyquist rate, these replicas overlap, preventing the original signal...

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.
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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.
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Updated: Jun 15, 2026

A Multimodal Wide-Field Fourier-Transform Raman Microscope

A Multimodal Wide-Field Fourier-Transform Raman Microscope

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High-accuracy Doppler-limited 10(6) samples Fourier transform spectroscopy.

|March 4, 2010

Summary

A new Fourier spectrometer offers enhanced resolution for meteorological and Doppler-limited measurements. This advanced instrument achieves sub-Doppler spectral analysis with a wide spectral range, improving scientific data acquisition.

Area of Science:

Spectroscopy
Instrumental Science
Atmospheric Science

Background:

Fourier spectrometers are crucial for high-resolution spectral analysis.
Previous generations had limitations in resolution and spectral range.
Meteorological and Doppler-limited measurements require advanced spectroscopic tools.

Purpose of the Study:

To report the contribution of a third-generation Fourier spectrometer to meteorological and Doppler-limited measurements.
To detail instrumental improvements and a novel triple-pass system.
To demonstrate the spectrometer's capability for sub-Doppler spectral analysis.

Main Methods:

Development and implementation of a third-generation Fourier spectrometer.
Incorporation of a tilt-compensated triple-pass system to enhance resolution.

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Characterization of the apparatus function width and spectral range.

Main Results:

The triple-pass system increases resolution by a factor of 3.
The apparatus function width is better than 10(-3) cm(-1) (27 MHz).
The spectrometer provides wide spectral range information with sub-Doppler resolution.

Conclusions:

The third-generation Fourier spectrometer significantly advances high-resolution measurements.
The instrument enables unprecedented analysis of sub-Doppler spectra.
This technology offers new possibilities for meteorological and atmospheric research.