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Properties of Fourier Transform I

Properties of Fourier Transform I

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...

Properties of Fourier Transform II

Properties of Fourier Transform II

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...

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...

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...

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...

Parseval's Theorem for Fourier transform

Parseval's Theorem for Fourier transform

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.
To understand Parseval's theorem, it is essential to first comprehend how signal energy is typically calculated. When considering a...

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Updated: Jan 25, 2026

Proton Transfer and Protein Conformation Dynamics in Photosensitive Proteins by Time-resolved Step-scan Fourier-transform Infrared Spectroscopy

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Published on: June 27, 2014

Broadband static Fourier transform mid-infrared spectrometer.

Michael H Köhler, Stefan S Naßl, Patrick Kienle

|May 3, 2019

Summary

Static Fourier transform infrared spectrometers (sFTS) offer a cost-effective solution for moderate spectral resolution needs. This paper introduces a novel sFTS design achieving broadband mid-infrared operation with high light throughput.

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Area of Science:

Spectroscopy
Optical Engineering
Instrument Design

Background:

Static Fourier transform infrared spectrometers (sFTS) provide a cost-effective alternative to scanning instruments for applications requiring moderate spectral resolution.
Traditional designs may face limitations in dispersion and operational range.

Purpose of the Study:

To present a novel static Fourier transform infrared spectrometer (sFTS) design.
To demonstrate its capability for broadband mid-infrared spectroscopy using standard optical components and an uncooled microbolometer array.

Main Methods:

Development of an sFTS utilizing a single-mirror interferometer with concave mirrors to minimize dispersion.
Integration of standard optical components and an uncooled microbolometer array.
Broadband operation from 2800 cm⁻¹ to 600 cm⁻¹ at 12 cm⁻¹ spectral resolution.

Main Results:

The proposed sFTS design achieves broadband operation in the mid-infrared range.
Minimized dispersion effects due to the use of concave mirrors.
High light throughput and potential for enhanced temperature stability.

Conclusions:

The presented sFTS design offers a cost-effective and capable solution for mid-infrared spectroscopy.
Experimental results validate the proof of principle for this novel instrument.
The design's features enable broadband operation with minimized dispersion.