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Related Concept Videos

Aliasing01:18

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...
Reconstruction of Signal using Interpolation01:10

Reconstruction of Signal using Interpolation

Signal processing techniques are essential for accurately converting continuous signals to digital formats and vice versa. When a continuous signal is sampled with a period T, the resulting sampled signal exhibits replicas of the original spectrum in the frequency domain, spaced at intervals equal to the sampling frequency. To handle this sampled signal, a zero-order hold method can be applied, which creates a piecewise constant signal by retaining each sample's value until the next sampling...
Bandpass Sampling01:17

Bandpass Sampling

In signal processing, bandpass sampling is an effective technique for sampling signals that have most of their energy concentrated within a narrow frequency band. This type of signal is known as a bandpass signal. The key principle of bandpass sampling involves sampling the signal at a rate that is greater than twice the signal's bandwidth to prevent aliasing.
A bandpass signal has a spectrum with a lower frequency limit, denoted as ω1, and an upper frequency limit, denoted as ω2. The spectrum...
Sampling Theorem01:15

Sampling Theorem

In signal processing, the analysis of continuous-time signals, denoted as x(t), often involves sampling techniques to convert these signals into discrete-time signals. This process is essential for digital representation and manipulation. A critical component in sampling is the train of impulses, characterized by the sampling interval and the sampling frequency. The relationship between these parameters and the original signal's properties dictates the success of the sampling process.
Trigonometric Fourier series01:17

Trigonometric Fourier series

Fourier series is a foundational mathematical technique that decomposes periodic functions into an infinite series of sinusoidal harmonics. This method enables the representation of complex periodic signals as sums of simple sine and cosine functions, facilitating their analysis and interpretation in various fields, including signal processing, acoustics, and electrical engineering.
The trigonometric Fourier series specifically expresses a periodic function with a defined period T using sine...
Upsampling01:22

Upsampling

Managing signal sampling rates is essential in digital signal processing to maintain signal integrity. A decimated signal, characterized by a reduced frequency range due to its lower sampling rate, can be upsampled by inserting zeros between each sample. This upsampling process expands the original spectrum and introduces repeated spectral replicas at intervals dictated by the new Nyquist frequency. To refine this zero-inserted sequence, it is passed through a lowpass filter with a cutoff...

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Spectrum constructing with nonuniform samples using least-squares approximation by cosine polynomials.

Cong Feng1, Jingqiu Liang, Zhongzhu Liang

  • 1State Key Laboratory of Applied Optics, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China.

Applied Optics
|December 24, 2011
PubMed
Summary
This summary is machine-generated.

This study introduces a stable algorithm for constructing spectra from non-uniformly sampled interferograms using least-squares approximation. The method achieves a low spectrum-constructing error of 0.03% in Fourier transform spectroscopy.

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Area of Science:

  • Spectroscopy
  • Optical Engineering
  • Applied Mathematics

Background:

  • Fourier transform spectroscopy (FTS) typically requires uniform sampling of interferograms.
  • Non-uniform sampling can arise in static FTS systems due to mechanical imperfections or specific experimental designs.
  • Developing robust algorithms for spectral reconstruction from non-uniform data is crucial for improving FTS efficiency and accuracy.

Purpose of the Study:

  • To develop and validate a stable algorithm for spectral reconstruction from simulated non-uniformly sampled interferograms.
  • To assess the accuracy and performance of the proposed method in a step-mirror-based static Fourier transform spectrometer.

Main Methods:

  • Utilizing the least-squares approximation of cosine polynomials for spectral reconstruction.
  • Simulating non-uniform sampling of interferograms acquired by a static Fourier transform spectrometer with a step-mirror mechanism.
  • Performing numerical and experimental validation of the developed algorithm.

Main Results:

  • The least-squares approximation algorithm demonstrated stability in spectral reconstruction.
  • A low spectrum-constructing error of 0.03% was achieved using the proposed method.
  • Both numerical simulations and experimental data confirmed the algorithm's effectiveness.

Conclusions:

  • The least-squares approximation of cosine polynomials provides a stable and accurate method for spectral reconstruction from non-uniformly sampled interferograms.
  • This algorithm is suitable for application in static Fourier transform spectrometers, particularly those with step-mirror configurations.
  • The findings contribute to advancing spectral analysis techniques in situations with non-ideal sampling conditions.