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Downsampling01:20

Downsampling

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When considering a sampled sequence with zero values between sampling instants, one can replace it by taking every N-th value of the sequence. At these integer multiples of N, the original and sampled sequences coincide. This process, known as decimation, involves extracting every N-th sample from a sequence, thereby creating a more efficient sequence.
The Fourier transform of the decimated sequence reveals a combination of scaled and shifted versions of the original spectrum. This...
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The probability of having two carbon-13 atoms next to each other is negligible because of the low natural abundance of carbon-13. Consequently, peak splitting due to carbon-carbon spin-spin coupling is not observed in spectra. However, protons up to three sigma bonds away split the carbon signal according to the n+1 rule, resulting in complicated spectra.
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Upsampling01:22

Upsampling

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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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Difference from Background: Limit of Detection01:05

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The limit of detection (LOD) is the smallest amount of analyte that can be distinguished from the background noise. The LOD value corresponds to the concentration at which the analyte signal is three times larger than the standard deviation of the blank signal. Below this value, the analyte signal cannot be differentiated from the background noise. It is calculated by dividing the calibration slope by 3 times the standard deviation of the blank signals.
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Spin systems where the difference in chemical shifts of the coupled nuclei is greater than ten times J are called first-order spin systems. These nuclei are weakly coupled, and their chemical shifts and coupling constant can generally be estimated from the well-separated signals in the spectrum.
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NMR Spectrometers: Resolution and Error Correction

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When magnetic nuclei in a sample achieve resonance and undergo relaxation, the signal detected in NMR is an approximately exponential free induction decay. Fourier transform of an exponential decay yields a Lorentzian peak in the frequency domain. Lorentzian peaks in an NMR spectrum are defined by their amplitude, full width at half maximum, and position, where the peak width is governed by the spin-spin relaxation time alone. In real experiments, however, the applied magnetic field is rendered...
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Noise reduction in spectroscopic detection with compressed sensing.

Junyan Sun1,2, Deran Zhang2, Ziqian Cheng2

  • 1Beijing Computational Science Research Center, Beijing 100193, China.

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|October 23, 2025
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Summary

Compressed sensing enhances temporal spectroscopy by reducing noise and measurements. This under-sampling technique, applied to pump-probe spectroscopy, offers a promising method for improved spectral data acquisition.

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Area of Science:

  • Spectroscopy
  • Signal Processing
  • Optics

Background:

  • Traditional spectroscopy uses uniform sampling, adhering to the Nyquist-Shannon theorem.
  • This theorem mandates high sampling rates, potentially increasing measurement time and noise.
  • Prior signal knowledge, like sparsity, can relax sampling constraints.

Purpose of the Study:

  • Investigate compressed sensing for temporal spectroscopic measurements.
  • Explore compressed sensing's potential to mitigate measurement and intrinsic noise.
  • Demonstrate noise reduction in single-shot pump-probe spectroscopy.

Main Methods:

  • Applied compressed sensing to single-shot pump-probe spectroscopic data.
  • Proposed an experimental scheme using a digital mirror device for temporal sampling.
  • Leveraged signal sparsity to relax traditional sampling rate requirements.

Main Results:

  • Demonstrated significant noise reduction in temporal spectroscopic measurements.
  • Validated the effectiveness of compressed sensing for under-sampling spectral data.
  • Showcased a feasible method for implementing compressed sensing in spectroscopy.

Conclusions:

  • Compressed sensing is effective for noise reduction in temporal spectroscopy.
  • This technique can decrease the number of required sample measurements.
  • The proposed digital mirror device scheme offers a practical approach for advanced spectroscopic sampling.