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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...

Upsampling

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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There can be more than one mode in a data set if multiple values have the same highest frequency. For instance, suppose that the Statistics exam scores of 20 students are: 50; 53; 59; 59; 63; 63; 72; 72; 72; 72; 72; 76; 78; 81; 83; 84; 84; 84; 90; 93. Here, the mode is 72, as it occurs most frequently, five times.
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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.
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¹H NMR Signal Multiplicity: Splitting Patterns

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When protons A and X are coupled, their nuclear spin energy levels are slightly modified. This is because the energy required to excite proton A to a spin state parallel to proton X is slightly different from the energy required for it to become anti-parallel to spin X. Consequently, there are two possible excitation frequencies for A (A1 and A2), depending on the spin state of X, and vice versa. The mutual nature of coupling implies that the difference between frequencies A1 and A2, indicated...

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A sample refers to a smaller subset representative of a larger population. In analytical chemistry, studying or analyzing an entire population is often impractical or impossible. Therefore, samples are used to draw inferences and generalize the whole population. The sampling method selects individuals or items from a population to create a sample. Standard sampling methods include random, judgemental, systematic, stratified, and cluster sampling.
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Sub-sampled modal decomposition in few-mode fibers.

Kyuhong Choi, Changsu Jun

|October 7, 2021

Summary

Sub-sampling beam images significantly speeds up modal decomposition, reducing calculation time by 100x. This data-efficient method maintains high accuracy for analyzing beam characteristics.

Area of Science:

Optical physics
Computational optics

Background:

Modal decomposition retrieves detailed beam information from its profile.
Numerical modal decomposition offers a simple method using only a measured profile and algorithm.
Current methods often require full pixel data, limiting efficiency.

Purpose of the Study:

To improve modal decomposition efficiency through data-efficient sub-sampling of beam images.
To investigate the impact of sub-sampling parameters (window size, pixel count, algorithm) on performance.
To demonstrate a faster and accurate modal decomposition technique.

Main Methods:

Developed a data-efficient sub-sampling strategy for beam images.
Investigated various sub-sampling parameters and algorithms, including a modified stochastic parallel gradient descent.

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Performed experiments on 3-mode and 6-mode beams with reduced pixel data.

Main Results:

Achieved a ~100x reduction in calculation time compared to full pixel modal decomposition.
Maintained a low error function level of approximately 10^-3.
Validated the effectiveness of sub-sampling for both 3-mode and 6-mode beams.

Conclusions:

Sub-sampling beam images is a highly effective method for accelerating modal decomposition.
This technique offers significant speed improvements without compromising accuracy.
The approach is adaptable to various numerical and AI algorithms for real-time beam analysis and control.