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Related Concept Videos

Trigonometric Fourier series01:17

Trigonometric Fourier series

820
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...
820
Convergence of Fourier Series01:21

Convergence of Fourier Series

429
The Fourier series is a powerful mathematical tool for representing periodic signals as an infinite sum of complex exponentials. In practice, this infinite series is truncated to a finite number of terms, yielding a partial sum. This truncation makes the approximation of the signal feasible but introduces certain challenges, particularly near discontinuities, known as the Gibbs phenomenon.
The Gibbs phenomenon refers to the persistent oscillations and overshoots that occur near discontinuities...
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Fast Fourier Transform01:10

Fast Fourier Transform

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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...
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Exponential Fourier series01:24

Exponential Fourier series

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In audio signal processing, the exponential Fourier series plays a crucial role in sound synthesis, allowing complex sounds to be broken down into simpler sinusoidal components. This decomposition process is fundamental in analyzing and reconstructing musical notes and other audio signals. The exponential Fourier series expresses periodic signals as the sum of complex exponentials at both positive and negative harmonic frequencies, providing a powerful tool for signal analysis.
Euler's identity...
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Properties of Fourier series I01:20

Properties of Fourier series I

772
The Fourier series is a powerful tool in signal processing and communications, allowing periodic signals to be expressed as sums of sine and cosine functions. A foundational property of the Fourier series is linearity. If we consider two periodic signals, their linear combination results in a new signal whose Fourier coefficients are simply the corresponding linear combinations of the original signals' coefficients. This property is crucial in applications like frequency modulation (FM) radio,...
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Properties of Fourier series II01:21

Properties of Fourier series II

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Time scaling of signals is a crucial concept in signal processing that affects the Fourier series representation without altering its coefficients. The process modifies the fundamental frequency, thereby changing how the series represents the signal over time. This principle is essential in various applications, including audio and image processing, where signal manipulation is frequent. Understanding function symmetries is fundamental to simplifying the Fourier series.
A function f(t) is...
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Related Experiment Video

Updated: Feb 10, 2026

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High-speed Fourier ptychographic microscopy based on programmable annular illuminations.

Jiasong Sun1,2,3, Chao Zuo4,5,6, Jialin Zhang1,2,3

  • 1School of Electronic and Optical Engineering, Nanjing University of Science and Technology, No. 200 Xiaolingwei Street, Nanjing, Jiangsu Province, 210094, China.

Scientific Reports
|May 18, 2018
PubMed
Summary

High-speed quantitative phase imaging (QPI) using annular illumination Fourier ptychographic microscopy (FPM) enables rapid, label-free cell analysis. This technique achieves video-rate imaging for live cell studies, overcoming previous throughput limitations.

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Area of Science:

  • Biophotonics
  • Cellular Imaging
  • Microscopy

Background:

  • Quantitative phase imaging (QPI) is crucial for label-free cell analysis, but conventional Fourier ptychographic microscopy (FPM) suffers from low temporal throughput due to large data requirements.
  • Understanding the theoretical basis and optimal illumination for FPM is essential for improving phase imaging accuracy.

Purpose of the Study:

  • To develop a high-speed FPM technique for rapid, high-content cellular phenotype characterization.
  • To optimize illumination schemes for accurate phase recovery in FPM.

Main Methods:

  • Developed a high-speed FPM technique utilizing programmable annular illuminations (AIFPM).
  • Analyzed the optical transfer function (OTF) to determine optimal LED placement for phase information recovery.
  • Acquired data using only 4 low-resolution images with tilted illuminations.

Main Results:

  • Achieved high-speed imaging of Hela cells undergoing mitosis and apoptosis at 25 Hz.
  • Obtained a full-pitch resolution of 655 nm with an effective NA of 0.8.
  • Covered a wide field-of-view (1.77 mm²) with a space-bandwidth-time product of 411 megapixels per second.

Conclusions:

  • Demonstrated the capability of AIFPM for high-speed, high-throughput imaging of live cells.
  • Achieved video-rate QPI performance across broad spatial and temporal scales.
  • Overcame limitations of conventional FPM for live cell imaging applications.