Related Experiment Video
Updated: Jul 7, 2026

13:44
Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
Published on: August 30, 2013
Image-scaling problem in the optical fractional Fourier transform
Applied Optics
|August 10, 1997
Summary
This study highlights the importance of scale factors in optical fractional Fourier transforms. Achieving exact and cascadable transforms requires specific input plane scaling and no quadratic phase at the spectrum plane.
Area of Science:
- Optics and Photonics
- Signal Processing
Background:
- The optical fractional Fourier transform (OFFT) is a powerful tool for signal processing.
- Practical implementation of OFFT faces challenges related to scale factors and cascading.
Purpose of the Study:
- To emphasize the significance of scale factors in OFFT.
- To detail requirements for exact and cascadable OFFT in practical applications.
Main Methods:
- Analysis of scale factor relationships in OFFT.
- Investigation of optical setups for OFFT implementation.
Main Results:
- Identified the critical role of scale factors for exact and cascadable OFFT.
- Determined that image scale must be the reciprocal of the input plane scale.
- Established the necessity of eliminating quadratic phase terms at the spectrum plane.
Conclusions:
- Proper control of scale factors is essential for practical OFFT.
- Achieving cascadable OFFT requires specific optical setup configurations.
Related Concept Videos
Properties of Fourier series II
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...
A function f(t) is...
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)log2N multiplications, offering a much faster performance.
The computational efficiency of the FFT becomes...
The computational efficiency of the FFT becomes...
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...
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...
If the sampling frequency is below the Nyquist rate, these replicas overlap, preventing the original signal...
Scaling
In designing and analyzing filters, resonant circuits, or circuit analysis at large, working with standard element values like 1 ohm, 1 henry, or 1 farad can be convenient before scaling these values to more realistic figures. This approach is widely utilized by not employing realistic element values in numerous examples and problems; it simplifies mastering circuit analysis through convenient component values. The complexity of calculations is thereby reduced, with the understanding that...
Convergence of Fourier Series
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...
The Gibbs phenomenon refers to the persistent oscillations and overshoots that occur near discontinuities...

