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Design Example

Design Example

The innovation of touch-tone telephony revolutionized the telecommunications industry by replacing the traditional rotary dial with a dual-tone multi-frequency (DTMF) signaling system. This system uses a matrix-style keypad with buttons arranged in four rows and three columns, creating 12 distinct signals each assigned to a pair of frequencies. Each button press results in a simultaneous generation of two sinusoidal tones – one from a low-frequency group (697 to 941 Hz) and one from a...

Active Filters

Active Filters

Active filters are electronic circuits that use operational amplifiers (op-amps), resistors, and capacitors to filter out unwanted frequency components from a signal. A first-order low-pass active filter is designed to pass signals with a frequency lower than a certain cutoff frequency and attenuate frequencies higher than that cutoff frequency. The transfer function for a first-order low-pass active filter is:

Time and frequency -Domain Interpretation of Phase-lead Control

Time and frequency -Domain Interpretation of Phase-lead Control

Phase-lead controllers are commonly used in various control systems to enhance response speed and stability. Adjusting the brightness on a television screen offers a practical example of phase-lead control. When contrast is enhanced, a phase-lead controller is employed. Mathematically, phase-lead control is identified when the first parameter is smaller than the second.
The design of phase-lead control involves the strategic placement of poles and zeros to balance steady-state error and system...

Geometry of Hyperbolas

Geometry of Hyperbolas

A hyperbola consists of all points where the absolute difference of distances to two fixed points, called foci, remains constant. The standard equation isEach branch extends infinitely and approaches two asymptotes, which guide the curve’s behavior. The parameters a and b define key features: a measures the distance from the center to each vertex along the transverse axis, while b influences the slopes of the asymptotes. The asymptotes have equationsA rectangle centered at the origin with...

Time and frequency -Domain Interpretation of Phase-lag Control

Time and frequency -Domain Interpretation of Phase-lag Control

Phase-lag controllers are widely used in control systems to improve stability and reduce steady-state errors. A dimmer switch controlling the brightness of a light bulb serves as a practical example of phase-lag control, gradually adjusting the bulb's brightness. Mathematically, phase-lag control or low-pass filtering is represented when the factor 'a' is less than 1.
Phase-lag controllers do not place a pole at zero, but instead influence the steady-state error by amplifying any finite,...

Transfer function and Bode Plots-II

Transfer function and Bode Plots-II

In the standard form, the transfer function is shown in constant gain, poles/zeros at origin, simple poles/zeros, and quadratic poles/zeros; each contributing uniquely to the system's overall response. The term represents the magnitude of the simple zero:

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Related Experiment Video

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Shaping the Amplitude and Phase of Laser Beams by Using a Phase-only Spatial Light Modulator

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Geometric approach for designing optical binary amplitude and binary phase-only filters.

M M Matalgah, J Knopp, L Eifler

|February 28, 2008

Summary

A new geometric solution designs optimal binary amplitude filters (OBAF) and optimal binary phase-only filters (OBPOF). This method maximizes correlation plane field strength, offering exact solutions for filter design and error bounding.

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Last Updated: Jul 7, 2026

Shaping the Amplitude and Phase of Laser Beams by Using a Phase-only Spatial Light Modulator

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Published on: January 28, 2019

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The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry

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Published on: August 12, 2013

Area of Science:

Optics
Image Processing
Computer Vision

Background:

Optimal filters are crucial for pattern recognition and image analysis.
While analytic solutions exist for real optimal filters, optimal binary phase-only filters lack a direct analytic design method.
Existing methods for optimal binary phase-only filters often rely on iterative or approximate algorithms like binning.

Purpose of the Study:

To establish a novel geometric solution for designing optimal binary amplitude filters (OBAF) and optimal binary phase-only filters (OBPOF).
To provide an exact and efficient method for filter design applicable to any object.
To offer insights into the filter design process and error analysis of related algorithms.

Main Methods:

Developing a geometric approach based on constructing a unique convex polygon from ordered phasors of the object's Fourier transform.
Utilizing the maximum distance across the convex polygon to partition phasors for OBAF and OBPOF design.
Applying the geometric solution to derive exact filter designs and analyze the binning process.

Main Results:

An exact geometric solution for designing OBAF and OBPOF is established.
The method maximizes field strength at the correlation plane's origin.
The convex polygon construction provides qualitative insights into design criticality and bounds errors in binning algorithms.

Conclusions:

The geometric solution offers a precise and insightful method for designing OBAF and OBPOF.
This approach enhances understanding of filter design principles and improves existing algorithms.
Demonstrated applications in computer simulations and fingerprint identification highlight the practical utility of the method.