Shift-and-scale-invariant pattern recognition using an elliptic coordinate-transformed phase-only filter

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...

Curvilinear motion characterizes the movement of a particle or object along a curved path, notably evident when envisioning a car navigating a winding road. If the car starts at point A, its position vector is established within a fixed frame of reference, where the ratio of the position vector to its magnitude signifies the unit vector pointing in the position vector's direction.
As the car advances, its position evolves over time. Quantifying the car's velocity involves computing the time...

Transformations modify the graphical representation of a function without changing its fundamental form. One common transformation is reflection, which flips the graph across a designated axis. When the vertical coordinates of all points are multiplied by the negative one, the entire graph is mirrored over the horizontal axis. This transformation reverses the vertical orientation of peaks and troughs, akin to signal inversion in electrical systems, where a waveform is flipped, but the timing of...

Understanding the working function of different types of controllers can be illustrated with practical analogies, such as adjusting a stereo's volume equalizer. Cranking up the bass involves a phase-lead controller, which functions as a high-pass filter, while increasing the treble uses a phase-lag controller, which acts as a low-pass filter. PD controllers, similar to high-pass filters, enhance the system's response to high-frequency components. PI controllers, akin to low-pass filters, manage...

Multiple integration is an important mathematical method used to calculate physical quantities distributed over a two-dimensional region, such as the total mass of an elliptical plate. In this process, the density function is evaluated throughout the entire region enclosed by the ellipse. The contributions from all points inside the boundary are then accumulated to determine the total mass.When integration is performed directly in rectangular coordinates, the elliptical boundary produces limits...

In signal processing, Discrete-Time Fourier Transforms (DTFTs) play a critical role in analyzing discrete-time signals in the frequency domain. Various properties of the DTFTs such as linearity, time-shifting, frequency-shifting, time reversal, conjugation, and time scaling help understand and manipulate these signals for different applications.
The linearity property of DTFTs is fundamental. If two discrete-time signals are multiplied by constants a and b respectively, and then combined to...