Integral transformIn mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed function can generally be mapped back to the original function space using the inverse transform. An integral transform is any transform of the following form: The input of this transform is a function , and the output is another function .
Z-transformIn mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation. It can be considered as a discrete-time equivalent of the Laplace transform (s-domain). This similarity is explored in the theory of time-scale calculus. Whereas the continuous-time Fourier transform is evaluated on the Laplace s-domain's imaginary line, the discrete-time Fourier transform is evaluated over the unit circle of the z-domain.
Cutoff frequencyIn physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced (attenuated or reflected) rather than passing through. Typically in electronic systems such as filters and communication channels, cutoff frequency applies to an edge in a lowpass, highpass, bandpass, or band-stop characteristic – a frequency characterizing a boundary between a passband and a stopband.
Linear filterLinear filters process time-varying input signals to produce output signals, subject to the constraint of linearity. In most cases these linear filters are also time invariant (or shift invariant) in which case they can be analyzed exactly using LTI ("linear time-invariant") system theory revealing their transfer functions in the frequency domain and their impulse responses in the time domain. Real-time implementations of such linear signal processing filters in the time domain are inevitably causal, an additional constraint on their transfer functions.
Harmonic analysisHarmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on the real line, or by Fourier series for periodic functions. Generalizing these transforms to other domains is generally called Fourier analysis, although the term is sometimes used interchangeably with harmonic analysis.
LinearityIn mathematics, the term linear is used in two distinct senses for two different properties: linearity of a function (or mapping ); linearity of a polynomial. An example of a linear function is the function defined by that maps the real line to a line in the Euclidean plane R2 that passes through the origin. An example of a linear polynomial in the variables and is Linearity of a mapping is closely related to proportionality. Examples in physics include the linear relationship of voltage and current in an electrical conductor (Ohm's law), and the relationship of mass and weight.
Negative frequencyIn mathematics, signed frequency (negative and positive frequency) expands upon the concept of frequency, from just an absolute value representing how often some repeating event occurs, to also have a positive or negative sign representing one of two opposing orientations for occurrences of those events. The following examples help illustrate the concept: For a rotating object, the absolute value of its frequency of rotation indicates how many rotations the object completes per unit of time, while the sign could indicate whether it is rotating clockwise or counterclockwise.
Prototype filterPrototype filters are electronic filter designs that are used as a template to produce a modified filter design for a particular application. They are an example of a nondimensionalised design from which the desired filter can be scaled or transformed. They are most often seen in regard to electronic filters and especially linear analogue passive filters. However, in principle, the method can be applied to any kind of linear filter or signal processing, including mechanical, acoustic and optical filters.
Butterworth filterThe Butterworth filter is a type of signal processing filter designed to have a frequency response that is as flat as possible in the passband. It is also referred to as a maximally flat magnitude filter. It was first described in 1930 by the British engineer and physicist Stephen Butterworth in his paper entitled "On the Theory of Filter Amplifiers". Butterworth had a reputation for solving very complex mathematical problems thought to be 'impossible'. At the time, filter design required a considerable amount of designer experience due to limitations of the .
Step responseThe step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the outputs of a general system when its inputs change from zero to one in a very short time. The concept can be extended to the abstract mathematical notion of a dynamical system using an evolution parameter.
Band-pass filterA band-pass filter or bandpass filter (BPF) is a device that passes frequencies within a certain range and rejects (attenuates) frequencies outside that range. In electronics and signal processing, a filter is usually a two-port circuit or device which removes frequency components of a signal (an alternating voltage or current). A band-pass filter allows through components in a specified band of frequencies, called its passband but blocks components with frequencies above or below this band.
Low-pass filterA low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter design. The filter is sometimes called a high-cut filter, or treble-cut filter in audio applications. A low-pass filter is the complement of a high-pass filter. In optics, high-pass and low-pass may have different meanings, depending on whether referring to the frequency or wavelength of light, since these variables are inversely related.
