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# Multidimensional filter design and implementation

Many concepts in one–dimensional signal processing are similar to concepts in multidimensional signal processing. [1]

## Algorithm

In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems.

## Bounded set

In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size.

## Cepstrum

A cepstrum is the result of taking the inverse Fourier transform (IFT) of the logarithm of the estimated spectrum of a signal.

## Chebyshev polynomials

In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively.

## Communication

Communication (from Latin commūnicāre, meaning "to share") is the act of conveying intended meanings from one entity or group to another through the use of mutually understood signs and semiotic rules.

## Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers.

## Complexity

Complexity characterises the behaviour of a system or model whose components interact in multiple ways and follow local rules, meaning there is no reasonable higher instruction to define the various possible interactions.

## Convolution

In mathematics (and, in particular, functional analysis) convolution is a mathematical operation on two functions (f and g) to produce a third function, that is typically viewed as a modified version of one of the original functions, giving the integral of the pointwise multiplication of the two functions as a function of the amount that one of the original functions is translated.

## CT scan

A CT scan, also known as computed tomography scan, makes use of computer-processed combinations of many X-ray measurements taken from different angles to produce cross-sectional (tomographic) images (virtual "slices") of specific areas of a scanned object, allowing the user to see inside the object without cutting.

## Data compression

In signal processing, data compression, source coding, or bit-rate reduction involves encoding information using fewer bits than the original representation.

## Dependent and independent variables

In mathematical modeling, statistical modeling and experimental sciences, the values of dependent variables depend on the values of independent variables.

## Design

Design is the creation of a plan or convention for the construction of an object, system or measurable human interaction (as in architectural blueprints, engineering drawings, business processes, circuit diagrams, and sewing patterns).

## Digital image processing

In computer science, Digital image processing is the use of computer algorithms to perform image processing on digital images.

## Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

## Dirac delta function

In mathematics, the Dirac delta function (function) is a generalized function or distribution introduced by the physicist Paul Dirac.

## Discrete cosine transform

A discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequencies.

## Discrete Fourier transform

In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency.

## Discrete Hartley transform

A discrete Hartley transform (DHT) is a Fourier-related transform of discrete, periodic data similar to the discrete Fourier transform (DFT), with analogous applications in signal processing and related fields.

## Discrete time and continuous time

In mathematics and in particular mathematical dynamics, discrete time and continuous time are two alternative frameworks within which to model variables that evolve over time.

## Fast Fourier transform

A fast Fourier transform (FFT) is an algorithm that samples a signal over a period of time (or space) and divides it into its frequency components.

## Filter (signal processing)

In signal processing, a filter is a device or process that removes some unwanted components or features from a signal.

## Finite impulse response

In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of finite duration, because it settles to zero in finite time.

## Frequency

Frequency is the number of occurrences of a repeating event per unit of time.

## Function application

In mathematics, function application is the act of applying a function to an argument from its domain so as to obtain the corresponding value from its range.

## Impulse response

In signal processing, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse.

## Infinite impulse response

Infinite impulse response (IIR) is a property applying to many linear time-invariant systems.

## Inverse transform sampling

Inverse transform sampling (also known as inversion sampling, the inverse probability integral transform, the inverse transformation method, Smirnov transform, golden ruleAalto University, N. Hyvönen, Computational methods in inverse problems. Twelfth lecture https://noppa.tkk.fi/noppa/kurssi/mat-1.3626/luennot/Mat-1_3626_lecture12.pdf) is a basic method for pseudo-random number sampling, i.e. for generating sample numbers at random from any probability distribution given its cumulative distribution function.

## Linear difference equation

In mathematics and in particular dynamical systems, a linear difference equation or linear recurrence relation equates 0 to a polynomial that is linear in the various iterates of a variable—that is, in the values of the elements of a sequence.

## Noise

Noise is unwanted sound judged to be unpleasant, loud or disruptive to hearing.

## Operation (mathematics)

In mathematics, an operation is a calculation from zero or more input values (called operands) to an output value.

## Parseval's theorem

In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform.

Radar is an object-detection system that uses radio waves to determine the range, angle, or velocity of objects.

## Seismic tomography

Seismic tomography is a technique for imaging the subsurface of the Earth with seismic waves produced by earthquakes or explosions.

## Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.

## Signal processing

Signal processing concerns the analysis, synthesis, and modification of signals, which are broadly defined as functions conveying "information about the behavior or attributes of some phenomenon", such as sound, images, and biological measurements.

## Signaling (telecommunications)

In telecommunication, signaling has the following meanings.

## Sonar

Sonar (originally an acronym for SOund Navigation And Ranging) is a technique that uses sound propagation (usually underwater, as in submarine navigation) to navigate, communicate with or detect objects on or under the surface of the water, such as other vessels.

## Spectral density estimation

In statistical signal processing, the goal of spectral density estimation (SDE) is to estimate the spectral density (also known as the power spectral density) of a random signal from a sequence of time samples of the signal.

## Spectrum

A spectrum (plural spectra or spectrums) is a condition that is not limited to a specific set of values but can vary, without steps, across a continuum.

## Standard array

In coding theory, a standard array (or Slepian array) is a q^ by q^ array that lists all elements of a particular \mathbb_q^n vector space.

## Thomas F. Quatieri

Thomas Francis Quatieri, Jr. is an American electrical engineer and Senior Technical Staff member at the MIT Lincoln Laboratory.

## Transfer function

In engineering, a transfer function (also known as system function or network function) of an electronic or control system component is a mathematical function giving the corresponding output value for each possible value of the input to the device.

## Value (mathematics)

In mathematics, value may refer to several, strongly related notions.

## Variable (mathematics)

In elementary mathematics, a variable is a symbol, commonly an alphabetic character, that represents a number, called the value of the variable, which is either arbitrary, not fully specified, or unknown.

## Vector (mathematics and physics)

When used without any further description, vector usually refers either to.

## Video processing

In electronics engineering, video processing is a particular case of signal processing, which often employs video filters and where the input and output signals are video files or video streams.

## Z-transform

In 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 representation.

## References

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