Editing Digital signal processing (section)

== Domains ==
DSP engineers usually study digital signals in one of the following domains: [[time domain]] (one-dimensional signals), spatial domain (multidimensional signals), [[frequency domain]], and [[wavelet]] domains. They choose the domain in which to process a signal by making an informed assumption (or by trying different possibilities) as to which domain best represents the essential characteristics of the signal and the processing to be applied to it. A sequence of samples from a measuring device produces a temporal or spatial domain representation, whereas a [[discrete Fourier transform]] produces the frequency domain representation.

=== Time and space domains ===
[[Time domain]] refers to the analysis of signals with respect to time. Similarly, space domain refers to the analysis of signals with respect to position, e.g., pixel location for the case of image processing.

The most common processing approach in the time or space domain is enhancement of the input signal through a method called filtering. [[Digital filter]]ing generally consists of some linear transformation of a number of surrounding samples around the current sample of the input or output signal. The surrounding samples may be identified with respect to time or space. The output of a linear digital filter to any given input may be calculated by [[convolution|convolving]] the input signal with an [[impulse response]].

=== Frequency domain ===
{{Main|Frequency domain}}

Signals are converted from time or space domain to the frequency domain usually through use of the [[Fourier transform]]. The Fourier transform converts the time or space information to a magnitude and phase component of each frequency. With some applications, how the phase varies with frequency can be a significant consideration. Where phase is unimportant, often the Fourier transform is converted to the power spectrum, which is the magnitude of each frequency component squared.

The most common purpose for analysis of signals in the frequency domain is analysis of signal properties. The engineer can study the spectrum to determine which frequencies are present in the input signal and which are missing. Frequency domain analysis is also called ''spectrum-'' or ''spectral analysis''.

Filtering, particularly in non-realtime work can also be achieved in the frequency domain, applying the filter and then converting back to the time domain. This can be an efficient implementation and can give essentially any filter response including excellent approximations to [[brickwall filter]]s.

There are some commonly used frequency domain transformations. For example, the [[cepstrum]] converts a signal to the frequency domain through Fourier transform, takes the logarithm, then applies another Fourier transform. This emphasizes the harmonic structure of the original spectrum.

===Z-plane analysis===
Digital filters come in both [[infinite impulse response]] (IIR) and [[finite impulse response]] (FIR) types. Whereas FIR filters are always stable, IIR filters have feedback loops that may become unstable and oscillate. The [[Z-transform]] provides a tool for analyzing stability issues of digital IIR filters. It is analogous to the [[Laplace transform]], which is used to design and analyze analog IIR filters.

===Autoregression analysis===
A signal is represented as linear combination of its previous samples. Coefficients of the combination are called autoregression coefficients. This method has higher frequency resolution and can process shorter signals compared to the Fourier transform.<ref name = "Marple">{{Cite book| publisher = Prentice Hall| isbn = 978-0-13-214149-9| last = Marple| first = S. Lawrence| title = Digital Spectral Analysis: With Applications| location = Englewood Cliffs, N.J| date = 1987-01-01}}</ref> [[Prony's method]] can be used to estimate phases, amplitudes, initial phases and decays of the components of signal.<ref name = "Ribeiro" /><ref name = "Marple" /> Components are assumed to be complex decaying exponents.<ref name = "Ribeiro">{{Cite journal| doi = 10.1006/mssp.2001.1399| issn = 0888-3270| volume = 17| issue = 3| pages = 533–549| last1 = Ribeiro| first1 = M.P.| last2 = Ewins| first2 = D.J.| last3 = Robb| first3 = D.A.| title = Non-stationary analysis and noise filtering using a technique extended from the original Prony method| journal = Mechanical Systems and Signal Processing| access-date = 2019-02-17| date = 2003-05-01| bibcode = 2003MSSP...17..533R| url = http://linkinghub.elsevier.com/retrieve/pii/S0888327001913998}}</ref><ref name = "Marple" />

===Time-frequency analysis===
A time-frequency representation of signal can capture both temporal evolution and frequency structure of analyzed signal. Temporal and frequency resolution are limited by the principle of uncertainty and the tradeoff is adjusted by the width of analysis window. Linear techniques such as [[Short-time Fourier transform]], [[wavelet transform]], [[filter bank]],<ref>{{Cite conference| last1 = So| first1 = Stephen| last2 = Paliwal| first2 = Kuldip K.| title = Improved noise-robustness in distributed speech recognition via perceptually-weighted vector quantisation of filterbank energies| book-title = Ninth European Conference on Speech Communication and Technology| date = 2005}}</ref> non-linear (e.g., [[Wigner–Ville transform]]<ref name = "Ribeiro" />) and [[autoregressive]] methods (e.g. segmented Prony method)<ref name = "Ribeiro" /><ref>{{Cite journal| doi = 10.1515/acgeo-2015-0012| issn = 1895-6572| volume = 63| issue = 3| pages = 652–678| last1 = Mitrofanov| first1 = Georgy| last2 = Priimenko| first2 = Viatcheslav| title = Prony Filtering of Seismic Data| journal = Acta Geophysica| date = 2015-06-01| bibcode = 2015AcGeo..63..652M| s2cid = 130300729| doi-access = free}}</ref><ref>{{Cite journal| doi = 10.20403/2078-0575-2020-2-55-67| issn = 2078-0575| issue = 2| pages = 55–67| last1 = Mitrofanov| first1 = Georgy| last2 = Smolin| first2 = S. N.| last3 = Orlov| first3 = Yu. A.| last4 = Bespechnyy| first4 = V. N.| title = Prony decomposition and filtering| journal = Geology and Mineral Resources of Siberia| access-date = 2020-09-08| date = 2020| s2cid = 226638723| url = http://www.jourgimss.ru/en/SitePages/catalog/2020/02/abstract/2020_2_55.aspx}}</ref> are used for representation of signal on the time-frequency plane. Non-linear and segmented Prony methods can provide higher resolution, but may produce undesirable artifacts. Time-frequency analysis is usually used for analysis of non-stationary signals. For example, methods of [[fundamental frequency]] estimation, such as RAPT and PEFAC<ref>{{Cite journal| doi = 10.1109/TASLP.2013.2295918| issn = 2329-9290| volume = 22| issue = 2| pages = 518–530| last1 = Gonzalez| first1 = Sira| last2 = Brookes| first2 = Mike| title = PEFAC - A Pitch Estimation Algorithm Robust to High Levels of Noise| journal = IEEE/ACM Transactions on Audio, Speech, and Language Processing| access-date = 2017-12-03| date = February 2014| s2cid = 13161793| url = https://ieeexplore.ieee.org/document/6701334}}</ref> are based on windowed spectral analysis.

===Wavelet===
[[File:Jpeg2000 2-level wavelet transform-lichtenstein.png|thumb|300px|An example of the 2D discrete wavelet transform that is used in [[JPEG2000]]. The original image is high-pass filtered, yielding the three large images, each describing local changes in brightness (details) in the original image. It is then low-pass filtered and downscaled, yielding an approximation image; this image is high-pass filtered to produce the three smaller detail images, and low-pass filtered to produce the final approximation image in the upper-left.]]
In [[numerical analysis]] and [[functional analysis]], a [[discrete wavelet transform]] is any [[wavelet transform]] for which the [[wavelet]]s are discretely sampled. As with other wavelet transforms, a key advantage it has over [[Fourier transform]]s is temporal resolution: it captures both frequency ''and'' location information. The accuracy of the joint time-frequency resolution is limited by the [[Uncertainty principle#Signal processing|uncertainty principle]] of time-frequency.

===Empirical mode decomposition===
Empirical mode decomposition is based on decomposition signal into [[intrinsic mode function]]s (IMFs). IMFs are quasi-harmonical oscillations that are extracted from the signal.<ref>{{Cite journal| doi = 10.1098/rspa.1998.0193| issn = 1364-5021| volume = 454| issue = 1971| pages = 903–995| last1 = Huang| first1 = N. E.| last2 = Shen| first2 = Z.| last3 = Long| first3 = S. R.| last4 = Wu| first4 = M. C.| last5 = Shih| first5 = H. H.| last6 = Zheng| first6 = Q.| last7 = Yen| first7 = N.-C.| last8 = Tung| first8 = C. C.| last9 = Liu| first9 = H. H.| title = The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis| journal = Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences| access-date = 2018-06-05| date = 1998-03-08| bibcode = 1998RSPSA.454..903H| s2cid = 1262186| url = http://rspa.royalsocietypublishing.org/cgi/doi/10.1098/rspa.1998.0193}}</ref>