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Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are a sequence of numbers that represent samples of a continuous variable in a domain such as time, space, or frequency. In digital electronics, a digital signal is represented as a pulse train, which is typically generated by the switching of a transistor.
A Fast Fourier Transform (FFT) is an algorithm that computes the Discrete Fourier Transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain to a representation in the frequency domain and vice versa. The DFT is obtained by decomposing a sequence of values into components of different frequencies. This operation is useful in many fields, but computing it directly from the definition is often too slow to be practical. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse factors. As a result, it manages to reduce the complexity of computing the DFT from , which arises if one simply applies the definition of DFT, to , where n is the data size. The difference in speed can be enormous, especially for long data sets where n may be in the thousands or millions. In the presence of round-off error, many FFT algorithms are much more accurate than evaluating the DFT definition directly or indirectly. There are many different FFT algorithms based on a wide range of published theories, from simple complex-number arithmetic to group theory and number theory.
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing signals, such as sound, images, potential fields, seismic signals, altimetry processing, and scientific measurements. Signal processing techniques are used to optimize transmissions, digital storage efficiency, correcting distorted signals, subjective video quality, and to also detect or pinpoint components of interest in a measured signal.
A discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequencies. The DCT, first proposed by Nasir Ahmed in 1972, is a widely used transformation technique in signal processing and data compression. It is used in most digital media, including digital images, digital video, digital audio, digital television, digital radio, and speech coding. DCTs are also important to numerous other applications in science and engineering, such as digital signal processing, telecommunication devices, reducing network bandwidth usage, and spectral methods for the numerical solution of partial differential equations.
The Kaiser window, also known as the Kaiser–Bessel window, was developed by James Kaiser at Bell Laboratories. It is a one-parameter family of window functions used in finite impulse response filter design and spectral analysis. The Kaiser window approximates the DPSS window which maximizes the energy concentration in the main lobe but which is difficult to compute.
In signal processing and statistics, a window function is a mathematical function that is zero-valued outside of some chosen interval. Typically, windows functions are symmetric around the middle of the interval, approach a maximum in the middle, and taper away from the middle. Mathematically, when another function or waveform/data-sequence is "multiplied" by a window function, the product is also zero-valued outside the interval: all that is left is the part where they overlap, the "view through the window". Equivalently, and in actual practice, the segment of data within the window is first isolated, and then only that data is multiplied by the window function values. Thus, tapering, not segmentation, is the main purpose of window functions.
In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time, as in time series. Put simply, a time-domain graph shows how a signal changes over time, whereas a frequency-domain graph shows how the signal is distributed within different frequency bands over a range of frequencies. A complex valued frequency-domain representation consists of both the magnitude and the phase of a set of sinusoids at the frequency components of the signal. Although it is common to refer to the magnitude portion as the frequency response of a signal, the phase portion is required to uniquely define the signal.
In mathematics the finite Fourier transform may refer to either
The Fourier transform of a function of time, s(t), is a complex-valued function of frequency, S(f), often referred to as a frequency spectrum. Any linear time-invariant operation on s(t) produces a new spectrum of the form H(f)•S(f), which changes the relative magnitudes and/or angles (phase) of the non-zero values of S(f). Any other type of operation creates new frequency components that may be referred to as spectral leakage in the broadest sense. Sampling, for instance, produces leakage, which we call aliases of the original spectral component. For Fourier transform purposes, sampling is modeled as a product between s(t) and a Dirac comb function. The spectrum of a product is the convolution between S(f) and another function, which inevitably creates the new frequency components. But the term 'leakage' usually refers to the effect of windowing, which is the product of s(t) with a different kind of function, the window function. Window functions happen to have finite duration, but that is not necessary to create leakage. Multiplication by a time-variant function is sufficient.
Stransform as a time–frequency distribution was developed in 1994 for analyzing geophysics data. In this way, the S transform is a generalization of the short-time Fourier transform (STFT), extending the continuous wavelet transform and overcoming some of its disadvantages. For one, modulation sinusoids are fixed with respect to the time axis; this localizes the scalable Gaussian window dilations and translations in S transform. Moreover, the S transform doesn't have a cross-term problem and yields a better signal clarity than Gabor transform. However, the S transform has its own disadvantages: the clarity is worse than Wigner distribution function and Cohen's class distribution function.
In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of discrete values.
Julius Ferdinand von Hann was an Austrian meteorologist. He is seen as a father of modern meteorology.
Thomas Shi-Tao Huang was a Chinese-born American computer scientist, electrical engineer, and writer. He was a researcher and professor emeritus at the University of Illinois at Urbana-Champaign (UIUC). Huang was one of the leading figures in computer vision, pattern recognition and human computer interaction.
The Hann function is named after the Austrian meteorologist Julius von Hann. It is a window function used to perform Hann smoothing. The function, with length and amplitude is given by:
Kamisetty Ramamohan Rao was an Indian-American electrical engineer. He was a professor of Electrical Engineering at the University of Texas at Arlington. Academically known as K. R. Rao, he is credited with the co-invention of discrete cosine transform (DCT), along with Nasir Ahmed and T. Natarajan due to their landmark publication, Discrete Cosine Transform.
A digital delay line is a discrete element in a digital filter, which allows a signal to be delayed by a number of samples. Delay lines are commonly used to delay audio signals feeding loudspeakers to compensate for the speed of sound in air, and to align video signals with accompanying audio, called audio-to-video synchronization. Delay lines may compensate for electronic processing latency so that multiple signals leave a device simultaneously despite having different pathways.
In digital signal processing (DSP), a normalized frequency is a ratio of a variable frequency and a constant frequency associated with a system. Some software applications require normalized inputs and produce normalized outputs, which can be re-scaled to physical units when necessary. Mathematical derivations are usually done in normalized units, relevant to a wide range of applications.
Nasir Ahmed is an Indian-American electrical engineer and computer scientist. He is Professor Emeritus of Electrical and Computer Engineering at University of New Mexico (UNM). He is best known for inventing the discrete cosine transform (DCT) in the early 1970s. The DCT is the most widely used data compression transformation, the basis for most digital media standards and commonly used in digital signal processing. He also described the discrete sine transform (DST), which is related to the DCT.
Georgios B. Giannakis is a Greek-American Computer Scientist, engineer and inventor. He has been an Endowed Chair Professor of Wireless Telecommunications, he was Director of the Digital Technology Center, and at present he is a McKnight Presidential Chair with the Department of Electrical and Computer Engineering at the University of Minnesota.
Maamar Bettayeb is a control theorist, educator and inventor. He is the author of publications on understanding the singular value decomposition and model order reduction. Bettayeb is also a promoter of scientific research.
References
- ↑ "Summer School 2009 Invited Speakers". University of Alberta HCDC Laboratory. Retrieved 2018-08-04.
- ↑ Harris, F. J. (January 1978). "On the use of windows for harmonic analysis with the discrete Fourier transform" (PDF). Proceedings of the IEEE. 66 (1): 51–83. Bibcode:1978IEEEP..66...51H. CiteSeerX 10.1.1.649.9880 . doi:10.1109/PROC.1978.10837. ISSN 0018-9219. S2CID 426548.
- ↑ "fredric j harris, Ph.D." Wireless Innovation Forum. 14 September 2016. Archived from the original on 2018-08-04. Retrieved 2018-08-04.
- ↑ "Society News". IEEE Signal Processing Magazine. 20 (2): 10–11. March 2003. Bibcode:2003ISPM...20...10.. doi:10.1109/MSP.2003.1184333. ISSN 1053-5888.
- ↑ "Johnsons Make Gift to Endow fred harris Chair in DSP". SDSU College of Engineering. San Diego State University. 2020-09-11. Archived from the original on 2021-06-13. Retrieved 2021-06-13.
