e-book Fourier Analysis on Finite Groups with Applications in Signal Processing and System Design

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Additionally, consideration is given to the polynomial expressions and decision diagrams defined in terms of Fourier transform on finite non-Abelian groups. A solid foundation of this complex topic is provided by beginning with a review of signals and their mathematical models and Fourier analysis. Next, the book examines recent achievements and discoveries in:. Among the highlights is an in-depth coverage of applications of abstract harmonic analysis on finite non-Abelian groups in compact representations of discrete functions and related tasks in signal processing and system design, including logic design.

All chapters are self-contained, each with a list of references to facilitate the development of specialized courses or self-study. With nearly illustrative figures and fifty tables, this is an excellent textbook for graduate-level students and researchers in signal processing, logic design, and system theory-as well as the more general topics of computer science and applied mathematics. Request permission to reuse content from this site.

Undetected location. NO YES. When the real and imaginary parts of a complex function are decomposed into their even and odd parts , there are four components, denoted below by the subscripts RE, RO, IE, and IO. And there is a one-to-one mapping between the four components of a complex time function and the four components of its complex frequency transform : [4]. The Fourier variants can also be generalized to Fourier transforms on arbitrary locally compact Abelian topological groups , which are studied in harmonic analysis ; there, the Fourier transform takes functions on a group to functions on the dual group.

This treatment also allows a general formulation of the convolution theorem , which relates Fourier transforms and convolutions. See also the Pontryagin duality for the generalized underpinnings of the Fourier transform. More specific, Fourier analysis can be done on cosets, [5] even discrete cosets. In signal processing terms, a function of time is a representation of a signal with perfect time resolution , but no frequency information, while the Fourier transform has perfect frequency resolution , but no time information. As alternatives to the Fourier transform, in time—frequency analysis , one uses time—frequency transforms to represent signals in a form that has some time information and some frequency information — by the uncertainty principle , there is a trade-off between these.

These can be generalizations of the Fourier transform, such as the short-time Fourier transform , the Gabor transform or fractional Fourier transform FRFT , or can use different functions to represent signals, as in wavelet transforms and chirplet transforms , with the wavelet analog of the continuous Fourier transform being the continuous wavelet transform.

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A primitive form of harmonic series dates back to ancient Babylonian mathematics , where they were used to compute ephemerides tables of astronomical positions. The classical Greek concepts of deferent and epicycle in the Ptolemaic system of astronomy were related to Fourier series see Deferent and epicycle: Mathematical formalism.

In modern times, variants of the discrete Fourier transform were used by Alexis Clairaut in to compute an orbit, [10] which has been described as the first formula for the DFT, [11] and in by Joseph Louis Lagrange , in computing the coefficients of a trigonometric series for a vibrating string. Historians are divided as to how much to credit Lagrange and others for the development of Fourier theory: Daniel Bernoulli and Leonhard Euler had introduced trigonometric representations of functions, [11] and Lagrange had given the Fourier series solution to the wave equation, [11] so Fourier's contribution was mainly the bold claim that an arbitrary function could be represented by a Fourier series.

The subsequent development of the field is known as harmonic analysis , and is also an early instance of representation theory. In signal processing , the Fourier transform often takes a time series or a function of continuous time , and maps it into a frequency spectrum. That is, it takes a function from the time domain into the frequency domain; it is a decomposition of a function into sinusoids of different frequencies; in the case of a Fourier series or discrete Fourier transform , the sinusoids are harmonics of the fundamental frequency of the function being analyzed.

When the function f is a function of time and represents a physical signal , the transform has a standard interpretation as the frequency spectrum of the signal. Fourier transforms are not limited to functions of time, and temporal frequencies. They can equally be applied to analyze spatial frequencies, and indeed for nearly any function domain. This justifies their use in such diverse branches as image processing , heat conduction , and automatic control.

From Wikipedia, the free encyclopedia. Fourier transforms Continuous Fourier transform Fourier series Discrete-time Fourier transform Discrete Fourier transform Discrete Fourier transform over a ring Fourier analysis Related transforms Branch of mathematics regarding periodic and continuous signals. Main article: Fourier transform. Main article: Fourier series. Main article: Discrete-time Fourier transform. Main article: Discrete Fourier transform. Further information: Time—frequency analysis.

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Criminalistics: An Introduction to Forensic Science. Theory and Application of Digital Signal Processing. Englewood Cliffs, NJ. Fourier Analysis on Coset Spaces. Rocky Mountain Journal of Mathematics. Indiscrete Thoughts. The Exact Sciences in Antiquity 2nd ed. Dover Publications. International Journal of Modern Physics E.

Fourier Analysis on Finite Groups with Applications in Signal Processing and System Design

Fourier Analysis on Finite Groups and Applications. Cambridge University Press. Basic Algebra. All we are left with is a sequence of numbers, and all signal processing manipulations, with their intended results, are independent of the way the discrete-time signal is obtained.

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The power and the beauty of digital signal processing lies in part with its invariance with respect to the underlying physical reality. This is in stark contrast with the world of analog circuits and systems, which have to be realized in a version specific to the physical nature of the input signals.

The following sequences are fundamental building blocks for the theory of signal processing. Unit Step. The unit step can be obtained via a discrete-time integration of the impulse see eq. Exponential Decay. The exponential decay is, as we will see, the free response of a discrete-time first order recursive filter. Complex Exponential. Special cases of the complex exponential are the real-valued discrete-time sinusoidal oscillations:.

Fourier Analysis On Finite Groups With Applications In Signal Processing And System Design

This can be expressed as an equation:. A sequence x [ n ], shifted by an integer k is simply:. The delay operator can be indicated by the following notation:. The sum of two sequences x [ n ] and w [ n ] is their term-by-term sum:.

Fourier Series Representation of Signal

Please note that sum and scaling are linear operators. The product of two sequences x [ n ] and w [ n ] is their term-by-term product.


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The discrete-time equivalent of integration is expressed by the following running sum:. Intuitively, integration computes a non-normalized running average of the discrete-time signal. A discrete-time approximation to differentiation is the first-order difference: 3. The signal reproducing formula is a simple application of the basic signal and signal properties that we have just seen and it states that.


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Any signal can be expressed as a linear combination of suitably weighed and shifted impulses. In this case, the weights are nothing but the signal values themselves. We define the energy of a discrete-time signal as. Obviously, the energy is finite only if the above sum converges, i. A signal with this property is sometimes referred to as a finite- energy signal. For a simple example of the converse, note that a periodic signal which is not identically zero is not square-summable. We define the power of a signal as the usual ratio of energy over time, taking the limit over the number of samples considered:.

Clearly, signals whose energy is finite, have zero total power i. Exponential sequences which are not decaying i. Note, however, that many signals whose energy is infinite do have finite power and, in particular, periodic signals such as sinusoids and combinations thereof. Due to their periodic nature, however, the above limit is undetermined; we therefore define their power to be simply the average energy over a period. Of course, in the practice of signal processing, it is impossible to deal with infinite quantities of data: for a processing algorithm to execute in a finite amount of time and to use a finite amount of storage, the input must be of finite length; even for algorithms that operate on the fly, i.

When a discrete-time signal admits no closed-form representation, as is basically always the case with real-world signals, its finite time support arises naturally because of the finite time spent recording the signal: every piece of music has a beginning and an end, and so did every phone conversation. In the case of the sequence representing the Dow Jones index, for instance, we basically cheated since the index did not even exist for years before , and its value tomorrow is certainly not known — so that the signal is not really a sequence, although it can be arbitrarily extended to one.

More importantly and more often , the finiteness of a discrete-time signal is explicitly imposed by design since we are interested in concentrating our processing efforts on a small portion of an otherwise longer signal; in a speech recognition system, for instance, the practice is to cut up a speech signal into small segments and try to identify the phonemes associated to each one of them. By describing one period graphically or otherwise , we are, in fact, providing a full description of the sequence.

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The complete taxonomy of the discrete-time signals used in the book is the subject of the next Sections ans is summarized in Table 2. To introduce a point that will reappear throughout the book, a finite-length signal of length N is entirely equivalent to a vector in C N. This equivalence is of immense import since all the tools of linear algebra become readily available for describing and manipulating finite-length signals. We can represent an N -point finite-length signal using the standard vector notation.


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Note the transpose operator, which declares x as a column vector; this is the customary practice in the case of complex-valued vectors. Alternatively, we can and often will use a notation that mimics the one used for proper sequences:. Here we must remember that, although we use the notation x [ n ], x [ n ] is not defined for values outside its support, i. Note that we can always obtain a finite-length signal from an infinite sequence by simply dropping the sequence values outside the indices of interest. Vector and sequence notations are equivalent and will be used interchangeably according to convenience; in general, the vector notation is useful when we want to stress the algorithmic or geometric nature of certain signal processing operations.