Abstarct
In this chapter we see how group theory enters in an explicit manner in the study of symmetry, and in what way the connections between group theory and symmetry are relevant to the study of signals and systems. The ideas discussed here will serve as a background for the study of modern harmonic analysis on groups, and of its applications in signal representation and processing.
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Notes
- 1.
In the theory of 2–D and multidimensional digital filters, these symmetries are put to use in simplifying design methods. See Antoniou [12].
- 2.
These techniques, in the form that they are in use today, are largely the outcome of intense and closely knit activities in mathematics and mathematical physics begun in the twenties. See the collection of papers in MacKey [13] for a very illuminating historical account of their development and role. On the general subject of symmetry, see the classic by Weyl [19].
- 3.
The method of symmetrical components of power systems, the Bartlett’s bisection theorem for 2–port networks, and the method of characterizing differential amplifiers in terms of common and differential mode gains are some of the well known examples of this. The notion of characteristic or image impedance of a 2–port, which is central to the classical theory of image parameter filters, is also another instance of the use of symmetry.
- 4.
A significant early contribution in this class, and perhaps amongst the first ones, to point out the role of group theory in the study of invariance and symmetry in circuit problems, is that of Howitt [9]. Although this paper drew attention to several significant avenues of work, including one that points to the state space approach, it does not appear to have received the kind of attention it deserved at the time. Note, in particular, the following comments in his concluding remarks: “What in electric circuit theory corresponds to the principal or normal coordinates in dynamic theory? …in the study of an electrical network …, one continually encounters many seemingly unrelated branches of mathematics, such as (1) continued fractions, …, (5) group theory, (6) Fourier series and transforms …etc.. It seems almost as if something were there, inarticulately trying to make itself understood. But perhaps it must await a modern Euler.”
Another interesting early paper to independently make a strong case for the use of group theory in the analysis and classification of networks was that of Gaertner [6]. Through his treatment of 2–port networks and their cascade connections, he put forth the point that the group theoretic approach “justifies itself not only by the results of its derivations but also by adding to the understanding and insight and thus providing ideas and suggestions on how to handle certain problems.” Also see in this connection the classical work by Brillouin [1, Chapter 9].
- 5.
- 6.
- 7.
This is equivalent to saying that if we rotate it around its centre by multiples of 2π ∕ 3 radians, or reflect it along the perpendicular from a vertex to the opposite side, it coincides with itself.
- 8.
Recall that for a relational structure, the most commonly encountered situation is of the type in which the pertinent relations are given on the ground set or powers of it. In more general cases, the relations may be over not just over the ground set but on several other external sets in addition to it. For us here, \({\mathbb{R}}^{+}\) is such an external set for the relation \(\mathcal{R}\). Admittedly, structures with external sets can not be clubbed with those without external sets. For the present purposes, however, we shall overlook this fact.
- 9.
Incidentally, the second condition, r 12 = r 21, is that of reciprocity, which means that symmetry in the linear 2–port implies reciprocity.
- 10.
We use here the standard result that if A is a group and G a finite nonempty subset of it that is closed under the group operation then G is a subgroup of A. This follows easily from considering any element a ∈ G and its powers, and showing that G contains a − 1 as well as e, the identity element of A. Consider the powers of a as a sequence of successively generated elements \({a}^{m} = {a}^{m-1}a\) for \(m = 2, 3,\ldots \). Since G is closed, all these elements are in G. But, since G is finite, the elements must begin to repeat after a finite number of steps (at most equal to n, the cardinality of G). Let us say a j = a k for j > k > 0. Then by the cancellation law of A, \({a}^{j-k} = e\). Thus the identity e, being a positive power of a, is in G. Moreover, \({a}^{j-k} = a{a}^{j-k-1} = e\) and \(j - k - 1 \geq 0\), i.e., \({a}^{j-k-1} = {a}^{-1} \in \mathbf{G}\).
- 11.
A function or a transformation is also a binary relation. So functions and transformations are equally well covered by this interpretation of symmetry.
- 12.
Besides being mathematically more tractable, the linear ones cover most of our needs of representing symmetry operations in physical problems that are of interest to us here. There is yet another angle from which the constraint of linearity is well justified. When the ground set is simply a set, every member stands on its own—x’s membership implies nothing about y’s membership of the set. On the other hand, if the ground set has additional structure, such as that of a vector space, membership of x and y implies membership of many other ‘relatives’ of theirs by definition (αx and x + y, for instance, for a vector space). For a map ϕ on the set, it is then pertinent to ask whether it is ‘nice’ enough to preserve this implied membership of the relatives. Specifically for a vector space, does it take αx into αϕ(x) and x + y into ϕ(x) + ϕ(y)? In other words, is it a linear map? So long as our practical needs are met by them, it is reasonable to confine our attention to the linear ones.
- 13.
Note that P is a subgroup of the group of all invertible linear transformations on V under composition.
- 14.
This would be in line with the ideas of Klein’s Erlanger Programme.
- 15.
By a permutation I mean a one-to-one onto function from a set to itself. For a permutation π, π−1 denotes the inverse permutation.
- 16.
For a purely mathematical justification for introducing such operators here, and for their role, see Edwards [3, vol. 1, pp. 16–17, pp. 57–59].
- 17.
If I is finite then it has the structure of what is known as a mixed radix number system, of which modulo arithmetic is a special case. The theory of transforms such as the DFT and its other variants and generalizations are very closely linked to this case. For details see Siddiqi and Sinha [16, 17].
- 18.
Under addition and composition, the set of all linear transformations forms a ring and with scaling also included, it forms an algebra. The class \(\mathcal{H}\) is a subring and a subalgebra in the respective settings. Note that all this is true whether the set P is a group or not. The fact that P is a group makes \(\mathcal{H}\) a very special kind of algebra with structural properties that are of central significance to us here.
- 19.
The term “invariant”, in the sense used here, is defined later in Section 5.5.
- 20.
A matrix is block–diagonal if it consists of square submatrices on the principal diagonal, and has zeros every where else. A diagonal matrix is also block–diagonal, with every diagonal block of size one.
- 21.
For extensive discussions on commutative matrices and their properties, see Suprunenko [18].
- 22.
Although intuitively clear, you need to formally check that this is true. See Hoffman and Kunze [8, Lemma, Section 6.6, p. 209].
- 23.
With respect to different bases, a transformation has different matrices. So one can not talk of a transformation being in a block–diagonal form. Rather, one says that the transformation is block–diagonalizable, i.e., there is a basis with respect to which its matrix is block–diagonal.
- 24.
In purely matrix terms, this means that we have here a procedure for simultaneously diagonalizing all members of a given abelian group of matrices.
- 25.
Equation (1.14) there is a matrix version of (4.9), in which the matrices Di form a cyclic group of order 4. We found in this case that there is a matrix W that simultaneously diagonalizes the matrices Di.
- 26.
I am essentially stating here a variation of what Gross [7] enunciates as the fundamental problem of harmonic analysis.
- 27.
We need not check for P 1 as it is the identity matrix for this case.
References
L. Brillouin. Wave Propagation in Periodic Structures. McGraw-Hill, New York, 1946.
John D. Dixon and Brian Mortimer. Permutation Groups. Springer, New York, 1996.
R.E. Edwards. Fourier Series: A Modern Introduction, volume 1. Springer-Verlag, New York, 1979.
R. Foote, G. Mirchandani, et al. A wreath product group approach to signal and image processing: part i–multiresolution analysis. IEEE. Trans. on Signal Processing, 48(1):102–132, 2000.
R. Foote, G. Mirchandani, et al. A wreath product group approach to signal and image processing: part ii–convolution, correlation, and applications. IEEE. Trans. on Signal Processing, 48(3):749–767, 2000.
Wolfgang Gaertner. The group theoretic aspect of four–pole theory. IRE National Convention, Circuit Theory, Part 2:36–43, 1954.
Kenneth I. Gross. On the evolution of noncommutative harmonic analysis. Amer. Math. Month., 85:525–548, 1978.
Kenneth Hoffman and Ray Kunze. Linear Algebra. Prentice–Hall (India), New Delhi, 1972.
Nathan Howitt. Group theory and the electric circuit. Physical Review, 37:1583–1595, 1931.
David M. Kerns. Analysis of symmetrical waveguide junctions. Jr. Res. Nat. Bur. Stand., 46:515–540, 1949.
Reiner Lenz. Group Theoretical Methods in Image Processing. Springer-Verlag, New York, 1990.
Wu-Sheng Lu and Andreas Antoniou. Two–Dimensional Digital Filters. Marcel Dekker, New York, 1992.
George W. Mackey. The Scope and History of Commutative and Noncommutative Harmonic Analysis, volume 5 of History of Mathematics. American Mathematical Society, 1992.
A.E. Pannenborg. On the scattering matrix of symmetrical waveguide junctions. Phillips Res. Rep., 7:168–188, 1952.
Gunther Schmidt and Thomas Ströhlein. Relations and Graphs. Springer-Verlag, New York, 1993.
M.U. Siddiqi and V.P. Sinha. Generalized walsh functions and permutation–invariant systems. IEEE EMC. Symp., Montreal, pages 43–50, 1977.
M.U. Siddiqi and V.P. Sinha. Permutation–invariant systems and their application in the filtering of finite discrete data. Proc. IEEE Int. Conf. on ASSP, pages 352–355, 1977.
D.A. Suprunenko and R.I. Tyshkevich. Commutative Matrices. Academic, London, 1968.
Hermann Weyl. Symmetry. Oxford University Press, New York, 1952.
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Sinha, V.P. (2010). Symmetries, Automorphisms and Groups. In: Symmetries and Groups in Signal Processing. Signals and Communication Technology. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9434-6_4
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