Abstarct
The notion of a signal, like that of weight or temperature, is a two-sided one. We commonly think and speak of signals as functions of some sort, with numerical values both for their domain and for their range. Yet, signals to begin with have to do with what we perceive of objects and events around us through our senses. We could thus, to be more explicit, say that the term signal carries within it two connotations, one empirical and one formal. The former refers to the physical world in which statements about its objects and events are true or false in the sense that they are empirically observed to be so. The latter, on the other hand, refers to abstractions that are members of a formally defined mathematical world in which statements about a class of its members are true or false in the sense that they do or do not logically follow from the initial definitions and axioms governing that class.
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Notes
- 1.
For more on this issue from the point of view of modeling, see Rosen [29, Chapters 2, 3]. The book as a whole makes profound reading, touching on several fundamental issues.
- 2.
They may be of a geometric nature, either directly, as in the case of a metric space, or through implied possibilities, as in the case of a vector space.
- 3.
Desirable though this practice is, it is not a necessary requirement from a purely logical point of view. Thus in geometry axiomatically formulated on modern lines, nothing is in principle lost if, as Hilbert is said to have remarked, wherever we say “points, lines, and planes”, we say instead, “tables, chairs and beer mugs”. For more on this, and on the axiomatic method as understood in the modern sense, see Kennedy [18] and Meschkowski [23, Chapter 8, pp. 63–71].
- 4.
Presumably, there was no need at this stage to consider space as a whole. How a set of straight lines, circles etc., were related to each other was for them a matter to be sorted out locally, without any reference to space in its entirety. The idea of coordinatizing space as a whole, and of referring to geometrical figures and curves in terms of coordinates of their points, originated in the works of Descartes and Fermat on what came to be known as analytic geometry. Of their independent works on the subject, the first to come out and to draw public attention to it was Descartes’ La Géométrie, published in the year 1637 as an appendix to his Discourses. An English translation of this fascinating piece of work is available [30]. See Boyer [3] for a detailed historical account.
- 5.
See Einstein [11] for a very lucid account of this point and for a general discussion on the origins of the concept of space.
- 6.
Gauss (1777–1855) was ahead of all others in seeing all this, but he kept his ideas to himself, fearing that his radical views would be adversely received. Public attention was first drawn to these ideas through the independent works of Lobatchevsky (1793–1856) and Bolyai (1802–1860), and it was subsequently the unifying work of Riemann (1826–1866) on the foundations of geometry that brought them from the fringes to the mainstream of mathematical thinking of the time. See Boyer [4, Chapter 24, pp. 585–590] for more historical details.
- 7.
Ernst Mach (1838–1916), in his Science of Mechanics published in 1883, is credited to have explicitly articulated for the first time this modern viewpoint. See von Mises [34], and also Hempel [15] and Wilder [35], for details on this, and for very illuminating discussions on the axiomatic method as it is understood today.
- 8.
- 9.
- 10.
Nicholas Bourbaki is, if one were to go by the works published in this name, one of the most outstanding mathematicians appearing on the scene in the twentieth century. In actual fact, however, the name is a collective pseudonym for a continually changing group of leading mathematicians, who have been at work since the 1930s on a unification drive for mathematics. As to the outcomes of their labours in this drive, these are perhaps too advanced for us to worry about here; being aware of their existence is just about enough. See Cartan [6] for more on the myth and reality concerning Bourbaki.
- 11.
- 12.
We assume here for our purposes that the input and output signals are members of the same signal space. They may, in principle, belong to different spaces. We ignore this situation.
- 13.
I assume that you are familiar with this convolutional formula and its basic properties.
- 14.
The first form makes it a vector space, and the second one a ring.
- 15.
While cascading is valid on any signal space whatever, summation and scaling are structure–specific; note the way the addition and scaling operations of the signal space enter their definitions.
- 16.
Such a structure is commonly referred to as an algebra.
- 17.
I should mention here as an aside that the two conditions together are equivalent to the single condition \(H(\alpha x + \beta y) = \alpha (Hx) + \beta (Hy)\) for any real α and β. Further, homogeneity is, at least partially, implied by additivity. For, H(1x) = 1(Hx), and assuming additivity, \(H(nx) = H\left [(n - 1)x + x)\right ] = H\left [(n - 1)x\right ] + H(x)\) . If homogeneity is true for (n − 1), then by mathematical induction, H(nx) = nH(x) for any positive integer n. With a little more work on the same lines, it can be shown that for any rational α, additivity implies homogeneity. This cannot be said, however, for all reals. There are interesting deep set theoretic matters involved here. See Hrbacek and Jech[16, Chapter 9, pp. 177–178].
- 18.
A unary operation on a set X maps a member x ∈ X into another member y ∈ X, so that the operation can be looked upon as a binary relation consisting of a subset of X × X; likewise, a binary operation can be looked upon as a ternary relation consisting of a subset of X × X × X.
- 19.
The various intermediate steps might seem like fussing over what is trivially obvious. But that is because of our familiarity with the properties of numbers under addition. You will appreciate their merit better if you study algebraic structures in general.
- 20.
In physical terms, an RLC circuit, for instance, produces the same response no matter when you apply the input, whether today or tomorrow.
- 21.
Recall that D n denotes the composition of the operator D applied n–times in cascade. For instance, by D 2 x we mean D(D(x)).
- 22.
Formally, it is an instance of a substructure, a term we shall properly examine in the next chapter.
- 23.
For \(a,b \in \mathbb{N}\), we say a ≤ b if and only if there is an \(c \in \mathbb{N}\) such that \(a + c = b\). The product ab is similarly defined in terms of addition. Of course, for our present purposes, the product operation has no place in the role of \(\mathbb{N}\) as an index set.
- 24.
A striking example of this is the so-called alternation theorem of minmax approximation theory. For the approximation of a real function by a polynomial of a fixed degree, it reads roughly as follows: the maximum deviation of the polynomial from the function is minimum over the interval if and only if the peaks of the error function alternate in sign as many times as the degree and are equal in magnitude. Filter designers make copious use of this result. See Rice [28, Chapter 3, p. 56] and Cheney [7, Chapter 3, p. 75] for the theorem, and Rabiner [27, Chapter 3, pp. 127–139] for filter applications.
- 25.
The argument I have used here is not exactly one of mathematical induction, and it can be faulted for not being really rigorous. But it will do for the present.
- 26.
A transform such as the Laplace or the Z–, with which I assume you are already familiar, maps the given functions into functions of a complex variable. The merit of this mapping is that, in addition to being linear, it replaces operations of differentiation, integration, shifting, and convolution of the original functions by ostensibly simpler algebraic operations in the transform domain. The idea of using addition and convolution as the basic operations on functions, and of using fractions of functions, was utilized by Jan Mikusiński [25] in the fifties to provide an alternative to the Laplace transform. For more on its applications to sequences, see Traub [33].
- 27.
Integers under addition and multiplication, and sequences under addition and convolution, are instances of a structure known as an integral domain.
- 28.
Strictly speaking, we should use different symbols to designate equality and addition here, because they are not the same as those for sequences. But that would make the notation cumbersome. So we use the same symbols in both the cases.
- 29.
As in the case of the convolution sign, we shall omit the product sign and write p ⋅ q simply as pq for fractions p and q.
- 30.
Such a structure is called in algebra a field.
- 31.
Complex numbers, treated as ordered pairs of reals, provide another familiar illustration of this strategy.
- 32.
If you are not familiar with this, see Harmuth [14] to get an idea.
- 33.
- 34.
Of course, in the chain of developments along these lines, one soon finds that there are problems peculiar to the 2-D theory, requiring fresh thinking to deal with them. A case in point is the problem arising from the fact that for the so-called fundamental theorem of algebra—an n-th degree polynomial of a single variable has precisely n roots in the complex plane, there is no counterpart in the two variable case. See Antoniou [20, p. 3] for more elaborate comments on this issue. To get an idea of the nuances of the theory of functions of several complex variables, see Grauert and Fritzsche [12].
- 35.
- 36.
See P1–P6 on page 16. In the language of algebra, I is a finite abelian group.
- 37.
We are treading on linear algebra here. The set of all signals under addition and scaling by reals has the structure of a finite dimensional vector space in this case. Saying that there is a unique expansion in terms of the δ’s for any x is equivalent to saying more formally that the δ’s are linearly independent, and that they constitute a basis for the space. Although not essential, it would be worthwhile brushing up your linear algebra at this stage. For a compact introduction, see Artin [1].
- 38.
In linear algebra a basis is commonly introduced as a linearly independent set of vectors that spans the space. We mean essentially the same thing, except that for a finite dimensional vector space, we have put it differently in terms of the notion of uniqueness. This appears to be a more parsimonious way of introducing the concept at this stage. Uniqueness of expansion is equivalent to linear independence.
- 39.
Suppose there were no finite index sets of the stipulated kind (finite abelian groups). In that case there would be nothing that our derivations, even though correct, would refer to. At least one such set, and preferably many such different sets, must therefore exist for the abstractions to have any significance. We will have more on this when we discuss abstract structures in general.
- 40.
Observe that D 2 and D 3 equal D 1 2 and D 1 3 respectively. The matrices form a special kind of abelian group–the so called cyclic group– of which one single member, raised to different powers, gives all other members.
- 41.
For a matrix A, A ′, \(\bar{A}\), \({A}^{{_\ast}}(\equiv \bar{ A}^\prime)\) will denote its transpose, (complex) conjugate and conjugate transpose respectively.
- 42.
Seeing transforms from this angle has given rise to noteworthy developments in two areas. One is the area of digital logic and fault diagnostics. For details, see Hurst [17]. The other is of multiresolution signal representation using wavelet transforms. See Meyer [24] and Strang [32] for a good introduction.
- 43.
Yes. The notation is rather puzzling at first glance. In using χ(a−1), and not χ(a), to denote the eigenvalue of Da, the motive is to keep in line with the action of Da on f.
- 44.
Invariant means and invariant integrals provide an important vantage point in modern harmonic analysis. For more on this, see Edwards [10, vol. 1, pp. 21–26].
- 45.
For a function f, \(\bar{f}\) denotes its complex conjugate.
- 46.
The second part can also be said to mean that the basis functions are normalized with respect to the invariant mean as inner product.
- 47.
Take note of the fact that \(M(f\bar{g})\) qualifies as what is called an inner product in linear algebra, making the signal space an inner product space.
- 48.
Use the familiar trick of Fourier series expansion. Multiply both sides by \(\bar{{\phi }}_{b}\), take the invariant mean and invoke orthogonality. We are indeed doing a generalized fourier analysis here.
- 49.
Having first called the transform of a function as spectrum, it might smack of ambiguity to use the term for an operator H. It is not really so because we are in effect referring to its impulse response h and its transform. Incidentally, there is an interesting historical coincidence about the origin of the term “spectrum” that is worth mentioning here. Independently of physics, it was introduced in mathematics for operators in general, as an extension of the notion of eigenvalues, to mean the set of values of a scalar λ such that (λI − T) is not invertible, where I is the identity operator. It was later recognized that its use in physics (and subsequently in engineering), when formally interpreted, coincided with that in mathematics. See Steen [31]. Calling \(=\hat{h}\) the spectrum of H is justified from this angle too.
- 50.
This simplistic approach of identifying the eigenvectors would admittedly be of little use for bigger index sets. As already mentioned, group representation theory provides us a general systematic technique for doing this. We will discuss it in Chapter 5.
- 51.
Observe that it is the 4–point DFT matrix.
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Sinha, V.P. (2010). Signals and Signal Spaces: A Structural Viewpoint. In: Symmetries and Groups in Signal Processing. Signals and Communication Technology. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9434-6_1
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