Abstract
We shall use the term signal to designate a function of time. Unless specified otherwise, the time shall be allowed to assume all real values. In order not to dwell on mathematical generalities that are of peripheral interest in engineering problems the mathematical functions which we shall employ to represent these signals will be restricted to those that are piecewise differentiable. We shall also be interested in treating a collection of such signals as a single entity, in which case we shall define these signals as components of a vector.
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Notes
- 1.
There are many other proofs of the Schwarz inequality. A particularly simple one is the following. For any two functions we have the identity
$${ \left \|f\right \|}^{2}{\left \|g\right \|}^{2} -{\left \vert \left (f,g\right )\right \vert }^{2} = 1/2\iint {\left \vert f(x)g(y) - f(y)g(x)\right \vert }^{2}dxdy.$$Since the right side is nonnegative (1.17) follows (from Leon Cohen, “Time-Frequency Analysis,” Prentice Hall PTR, Englewood Cliffs, New Jersey (1995) p. 47).
- 2.
U H U = I NN
- 3.
The term energy as used here is to be understood as synonymous with the square of the signal norm which need not (and generally does not) have units of energy.
- 4.
Mathematically it is immaterial which set is designated as the direct and which as the reciprocal basis.
- 5.
Another set of orthogonal polynomials can be constructed using the eigenvectors of the Gram matix of t k. The orthogonal polynomials are then given by (1.62).
- 6.
Here we distinguish a random variable by an underline and denote its expectation (or ensemble average) by enclosing it in < >
- 7.
Formulas (1.201) and (1.202) assume a more esthetically pleasing form if we assume an orthonormal set for then Q n = 1. Even though this can always be realized by simply dividing each expansion function by \(\sqrt{Q_{n}}\), it is not customary in applied problems and we shall generally honor this custom.
- 8.
The reader may be puzzled by the change in the argument from t − t ′, which we employed in the preceding subsection for the delta function, to t ′ − t. We do this to avoid distinguishing between the derivative with respect to the argument and with respect to t ′ in the definition of \({\delta }^{(k)}\left (t\right )\).
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Wasylkiwskyj, W. (2013). Signals and Their Representations. In: Signals and Transforms in Linear Systems Analysis. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3287-6_1
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DOI: https://doi.org/10.1007/978-1-4614-3287-6_1
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