Abstract
Fourier transform methods find application in problems formally similar to those for which Laplace transform techniques are a suitable tool. Such applications include integral equations and partial and ordinary differential equations. The formal difference between the two classes of problems is that Laplace transforms are applied to functions defined on a half-line, while Fourier transforms apply to functions whose domain is the entire real axis. As a consequence, Laplace transforms are associated with initial value problems (transient responses), while Fourier transforms find more common application in input-output (forced response) models.
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- 1.
- 2.
The class of transformable functions and sense of convergence of the inversion integral is discussed in the following Section 7.5.
- 3.
This correspondence is one reason for defining the Fourier transform as we have above (alternatives displace the \(\frac{1} {2\pi }\) inversion factor, or introduce minus signs).
- 4.
Multiplication of a transformable function by real exponent exponential will often produce a function which has no transform in the sense described above.
- 5.
By an appeal to the transform pair argument, the differentiation rule may be regarded as a version of the same result.
- 6.
For purposes of exposition, we refer to the independent variable as “time.” The discussion obviously applies as well to “spatially invariant” effects.
- 7.
There exist examples of equations of this form for which the formal calculations are not justifiable.
- 8.
See the following section and references therein.
- 9.
Such functions are said to be “of class L 1.”
- 10.
Justified since the double integral exists.
- 11.
The result in question is the Dominated Convergence Theorem for Lebesgue integrals.
- 12.
As mentioned above, there are other choices of “approximate delta functions” which may be employed in such arguments. Many of these choices are related to the idea of a “summation method” for the (possibly divergent) inversion integral.
- 13.
The terminology is that f belongs to the class L 2.
- 14.
This is in consonance with the usual Laplace transform inversion, and naturally introduces the 2π i factor associated with the Residue Theorem.
- 15.
This amounts to the evaluation of the principal value at infinity of the indicated improper integral.
- 16.
From this view the result is analogous to some classical results concerning differentiation of series. Modulo a side condition, attempt the calculation; if a sensible result is obtained, it is the desired answer.
- 17.
In this case the integral should be interpreted as a “principal value at infinity,” that is,\(\mathop{\mbox{ lim}}\limits_{B \rightarrow \infty }\int _{-B}^{B}\hat{f}(\omega )\,d\omega\).
- 18.
This is essentially the class for which we originally defined the Laplace transform in Chapter 6
- 19.
In fact, this may be taken as the “usual form” of the inversion, with the implicit understanding that (7.17) is to be employed in the case of L 2 convergence.
- 20.
The integrals encountered are convergent since
$$\displaystyle{\biggm |\int _{0}^{T}e(\tau )\,e^{-s\tau }\,d\tau \biggm | \leq \left (\int _{ 0}^{T}\vert e(\tau )\vert ^{2}\,d\tau \right )^{\frac{1} {2} }\left (\int _{ 0}^{T}\vert e^{-s\tau }\vert ^{2}\,d\tau \right )^{\frac{1} {2} }}$$while e −s τ < 1 for Re(s) > 0.
- 21.
In the finite-dimensional case, it is usual to show that V ′ is isomorphic to V. A common reaction to this result is to wonder why the subject was mentioned at all. However, for infinite-dimensional V, the V ′ is typically quite distinct from V. This distinction is the source of distribution theory.
- 22.
This means that a notion of closeness of elements of V is defined. In short, V has a topology, and is a topological vector space.
- 23.
We leave the definition of a topology on S and verification of continuity of δ for a more comprehensive treatment of the subject.
- 24.
The apparent simplicity at this stage is due to the effort (alluded to above) involved in careful analysis of the vector space S.
- 25.
Strictly speaking, the step moving derivatives outside of the pairing, although intuitively clear, requires more justification than we have provided.
- 26.
Certain initial value problems may also be treated (with some care) by including distributions (generalized functions) among the admissible forcing functions.
- 27.
The procedure is essentially similar to the process of finding periodic solutions by use of Fourier series. If a solution of the required properties exists, the method discovers it.
- 28.
Recall that we assume p(i β) ≠ 0, so that both exist.
- 29.
In the case that the network in question has only Fourier transformable solutions in response to transformable inputs, W(i ω) is referred to as the frequency response of the network. In the contrary case, Fourier transforms are inapplicable, and the transient analysis of Section 6.8 should be employed.
- 30.
The physical interpretation of this model is that the physical extent is so large that “end effects” are negligible.
- 31.
These manipulations assume sufficient smoothness of the solution sought. We make such assumptions without further apology, since they are required to carry out the formal calculations of the solution process.
- 32.
One might attempt to use a Laplace transform with respect to the y variable, but in view of the ultimate answer, that procedure leads to unfamiliar transforms. In principle it would work.
- 33.
This is a special case of the linear, constant coefficient problems of Section 6.6
- 34.
Even though we seek U only on the interval [0, 1], it is convenient to extend the solution beyond x = 1 in order to use the transform.
- 35.
This expression apparently contains a branch cut, when regarded as a function of the complex variable s. Consideration of the power series shows, however, that the result is actually an entire function of s with no branch cut required.
- 36.
Strictly speaking, in order to apply L’Hopital’s rule, we define \(\sqrt{s}\) as an analytic function by installing a branch cut in the s-plane, avoiding the negative real axis.
- 37.
The validity of the procedure relies on considerations of the sort used in Section 7.6, substantially complicated by the “x” dependence of the integrand. As the computations of this section are being made on a formal basis, we casually sidestep this issue.
- 38.
This simple model is in practice augmented by models for stochastic noise (interference) and channel distortion. In such models, the messages are also modeled as stochastic (rather than deterministic) functions.
- 39.
This is double sideband suppressed carrier amplitude modulation (DSBSC-AM). Commercial AM transmits (1 + m f(t))cos(ω c t), and so contains a term at the carrier frequency.
- 40.
The function w is referred to as the impulse response of an “ideal low pass filter.” Since (7.46) as it stands requires future values of the received signal r to compute the present message value, the processing in practice is replaced by a physically realizable approximation to the ideal low pass filter.
- 41.
Examples include digital telephony, voice compression, and digital media recordings.
Further Reading
A.B. Carlson, Communication Systems: An Introduction to Signals and Noise in Electrical Communication, 2nd edn. (McGraw-Hill, New York, 1975)
I.N. Gelfand, C.B. Shilov, Generalized Functions, vol. 1 (Academic, New York, 1964)
I.N. Gelfand, C.B. Shilov, Generalized Functions, vol. 2 (Academic, New York, 1968)
R.R. Goldberg, Fourier Transforms (Cambridge University Press, Cambridge, 1962)
J. Horvarth, Topological Vector Spaces and Distributions (Addison-Wesley, Reading, 1966)
E.M.J. Lighthill, Introduction to Fourier Analysis and Generalized Functions (Cambridge University Press, Cambridge, 1964)
J.F. Marsden, Basic Complex Analysis (W.H. Freeman and Company, San Francisco, 1975)
A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962)
J.G. Proakis, Digital Communications, 3rd edn. (McGraw-Hill, New York, 1995)
H. Schwartz, W.R. Bennett, S. Stein, Communication Systems and Techniques (McGraw-Hill, New York, 1966)
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Davis, J.H. (2016). Fourier Transforms. In: Methods of Applied Mathematics with a Software Overview. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-43370-7_7
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