2.12: Complex Amplitudes; Fourier Amplitudes and Fourier Transforms

The notion of a continuum network fs introduced for the first time in the next section. The associated transfer relations illustrated there are a theme throughout the chapters which follow. Among several reasons for their use is the organization they lend to the representation of complicated, largely linear, systems. In this chapter, the continuum networks represent electromagnetic fields. Later, they represent fluid and (to some degree) solid mechanics, heat and mass transfer, and electromechanical continua in general. These networks make it possible to set aside one part of a given problem, derive the associated relations once and for all and accumulate these for later use. Such relations will be picked up over and over in solving different problems and, properly understood, are a useful reference.

Complex Amplitudes

In many practical situations, excitations are periodic in one or two spatial directions, in time or in space and time. The complex amplitude representation of fields, useful in dealing with these situations, is illustrated by considering the function \(\phi (z,t)\) which has dependence on \(z\) given explicitly by

\[ \phi(z,t) = R e \tilde{\phi}(t) e^{-jkz} \label{1} \]

With the wavenumber \(k\) real, the spatial distribution is periodic with wavelength \(\lambda = 2 \pi/k \) and spatial phase determined by the complex amplitude \(\tilde{\phi}\). For example, if \(\tilde{\phi} = \tilde{\phi_o}(t) \) is real and \(k\) is real, then \(\phi(z,t) = \phi_o(t) cos kz\).

The spatial derivative of \(\phi\) follows from Equation \ref{1} as

\[ \frac{\partial{\phi}}{\partial{z}} = Re \Big[-jk \tilde{\phi} (t) e^{-jkz} \Big] \label{2} \]

The following identifications can therefore be made:

\[ \Big[ \phi(z,t), \quad \frac{\partial{\phi}}{\partial{z}} (z,t) \Big] \Leftrightarrow \Big[ \tilde{\phi}(t), \quad -jk\tilde{\phi}(t) \Big] \label{3} \]

with it being understood that even though complex amplitudes are being used, the temporal dependence is arbitrary. There will be occasions where the time dependence is specified, but the space dependence is not. For example, complex amplitudes will take the form

\[ \phi(z,t) = Re \hat{\phi} (z) e^{j \omega t} \label{4} \]

where \(\hat{\phi} (z)\) is itself perhaps expressed as a Fourier series or transform (see Sec. 5.16).

Most often, complex amplitudes will be used to represent both temporal and spatial dependences:

\[ \phi(z,t) = Re \hat{\phi} (z) e^{j(\omega t - kz)} \label{5} \]

The (angular) frequency \(\omega\) can in general be complex. If \(\phi\) is periodic in time with period \(T\), then \(T = 2 \pi / \omega\). For complex amplitudes \(\hat{\phi}\), the identifications are:

\[ \Big[ \phi(z,t), \quad \frac{\partial{\phi}}{\partial{z}} (z,t), \quad \frac{\partial{\phi}}{\partial{t}} (z,t) \Big] \Leftrightarrow \Big[ \hat{\phi}, \quad -jk\hat{\phi}, \quad j \omega \hat{\phi} \Big] \label{6} \]

If \(\omega\) and \(k\) are real, Equation \ref{5} represents a traveling wave. At any instant, its wavelength is \(2 \pi /k\), at any position its frequency is \(\omega\) and points of constant phase propagate in the \(+z\) direction with the phase velocity \(\omega/k\).

Fourier Amplitudes and Transforms

The relations between complex amplitudes are identical to those between Fourier amplitudes or between Fourier transforms provided that these are suitably defined. For a wide range of physical situations it is the spatially periodic response or the temporal sinusoidal steady state that is of interest. Simple combinations of solutions represented by the complex amplitudes then suffice, and there is no need to introduce Fourier concepts. Even so, it is important to recognize at the outset that the spatial information required for analysis of excitations with arbitrary spatial distributions is inherent to the transfer relations based on single-complex-amplitude solutions.

The Fourier series represents an arbitrary function periodic in \(z\) with fundamental periodicity length \(l\) by a superposition of complex exponentials. In terms of complex Fourier coefficients \(\hat{\phi_n}(t)\), such a series is

\[ \phi(z,t) = \sum_{n= -\infty}^{\infty} \tilde{\phi}_n (t)e^{-jk_nz}; \quad k_n \equiv 2n \pi/l; \quad \tilde{\phi_n^{*}} = \tilde{\phi}_{-n} \label{7} \]

where the condition on \(\tilde{\phi}_n\) insures that \(\tilde{\phi}\) is real. Thus, with the identification \(\tilde{\phi} \rightarrow \tilde{\phi}_n\) and \(k \rightarrow k_n\), each complex exponential solution of the form of Equation \ref{1} can be taken as one term in the Fourier series. The mth Fourier amplitude \(\tilde{\phi}_m\) follows by multiplying Equation \ref{7} by the complex conjugate function \(exp(jk_mz)\) and. integrating over the length \(l\) to obtain only one term on the right. This expression can then be solved for \(\tilde{\phi}_m\) to obtain the inverse relation

\[ \tilde{\phi}_m = \frac{1}{l} \int_{z}^{z+l} \phi(z,t) e^{jk_mz} dz \label{8} \]

If the temporal dependence is also periodic, with fundamental period \(T\), the Fourier series can also be used to represent the time dependence in Equation \ref{7}:

\[ \phi(z,t) = \sum_{m=-\infty}^{+\infty} \sum_{n=-\infty}^{+\infty} \hat{\phi}_{mn} e^{j(\omega_m t - k_nz)}; \quad \hat{\phi}^{*}_{mm} = \hat{\phi}_{-m-n} \label{9} \]

where the condition on the amplitudes insures that \(\phi(z,t)\) is real. One component out of this double summation is the traveling-wave solution represented by the complex amplitude form, Equation \ref{5}. The rules given by Eqs. \ref{3} and \ref{6} pertain either to the complex amplitudes or the Fourier coefficients.

The Fourier transform is convenient if the dependence is not periodic. With the Fourier transform \(\tilde{\phi}(k,t)\) given by

\[ \tilde{\phi}(k,t) = \int_{-\infty}^{+\infty} \phi(z,t) e^{jkz} dz \label{10} \]

the functional dependence on \(z\) is a superposition of the complex exponentials

\[ \phi(z,t) = \int_{-\infty}^{+\infty} \tilde{\phi}(k,t) e^{-jkz} \frac{dk}{2 \pi} \label{11} \]

The relation between the transform and the transform of the derivative can be found by taking the transform of \(\partial{\phi}/\partial{z}\) using Equation \ref{11} and integrating by parts. Recall that \(\int v du = uv - \int u dv\) and identify \(du \rightarrow \partial{\phi}/\partial{z} dz\), and \(v \rightarrow exp (jkz) \)it follows that

\[ \int_{-\infty}^{+\infty} \frac{\partial{\phi}}{\partial{z}} e^{jkz} dz = \phi e^{jkz} \Big\vert_{-\infty}^{+\infty} -jk \int_{-\infty}^{+\infty} \phi e^{jkz} dz \label{12} \]

For properly bounded functions the first term on the right vanishes and the second is \(-jk \tilde{\phi}(k,t)\). The transform of \(\partial{\phi}/\partial{z}\) is simply \(-jk \tilde{\phi}\) and thus the Fourier transform also follows the rules given with Equation \ref{3}.

Extension of the Fourier transform to a second dimension results in the transform pair

\[ \begin{align} \phi(z,t) &= \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \hat{\phi} (k,\omega) e^{j(\omega t - kz)} \frac{dk}{2 \pi} \frac{d \omega}{2 \pi} \nonumber \\ \hat{\phi}(k,\omega) &= \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \phi(z,t) e^{-j (\omega t - kz)} dt dz \nonumber \end{align} \label{13} \]

which illustrates how the traveling-wave solution of Equation \ref{5} can be viewed as a component of a complicated function. Again, relations between complex amplitudes are governed by the same rules, Equation \ref{6}, as are the Fourier amplitudes \(\hat{\phi} (k,\omega)\).

If relationships are found among quantities \(\tilde{\phi}(t)\), then the same relations hold with \(\tilde{\phi} \rightarrow \hat{\phi} \) and \(\partial{()}/\partial{t} \rightarrow j \omega \), because the time dependence \(exp(j \omega t)\) is a particular case of the more general form \(\tilde{\phi}(t)\).

Averages of Periodic Functions

An identity often used to evaluate temporal or spatial averages of complex-amplitude expressions is

\[ \Bigg \langle Re \tilde{A} e^{-jkz} Re \tilde{B} e^{-jkz} \Bigg \rangle_z = \frac{1}{2} Re \tilde{A} \tilde{B}^{*} \label{14} \]

where \(\langle \rangle_z\) signifies an average over the length \(2 \pi/k\) and it is assumed that \(k\) is real. This relation follows by letting

\[ Re \tilde{A} e^{-jkz} Re \tilde{B} e^{-jkz} = \frac{1}{2} \Bigg[ \tilde{A} e^{-jkz} + \tilde{A}^{*} e^{jkz} \Bigg] \frac{1}{2} \Bigg[ \tilde{B} e^{-jkz} + \tilde{B}^{*} e^{jkz} \Bigg] \label{15} \]

and multiplying out the right-hand side to obtain

\[ \frac{1}{4} \Bigg[ \tilde{A} \tilde{B} e^{-2jkz} + \tilde{A}^{*} \tilde{B}^{*} e^{2jkz} \Bigg] + \frac{1}{4} \Bigg[\tilde{A} \tilde{B}^{*} + \tilde{A}^{*} \tilde{B} \Bigg] \label{16} \]

The first term is a linear combination of \(cos(2kz)\) and \(sin(2kz)\) and hen e averages to zero. The second term is constant and identical to the right-hand side of Equation \ref{14}.

A similar theorem simplifies evaluation of the average of two periodic functions expressed in the form of Equation \ref{7}:

\[ \begin{align} \Bigg \langle AB \Bigg \rangle_z &= \Bigg \langle \sum_{n=-\infty}^{+\infty} \tilde{A}_n (t) e^{-jk_n z} \quad \sum_{m=-\infty}^{+\infty} \tilde{B}_m (t) e^{-jk_m z} \Bigg \rangle_z \nonumber \\ &= \sum_{n=-\infty}^{+\infty} \tilde{A}_n \tilde{B}_{-n} = \sum_{n=-\infty}^{+\infty} \tilde{A}_n \tilde{B}_{n}^{*} \nonumber \end{align} \label{17} \]

Of course, either the complex amplitude theorem of Equation \ref{14} or the Fourier amplitude theorem of Equation \ref{17} applies to time averages with \(kz \rightarrow -\omega t\).