On Some “Paradoxes” Related to Spectral Expansions

S. M. Rytov

The article considers certain apparent contradictions in the field of spectral expansions which, as practice shows, are very often encountered by those working in radio and spectroscopy. Errors connected with a misunderstanding of these “paradoxes” have found their way into a considerable number of works and into the specialized literature. The correct approach to this kind of question was given by Academician L. I. Mandelstam, who also brought some of them to complete clarity. Part of the article (§§ 2 and 3) is therefore devoted to presenting the corresponding considerations of Academician L. I. Mandelstam.

1. The “Contradiction” Between Mathematics and Experiment

Spectral expansions, widely used in the theory of linear oscillations (where the principle of superposition holds), are based on Fourier’s theorem. This theorem asserts that, under certain restrictions, a function \(f(t)\), generally speaking complex, can be represented either by the integral

\[ f(t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} g(\omega)e^{i\omega t}\,d\omega,\qquad g(\omega)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} f(t)e^{-i\omega t}\,dt, \tag{1} \]

if it is nonperiodic, or by the series

\[ f(t)=\sum_{n=1}^{\infty} c_n e^{i\omega_n t},\qquad c_n=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{+T} f(t)e^{-i\omega_n t}\,dt, \tag{2} \]

if it is almost periodic, or by the series

\[ f(t)=\sum_{-\infty}^{+\infty} c_n e^{i\omega t},\qquad c_n=\frac{\omega}{2\pi}\int_{-\frac{\pi}{\omega}}^{+\frac{\pi}{\omega}} f(t)e^{-in\omega t}\,dt, \tag{3} \]

if it is periodic. In all cases \(f(t)\) must be specified over the entire interval of \(t\) from \(-\infty\) to \(+\infty\); for a periodic

…of the function in the interval \(\left(-{\pi\over\omega},+{\pi\over\omega}\right)\) is equivalent to this. Thus, in Fourier expansions conveying the behavior of \(f(t)\) at all times, the spectral amplitudes \(c_n\), or the spectral densities \(g(\omega)\), do not depend on \(t\), i.e. the harmonic spectrum is unchanging, fixed once and for all.

Quite often this immutable mathematical fact comes into conflict with notions drawn from the most, it would seem, obvious phenomena, which speak of a change of the harmonic spectrum in time. Indeed, there can be no doubt that the spectrum observed in a spectroscope disappears at once as soon as we extinguish the spark or arc under study, or that, when the operation of a radio station ceases, both the carrier and the sidebands disappear. Moreover, the spectrum of a radio station changes continuously during the transmission itself—from word to word, from signal to signal; and when an opera is transmitted in the evening it is undoubtedly different from that during the transmission of news in the morning. How, then, do matters stand with Fourier’s theorem in such a case?

This apparent contradiction is easily resolved if one analyzes more deeply what is actually meant when one speaks of a changing spectrum. The analysis in question, and which brings clarity to the matter, is based on considerations repeatedly expressed by the late Academician L. I. Mandelstam in connection with certain other—generally analogous—“paradoxes” from the field of spectral expansions. Since these considerations thus have a direct bearing on the question posed above, and since they are not sufficiently widely known,* I shall begin with a brief account of them.

2. WHAT IS A “SINUSOID WITH VARIABLE AMPLITUDE OR FREQUENCY”?

A harmonic or sinusoidal oscillation is, by definition, the function \(A\cos(\omega t+\varphi)\), or, in complex form, the function

\[ Ae^{i(\omega t+\varphi)}, \tag{4} \]

defined on the entire infinite interval of \(t\) from \(-\infty\) to \(+\infty\), with the amplitude \(A\), frequency \(\omega\), and initial phase \(\varphi\) being constant quantities. Therefore, from the mathematician’s point of view, such expressions as “a sinusoid with variable amplitude” are meaningless, contradic-

* L. I. Mandelstam dwelt on these questions in his lectures and seminars, which have not yet been published. The statements of L. I. Mandelstam referred to here have been reflected only in special studies by his pupils, for example in the work of G. S. Gorelik, “Resonance phenomena in systems with periodically varying parameters” (ZhTF, 4, 1783, 1934; 5, 195 and 489, 1935), and in the author’s work “Modulated oscillations and waves” (Proceedings of FIAN, 2, issue 1, 1938).

which are alien to the very definition of a sinusoid. Nevertheless, physicists and engineers make daily and successful use of the concepts of sinusoidal oscillation, monochromatic light, musical tone, etc., despite the fact that real processes are never sinusoidal, if only because they have a beginning and an end. Evidently, there exist such conditions under which oscillations deviating from sinusoidality may, with the required accuracy, be regarded as sinusoidal. What do these conditions consist in?

Any change in at least one of the quantities \(A\), \(\omega\), and \(\varphi\) in (4) is equivalent to the presence of a frequency spectrum, the presence of a set of sinusoidal oscillations. Thus, for example, with amplitude modulation of the form

\[ A=A_0(1+k\cos\Omega t) \tag{5} \]

we obtain (putting, for simplicity, \(\varphi=0\)):

\[ A_0(1+k\cos\Omega t)e^{i\omega t} = A_0e^{i\omega t} + \frac{kA_0}{2}e^{i(\omega+\Omega)t} + \frac{kA_0}{2}e^{i(\omega-\Omega)t}, \tag{6} \]

i.e. a set of three sinusoidal oscillations with frequencies \(\omega\) (the carrier) and \(\omega\pm\Omega\) (the side frequencies).

The right- and left-hand sides of (6) are mathematically identical—they are one and the same thing. To ask what we have in reality—whether a variable amplitude \(A\) in the oscillation \(e^{i\omega t}\), or three sinusoidal oscillations—is utterly meaningless. However, only 15 years ago such an authoritative scientist and radio engineer as Fleming opened a controversy in the pages of Nature ) claiming that only the “modulated carrier” is real, while the side frequencies or bands are merely a convenient computational device, a mathematical fiction to which nothing corresponds in reality. The conclusions from this concerned not at all lofty matters, but the most urgent practical questions. Since in reality, wrote Fleming, we have one single frequency \(\omega\), there are no grounds for establishing frequency intervals between radio stations, and the density of occupancy of the ether is limited only by the selectivity of receivers... Of course, Fleming could not consistently defend his point of view, which declared the left-hand side of (6) real and the right-hand side, identical to it, unreal *). For it is difficult to answer the question of what is real: 10, \(5+5\), or \(2+8\).

However, if, having 10 rubles, you must pay 2 rubles, then among all possible ways of breaking 10 into summands one is singled out

) See Nature, 125, 93, 198, 271, 306, 1930. Fuller literary data are contained in my work “Modulated Oscillations and Waves,” pp. 78 ff. (Proceedings of FIAN, 2*, issue I, 1938).

**) One should not think that Fleming’s intervention was an isolated fact. The discussion of the “reality” of side frequencies has quite a long history, on which, however, we cannot dwell here. Erroneous statements on this question occur in the literature even after the discussion with Fleming.

namely the decomposition into 2 and 8. In other words, the concrete circumstances single out, from among all mathematically equivalent modes of representation, the mode most suitable, most expedient under the given conditions. As L. I. Mandelstam pointed out, the same applies to the question under consideration.

One should not confuse the question of the reality of existence—which, if posed, is in any case resolved in the same way for all identical representations—with the question of the expediency of one or another of these representations. Speaking of “reality” or “unreality,” what is actually meant is precisely this latter question. But it acquires meaning and content only when it is indicated what we intend to do with the oscillation (6), how and by means of what to investigate it. Apart from the properties of the perceiving or analyzing apparatus, this question hangs in the air.

As an illustration, L. I. Mandelstam also gave the following example. Suppose that before us is a mixture of large and small, iron and copper balls. The question is asked whether we have iron and copper balls or large and small ones. Being unable to solve this question in general, we take a concrete analyzer—a sieve—and sift the mixture. As a result it turns out that it consists of large and small balls. However, another analyzing instrument—a magnet—shows us that in reality the mixture contains iron and copper balls.

In our case we are specifically interested in the action of an oscillation (say, of an electromotive force) of the form (6) on a selective device, on a linear receiver possessing a definite pass band, for example on a simple resonant circuit. The expediency of one or another representation of the emf is therefore determined by the properties of such a circuit. For the current \(I\) in the circuit we have the equation

\[ \ddot I + 2h\dot I + \omega_0^2 I = Ae^{i\omega t} = A_0(1+k\cos\Omega t)e^{i\omega t}. \tag{7} \]

Let the circuit be tuned to the carrier \((\omega_0=\omega)\), and let the damping coefficient \(h \ll \omega\) and the modulation frequency \(\Omega \ll \omega\), as is the case under practical conditions. Then the forced oscillation, to terms of order \(h/\omega\) and \(\Omega/\omega\), will be:

\[ I=A_0\left[1+\frac{k}{\sqrt{1+\left(\frac{\Omega}{h}\right)^2}}\cos(\Omega t-\alpha)\right]\frac{e^{i\omega t}}{2i\omega h}, \qquad \tg \alpha=\frac{\Omega}{h}. \tag{8} \]

The course of \(I(t)\) is essentially different depending on the magnitude of the ratio \(\Omega/h\), which can be interpreted in two ways. In the spectral approach we shall say that this is the ratio of the frequency interval between the carrier and the side frequency to the half-width of the resonance curve of the circuit. If, however, we wish to speak not of the spectrum but of the course of the forced oscillation in time, then \(\Omega/h\) represents (multi-

… relation (multiplied by \(2\pi\)) of the settling time of the oscillations in the circuit \(1/h\) to the modulation period \(\frac{2\pi}{\Omega}\). Of course, the two interpretations are completely equivalent.

If \(\Omega \ll h\), i.e. the modulation is slow in comparison with the settling time, or, speaking in spectral language, the side frequencies \(\omega \pm \Omega\) are displaced from the carrier \(\omega\) by much less than the width of the resonance curve, then (8) gives:

\[ I=\frac{Ae^{i\omega t}}{2i\omega h} =\frac{A_0(1+k\cos\Omega t)e^{i\omega t}}{2i\omega h}. \]

Thus, in this case, when the circuit does not separate the harmonic components of the emf, we obtain an undistorted quasi-stationary reproduction of the modulation. At each given instant the oscillation \(I\) in the circuit is as if the force \(Ae^{i\omega t}\) were indeed sinusoidal, i.e. as if \(A=\mathrm{const}\). This quasi-stationarity, which makes it possible to take account of the change of amplitude (or frequency, or phase) simply in the final formulas obtained and strictly valid for harmonic oscillation, is precisely what, and only what, can be contained in the words “a sinusoid with variable amplitude” (frequency, phase).

As \(\Omega/h\) increases, the modulation of \(I\) becomes more and more smoothed out (the coefficient of depth of current modulation is

\[ \frac{k}{\sqrt{1+\left(\frac{\Omega}{h}\right)^2}}, \]

and we come to the other limiting case \(\Omega \gg h\), when a true sinusoid is obtained in the circuit

\[ I=\frac{A_0e^{i\omega t}}{2i\omega h}. \]

Under these conditions, i.e. with selectivity that allows one to separate the carrier or the side frequencies from the spectrum, there is no sense in speaking of a “sinusoid with variable amplitude”; on the contrary, if Fleming’s terminology is used, the “reality” of the carrier and side frequencies appears with complete clarity. In exactly the same way, in acoustics it makes sense to speak of beats only so long as our ear cannot distinguish two tones of different pitch, but hears one tone of variable loudness. Here the receiving apparatus is the ear, and the criterion is its resolving power.

The simplicity of the quasi-stationary treatment, which transfers results obtained for harmonic oscillations to nonharmonic oscillations, has more than once made one forget that it is valid only for modulations slow in comparison with the settling time of the receiving apparatus, i.e. in the absence of separation

(as opticians say, resolution) of individual spectral components by these instruments.

Naturally, a sinusoid with variable frequency was not left aside either. At one time in radio engineering there existed the opinion that frequency modulation, unlike amplitude modulation, solves the problem of “crowding” on the air, and even a corresponding patent was issued to Robinson. The reasoning was as follows.

With amplitude modulation we have a wave band whose width is set by the transmitted sound frequencies and, consequently, cannot be reduced at our discretion. With frequency modulation we have “one single” frequency

\[ \omega(t)=\omega_0(1+k\cos \Omega t), \]

oscillating in the interval from \(\omega_0(1-k)\) to \(\omega_0(1+k)\), whose width \(2\omega_0 k\) can be made arbitrarily small by reducing \(k\), without any restrictions with respect to the modulation frequency \(\Omega\). If we now take a very selective receiver and tune it so that the interval \(2\omega_0 k\) lies on the slope of the resonance curve, then the amplitude of oscillations in the circuit will vary along the resonance curve in step with the change of \(\omega(t)\). Frequency modulation will be transformed into amplitude modulation, and all the deeper the steeper the resonance curve, i.e. the higher the selectivity of the receiver...

In reality everything will be exactly the opposite. The resonance curve is the curve of stationary amplitudes established under the action of a sinusoidal emf of one frequency or another. If the frequency changes, then the amplitude values prescribed by the resonance curve will not have time to establish themselves, and the smoothing of the modulation will be the stronger, the faster the modulation is in comparison with the time of establishment. In the spectral approach this means that, expanding the force in a harmonic series, we obtain in this case the carrier \(\omega_0\) and the side frequencies \(\omega_0 \pm \Omega\), \(\omega_0 \pm 2\Omega \ldots\), and, using—now quite legitimately—the resonance curve, we obtain the better reproduction of the modulation the wider the resonance curve (\(h \gg \Omega\)), i.e. the lower the selectivity of the circuit. Conversely, a circuit with very high selectivity (\(h \ll \Omega\)) will take us out of the quasi-stationary conditions, responding to each harmonic component of the spectrum separately.

3. CAN A SIGNAL BE HEARD BEFORE IT ARRIVES?

If a sufficiently selective apparatus can pick out individual harmonic oscillations of a spectrum, i.e., while under the action of a nonsinusoidal force, perform a harmonic oscillation with the frequency to which it is tuned, then we encounter a new “paradox.” A sinusoidal oscillation in an apparatus means an eternal oscillation, lasting in particular from time immemorial. How, then, is this to be reconciled with the fact that the acting force can

be a signal, for example a telegraph signal, beginning only from the moment \(t=0\)? It turns out that our sufficiently selective resonator can sense the signal before its arrival and is thus capable of predicting the future... Why, then, in reality, does no arbitrarily high selectivity of the resonator make it possible in such cases to pick out from the Fourier expansion an eternally lasting separate sinusoidal component?

The point is that a force which begins to act from some moment is thereby a nonperiodic force, and therefore its spectrum contains no separate harmonics, but is continuous. It is not a series, but a Fourier integral, i.e. a continuous set of sinusoids, from which any arbitrarily narrow pass band will always cut out not one, but an infinite set of sinusoids. In order for a separate harmonic to be singled out, the force must be expressed by a Fourier series, i.e. be periodic or almost periodic. But then no contradiction arises, since the force itself acts, repeating (or almost repeating), from \(t=-\infty\). There is nothing surprising in the fact that, say, periodically repeated telegraph signals cause a sufficiently selective (slowly settling) circuit to “ring” continuously. The paradox arises only when the force did not act at all before the moment \(t=0\), but, as has been said, in this case it is impossible to single out a separate sinusoid, since there are no such sinusoids in the spectrum.

True, even here there is as yet no exhaustive answer. The following remains unclear. In the continuous set of sinusoids in the spectrum of the signal \(f(t)\):

\[ f(t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} g(\omega)e^{i\omega t}d\omega, \tag{9} \]

the spectral densities \(g(\omega)\) are precisely such that all these sinusoidal oscillations mutually cancel up to the moment \(t=0\). Our apparatus, having passability \(\psi(\omega)\), gives the response:

\[ f_1(t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} g(\omega)\psi(\omega)e^{i\omega t}d\omega, \]

where the function \(\psi(\omega)\) may be arbitrarily sharp*). It seems very strange that the mutual cancellation of the sinusoids occurs not only for specially calculated \(g(\omega)\), but also for any
\(g_1(\omega)=g(\omega)\psi(\omega)\). It remains to prove that this is indeed so.

*) Generally speaking, the function \(\psi(\omega)\) is complex, i.e. \(\psi(\omega)=A(\omega)e^{i\varphi(\omega)}\). Radio engineers call \(A(\omega)\) and \(\varphi(\omega)\) the frequency and phase characteristics of the apparatus. This terminology is generally accepted, but somewhat illogical, since for \(A(\omega)\) the name is taken from the argument \(\omega\), and for \(\varphi(\omega)\) from the function \(\varphi\). It would be more natural to call \(A(\omega)\) the amplitude characteristic.

Let us again take a simple resonant circuit in which the force \(e^{i\omega t}\) produces the oscillation

\[ \psi(\omega)e^{i\omega t}=\frac{e^{i\omega t}}{\omega_0^2-\omega^2+2i\omega h}, \]

and hence the force (9) produces the oscillation

\[ f_1(t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} \frac{g(\omega)e^{i\omega t}\,d\omega}{\omega_0^2-\omega^2+2i\omega h}. \tag{10} \]

It is necessary to show that if, for \(t<0\), \(f(t)=0\), then also \(f_1(t)=0\), whatever \(\omega_0\) and \(h\) may be.

For this purpose let us turn to the plane of the complex variable \(\omega\) (Fig. 1). The path of integration is the real axis. The integrand in (10) has two poles:

\[ \omega=ih\pm\sqrt{\omega_0^2-h^2}, \]

which, for every \(h>0\), lie in the upper half-plane. We can close the path of integration by an infinitely large semicircle, if we choose the latter so that the integral along it adds nothing to (10). This choice means either closing above, where \(\operatorname{Im}\omega>0\), or below, where \(\operatorname{Im}\omega<0\). To make this choice, substitute in (10) the expression for \(g(\omega)\):

\[ g(\omega)=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty} f(\xi)e^{-i\omega \xi}\,d\xi. \tag{11} \]

We have at once taken here integration only over \(\xi>0\), since for \(\xi<0\), \(f(\xi)=0\). Substitution in (10) gives:

\[ f_1(t)=\frac{1}{2\pi}\int_{0}^{\infty} f(\xi)\,d\xi \int_{-\infty}^{+\infty} \frac{e^{i\omega(t-\xi)}\,d\omega}{\omega_0^2-\omega^2+2i\omega h}. \tag{12} \]

In the exponent of the exponential in (12) there stands

\[ i\omega(t-\xi)=i(\operatorname{Re}\omega+i\operatorname{Im}\omega)(t-\xi) =-\operatorname{Im}\omega(t-\xi)+i\operatorname{Re}\omega(t-\xi). \]

Thus, when \(t-\xi>0\), the exponential decreases on going into the upper half-plane \((\operatorname{Im}\omega>0)\), and when \(t-\xi<0\)—into the lower one.

We are interested in \(f_1(t)\) for \(t<0\). But if \(t<0\), then certainly \(t-\xi<0\), since \(\xi>0\). Hence we must close the path of integration—

of integration in the lower half-plane. Since it does not enclose any poles in this case, \(f_1(t)=0\), as was required to prove.

If \(t>0\), then for \(\xi>t\) we must again close the path in the lower half-plane, and this part of the integral with respect to \(\xi\) (from \(t\) to \(\infty\)) gives zero. There remains the part for which \(\xi<t\), and the path of integration must be closed above. It then encloses both poles of the integrand, and we obtain:

\[ t>0,\quad f_1(t)=\frac{1}{2\pi}\int_0^t f(\xi)\,d\xi \oint \frac{e^{i\omega(t-\xi)}\,d\omega}{\omega_0^2-\omega^2+2i\omega h} = \]

\[ =\frac{1}{2\pi}\int_0^t f(\xi)\,d\xi\, \frac{1}{2\sqrt{\omega_0^2-h^2}} \left\{ \oint \frac{e^{i\omega(t-\xi)}\,d\omega} {\omega-ih+\sqrt{\omega_0^2-h^2}} - \right. \]

\[ \left. - \oint \frac{e^{i\omega(t-\xi)}\,d\omega} {\omega-ih-\sqrt{\omega_0^2-h^2}} \right\}. \]

By Cauchy’s theorem the residues of the two integrals are respectively

\[ 2\pi i e^{\,i\left(ih-\sqrt{\omega_0^2-h^2}\right)(t-\xi)} \quad\text{and}\quad 2\pi i e^{\,i\left(ih+\sqrt{\omega_0^2-h^2}\right)(t-\xi)} . \]

Consequently,

\[ f_1(t)=\frac{1}{\sqrt{\omega_0^2-h^2}}\int_0^t f(\xi)\,d\xi\, \frac{i}{2} \left\{ e^{-\left(h+i\sqrt{\omega_0^2-h^2}\right)(t-\xi)} -\right. \]

\[ \left. - e^{-\left(h-i\sqrt{\omega_0^2-h^2}\right)(t-\xi)} \right\} = \]

\[ =\frac{1}{\sqrt{\omega_0^2-h^2}} \int_0^t f(\xi)e^{-h(t-\xi)} \sin\!\left[\sqrt{\omega_0^2-h^2}\,(t-\xi)\right]\,d\xi, \]

i.e. for \(t>0\) we have obtained that form of the solution which is usually obtained for the equation

\[ \ddot r+2h\dot r+\omega_0^2 r=f(t) \]

by the method of variation of arbitrary constants under the initial conditions

\[ r(0)=0,\quad \dot r(0)=0. \]

Of course, it cannot be proved—and in fact this is not so—that \(f_1(t)=0\) after the end of the signal. On the contrary, the more selective the resonator, the longer it will oscillate after the action of the force has ceased. But since this occurs after the action of the force, no contradiction can arise.

We have considered the special case of a resonant circuit, but from the proof it is clear that only one of its properties, playing the decisive role, was essential: the poles of the admittance \(\Phi(\omega)\) lay in the upper half-plane, i.e. the damping exponent \(h\) was posi-

is positive, the system was dissipative. Obviously, for any dissipative system the same will be true: all the attenuation exponents will be positive. If even one of them is negative, then the whole proof loses its force, and such a system may oscillate before the signal arrives. A physicist will at once say that it has simply self-excited because of regeneration. But the integral, of course, is not obliged to know whether these oscillations are a response to a signal that has not yet arrived or not. For it, causes and effects do not exist, and it says simply that in this case it is possible that \(f_1(t)\ne 0\) both for \(t>0\) and for \(t<0\), when \(f(t)=0\).

4. IDEAL FILTER

Let us now dwell on one question, which shows that we have not yet completely got rid of surprises.

We spoke of transmittances \(\psi(\omega)\), which for systems with \(n\) degrees of freedom are expressed as the reciprocal of a polynomial of degree \(2n\) in \(\omega\), and therefore possess \(2n\) poles. But why should we restrict ourselves to such \(\psi(\omega)\), especially since calculating them for a complex system is rather cumbersome? Would it not be simpler, instead of writing the differential equations of a complex circuit, to specify \(\psi(\omega)\), as practitioners usually do, by some convenient function that approximates the experimental frequency and phase characteristics of the apparatus sufficiently well. Let us take, in particular, an ideal filter, for which

\[ \psi(\omega)= \begin{cases} 1 & \text{in the frequency band from } \omega_0-\Delta \text{ to } \omega_0+\Delta,\\ 0 & \text{everywhere outside this band.} \end{cases} \]

The response of such a filter to the force (9) will be

\[ f_1(t)=\frac{1}{\sqrt{2\pi}}\int_{\omega_0-\Delta}^{\omega_0+\Delta} g(\omega)e^{i\omega t}\,d\omega . \tag{13} \]

Can it be proved that this response is equal to zero before \(t=0\), i.e. before the beginning of the signal? Not at all; on the contrary, it is easy to verify the opposite.

Indeed, substituting in (13) expression (11) for \(g(\omega)\) and carrying out the integration with respect to \(\omega\), we obtain

\[ f_1(t)=\frac{1}{\pi}\int_0^\infty f(\xi)e^{i\omega_0(t-\xi)} \frac{\sin \Delta(t-\xi)}{t-\xi}\,d\xi . \]

Suppose, for definiteness, that the signal beginning at the instant \(t=0\) [this has already been taken into account in (11)] is an oscillation

\(f(t)=e^{i\omega_0 t}\), lasting until the moment \(t=T\). Then \(f(\xi)=e^{i\omega_0 \xi}\) in the interval \((0,T)\), and \(f_1(t)\) takes the form:

\[ f_1(t)=\frac{e^{i\omega_0 t}}{\pi}\int_0^T \frac{\sin\Delta(t-\xi)}{t-\xi}\,d\xi = \frac{e^{i\omega_0 t}}{\pi} \int_{\Delta(t-T)}^{\Delta t}\frac{\sin x}{x}\,dx. \tag{14} \]

The last expression was obtained by the change of variable of integration \(\Delta(t-\xi)=x\).

It follows from (14) that \(f_1(t)\) tends to zero both as \(t\to\infty\) and as \(t\to-\infty\), but, generally speaking, for \(t<0\), i.e. before the beginning of the signal, the response \(f_1(t)\ne0\). Thus, our filter receives the signal before its arrival. What is the matter here?

The answer is already contained in the very formulation of the question. We have invented an “apparatus” that patently cannot be realized by means of linear devices, i.e. devices described by linear differential equations. If, however, we wish to resort to nonlinear systems (say, of the type of autoparametric filters, or to systems containing relays), then we must immediately forget about Fourier expansions and, in general, about any decompositions whatever of the force, since in nonlinear systems there is no superposition principle. The transmittance \(\psi(\omega)\) may in this case be understood as the set of those stationary regimes which occur in our system under the action of a sinusoidal force of one or another frequency \(\omega\), but knowledge of \(\psi(\omega)\) makes it possible to draw no conclusions about what will result from the simultaneous action on the apparatus of even two sinusoids, not to mention a force with a continuous spectrum. Thus, a linear apparatus acting according to (13) is indeed a mathematical fiction, like the advanced potentials in electrodynamics.

5. WHAT IS A “SPECTRUM” THAT CHANGES WITH TIME

Suppose that a radio station emits the modulated oscillation (6), but not forever, rather from the moment \(t=0\) to \(t=T\). What do we mean when we say that the spectrum (6) exists only during the operation of the station, while before and after this interval of time it does not exist? By this we understand that whatever selective receiver, easily separating the carrier and the side frequencies, we might take, with the station not operating we shall detect neither this carrier nor the side frequencies. Similarly, when we say that the spectrum of a spark disappears together with the extinction of the spark itself, we imply the disappearance of that pattern of spectral lines which we observe in a spectroscope. Thus, what is at issue is the response of the receiving apparatus, and not the Fourier expansion of the force acting on this apparatus. But then at once there emerges a distinction between those spectral components of the force which our apparatus separates-

resolves, i.e., which are separated in frequency by more than the width of its resonance curve, and those which it does not resolve.

Let us first take an example in which only periodic modulations of the intensity occur. Suppose that in the radiation of the radio station (6), in addition to modulation with the sound frequency \(\Omega\), there occurs (say, because of the power supply) also a very slow modulation with frequency \(\Omega'\), corresponding, for example, to one period per minute. Then the oscillation of the emf acting on the receiver will be

\[ \begin{aligned} &A_0(1+k'\cos\Omega't)(1+k\cos\Omega t)e^{i\omega t}=\\ &\quad =A_0(1+k'\cos\Omega't)\left\{e^{i\omega t}+\frac{k}{2}e^{i(\omega+\Omega)t}+\frac{k}{2}e^{i(\omega-\Omega)t}\right\}=\\ &\quad =A_0e^{i\omega t}+\frac{k'A_0}{2}e^{i(\omega+\Omega')t} +\frac{k'A_0}{2}e^{i(\omega-\Omega')t} +\frac{kA_0}{2}e^{i(\omega+\Omega)t}+\\ &\qquad +\frac{k'kA_0}{4}e^{i(\omega+\Omega+\Omega')t} +\frac{k'kA_0}{4}e^{i(\omega+\Omega-\Omega')t} +\frac{kA_0}{2}e^{i(\omega-\Omega)t}+\\ &\qquad +\frac{k'kA_0}{4}e^{i(\omega-\Omega+\Omega')t} +\frac{k'kA_0}{4}e^{i(\omega-\Omega-\Omega')t}. \end{aligned} \tag{15} \]

The last two lines constitute the Fourier expansion of our emf, consisting of 9 true sinusoids. As a result of the very slow modulation \(\Omega'\), the carrier \(\omega\) and the side frequencies \(\omega\pm\Omega\) are each split into three very close frequencies. Fig. 2 gives an idea of the form of the spectrum, but, of course, the scales in it are not maintained. Indeed, \(\Omega\) corresponds, say, to 1000 cycles, while \(\Omega'\) to \(1/60\) of a cycle.

Fig. 2.

The circuit with which we wish to obtain good reproduction of the sound modulation must have a resonance curve broader than \(2\Omega\). The oscillation in such a circuit will then be quite quasi-stationary, i.e., will be described by the upper line of (15)—a sinusoid whose amplitude changes with the frequencies \(\Omega\) and \(\Omega'\). Intending to separate the carrier \(\omega\) and the side frequencies \(\omega\pm\Omega\), we take a more selective circuit, whose resonance curve is much narrower than \(\Omega\). But if this curve is much broader than \(\Omega'\)—and in practice this will be so, since we cannot construct circuits with decrements of order \(10^{-7}\) and smaller—then the response of such a circuit will correspond to the second line of (15). We have singled out the carrier \(\omega\) or the side frequencies \(\omega\pm\Omega\), but the amplitude of each of these “spectral components” will quasi-stationarily repeat the very slow modulation \(\Omega'\).

Similarly, the situation is the same in the case when the oscillation (6) is modulated by the function

\[ F(t)= \begin{cases} 1 & \text{for } 0<t<T,\\ 0 & \text{outside the interval }(0,T). \end{cases} \tag{16} \]

The harmonic expansion of this function is given by the Fourier integral

\[ F(t)=\frac{1}{2\pi i}\int_{-\infty}^{+\infty}\frac{1-e^{-i\alpha T}}{\alpha}e^{i\alpha t}\,d\alpha, \]

whose density has a maximum at the point \(\alpha=0\), equal to \(\dfrac{T}{2\pi}\), and is the sharper the larger \(T\) is. The oscillation (6), lasting from \(t=0\) to \(t=T\), can therefore be represented in the form:

\[ \begin{aligned} &A_0F(t)\left(1+k\cos\Omega t\right)e^{i\omega_0 t} =A_0F(t)\left\{e^{i\omega_0 t}+\frac{k}{2}e^{i(\omega_0+\Omega)t} +\frac{k}{2}e^{i(\omega_0-\Omega)t}\right\}=\\ &=\frac{A_0}{2\pi i}\int_{-\infty}^{+\infty} \frac{1-e^{-i(\omega-\omega_0)T}}{\omega-\omega_0}e^{i\omega t}\,d\omega +\frac{kA_0}{4\pi i}\int_{-\infty}^{+\infty} \frac{1-e^{-i(\omega-\omega_0-\Omega)T}}{\omega-\omega_0-\Omega}e^{i\omega t}\,d\omega+\\ &\quad+\frac{kA_0}{4\pi i}\int_{-\infty}^{+\infty} \frac{1-e^{-i(\omega-\omega_0+\Omega)T}}{\omega-\omega_0+\Omega}e^{i\omega t}\,d\omega. \end{aligned} \tag{17} \]

In the integrals we have made a change of the variable of integration, putting \(\omega=\omega_0+\alpha\) in the first, \(\omega=\omega_0+\Omega+\alpha\) in the second, and \(\omega=\omega_0-\Omega+\alpha\) in the third.

The lower expression is the Fourier expansion of our oscillation, representing a superposition of three continuous spectra with maxima at the frequencies \(\omega_0\) and \(\omega_0\pm\Omega\). Thus, the presence of a beginning and an end in oscillation (6) has led to the fact that the individual spectral lines \(\omega_0\) and \(\omega_0\pm\Omega\) have been smeared out, turning into maxima of a continuous spectrum. The width of these maxima is of the order \(1/T\) and, consequently, is approximately 1 cycle if the oscillation lasts for \(T=1\) sec. Once again, a receiver with realistically attainable selectivity will be able to separate the maxima from one another, but will pass entirely the whole essential frequency band around each maximum. And this means that the receiver will reproduce the modulation \(F(t)\) quasi-stationarily and, consequently, the carrier \(\omega_0\) and the side frequencies \(\omega_0\pm\Omega\) will appear at the instant \(t=0\) and disappear at the instant \(t=T\) [the second line of (17)]. By what was proved in the preceding section, we may assert even more: no matter what the selectivity of the receiver, neither the carrier nor the side frequencies will produce a response before \(t=0\). However, after \(t=T\) the response will last the longer, the higher the selectivity of the receiver.

When it is said that the radiation spectrum of a radio station changes from one word uttered before the microphone to another, and even from syllable

with respect to the syllable, we imagine that the response of a filter selecting some narrow portion of the side band of the spectrum is constantly “breathing” in step with the transmitted speech or music. This means that our filter, passing a band, for example, from 1000 to 1100 cycles, is not capable of separating spectral components that are less than 100 cycles apart. Since the change of syllables occurs with frequencies on the order of 10 cycles, at the output of the filter, at a frequency of about 1000 cycles, we obtain a quasi-stationary imprint of the change of syllables.

In conclusion, a few words about the spectral optical apparatus. Here everything that has been said appears especially vividly, since a spectroscope gives a picture in which the frequency scale is presented all at once in the form of a spatial scale.

Diffraction at the objective of the tube leads, as is known, to the fact that even with ideally monochromatic light and an infinitely narrow collimator slit we obtain, in the focal plane of the tube, at the point corresponding to the light frequency \(\omega\), not a separate line, but only the principal maximum of a continuous diffraction pattern. The spectroscope resolves only sufficiently separated frequencies, whose principal maxima do not overlap to a significant degree. For closer frequencies we have a superposition of oscillations at one and the same place in the spectral pattern and, consequently, observe at this place the result of the addition of these oscillations.

Because the source under investigation sends light into the spectroscope not eternally, but from the moment \(t = 0\) to \(t = T\), each monochromatic component of the radiation (if it is assumed that such components exist at all) is modulated by the function (16). If \(T\) is greater than \(10^{-11}\) seconds, then modern spectroscopes are not capable of detecting the broadening of lines arising from this modulation. The principal maxima corresponding to sinusoidal components of the oscillation \(F(t)e^{i\omega t}\) practically lie on top of one another, and we shall see a single maximum at the frequency \(\omega\), modulated by the function \(F(t)\); i.e., appearing and disappearing together with the ignition and extinction of the source.

Thus, the “variable spectrum” is a Fourier expansion, not one carried through to the end. Its spectral amplitudes depend on time—varying quasi-stationarily in accordance with those frequency intervals that the given apparatus does not resolve. Fourier’s theorem gives expansions that physically could be realized only by an apparatus with infinitely great resolving power; real apparatuses, however, possess finite resolving power, and our spectral intuition, drawn from the behavior of such apparatuses, refers to “undecomposed” spectra consisting of “sinusoids with variable amplitudes, frequencies, or phases.”