DEVICE PERFORMING THE FOURIER TRANSFORM

Max Born\(^{1,2}\) is credited with the idea of a very elegant device, which is a special development of the photoelectric harmonic analyzer proposed in 1938 by Montgomery.\(^{3}\) An almost identical device was constructed almost simultaneously, but apparently independently, by Brown and Littleton,\(^{4}\) and the two designs proved to be very similar. The device in question automatically performs the Fourier transform for an arbitrary function \(f(x)\) specified on a finite interval, when this function is represented either graphically or “photographically,” i.e., by a variation in the blackening density of a photographic film.

The Fourier transform for a function \(f(x)\) is expressed, as is known, by the relation

\[ g(y)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} f(x)e^{-yx}\,dx, \tag{1} \]

where \(g(y)\) is the conjugate Laplace function. The transform itself is sometimes also called the Laplace transform.

The actual computation of the integral (1) is carried out analytically only in the simplest cases, while graphically, for any rather complicated function \(f(x)\), it involves a great expenditure of time. The device described in the papers under review performs the required transform in a few seconds.

The principle of its operation is very simple and is based on the following. Integral (1) can be divided into a real and an imaginary part; in the first, \(e^{-yx}\) gives \(\cos yx\), and in the second \(\sin yx\), or \(\cos\left(yx+\frac{\pi}{2}\right)\). It is assumed, with respect to the function \(f(x)\), that it is real and is specified only on a finite interval \((a,b)\). For most practical cases the latter circumstance presents no inconvenience.

The problem of finding the function

\[ g(y)=h(\cos y)+ik(\sin y) \]

is thus reduced to finding two functions of the form

\[ \psi(y)=\int_a^b f(x)\cos(yx+\delta)\,dx \tag{2} \]

for the values \(\delta=0\) and \(\delta=\frac{\pi}{2}\). This operation is performed by the device.

For this purpose, first of all, the function \(f(x)\) is drawn on a definite scale, and a transparency is then cut out along its contour from black paper so that the area lying below the contour of the function remains transparent. The prepared transparency (or, if the function is represented not graphically but photographically, the corresponding frame of film, varying in density) is placed in front of the window of the device. If a uniform beam of light is directed onto the window, then the light flux transmitted by the transparency (or film) will be proportional to

\[ \int_a^b f(x)\,dx; \]

the limits of integration \((a,b)\) are determined, on the adopted scale, by the width of the window. If, however, the beam of light is not uniform, but varies sinusoidally in intensity along the \(x\)-axis with pe-

period \(\lambda=\frac{2\pi}{y}\), then the transmitted light flux will be proportional to the value of integral (2) for the given value of \(y\). In reality, it is necessary to take into account, first, that the depth of the sinusoidal “modulation” of the light beam is not equal to 100%, and, second, that in the case of functions \(f(x)\) that also take negative values, the transparency represents the function \(f(x)+\mathrm{const.}\), where the constant is equal to the maximum negative value of the function. This leads to the appearance of constant terms in expression (2) and, consequently, to a nonproportionality of the transmitted light flux to the function \(\psi(y)\). The corresponding additive constant must be introduced into the final readings of the instrument.

By the method explained below, which distinguishes the instrument described both from the original design of Montgomery, as well as from those in \(^{1,2}\) and in \(^{4}\), the spatial period \(\lambda\) of the illuminating beam can be varied continuously from \(\lambda=\lambda_{\min}\) \((y=y_{\max})\) to \(\lambda=\infty\) \((y=0)\). The light flux changing in this process behind the transparency represents (with the indicated limitations) the function \(\psi(y)\) as it changes from \(\psi(0)\) to \(\psi(y_{\max})\). The receiver of the light flux is a photoelectron multiplier. The pre-amplified photocurrent of the multiplier is fed to the vertical plates of a cathode-ray oscillograph, to the horizontal plates of which the corresponding time sweep is applied (in the Brown and Littleton instrument the connection to the oscillograph is somewhat different). With sufficiently rapid periodic variation of \(y\), the oscillograph beam draws directly on the screen the graph of the function \(\psi(y)\), which can be photographed and measured.

The figure explains the optical scheme of the instrument. Behind the light source \(A\) and the condenser \(B\) is located the “sine screen” \(C\), i.e., a screen whose transparency varies sinusoidally in the direction of a certain axis. By means of the adjusting screw \((D)\), the “sine screen” can be displaced along this axis in its annular mount \(E\), which makes it possible to change the angle \(\delta\) in formula (2). The ring \(E\) rotates, and the axis of rotation coincides with the optical axis of the instrument passing exactly through the center of the narrow slit \(G\), on which the lens \(F\) focuses the image of the “sine screen.” Along the length of the slit (whose width must be \(\ll \lambda_{\min}\)) a sinusoidal distribution of illumination is obtained, with a spatial period that changes when the “sine screen” is rotated from \(\lambda_{\min}\) to \(\infty\). The cylindrical lens \(H\) gives in its fo-

cal plane the light distribution in the form of vertical bands, illuminating the transparency located here. The bands are periodically compressed and expanded, following the rotation of the “sine screen,” and during one half-period of such variation the magnitude of the luminous flux gives the function \(\phi(y)\).

With the aid of lens \(J\), the luminous flux that has passed through the transparency is concentrated on the photocell (photomultiplier) \(K\), and the photocurrent, as already indicated, is fed to the oscillograph. Owing to the rotational motion of the “sine screen,” the graph of the function \(\phi(y)\) on the oscillograph screen is arranged symmetrically on both sides of the ordinate axis \(y=0\), i.e., it gives both \(\phi(y)\) and \(\phi(-y)\). To obtain a linear \(y\)-axis on the screen, the time sweep must be sinusoidal.

In the article² a number of photographs of the oscillograms obtained for several simple cases are presented. Comparison with the computed data gives a quite satisfactory result. The accuracy of the instrument can probably be increased without particular difficulty.

The instrument constructed by Brown and Littleton⁴ differs from the one described in that it is intended for the analysis of sound films. Here the photocurrent controls not the deflection of the oscillograph beam, but the brightness of the spot on the screen. With the aid of a sweep, the spot is stretched into a strip of the spectrum. As the film is drawn past the film gate, this spectrum changes continuously, and the resulting picture is photographed on synchronously moving motion-picture film. Thus, the harmonic analysis of a certain sound recorded on the film of a sound motion picture is carried out continuously. Such a method obviously has considerable practical value. Only one comment must be made. If the Boltz instrument obtains a decomposition that should, in principle, be regarded as exact, for \(\Phi\) is by definition equal to \(f(x)\) in \((a, b)\) and to zero outside this interval, then Brown and Littleton are dealing with a function which, generally speaking, is not zero outside the interval cut out by the window of the instrument. This means that their instrument admits an error, the greater the narrower the window and the lower the frequency of the given harmonic. Here there is manifested the familiar relation of uncertainty from the theory of the Fourier integral. If one strives to widen the window, i.e., the interval \((a, b)\), then the instrument will more accurately reproduce the true picture of the harmonics, but at the same time will lose in “resolving power,” in the sense of the possibility of assigning the spectrum to some definite interval of time.

It may be mentioned that the attempt made by the authors just mentioned to obtain the acoustic spectrum of a continuous sound process is not the only one. In particular, an analogous problem is solved by an installation constructed in the laboratory of the Bell company⁵. Here the principle of the heterodyne harmonic analyzer is used, and the recording is not made on film, but is continuously displayed on a small screen. The authors consider one application of their method to be the possibility of “visual hearing”: thus, a deaf person, after appropriate training, can read with his eyes a sound whose spectrum passes before him on the screen.

B. Bagaryatsky

CITED LITERATURE

1. Max Born, R. Fürth and R. W. Pringle, Nature, 156, 756 (1945).
2. R. Fürth and R. W. Pringle, Phil. Mag., 37, 1 (1946).
3. H. Montgomery, Bell Sist. Techn. Journ., 17, 406 (1938).
4. D. Brown and J. W. Littleton, Nature 160, 709 (1947).
5. Journ. Acoust. Soc. Amer., 18, 1–89 (1946).