An Optical Harmonic Analyzer * 



By H. C. MONTGOMERY 



An instrument which makes a Fourier Series Analysis of a func- 

 tion by optical means has recently been completed. The function 

 to be analyzed is supplied in the form of a variation in the density or 

 in the width of the transparent portion of a photographic film. The 

 analysis is performed by a direct evaluation of the integrals which 

 form the coefiicients in a Fourier Series, and the results are theo- 

 retically exact in the sense that the measurement of each harmonic 

 is independent of the other harmonics which may be present in the 

 function. The operation of the instrument is largely automatic, 

 and is rapid enough so that 30 harmonics can be measured in 

 about a minute and a half.^ 



A PERIODIC function can be represented for all values of the var- 

 ■^ ^ iable by a Fourier Series. A function which is not periodic can 

 be so represented between any finite limits, although the series may 

 be entirely unlike the function beyond these limits. If a function is 

 approximately periodic, the Fourier Series representing adjacent 

 portions of it will generally be approximately alike. 



Although in general an infinite number of terms is required to 

 represent a function exactly, it is common experience that a great 

 many functions of practical interest can be closely approximated by a 

 series of from ten to thirty terms. - 



*» 

 Principle of Operation 



The principle of this analyzer was suggested by E. C. Wente of 

 these Laboratories.^ It may be outlined as follows. 



The Fourier Series expansion of a function is given by either of the 

 following equivalent expressions.^ 



* Presented at Meeting of Acoustical Society of America, Washington, D. C, 

 May 3, 1938. 



1 For comparison, analysis to 30 harmonics on the Henrici type instrument re- 

 quires five or six hours. A resonance analyzer, such as the vibrating reed type, can 

 complete an analysis in a few seconds, but the phases will not be given, and if the 

 function is provided in graphical form it must be converted into an electrical or 

 acoustic wave form repeated enough times for the resonant elements to reach a steady 

 state response. 



- A description of a number of the more important methods of harmonic analysis, 

 together with a bibliography, is contained in "Sound Analysis," H. H. Hall, Jour. 

 Acous. Soc. Anier., vol. 8, pp. 257-262, April 1937. 



2 U. S. Patent No. 2,098,326. 



* The expressions in this form appi)' when the fundamental period is 2ir. There is 

 no loss of generality, as the scale of abscissa can alwaj's be so chosen as to conform 



406 



