362 BELL SYSTEM TECHNICAL JOURNAL 



The process of finding the frequency spectrum of a complex wave from its 

 magnitude-time function has a simple mathematical basis. It depends on 

 the fact that the square of a sinusoidal wave has a positive average value 

 over any interval of time, whereas the product of two sinusoidal waves of 

 different frequencies will average zero over a properly chosen interval of 

 time."* 



In theory then, as the magnitude-time function of a complex wave is the 

 sum of all the components of the frequency spectrum, we have only to mul- 

 ti])ly this magnitude-time function by a sinusoidal wave of the desired 

 frccjuency and then average the product over the proper time interval to 

 find the component of the spectrum at this frequency.^ 



One physical interpretation of this procedure can be given in terms of 

 modulation theory. The product of the magnitude-time function with a 

 sinusoidal wave will produce the beat or sum and difference frequencies be- 

 tween the frequency of the sinusoid and each component of the frequency 

 spectrum. Thus, if the spectrum contains the same frequency, a zero beat 

 or dc term is produced, and this term may be evaluated by averaging the 

 product over an interval that is of the proper length to make all the ac 

 components vanish. 



The application of this principle for spectrum analysis is simple when the 

 magnitude of the wave in question is a periodic function of time. The very 

 fact that the wave is periodic is sufficient proof that the only frequencies 

 that can be present in the wave are those corresponding to the basic repeti- 

 tion rate and its harmonics. Thus the frequency spectrum is confined to 

 these specific frequencies and so it takes the form of a Fourier series. Know- 

 ing that the possible frequencies are restricted in this way, the problem of 

 finding the frequency spectrum of a complex periodic wave is reduced to one 

 of performing the above averaging process at each possible frequency. The 

 period of the envelope of the Complex Wave is the proper time interval for 

 averaging, and the integral formulation for obtaining this average is that 

 for determining the coefficients in a Fourier series. 



The principle holds equally well when the magnitude-time function is non- 

 periodic, but the concept is complicated by the fact that the frequency 

 spectrum in such cases is transformed from one having a discrete number of 

 components of harmonically related frequencies to one having a continuous- 

 band of frequencies.*' Such s]:)ectra contain infinite numbers of sinusoidal 



■• The i)roper time interval is generally some integral multiple of the period correspond- 

 ing to the difference in frequency of the two sinusoid waves. 



* In practice it is generally necessary to multiply by both sine and cosine functions 

 because of i)ossible phase differences. 



8 One exception to this statement is the fact that any wave made up of two or more 

 incommensurate frequencies is nonperiodic. Yet such waves will have a discrete spectrum 

 if the number of components is finite. This incommensurate case is neglected throughout 

 the discussion. 



