SPECTRUM ANALYSIS OF WAVES 363 



components, each of infinitesimal amplitude and so close together in fre- 

 quency as to cover the entire frequency range uniformly. 

 ' The continuous band type of frequency spectrum is just as characteristic 

 of non-periodic waves as the discrete spectrum is of periodic waves. This 

 can be shown as a logical extension of the Fourier series representation of 

 periodic waves. The transition from a frequency spectrum consisting of a 

 series of discrete frequencies to one consisting of a continuous band of fre- 

 quencies can be made by treating the non-periodic function as a periodic 

 function in which the period is allowed to become very large. As the period 

 approaches infinity the fundamental recurrence rate approaches zero, so 

 that the harmonics merge into a continuous band of frequencies. 



This does not of course change the basic realtionship between the fre- 

 quency spectrum of a wave and its magnitude-time function. The mag- 

 nitude-time function is still the sum of the components of the frequency 

 spectrum. Also the frequency spectrum can still be obtained frequency by 

 frequency, by averaging the product of the magnitude-time function and a 

 unit sinusoid at each frequency. However, the actual transformations 

 in the case of the non-periodic functions require summations over infinite 

 bands of frequencies and over infinite periods of time and so fall into the 

 realm of the Fourier and similar integral transforms. 



However, in any case the problem of spectrum analysis reduces to an 

 averaging process. The process can be performed by mathematical inte- 

 gration in all cases where a satisfactory analytical expression for the mag- 

 nitude-time function is available. Fourier analysis provides a very powerful 

 technique for setting up the necessary integrals in such cases. 



This averaging process can also be done graphically. It is apparent from 

 the theory that if the product of the magnitude-time function and the 

 sinusoid is sampled at a sufficient number of points, spaced uniformly over 

 the proper time interval, then the average of the samples gives the desired 

 value. This technique is fully treated elsewhere" so that it will not be con- 

 sidered in detail here. However, use will be made of it in a qualitative way 

 to augment the physical picture. 



Non-Linear Aspects 



The use of the frequency spectrum in transmission studies is generally 

 limited to cases where the system in question is linear; that is, where the 

 transmission is independent of the amplitude of the signal. However, the 

 same techniques can still be used on systems employing successive linear 

 and non-linear components, in cases where the transmission through the 

 non-linear elements is independent of frequency. Under these conditions, 

 the magnitude-time representation of the wave can be used in computing 



'Whittaker and Robinson, Calculus of Observations. 



