STEADY STATE DEL A V AND APERIODIC SIGNALS 225 



envelope delay, this disturbance differs in that, in any finite range of fre- 

 quency, there are an infinity of sinusoids, the amplitudes of which need 

 not all be the same. For simplicity, we assume the actual filter to be 

 replaced by an idealized one in which there is no distortion within the band 

 and no transmission outside it. If the signal as a whole be represented by 

 a Fourier integral, we may obtain the desired disturbance, for an angular 

 frequency, oji, by integrating from wi — 5 to coi + 8. The disturbance may 

 be represented by ■ 



f(i) = real part oi M I exp [-a + i(o)t - d)] doo, (2) 



where M is a constant dependent on the magnitude of the signal and a 

 and 6 are functions of frequency and position which describe the spectrum 

 of the signal at various points in the system. 



The first step is to perform the indicated integration and express the 

 resulting function of time in a convenient form. For this we let 



€ = CO — COl . 



Since we are interested only in small values of e we may replace a by 



a — ai -\- aie, 



where ai and ai are the values of a and -— at coi . Similarly, 



oco 



e = di-\- d[e. 

 We define an instant, Tg , by 



T, = d[ , (3) 



and a time, t, by 



T= t- Te. (4) 



Substituting these in (2) and performing the integration, we get 



fit) = real part of 2M exp [ — ai + i{<j>iT — (di — coi^i)] 



{-a[-\-ir) 



If we introduce the angles, 



jS = arc tan , 



T 



and 



tanh ( — 8ai) 



7 = arc tan 



tanSr 



