598 
28 
or capacitance, on the frequency re- 
sponse, then the response to a square 
wave may be "stopped" on the screen 
of a cathode ray oscillograph, and 
the change in the response can be 
observed visually and continuously 
as the resistance, or capacitance, 
is altered. Thus the square-wave 
method is valuable to anyone build- 
ing an amplifier. Once the ampli- 
fier is built it may be necessary to 
determine the response to a given 
input transient signal. It is dif- 
Figure 17 - Illustration of the ficult to derive mathematically the 
Duhamel Integral response to a transient signal in 
terms of a known response to a square 
wave (23). This question may, however, be answered very conveniently if the 
response to a unit pulse is known, and for studying distortion, the square wave 
of adequate duration is sufficiently equivalent to it. The following analysis 
(24) shows how the response of any amplifier to a given signal may be calcu- 
lated provided the response to a unit pulse is known. Let 
E(t) be 1 for t 20 
' 
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t 
; the unit pulse, 
be 0 for t < 0 
A(t) be the response to the unit pulse, usually called 
the indicial admittance, 
S(t) be the input or excitation signal, and 
R(t) be the response of the amplifier to S(t). 
It will be assumed that the amplifier is a linear system, i.e., it 
is describable by linear differential equations. In that case the principle 
of superposition may be used. It is also necessary to assume that the coef- 
ficients of the differential equations of the system (amplifier) are con- 
stants. This means that the response is only a function of (t - 7) where T 
is the time at which the unit pulse is applied, and ¢ is the time at which 
the response is desired. 
Consider the input signal S as represented in Figure 1/. The sig- 
nal is to be considered as the sum of many rectangular pulses. Each pulse 
starts at a time Ar after the previous one. The height of the pulse is given 
by 
