ELIMINATION OF REFLECTIONS 



47 



fore they can be considered useful only in establish- 

 ing the general shape of the frequency-response curve 

 but not in establishing its absolute level. Even the 

 shape may be in considerable error. The greatest care 

 is therefore required in using these methods to be 

 sure that one is actually measuring the quantity 

 desired. 



5.3.6 



Electric Signal Methods: Pulses 



A more satisfactory method of eliminating the ef- 

 fects of reflected waves in measurements is by the use 

 of pulses. Instead of a steady signal, a pulse corre- 

 sponding to a sinusoidal signal of finite duration is 

 emitted. This pulse reaches the receiver by the direct 

 path before the reflected pulses arrive. If the response 

 of the receiver can be measured in the interval before 

 their arrival, their effect is entirely eliminated. Since 

 the front of the earliest reflected pulse arrives at a 

 time AL/c after the beginning of the direct pulse, 

 the response measurement must be completed in a 

 time t„, < AL/c after the arrival of the beginning of 

 the direct pulse, where AL is the smallest difference 

 in path between a direct and reflected signal. It is 

 noted immediately that the pulse method has an im- 

 portant advantage over the warbled-signal and noise- 

 band methods in that the reflected waves play no part 

 in the measurement of response, whereas in the latter 

 methods a composite sum of the direct and reflected 

 signals is measured. 



The limitations of the pulse method are indicated 

 by the resolving power of the method. The Fourier 

 spectrum of a pulse of finite length At can be shown 

 to contain, essentially, frequencies covering a band of 

 width 



' At 



power would be that corresponding to a pulse dura- 

 tion of j ... or 



rp = j Tm <; 



/AL 



(23) 



That this is actually the case will be shown now from 

 other considerations. 



When a sinusoidal signal is applied to any electri- 

 cal circuit containing inductances or capacitances or 

 both, steady values of the currents and voltages are 

 not immediately attained. At first there are present, 

 in addition to the steady-state voltages and currents, 

 transient voltages and currents which gradually die 

 out with time. The time required for the transient 

 essentially to disappear is known as the time constant 

 of the circuit. For simple circuits this time constant t 

 is independent of frequency, but for more compli- 

 cated ones this may not be the case. For example, for 

 a capacity C and a resistance R in series or parallel 



RC. 



For an inductance L and a resistance R in series or 

 parallel 



R 



For a resonant circuit containing an inductance L, a 

 capacitance C, and a resistance R 



2L 



(21) 



when 



and 



R 



_L> *i 



LC 4L 2 ' 



centered at the signal frequency of the pulse. The re- 

 solving power thus becomes 



RP 



A/ 



f&t 



which, it would appear, could be indefinitely in- 

 creased by extending the duration of the pulse. This 

 is illusory, however, since what is measured at a time 

 t„, < AL/c must be independent of how long the 

 pulse continues after the period AL/c has elapsed. 

 One would perhaps guess that the actual resolving 



\4L- LC 



R_ 



1L 



(22) when 



_L<*i. 



LC 4L- 



For simple resonant circuits it is convenient to in- 

 troduce a quantity Q. = ^foL/R, where / is the reso- 

 nant frequency. Q essentially measures the number of 

 cycles at resonance required for the transient to die 



