730 BELL SYSTEM TECHNICAL JOURNAL 



Ratio of Noise to Signal 



It may be helpful, in conclusion, to attempt a theoretical estimate of the 

 ratio of noise to signal in the audio output under the condition of severe 

 impulsive interference. , 



The ratio of the peak value of the pulse to the signal amplitude at the 

 frequency detector output is given by the ratio of the peak values of /(/), 

 as plotted in figure 5, to the values of ^i in the table above. To obtain 

 a result of practical significance, however, the effect of the audio circuit 

 should be taken into account. In the absence of specific information on 

 the structure of this circuit, we assume that the peak value of a pulse at 

 its output is equal to the area, or moment, of the pulse at its input times 

 twice the audio cutoff frequency. This is true for an ideal ''square cutoff" 

 filter and not seriously in error for actual circuits. The area of the largest 

 pulse at the frequency detector output is approximately 2A/{a + A ) and 

 the value of Ai, the signal fundamental amplitude, can be approximated by 



a^ + A^ 



(For the example above, this approximation gives Ai = .8, .5 and .2 as 

 compared to the exact values, .848, .478 and .191.) In this way we arrive 

 at the following estimate of the peak ratio of noise to signal in the audio 

 output: 



Max. value of largest pulse _ oja fco\ 



Signal amplitude ir x q 



where Wo is the cutoff-frequency of the audio circuit, q the signal frequency 

 and X the frequency deviation ratio. Then xq is the "frequency swing" of 

 the transmitter, i. e., the maximum departure of the instantaneous frequen- 

 cy from its mean value. It is to be noted that this formula is free from the 

 detector circuit parameters, a, A, R, and indicates that, to a first approxi- 

 mation, at least, the maximum ratio of noise to signal depends only upon 

 the audio circuit cutoff frequency and the FM swing. Furthermore, this 

 establishes a ceiling for the interference that will not be exceeded no mat- 

 ter how large the impulses may be. 



Bibliography 



1. Variable Frequency Electric Circuit Theory with Application to the Theory of Fre- 



quency-Modulation, John R. Carson and Thornton C. Fry in Bell System Technical 

 Journal, Vol. XVI, No. 4, October 1937. 



2. The Detection of Frequency Modulated Waves, J. G. Chaffee in Proc. LR.E., Vol. 23, 



May 1935. (Bell Tel. Sys. Monograph B-863) 



3. Effects of Tuned Circuits upon a Frequency Modulated Signal, Hans Roder in Proc. 



LR.E., Vol. 25, December 1937. 



4. The Reception of Frequency-Modulated Radio Signals, Victor J. Andrew in Proc. 



LR.E., Vol. 20, May 1932. 



