Response of a Linear Rectifier to Signal and Noise* 



By W. R. BENNETT 



WHEN the input to a rectifier contains both signal and noise com- 

 ponents, the resultant output is a complicated non-linear function of 

 signal and noise. Given the spectra of the signal and noise input waves, 

 the law of rectification, and the transmission characteristics of the input 

 and output circuits of the rectifier, it should, in general, be possible to 

 describe the spectrum of the resultant output wave. Before discussing the 

 solution of the general problem, we shall derive some results of a simpler 

 nature, which do not require a consideration of the distribution of the signal 

 and noise energies as functions of frequency. 



I. Direct-Current Component of Output 



A quantity of considerable importance is the average value of the output 

 amplitude. This is the quantity which would be read by a direct-current 

 meter. Calculation of the average or d-c response can be performed in terms 

 of the distribution of instantaneous output amplitudes in time. The dis- 

 tribution of output amplitude can be computed from the distribution of 

 instantaneous input amplitudes and the law of rectification. 



As an example, we shall compute the average current obtained from a 

 linear rectifier when the input to the rectifier consists of a sinusoidal signal 

 with random noise superposed upon it. The probability density function 

 of the signal voltage is first determined, and the result given in (3). The 

 corresponding probability density for the voltage of the noise is well known 

 and is given in (4). The distribution of occurrence of the resultant in- 

 stantaneous amplitudes of the combined noise and signal voltages is then 

 computed by the rules of mathematical probability, and the result is shown 

 in (7). The assumption that the rectifier is linear then leads directly to an 

 integral which yields the average current obtained from the rectifier. 



Let the signal voltage, £«, be given by 



Es = Po cos co/. (1) 



The possible angular values of oit are uniformly distributed throughout the 

 range to lir. The range Es to E^ + dE^ corresponds to the range of values 

 of w/ comprised in the interval. 



Ea ^ , ^ E, -\- dEs ,^. 



arc cos — < a/ < arc cos (2) 



■to ^o 



•Published in Acous. Soc. Anier. Jour., Jan., 1944. 



97 



