5-7] 



NARROW BAND NOISE 



263 



graph 5-6 in connection with the discussion of a square-law device. Using 

 the results of that paragraph, the autocorrelation of r- is computed as 

 follows : 



ri^ra^ 



a'^ + la^Xx + la^xi + arx^- + arx-^- -f- a?-y ^ + a'-y-^ + 1ax\X'^ 



+ .yi-j'2- + 4^2—2 (5-81) 



= (^2 + 2cr2)2 + 4<72((72p2 + ^2^) 



= (2^2)2(1 + ^/yV)2 + (2a2)2[p2 + 2(.S'/A^)p]. 



The spectrum of the video signal plus noise has three components: (a) an 

 impulse at zero frequency representing the d-c, (b) a continuous portion of 

 the same shape as the spectrum of the component x and y processes repre- 

 senting beats between the signal and the noise, (c) a continuous portion 

 somewhat wider than the spectrum of the x and y processes representing 

 beats between various parts of the noise spectrum. Fig. 5-12 illustrates the 



Impulse 



Signal Power = a^/2 

 Noise Power ■=2D\N=u 

 S/N= a2/2(7" 



27r(ay4) 

 Impulse 



27rW' 



47r W 



Signal Plus Noise 



Angular ^ 27rW"^ 1^ 



Frequency 



Noise Alone 



Angular Frequency 



Fig. 5-12 Power Density Spectra of the Square of the Envelope of a Sinusoidal 

 Signal Plus Narrow-Band Noise. 



forms of the various spectra in a typical case. The spectrum of the x and y 

 processes is assumed rectangular with bandwidth W. The density of the 

 positive and negative portions of the narrow band spectrum is denoted by 

 D. The d-c level is equal to twice the sum of the signal and noise powers. 

 The portion of the continuous spectrum corresponding to (b) is rectangular, 

 of half the width of the narrow band spectrum (considering only positive 



