5-7] NARROW BAND NOISE 259 



We suppose that the noise power is concentrated in the neighborhood of a 

 carrier frequency coc. Such a noise process can be constructed by modulating 

 a relatively low-frequency noise process by the carrier frequency. The 

 carrier frequency signal can be represented by either the in-phase or 

 quadrature component, and, in general, the narrow band noise will be 

 composed of both components. Denoting the low-frequency noise processes 

 corresponding to the in-phase and quadrature components about the carrier 

 by x{t) and y{t), the narrow band noise process denoted by z{t) can be 

 represented by 



z{t) = x{t) cos coc/ + y{t) sin oij. (5-67) 



In general, x{t) and jy(/) could be correlated and also might have dissimilar 

 features. But in most problems of practical interest they will be independ- 

 ent and have identical spectra and other statistical characteristics. If 

 the X and y processes did not have the same spectra and autocorrelation 

 functions, the narrow band process would depend upon time, as is apparent 

 in Equation 5-68 below. Requiring x and y to be independent makes the 

 spectrum of the narrow band process symmetrical about the carrier fre- 

 quency coc- We assume that x and y are independent and have identical 

 spectra. The autocorrelation function of the z process is computed as 

 follows : 



(Pz{t) = [xi cos ixiJi + y\ sin Wct]\[x2 cos 0)^/2 + y^ sin coo/2] 



= (i) XiXi [cos OOcT -j- cos C0c(2/ + t)] 



+ (I) Jl3'2 [cos WcT — cos C0c(2/ -(- t)] 



+ {h) ^ [sin coeT + sin co.(2/ + r)] (5-68) 



— (I) yiXi [sin cocT — sin coc(2/ -{■ r)] 



= (p{t) cos WcT 



where ^(t) denotes the autocorrelation function of the x and y processes. 

 The autocorrelation function ^z(r) is of exactly the same form as that of the 

 output of a product demodulator discussed in the preceding paragraph and 

 given in Equation 5-62. Thus the Fourier transform of .^^(t) giving the 

 power spectrum of the z process will be related to the spectrum of the x and 

 y processes, A^(co), in a manner similar to that indicated in Equation 5-63: 



A^.('^) = ihWio: - CO.) + A^(co -F COe)]. (5-69) 



From this expression, we see that the spectrum of narrow band symmetric 

 noise has the same form as the low-frequency modulating functions, but is 

 shifted to the vicinity of the carrier frequency. 



In a large class of radar systems, the transmitted and received signals 

 have the form of an RF carrier amplitude modulated by a low-frequency 

 waveform. In the majority of these systems, the modulation consists of a 



