18 



FUNDAMENTAL RELATIONS 



noise figure is used. The noise figure of a system 

 (taken here to be a receiver, for definiteness) is 

 defined as 



Pso/Psi' 



(34) 



where P„i = noise power (kTAj) from the antenna 

 which is being delivered to the receiver. 

 Pno = noise power at the output of the re- 

 ceiver, that is, the noise after the 

 amplifications and additions arising in 

 the receiver circuit. 

 Pjj = signal power from the antenna which is 



being delivered to the receiver. 

 P,o = signal power at the output of the 

 receiver, that is, the signal power after 

 detection and amplification have taken 

 place. 

 The ratio Ps./Pii is called the receiver gain. This 

 quantity is called g and must not be confused \x\ih 

 antenna gain G. Using equation (.32), equation (.34) 

 may be written 



The bandwidth A/ is measured by finding the area 

 under a curve of power-gain versus frequency and 

 equating this area to the area of a I'ectangle whose 

 width is interpreted as A/ and whose height corre- 

 sponds to the gain at the frequency at which the 

 gain is a maximum. 



:2.3.3 



Receiver Sensitivity 



Frequently receiver sensiti\'ity is defined by the 

 assumption that a received signal can be discrim- 

 inated when its output power is equal to the noise 

 output power. This assumption, while true for a 

 large class of receivers, is too rough for radar re- 

 ceivers. The method given here will explain the 

 procedure used for calculating the minimum dis- 

 cernible power of receivers for which the assumption 

 is true. The sensitivity of radar receivers is con- 

 sidered in Section 2.3.5. 



Referring to equation (34), the assumption that 

 signal output power is equal to noise output power 

 means that Pj„ = P„o. Hence 



But P,i is, on the as.sumption discussed above, just 

 the minimum discernible signal power, Pmin, at 

 the recei\'er input, that is, before amplification. 

 Hence, using equations (32) and (33), 



P„i„ = ATA/ . Fn^AX 10"^'A/ . F„ watt. (37) 



2.3.4 Measurement of the Noise Figure 



Remembering that the cases under discussion are 

 those for which the minimum discernible signal is 

 equal to the noise output power, equation (37) gives 

 an estimate of the minimum detectable power from 

 a measurement of the noise figure F, which may be 

 oljtained as follows. 



An antenna (or other signal generator) whose 

 impedance is matched to the receiver is connected 

 to the receiver. With the signal output reduced to 

 zero (so that the antenna fiu-nishes only noise power 

 to the receiver), the receiver gain is increased until 

 the noise gives a measurable output and the output 

 noise power is measured with a power meter. Now 

 a signal is impressed on the antenna and increased 

 to a point where the receiver output power is doubled, 

 and the input signal power is measured. Thus, 

 referring to eciuation (34), 



P P ■ 



Pso = Pno and F,, 



so that the measurement of the impressed signal 

 power indicated here gives P„. 



If the receiver consists of se\'eral elements in 

 cascade, including attenuators, amplifiers, and con- 

 verters, the overall noise figui'e can be compounded 

 from the noise figures and gains of the indi\'idual 

 components by means of the following equation : 



F = F 



F,„-l , P„3-l 



(38) 



F 



(36) 



gi gig2 



where F,, = overall noise figure, 



F„i, = noise figure of the A'th element, 

 (jt: = gain of the A'th element. 



In using this equation it is understood that the suc- 

 cessive stages are matched. 



It is clear from equation (38) that most of the 

 noise comes from the early stages of reception; in 

 high-frecjuency radar sets, it comes from the crystal 

 mixer and the first intermediate-frequency (i-f) 

 stage. This means of course that noise picked up at 



