FUNDAMENTAL RELATIONS 341 
receiver will be 
VAR, 
oo Nal Ld 31 
(Ra + R,)? } 
where V, is the rms value of the noise generated in 
the antenna. 
The noise power is maximum when the receiver is 
matched to its input; this maximum is 
V.2 
P, = —=kTAf watts (32) 
‘a 
by equation (30). Assuming equivalent temperature 
T = 290 degrees absolute, and measuring Af in 
megacycles, 
P,=4 X10 Af watt. (33) 
This result means that in an idealized receiver, 
noise is the thermal noise of an antenna of equivalent 
temperature 7’ = 290 degrees absolute, and the 
minimum detectable signal would be approximately 
equal to 4 X 10° Af watt. 
Noise Figure 
The sensitivity of a set cannot be described in 
terms of the thermal noise alone, because the set 
noise is usually several times the purely thermal 
noise. For this purpose another quantity called the 
noise figure is used. The noise figure of a system 
(taken here to be a receiver, for definiteness) is 
defined as 
zs Prof Pri 
z IDPs 
where P,,; = noise power (k7'Af) from the antenna 
which is being delivered to the receiver. 
Po = noise power at the output of the re- 
ceiver, that is, the noise after the 
amplifications and additions arising in 
the receiver circuit. 
P;; = signal power from the antenna which is 
being delivered to the receiver. 
P;, = signal power at the output of the 
receiver, that is, the signal power after 
detection and amplification have taken 
place. 
The ratio P;,/P;; is called the receiver gain. This 
quantity is called g and must not be confused with 
antenna gain G. Using equation (82), equation (34) 
may be written 
(34) 
os (35) 
The bandwidth Aj is measured by finding the area 
under a curve of power-gain versus frequency and 
equating this area to the area of a rectangle whose 
width is interpreted as Af and whose height corre- 
sponds to the gain at the frequency at which the 
gain is a maximum. 
Receiver Sensitivity 
Frequently receiver sensitivity 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 on page 342. 
Referring to equation (34), the assumption that 
signal output power is equal to noise output power 
means that P;, = Py»). Hence 
[f= ale (36) 
But P;; is, on the assumption discussed above, just 
the minimum discernible signal power, Pmin, at 
the receiver input, that is, before amplification. 
Hence, using equations (32) and (33), 
Prin= kKTAf- F, 4 X 10° Af- F, watt. (37) 
Measurement of the Noise Figure 
Remembering that the cases under discussion area 
those for which the minimum discernible signal is 
equal to the noise output power, equation (87) gives 
an estimate of the minimum detectable power from 
a measurement of the noise figure F,, which may be 
obtained 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 furnishes 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 tbe input signal power is measured. Thus, 
referring to equation (34), 
Ieee 1s 
Ps a Sl ey 
Py;  kTAf 
n and F, = 
so that the measurement of the impressed signal 
power indicated here gives F,. 
If the receiver consists of several elements in 
cascade, including attenuators, amplifiers, ahd con- 
verters, the overall noise figure can be compounded 
from the noise figures and gains of the individual 
components by means of the following equation: 
Fiy2—1 Fiy3—1 
f=) 
91 9192 
F, = Frat (38) 
