80 Chapter
4
The Ideal Detector and Noise Limitations
currents, we shall again use/to distinguish it from optical or infrared
frequencies. The energy per mode from Eq. (1.8) becomes kT since hf
« kT, and the energy per unit length becomes
du = kTdN IL = (2kT/c)df (4.7)
Onehalf of this energy is flowing to the right and the other half to the
left at a velocity c, so that the respective power flows, dP+ and dP, are
each given by
cdu^

kTdf.
If we denote B as the effective electrical
bandwidth of a circuit, then the resistor must be emitting and absorbing
a total thermal power of
kTB.
Now this power is not constant, since we
know from Eq. (1.1) that the probability distribution of expected energies
on the transmission line obeys the Boltzmann factor. Since the prob
ability of observing a specific energy is Boltzmann distributed so also is
the instantaneous power P. We may thus write
V(P) =
le''l^
(4.8)
P
and the mean square fluctuation of P, from Eq. (4.3), becomes
oo
oo
—PIP
*^
p^
=
j p^p(P)dP =JP^ ^^=^
dP =
(P)^
J x^e'^'dx
=
2(W (4.9)
0 0 0
:.AP'^
=(P)^
Therefore the rms power fluctuation is equal to the mean or kTB. This
fluctuation is the available noise power from the resistor, and it is a
simple circuit exercise to show that the equivalent shunt noise current is
given by
.1 = ^ (4.10)
The exponential distribution of expected power corresponds to a
Gaussian distribution of expected current and the thermal noise current