frequency in Hz.
In view of the above we can write
dP
n
=
hf
e
hf/KT

1
df
hf
KT
→
KTdf
for power available over an infinitesimal bandwidth
df
. Hence, over a finite bandwidth
B
in the frequency range
f
KT/h
, where
dP
n
is independent of
f
, the available noise
power of a resistor is
P
n
=
KTB.
It is convenient to define a noise phasor
V
n
, with a meansquare denoted as

V
n

2
,
such that
P
n
=
KTB
≡

V
n

2
8
R
B.
The noise phasor
V
n
represents a cosinusoid that carries the same average power as
noise voltage
v
(
t
)
over a unit bandwidth
8
, and can be treated as a regular voltage phasor
8
Any bandwidth unit can be employed here, e.g., Hz, kHz, or even
μ
Hz, as long as the same unit is
used to express
B
in the formula for
P
n
.
120
4 Radiation, antennas, links, imaging
in circuit calculations in superposition with phasors defined for the same frequency
f
.
Noise phasors
V
n
and
V
m
due to di
ff
erent resistors in a circuit are assumed to have (by
definition) independent random phases so that

V
n
+
V
m

2
=

V
n

2
+

V
m

2
is true
(see Example 1 below).
So far we talked about resistor noise, avoiding the subject in the title of this section,
namely, antenna noise. It turns out, however, that all the formulae above for
P
n
,
dP
n
, as
well as noise phasor
V
n
, also apply to the noise output of antennas immersed in blackbody
radiation.
To see how and why, assume that an antenna, terminated by a matched load
R

jX
, is
in thermal equilibrium with a background described by Planck’s blackbody distribution
E
(
f, T
)
df
=
8
π
c
3
hf
3
e
hf/KT

1
df
hf
KT
→
8
π
c
3
KTf
2
df
for electromagnetic energy density
E
. The antenna load resistance
R
at the blackbody
temperature
T
will produce and deliver an average noise power
KTB
to its matched
antenna over a bandwidth
B
, only be radiated out into the noisy background (assuming a
lossless antenna for simplicity). This, of course, would lead to a cooling of the load
unless
the antenna picks up and delivers an equal amount of noise power,
KTB
, back to the
load. Since a net heat exchange is not possible between systems at equal temperatures —
the load and the background, in this case — it follows that the antenna must indeed pick
up an available noise power of
KTB
from incident blackbody radiation over a bandwidth
B
[e.g.,
Burgess
, 1941].
The available noise power of antennas within a bandwidth
B
is always expressed as
P
n
=

V
n

2
8
R
B
=
KTB
whether or not the antenna is actually immersed in blackbody radiation. Hence,
T
in
the formula above represents an
equivalent antenna temperature
which accounts for the
radiation picked up from the environment, a sum of a large number independent random
cosinusoids with a mean square amplitude

V
n

2
in each unit bandwidth. We envision
an open circuit voltage phasor
V
1
+
V
2
+
V
3
+
· · ·
where each component is a randomphased voltage phasor due to an independent ra
diation source, sources like radio stars, galaxies, our galactic core, etc., (all of which
constitute signals for a radio astronomer), as well as atmospheric lightning and thermal
emissions from atmospheric gases, and so on. The meansquared value of the large sum