Noise Figure Notes
The drawing below illistrates the model used for the definition
and
use of "Noise Ratio" (NR) or "Noise Figure" (NF, which is "Noise Ratio"
expressed in dB) for a functional stage in a receiver. The input
signal power is given as Si, and is assumed to be corrupted
by thermal noise (kT0B) such that the signal to noise ratio
at the input is Si/kT0B. From the diagram,
it can be seen that the total noise at the output is the sum of two
contributions:
(1) the amplified noise which accompanied the signal, and (2) the noise
added by the stage and amplified by it. The total noise at the
output
is thus:
No = (AP)(kT0B)
+
(AP)(NR-1)(kT0B) = (AP)(NR)(kT0B)
or, expressed in dB:
No (dBm) = kT0B(dBm)
+
G(dB) + NF(dB)
The signal power at the output is simply the input power times the
stage
gain:
So = (AP)(Si)
in dB:
So(dBm) = Si(dBm) +
G(dB)

Thus the signal to noise ratio at the output is equal to the signal
to noise ratio at the input divided by NR, or decreased
(degraded)
by NF if dB are being used.
Cascading stages, Friis' Formula
If two stages are cascaded, we consider the output of the first stage (No
+ So) as the input (Si) to the succesive
stage.
We find that signal to noise ratio is degraded through stages 1 and 2
by
a factor:
NR(1, 2) = NR(1) +
(NR(2)
- 1) / AP(1)
This process can be repeated and generalized for several stages:
NR(1. . . k) = NR(1) +
(NR(2) - 1) / (AP(1)) + (NR(3) - 1) / (AP(1)AP(2))+
.
. .
. . . + (NR(k) - 1) /
(AP(1)AP(2)..AP(k-1))
The forgoing is known as Friis' Formula. It is crucial to note that
whereas gains and noise figures are usually given in dB units, the
computation
above must be performed on ratios converted from the decibel notation.
Passive Stages with Loss and Unspecified Noise Figure
When encountering a passive stage ( i ) with loss Li (dB)
=
-10 log [Ai ], Ai < 1 , no additional
noise
is encountered besides the kT0B noise associated with the
stage's
output resistance. In general, noise figure is not specified for
such a stage. In order for the model to apply, we must determine what
value
to use for NRi in the model in order to give a
consistent
result. This will be satisfied when we set NRi
= 1/Ai . In dB this reduces to Gi (dB)
=
-Li (dB) and Fi(dB) = Li(dB)
.
As an example, let the loss (Li) of a stage be 6 dB.
Then, Gi = -6 dB and Fi = 6 dB. For the
model,
Ai = 10-L/10 = 10-0.6
=
.25
NRi = 10L/10 = 1/ Ai
=
100.6 = 4
When these values are used in the model, we see that the signal
power
So at the output is equal to Si times the gain
(0.25),
and No will equal kT0B, QED.
If a stage with loss L1(dB) = -10 log[A1]
is
followed by a stage with specified gain G(dB) = 10 log[AP]
and
noise figure NF = 10 log[NR], and we apply Friis' formula:
NRTot = (1/A1 ) +
(NR - 1) / A1 = NR / A1
expressed in dB, we now have:
NF(dB)Tot = NF(dB) + L1(dB)
and
G(dB) Tot = G(dB) - L1(dB)
This is a very useful result. This means that when a device
with
specified gain and noise figure is immediately preceeded by one or more
stages characterized by passive losses and unspecified noise figures,
then
one can combine the stages into a single block having the following
properties:
-
The combined total gain in dB is equal to the specified device gain in
dB minus the preceeding loss in dB
-
The combined total noise figure in dB is equal to the specified device
noise figure in dB plus the preceeding loss in dB
This is a very simple computation when done in dB, and it reduces the
total
number of stages which must have their gains and noise figures
converted
from dB when Friis' formula is applied. It also reduces the number of
terms
in Friis' formula when the computation is performed.
Stages With Loss and Specified Noise Figure
When noise figure is specified for a stage with loss, three
possibilities
must be examined:
-
If specified noise figure NF(dB) is the same as the specified loss
L(dB),
then the considerations described above apply and the stage can be
combined
with the succeeding stage as above.
-
If specified noise figure NF(dB) is the greater than the specified loss
L(dB), then the stage must have some active devices which are adding
noise
to the system. The stage must be handled independently, and
must
be represented by an additive term when Friis' formula is applied.
-
If specified noise figure NF(dB) is less than the specified loss L(dB),
then we have a dubious situation. If we apply such a case to the
model above, the result will be an output noise power which is less
than
kT0B (a physical impossibility). The most likely
interpretation
is that the specification is implying excess noise, and that the value
for noise figure used in computing stage noise contribution should be
the
specified noise figure plus the loss (in dB). This case now
resolves
to case 2 above.
The following table illustrates noise propagation through a typical
reciever structure, illustrating the use of Friis' formula and the
combination of lossy stages:
