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 S_i, and is assumed to be corrupted by thermal noise (kT₀B) such that the signal to noise ratio at the input is S_i/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:

N_o = (A_P)(kT₀B) + (A_P)(NR-1)(kT₀B) = (A_P)(NR)(kT₀B)

or, expressed in dB:

N_o (dBm) = kT₀B(dBm) + G(dB) + NF(dB)

The signal power at the output is simply the input power times the stage gain:

S_o = (A_P)(S_i)

in dB:

S_o(dBm) = S_i(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 (N_o + S_o) as the input (S_i) 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) / A_P(1)

This process can be repeated and generalized for several stages:

NR(1. . . k) = NR(1) + (NR(2) - 1) / (A_P(1)) + (NR(3) - 1) / (A_P(1)A_P(2))+ . . .
. . . + (NR(k) - 1) / (A_P(1)A_P(2)..A_P(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 L_i(dB) = -10 log [A_i], A_i< 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 NR_i in the model in order to give a consistent result. This will be satisfied when we set NR_i = 1/A_i. In dB this reduces to G_i(dB) = -L_i(dB) and F_i(dB) = L_i(dB) . As an example, let the loss (L_i) of a stage be 6 dB. Then, G_i = -6 dB and F_i = 6 dB. For the model,

A_i = 10^-L/10 = 10^-0.6= .25

NR_i = 10^L/10 = 1/ A_i= 10^0.6 = 4

When these values are used in the model, we see that the signal power S_o at the output is equal to S_i times the gain (0.25), and N_o 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[A_P] and noise figure NF = 10 log[NR], and we apply Friis' formula:

NR_Tot = (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.