Lecture 22 - Noise Figure
In this lecture, we will continue our discussion of noise in electronic circuits. Building upon the observations about thermal noise in the previous lecture, we will consider how noise accumulates along a cascade of circuits, and what design criteria we can apply in order to manage the noise end-to-end.
Motivation in Course Context
We continue to focus on the receiver circuits because the desired received signal can be relatively weak due to propagation effects, and the thermal noise in the receiver can become significant relative to the received signal.
As the figure above suggests, there are many sources of thermal noise in the receiver circuits. We need to come up with tools for modeling how the noise accumulates, and how we prevent it from overwhelming the received signal through updates to the receiver design.
Power Gain / Loss
Each device of the radio receiver will be modeled as a gain or loss in power from the input to the output. Specifically, if the input power is \(P_i\) and the output power is \(P_o\), then the power transfer relationship will be defined as
\[ G = \frac{P_o}{P_i} \]
If \(G < 1\), the device will exhibit a power loss, while if \(G > 1\), the device will exhibit a power gain. On a dB scale, gains will be positive values, and losses with be negative values.
For filters, we will primarily focus on the loss in the passband. Terms such as insertion loss are commonly associated with passive filters, and we naturally want to keep the insertion loss as small as possible.
For mixers, we will primarily focus on the gain / loss of the translation from the input band to the output band. Terms such as conversion gain and conversion loss are commonly associated with mixers.
Noise Figure
Suppose a device with gain / loss \(G\) has at its input the output of a signal generator with signal power \(P_s\) and a noise power \(N_s\). The input signal-to-noise ratio (SNR) is then
\[ \mathrm{SNR}_i = \frac{P_s}{N_s} \]
The gain or loss of the component affects the input signal and noise in the same way. However, the component may exhibit internal thermal noise that contributes to the output signal.
Assuming the internal thermal noise has power \(N_d\) and is independent of the input signal and the input noise, the output noise power will be
\[ N_o = G\cdot N_s + N_d \]
so that the output SNR will be
\[ \mathrm{SNR}_o = \frac{G\ P}{G\ N_s + N_d} \]
The noise figure of the component is defined as the ratio of the input SNR to the output SNR, i.e.,
\[ \mathrm{NF} := \frac{\mathrm{SNR}_i}{\mathrm{SNR}_o}=\frac{1}{G}\frac{N_o}{N_s} \]
Solving for \(N_o\), we have
\[ N_o = \mathrm{NF}\cdot G \cdot N_s = G\cdot N_s + N_d \]
so that
\[ N_d = (\mathrm{NF}-1)\ G\ N_s \]
The scenario and parameterization of the noise figure is illustrated in the figure below.
We stress a few items at this point:
The relationships above relate the power of the internal noise to the power of the signal generator noise; they are not saying that the signals themselves are related, and in fact, they are modeled as indepedent random processes.
The gain and noise figure are defined relative to a specific signal generator noise, often thermal or Johnson noise for which \(N_s = kT B\) where \(k\) is Boltzman’s constant, \(T\) is the temperature in Kelvin, and \(B\) is the (baseband) bandwidth in Hz.
When we consider multiple devices in the following sections, all of their noise figures must be defined relative to a common signal generator noise input, and their internal noise powers are computed relative to the same input noise source.
Cascaded Components
Two Components
Suppose we cascade two components with gains / losses \(G_1\) and \(G_2\) and noise figures \(\mathrm{NF}_1\) and \(\mathrm{NF}_2\), respectively, as shown in the figure below. What is the gain / loss \(G_{12}\) as well as the noise figure \(\mathrm{NF}_{12}\) of the cascade?
First of all, we clearly have
\[ G_{12}=G_1 \cdot G_2 \]
To determine the cascade noise figure, let \(N_1\) and \(N_2\) denote noise powers at the respective device outputs, and let
\[ \begin{align} N_{d_1} &= (\mathrm{NF}_1 - 1) G_1 N_s \\ N_{d_2} &= (\mathrm{NF}_2 - 1) G_2 N_s \end{align} \]
be the source-input-referred internal noises of the two devices. We stress the importance of referencing these internal noises to the signal generator input noise, the idea being that the noise figure of each device individually is defined as if the device itself was connected ot the signal generator intead of being cascaded.
From above, we have
\[ N_1 = \mathrm{NF}_1\cdot G_1 \cdot N_s \]
and
\[ N_2 = G_2\cdot N_1 + N_{d_2} = G_2\cdot N_1 +(\mathrm{NF}_2-1) G_2 \cdot N_s \]
The first term represents the component of the output noise due to the source input noise, and the second term represents the component of the output noise due to the internal noise of the second device.
Substituting the equation for \(N_1\) into the equation for \(N_2\), we have
\[ N_2 = \left[\mathrm{NF}_1 + \frac{\mathrm{NF}_2-1}{G_1} \right] G_1 G_2 N_i = \mathrm{NF}_{12}\cdot G_{12}\cdot N_i \]
Thus, the noise figure of the cascade satisfies
\[ \mathrm{NF}_{12} = \mathrm{NF}_1 + \frac{\mathrm{NF}_2-1}{G_1} \]
Multiple Components
More generally, suppose we cascade multiple devices with gains / losses \(G_1,G_2,\ldots,G_K\) and noise figures \(\mathrm{NF}_1,\mathrm{NF}_2,\ldots,\mathrm{NF}_K\), respectively, as illustrated in the figure below.
Let \(G_{\{1,\ldots,k\}}\) denote the gain / loss and \(\mathrm{NF}_{\{1,\ldots,k\}}\) denote the noise figure of the cascade of the first \(1 \le k \le K\) components, respectively.
Applying the result for the cascade of two devices recursively, we have
\[ G_{\{1,\ldots,k\}} = \prod_{i=1}^{k} G_i \]
and
\[ \begin{align} \mathrm{NF}_{\{1,\ldots,K\}} &= \mathrm{NF}_{\{1,\ldots,K-1\}} + \frac{\mathrm{NF}_K - 1}{G_{\{1,\ldots,K-1\}}} \\ &= \mathrm{NF}_{\{1,\ldots,K-2\}} + \frac{\mathrm{NF}_{K-1}-1}{G_{\{1,\ldots,K-2\}}} + \frac{\mathrm{NF}_K-1}{G_{\{1,\ldots,K-1\}}} \end{align} \]
and so forth.
In particular, for \(K=3\), we have
\[ \mathrm{NF}_{\{1,\ldots,3\}} = \mathrm{NF}_{1} + \frac{\mathrm{NF}_2 - 1}{G_1} + \frac{\mathrm{NF}_3 - 1}{G_1\cdot G_2} \]
Adding a Low-Noise Amplifier (LNA)
The cascaded noise analysis above places significant importance on the first few devices having both low noise figure and high gain. It motiviates the addition of a device called a low-noise amplifier (LNA) to the front end of our radio receiver, as shown in the figure below.
As we will see in lab, it will make sense to add additional amplifiers in the receive chain to overcome the noise figure of the analog-to-digital converter as well.
Refences
H. T. Friis, “Noise Figures of Radio Receivers,” in Proceedings of the IRE, vol. 32, no. 7, pp. 419-422, July 1944. doi: 10.1109/JRPROC.1944.232049
Henry W. Ott, Noise Reduction Techniques in Electronic Systems, 2nd Edition, John Wiley & Sons, 1988.