How to determine noise figure: part 3

Noise factor and noise figure as defined in an IEEE standard can be derived from a two-port device’s equivalent noise temperature.

In part 1 and part 2 of this series we discussed several ways to indicate the noise performance of a device under test (DUT). We first introduced the concept of noise factor based on the work of Danish-American radio engineer Harald Friis. We’ll call this noise factor F_SNR, because it equals a DUT’s input signal-to-noise ratio (SNR) divided by the output SNR:

where S_IN, S_OUT, N_IN, and N_OUT are the input and output signal levels and the input and output noise levels, respectively.

We also noted that noise figure (NF) is generally expressed in decibels:

where S_IN(dB), S_OUT(dB), N_IN(dB), and N_OUT(dB) are the input and output signal and noise levels in decibels.

Then we introduced a noise factor based on temperature. We’ll call that one F_STD because it’s defined in a standard developed by the IRE, a precursor to the IEEE:

Here, N_ADD is the noise added by the DUT, G is the DUT gain, k is Boltzmann’s constant, and B is the measurement bandwidth. The corresponding noise figure is commonly expressed in dB:

Finally, we introduced a parameter called equivalent noise temperature:

So that’s basically three parameters for describing noise performance, two of which can be expressed in either linear or decibel terms. Why so many, and can we convert between them?
Before answering that question, let’s take a closer look at noise temperature. We described it in terms of mimicking the power output of a noise source (Figure 1a) using a resistor held at a fixed temperature T_E (Figure 1b), as shown in Eq. 5.

Figure 1. A noise source generates a noise power *N_ADD* (a), which equals the same noise power a resistor would generate when maintained at the equivalent noise temperature *T_E* (b).

By the way, what might be a typical value for T_E?
Let’s say our noise source in Figure 1a is adding a modest -160 dBm of noise in a 1-Hz bandwidth, which is equivalent to N_ADD equaling 100 × 10^-21 W. We know that Boltzmann’s constant is 1.38 × 10^-23 Joules/Kelvin, or 1.38 × 10^-23 watt-seconds/Kelvin, so we can calculate T_E as follows:

So we are pretty close to 6,000°C, and you can see why we might buy a commercial noise source rather than just using a resistor we have lying around. But to get back to your earlier question, can we convert between T_E and noise factor? The short answer, based on Eq. 5, is no. After all, noise factors and noise figures are based on gains or input and output SNRs. Since the noise source in Figure 1a is a single-port network, the concept of a ratio of input to output SNRs makes no sense.

Can we extend the noise-temperature concept to two-port networks?
Yes. Figure 2a shows a resistor maintained at reference temperature T₀= 290 K driving a power amplifier with gain G. The resistor generates a noise power of kT₀B. The amplifier amplifies kT₀B byG and adds its own noise contributionN_AD, with the total amplifier output equaling N_ADD + kT₀BG, as shown by the ideal power sensor and meter #1. Figure 2b shows a similar setup but with an ideal amplifier that contributes no noise of its own. To compensate, we increase the temperature of resistor R by T_E, resulting in the power-meter reading for power meter #2 in Figure 2b of k(T₀ + T_E)BG, where T_E represents the equivalent noise temperature of the two-port real-world amplifier in Figure 2a.