# Understanding the RF Noise Figure Specification

## Take a closer look at the definition of the RF noise figure and discuss some subtleties to help avoid erroneous interpretations of this specification.

In RF applications, we usually deal with very weak signals that can be easily obscured by the noise generated within our circuitry. The noise level ultimately determines the minimum signal that can be reliably detected by the receiver. Therefore, noise characterization of RF components and systems is paramount. In our introductory article about a noise figure, we learned how this metric is used to characterize the noise performance of RF components.

Now that we’re familiar with the basic concepts, we can take a closer look at the definition of the noise figure and discuss some subtleties that are not sometimes highlighted enough. This should help you avoid erroneous interpretations of this specification.

### The Noise Figure Definition and Noise Factor Equation

The noise factor (F) of a circuit can be defined as:

\[F=\frac{N_o}{GN_i}\]

**Equation 1.**

**Equation 1.**

Where:

- N
_{o}is the total noise at the output, including the effect of the internal noise sources within the circuit and the noise from the source impedance - N
_{i}is the noise the source impedance produces at the circuit's input - G is the power gain of the stage

Although this explanation is correct, and actually similar explanations are provided in some references, such as the 4th edition of the widely used textbook “Analysis and Design of Analog Integrated Circuits” by Paul R. Gray, it doesn’t provide all the details of the noise figure definition. According to the IEEE definition, N_{i} is the available thermal noise power of the source resistor at a temperature of T_{0 }= 290 K° (or 16.85 °C). This temperature is a little colder than a comfortable room temperature; however, it is sometimes referred to as the room temperature in RF work.

In addition, the IEEE definition states that N_{o} is the available noise power at the output of the device, and G is the available power gain of the device. The key points here are the reference temperature of the specification T_{0 }= 290 K°, as well as the descriptor “available” used to describe all three parameters of the equation N_{i}, N_{o}, and G. In the rest of the article, we’ll discuss in detail what the implications of the “available noise power at T_{0 }= 290 K" are.

### Maximum Available Noise Power

The random motion of thermally excited charge carriers manifests itself as noise in resistors. A noisy resistor can be modeled by adding a noise voltage source in series with the noiseless resistor, as shown below in Figure 1.

**Figure 1. **Example diagram of a noise resistor and adding a noise voltage source in series with a noiseless resistor.

**Figure 1.**Example diagram of a noise resistor and adding a noise voltage source in series with a noiseless resistor.

The noise voltage source has a PSD (power spectral density) of \( \overline {V_n^2} = 4 \space kTRB\), where:

- k is Boltzmann’s constant (1.38 × 10
^{-23}Joules/Kelvin) - T is the temperature in Kelvin
- B is the considered bandwidth (Hertz)

In the noise figure definition, N_{i} is the maximum available noise power of the source resistor. The question now is, what is the maximum noise power that the circuit in Figure 1(b) can provide? From basic circuit theory, we know that the maximum power is transferred when the load resistor is equal to the source resistor. Therefore, the following circuit (Figure 2) can be used to find the maximum available noise power of the source resistor, R_{S}.

**Figure 2. **Diagram of a circuit used to find the maximum available noise power of the source resistor.

**Figure 2.**Diagram of a circuit used to find the maximum available noise power of the source resistor.

Keep in mind that we have used the RMS (root-mean-square) value of the noise source in the above diagram. Since half of the noise voltage appears across the load, the noise power delivered to the matched load, R_{L }= R_{S}, can be found by:

\[\begin{eqnarray}

P_{L} = \frac{V_{L}^2}{R_L} &=& \Big ( {\frac{V_{n,rms}}{2}} \Big ) ^2 \times \frac{1}{R_L} \\

&=& \frac{4kTR_SB}{4} \times \frac{1}{R_S} \\

&=& kTB

\end{eqnarray}\]

**Equation 2.**

**Equation 2.**

This is an important result for noise figure calculations. Note that the available noise power is independent of the resistor value. Whether it is a 1 mΩ resistor or a 1 MΩ resistor, the available noise power is kTB. In a 1 Hz bandwidth, the available noise power is kT. The noise figure definition is based on the available noise power at T_{0 }= 290 K. Expressing kT_{0} in dB, the available noise power at this reference temperature works out to -174 dBm/Hz, as calculated below:

\[10 log \big ( kT_0 \big )=10 log \big ( 1.38 \times 10^{-23} \times 290 \big ) \approx -174 \text{ } dBm\]

### Noise Figure Specifies the Relative Amount of Added Noise

Since the noise figure definition is based on N_{i }= kT_{0}B, it specifies the *relative* amount of noise that is being added to the signal with respect to N_{i}. Consider the following noise figure equation that we derived in the previous article:

\[F=1+\frac{N_{o(added)}}{N_{o(source)}}\]

Here, N_{o(source)} is part of the output noise that stems from the source impedance; and N_{o(added)} is part of the output noise that is produced within the circuit itself—excluding the source resistance contribution. Noting that N_{o(source) }= kT_{0}BG, we get Equation 3:

\[F=1+\frac{N_{o(added)}}{kT_0BG}\]

**Equation 3.**

**Equation 3.**

To better visualize the noise terms of the above equation, consider the following plot in Figure 3, which is sometimes referred to as the “noise line”.

**Figure 3. **Plot showing a noise line.

**Figure 3.**Plot showing a noise line.

In the above diagram, the total output noise, N_{o}, is plotted against the source resistance temperature, T. If R_{S} were noiseless (or at T = 0 K), the only noise that appeared at the output would have been that of the device under test or N_{o(added)}. As we increase the temperature of R_{S}, its noise contribution rises. The noise figure metric, which corresponds to T = T_{0}, actually specifies the ratio of the output noise contributed by R_{S} at T_{0}—i.e., kT_{0}BG—to that of the device under test (N_{o(added)}). For instance, if the noise factor of a system is F = 2 (or NF = 3 dB), we know that N_{o(added)} is equal to kT_{0}BG.

As the graph clearly shows, the ratio of the R_{S} noise and N_{o(added)} is not constant and changes with T. Therefore, if R_{S} is at a temperature other than T_{0}, we cannot directly use the noise figure equation to find the output noise. Instead, we should first find the noise coming from the DUT (device under test), add the noise from R_{S} at the temperature of interest, and finally calculate the total output noise.

We can also express Equation 3 in terms of the input-referred noise values by dividing both the numerator and denominator of the fraction by the power gain of the stage. This produces Equation 4:

\[F=1+\frac{N_{i(added)}}{N_{i}}\]

**Equation 4.**

**Equation 4.**

In this equation, N_{i(added)} is the input-referred noise contributed by the DUT, and N_{i} is the available noise power of the source at 290 K. Again, if F = 2, the input-referred noise contributed by the DUT is equal to N_{i }= kT_{0}B. Let’s look at an example to clarify these concepts.

### Example: Using the Noise Figure Equation

The noise figure, bandwidth, and gain of an amplifier are, respectively:

- NF = 2.55 dB
- B = 10 MHz
- Gain = 5.97 dB

Assuming that the available input noise is kT_{A}B, find the output noise for two different cases: 1 - T_{A }= 290 K and 2 - T_{A }= 150 K.

We first find the linear values of the noise figure and gain:

\[F=10^{\frac{NF}{10}}=10^{0.255}=1.8\]

\[G=10^{\frac{Gain}{10}}=10^{0.597}=3.95\]

Since the definition of noise figure assumes that the input noise is the available noise power at T = 290 K, we can find the output noise at this temperature directly from Equation 1:

\[N_{o}=N_{i}FG=kT_{0}B\times FG\]

Expressing the right-hand side in decibels, we have:

\[\begin{eqnarray}

N_o &=& 10 log(kT_0) + 10log(BFG) \\

&=& -174 \text{ }dBm/Hz + 10 log(10 \times 10^{6} \times 1.8 \times 3.95) \\

&=& -95.48 \text{ } dBm

\end{eqnarray}\]

For T_{A }= 150 K, we cannot directly use the noise figure equation. However, the noise figure equation can be used to calculate the noise contributed by the system. Substituting N_{i }= kT_{0}B into Equation 4, the input-referred noise contributed by the system is found as:

\[N_{i(added)}=(F-1)kT_0B\]

With F = 1.8, we obtain N_{i(added) }= 0.8kT_{0}B. Therefore, the total noise at the input is:

\[N_{i(total)}=N_{i(added)}+kT_{A}B\\ =0.8kT_{0}B+kT_{A}B\]

Multiplying this value by the gain of the system, G, gives us the total output noise power. In the following equation, I write T_{A} in terms of T_{0} to simplify the calculations:

\[\begin{eqnarray}

N_{o(total)}&=&G \big ( 0.8kT_0B+kT_A \frac{T_0}{T_0}B \big ) \\

&=& GkT_0B(0.8 + \frac{150}{290})

\end{eqnarray}\]

Expressing the right-hand side in decibels, we have:

\[\begin{eqnarray}

N_{o(total)} &=& 10 log(kT_0) + 10log(BG \times 1.317) \\

&=& -174 \text{ }dBm/Hz + 10 log(10 \times 10^{6} \times 3.95 \times 1.317) \\

&=& -96.84 \text{ } dBm

\end{eqnarray}\]

Without paying attention to the definition of the noise figure, one might substitute N_{i }= k × 150 × B into Equation 1, which produces the incorrect result of N_{o }= -98.34 dBm.

### Physical Temperature or Noise Temperature

In the above discussion, we highlighted the impact the physical temperature of the source resistance, R_{S}, can have on our NF calculations. More often than not, the driving-point impedance (R_{S}) is at the same physical temperature as the DUT; however, the input noise power that the circuit receives is higher than kT_{0}B. This commonly occurs in cascaded systems where the noise floor is increased by each block in the signal chain. As a result, the input noise of the downstream stages in a cascade normally exceeds kT_{0}B. In these cases, again, we cannot find the output noise level by directly applying the noise figure equations. Instead, we can first use the NF equation to find the noise the circuit produces (N_{i(added)}) and then use that information along with the input noise level to find the total output noise. Additionally, it is also helpful to define an equivalent noise temperature T_{e} for the input noise. This is the temperature where the available thermal noise power (kT_{e}B) is equal to the input noise power. If the input noise power is N_{1}, its equivalent noise temperature is:

\[T_{e}=\frac{N_1}{kB}\]

The noise figure and equivalent noise temperature are interchangeable characterizations of the noise properties of a component. In the next article, we’ll look at examples of using the noise temperature concept.

### Noise Figure Specifies the SNR Degradation

The noise figure is a direct measure of the SNR (signal-to-noise ratio) degradation caused by the circuit. This statement is correct; however, it deserves a bit more explanation. Let’s consider the example we discussed above one more time. There we assumed that the noise figure and gain of the system are NF = 2.55 dB and Gain = 5.97 dB, respectively, and assume that the input signal power is -40 dBm. When R_{S} is at T_{A }= 290 K, the input noise power is:

\[\begin{eqnarray}

N_i &=& 10 log(kT_0) + 10log(B) \\

&=& -174 \text{ }dBm/Hz + 10 log(10 \times 10^{6}) \\

&=& -104 \text{ } dBm

\end{eqnarray}\]

From the example's results, we know that the output noise power is -95.48 dBm. The signal and noise powers at the input and output for this example are summarized in Figure 4.

**Figure 4. **Summarization of the signal and noise powers at the input and output for our earlier example.

**Figure 4.**Summarization of the signal and noise powers at the input and output for our earlier example.

The output signal power is found by multiplying the input signal by the power gain of the amplifier. Figure 4 also provides the input and output SNR, as well as the SNR degradation. Note that the ratio SNR_{i }/ SNR_{o} is equal to the noise figure NF = 2.55 dB, which is no big surprise because we know that this ratio is actually the definition of the noise figure. However, what about the case of T_{A }= 150 K? In this case, the input noise is N_{i }= -106.86 dBm. The results from the previous example are summarized in Figure 5.

**Figure 5. **Another summarization of results for the above example.

**Figure 5.**Another summarization of results for the above example.

As you can see, the SNR degradation (SNR_{i }/ SNR_{o}) is now greater than the NF. This is because the input noise is less than the standard value, making the noise contribution of the amplifier more significant. Therefore, the noise figure determines the SNR degradation when the input noise is kT_{0}B. For example, if a circuit has a noise figure of 7 dB and the input noise power of the block is kT_{o}B, then the SNR at the output of that block is 7 dB less than the input SNR.

In the next article, we’ll continue this discussion and learn about the “available gain power” that appears in the NF definition.

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