# Noise & Receiver Performance

In this article, I’m going to cover Noise Figure, Noise Factor, Noise Power, Receiver Bandwidth, and how they can affect overall radio receiver performance. Contrary to popular belief, these metrics have nothing to do with man-made interference.

### Noise Power

Noise Power is the integral of the noise power spectral density over a given bandwidth, usually measured in W/Hz or dBm/Hz.

Calculating the Noise Power is done using **k** (Boltzmann’s Constant, in joules per degrees Kelvin), **T** (the temperature, in degrees Kelvin), and **B** (the given bandwidth in Hertz). It is common to normalize the Noise Power to 1 Hz of bandwidth, making power spectral density (PSD), and Noise Power calculations easier.

For example, in 1 Hz of bandwidth, the Noise Power is 4.004E-21 W/Hz, or -174 dBm/Hz:

k = 1.38065\times 10^{-23}

T = 290\text { °K}

B = 1\text{ Hz}

1.38065\times 10^{-23} \times 290 \times 1 = 4.004\times 10^{-21}\text{ W/Hz}

Converting to mW:

4.004\times 10^{-21} \div 0.001 = 4.004\times 10^{-18}

Converting from mW to dBm in a 50 ohm system:

10\times log_{10}(4.004\times 10^{-18}) = -173.985 \text{ dBm/Hz}

This is now the Noise Power within 1 Hz of bandwidth, in a 50 ohm system.

Lets combine the calculations above to come up with the amount of Noise Power in 500 Hz of bandwidth. If we consider a perfect rectangular filter with a 500 kHz pass band (for CW Morse code reception, for example) the minimum achievable noise floor is -147 dBm, since the noise integration calculation over 500 kHz of the filter bandwidth is easy to do now, and is equal to this number.

10 \times log_{10}\frac{(1.38065 \times10^{-23} \times 290 \times 1 \times 500)}{0.001} = -146.985 \text{ dBm}

Try a few numbers in the calculator above. For 500 Hz, the Noise Power is very low, whereas for a 802.11b Wi-Fi channel, where the bandwidth is 22,000,000 Hz (22 MHz), the Noise Power is much higher. You can see how the maximum receiver sensitivity can be limited by the received Noise Power. Even if we consider a theoretically perfect (impossible) Wi-Fi receiver, noiseless in its design, it would not be able to discern a signal below the noise floor within that 22 MHz bandwidth.

Changing the temperature will clarify why it is advantageous for amplifiers, and other electronics to be cooled to very low temperatures in order to receive very small signals. Cooling them lowers the noise power *within the system* dramatically. For example, earth-based LNA’s (low noise amplifiers) are cooled to very low temperatures to improve their performance when a receiver has to discern extremely weak signals from deep space. Liquid helium is often used for this, as its temperature is 4 degrees Kelvin.

Of course, this is not as effective as it seems on the surface. 290 degrees Kelvin is commonly used in Noise Power calculations since it’s considered to be the average temperature of the planet. The only way you would be able to realize the huge improvement shown in the calculator would be to cool the entire earth using liquid helium.

There is not much we can do about thermal Noise Power. You can thank the above calculations, the big bang, and cosmic microwave background radiation for this. So, onward to Noise Figure.

### Noise Figure & Noise Factor

Noise Figure (NF) and Noise Factor (F) are both a measure of the degradation of the signal-to-noise ratio (SNR) resulting from components in a signal chain. These components could be passive components like resistors, capacitors, cables, connectors, and also active components like transistors, and integrated circuits like amplifiers, etc.

The Noise Factor (F) is the ratio of the SNR at the input to the SNR at the output, so:

\large{F = \frac{ (\frac{\text{signal}_{in}} {\text{noise}_{in}}) }{ (\frac{\text{signal}_{out}} {\text{noise}_{out}}) }}

The Noise Figure (NF) is the Noise Factor expressed in dB, so:

\large{NF_{dB} = 10\times log_{10}(F)}

You can visualize this noise entering the receiver, being limited by the receiver filter bandwidth (which actually adds to the noise figure, and further into the receiver it passes through various electronics, all increasing the noise figure, bit by bit.

The noise figure of a passive device is the same as its attenuation. For example, if you have a coaxial cable which attenuates the desired signal by 3 dB, then its noise figure is 3 dB.

### Calculating The Noise Factor

Calculating the Noise Factor (F) of a system is done via a cascade Friis formula. The same is true of the Noise Figure, since {NF_{dB} = 10\times log_{10}(F)} .

\large{F_{tot}=F1+\frac{F2-1}{G1}+\frac{F3-1}{G1G2}+\frac{F4-1}{G1G2G3}+\cdot\cdot\cdot +\frac{Fn-1}{G1G2G3\cdot\cdot\cdot Gn-1}}

Where:

F_{tot} is the total Noise Factor of n stages in the cascade.

F1 is the Noise Factor of stage 1

G1 is the gain of stage 1

F2 is the Noise Factor of stage 2

G2 is the gain of stage 2

F3 is the Noise Factor of stage 3

G3 is the gain of stage 3

Fn is the Noise Factor of the nth stage

Gn is the gain of the nth stage

The Noise Factor (and Figure) of a receiver is dominated by the Noise Factor of the initial gain stage of the system, so it is important for the front-end Low Noise Amplifier (LNA) in a receiver to have a low Noise Figure, else this will negatively impact the Noise Figure further down the chain. It is called a Low Noise Amplifier for a good reason.

Subsequent stages will have a lesser impact on Noise Figure, but will still cause deterioration to the overall signal to noise ratio and IP3.

For example, consider a signal chain consisting of several stages of Low Noise Amplifiers and band pass filters:

Stage | NF (dB) | Gain (dB) | Overall NF (dB) | Overall Gain (dB) |

Antenna Cable | 4 | -4 | 6 | -4 |

LNA 1 | 2 | 16 | 6.07 | 12 |

BPF 1 | 3 | -3 | 6.15 | 9 |

LNA 2 | 2 | 16 | 6.15 | 25 |

BPF 2 | 1.5 | -1.5 | 6.15 | 23.5 |

LNA 3 | 2 | 10 | 6.15 | 33.5 |

BPF 3 | 3 | -3 | 6.15 | 30.5 |

The overall NF of this system would be 6.15 dB.

Increasing the NF of LNA 1 to by 2 dB to 4 dB would increase the overall system NF to 8.10 dB.

However, if we leave the LNA 1 NF at 2 dB, and increase LNA 2’s NF from 2 dB to 4 dB, the overall system NF would increase slightly to 6.27 dB.

Such is the nature of the cascade calculation.

Remembering that the noise figure of a passive device is the same as its attenuation, do not lose sight of the impact of the Noise Figure of any lossy cables and connectors used between the antenna and the first LNA.

For example, in the table above, if we reduce the antenna cable loss from 4 dB to 1 dB, the overall system NF will decrease from 6.15 dB to 3.15 dB. Alternatively, increasing the NF of LNA 3 from 2 dB to 10 dB would only increase the overall system NF to 6.19 dB, again illustrating the knock-on effect in the cascade calculation.

When designing receiver front-ends, or RF amplifier/filter chains, it is important to understand the trade-offs in NF vs gain when deciding whether to put a lossy passive stage before an active gain stage, or the other way around.

### Still to come…

- Equivalent Noise Bandwidth (ENBW).
- Calculating the overall receiver sensitivity using thermal noise, ENBW, and NF.

### AD5GG

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