# Communication Systems/Noise Figure

This page is going to talk about the effect of noise on transmission systems.

## Types of Noise

Most man made electro-magnetic noise occurs at frequencies below 500 MHz. The most significant of these include:
       • Hydro lines
       • Ignition systems
       • Fluorescent lights
       • Electric motors
Therefore deep space networks are placed out in the desert, far from these sources of interference.
There are also a wide range of natural noise sources which cannot be so easily avoided, namely:
       •Atmospheric noise - lighting < 20 MHz
       •Solar noise - sun - 11 year sunspot cycle
       •Cosmic noise - 8 MHz to 1.5 GHz
       •Thermal or Johnson noise. Due to free electrons striking vibrating ions.
       •White noise - white noise has a constant spectral density over a specified range of frequencies. Johnson noise is an example of white noise.
       •Gaussian noise - Gaussian noise is completely random in nature however, the probability of any particular amplitude value follows the normal distribution curve. Johnson noise is Gaussian in nature.
       •Shot noise - bipolar transistors
             $i_{n}={\sqrt {2qI_{dc}\Delta f}}$
             where q = electron charge 1.6 x 10$^{-19}$ coulombs
       •Excess noise, flicker, 1/f, and pink noise < 1 KHz are Inversely proportional to frequency and directly proportional to temperature and dc current
       •Transit time noise - occurs when the electron transit time across a junction is the same period as the signal.
Of these, only Johnson noise can be readily analysed and compensated for. The noise power is given by:
             $P_{n}=kTB$
Where:
       k = Boltzmann's constant (1.38 x 10$^{-23}$ J/K)
       T = temperature in degrees Kelvin
       B = bandwidth in Hz
This equation applies to copper wire wound resistors, but is close enough to be used for all resistors. Maximum power transfer occurs when the source and load impedance are equal.

### Combining Noise Voltages

The instantaneous value of two noise voltages is simply the sum of their individual values at the same instant.
       $v_{total\;inst}=v_{1\;inst}+v_{2\;inst}$
This result is readily observable on an oscilloscope. However, it is not particularly helpful, since it does not result in a single stable numerical value such as one measured by a voltmeter.
If the two voltages are coherent [K = 1], then the total rms voltage value is the sum of the individual rms voltage values.
       $v_{total\;rms}=v_{1\;rms}+v_{2\;rms}$
If the two signals are completely random with respect to each other [K = 0], such as Johnson noise sources, the total power is the sum of all of the individual powers:
       $P_{total\;random\;noise}=P_{n1\;random}+P_{n2\;random}$
A Johnson noise of power P = kTB, can be thought of as a noise voltage applied through a resistor, Thevenin equivalent.
An example of such a noise source may be a cable or transmission line. The amount of noise power transferred from the source to a load, such as an amplifier input, is a function of the source and load impedances.
If the load impedance is 0 $\Omega$, no power is transferred to it since the voltage is zero. If the load has infinite input impedance, again no power is transferred to it since there is no current. Maximum power transfer occurs when the source and load impedances are equal.
       $P_{L\;\max }={\frac {e_{s}^{2}}{4R_{s}}}$
The rms noise voltage at maximum power transfer is:
       $e_{n}={\sqrt {4RP}}={\sqrt {4RkTB}}$
Observe what happens if the noise resistance is resolved into two components:
       $e_{n}^{2}=4RkTB=4\left({R_{1}+R_{2}}\right)kTB=e_{n1}^{2}+e_{n2}^{2}$
From this we observe that random noise resistance can be added directly, but random noise voltages add vectorially:
If the noise sources are not quite random, and there is some correlation between them [0 < K < 1], the combined result is not so easy to calculate:
       $P_{Total\;\left({{\rm {not}}\;{\rm {quite}}\;{\rm {random}}}\right)}={\frac {E_{1}^{2}+E_{2}^{2}+2KE_{1}E_{2}}{R_{0}}}=P_{1}+P_{2}=2K{\sqrt {P_{1}+P_{2}}}$
       where
       K = correlation [0 < K < 1]
       R$_0$ = reference impedance

## Noise Temperature

The amount of noise in a given transmission medium can be equated to thermal noise. Thermal noise is well-studied, so it makes good sense to reuse the same equations when possible. To this end, we can say that any amount of radiated noise can be approximated by thermal noise with a given effective temperature. Effective temperature is measured in Kelvin. Effective temperature is frequently compared to the standard temperature, $T_{o}$, which is 290 Kelvin.
In microwave applications, it is difficult to speak in terms of currents and voltages since the signals are more aptly described by field equations. Therefore, temperature is used to characterize noise. The total noise temperature is equal to the sum of all the individual noise temperatures.

## Noise Figure

The terms used to quantify noise can be somewhat confusing but the key definitions are:
       Signal to noise ratio: It is either unitless or specified in dB. The S/N ratio may be specified anywhere within a system.
             ${\frac {S}{N}}={\frac {{\rm {signal}}\;{\rm {power}}}{{\rm {noise}}\;{\rm {power}}}}={\frac {P_{s}}{P_{n}}}$
             $\left({\frac {S}{N}}\right)_{dB}=10\log {\frac {P_{s}}{P_{n}}}$
       Noise Factor (or Noise Ratio): $F={\frac {\left({\frac {S}{N}}\right)_{in}}{\left({\frac {S}{N}}\right)_{out}}}$ (unit less)
       Noise Figure: $NF=10\log F=SNR_{in}-SNR_{out}$ dB
This parameter is specified in all high performance amplifiers and is measure of how much noise the amplifier itself contributes to the total noise. In a perfect amplifier or system, NF = 0 dB. This discussion does not take into account any noise reduction techniques such as filtering or dynamic emphasis.

### Friiss' Formula & Amplifier Cascades

It is interesting to examine an amplifier cascade to see how noise builds up in a large communication system.
       $F={\frac {\left({\frac {S}{N}}\right)_{in}}{\left({\frac {S}{N}}\right)_{out}}}={\frac {S_{in}}{N_{in}}}\times {\frac {N_{out}}{S_{out}}}$
Amplifier gain can be defined as: $G={\frac {S_{out}}{S_{in}}}$
       Therefore the output signal power is: $S_{out}=GS_{in}$
       and the noise factor (ratio) can be rewritten as: $F={\frac {S_{in}}{N_{in}}}\times {\frac {N_{out}}{GS_{in}}}={\frac {N_{out}}{GN_{in}}}$
       The output noise power can now be written: $N_{out}=FGN_{in}$
From this we observe that the input noise is increased by the noise ratio and amplifier gain as it passes through the amplifier. A noiseless amplifier would have a noise ratio (factor) of 1 or noise figure of 0 dB. In this case, the input noise would only be amplified by the gain since the amplifier would not contribute noise.
       The minimum noise that can enter any system is the Johnson Noise:
             $N_{in\left({\rm {minimum}}\right)}=kTB$
       Therefore the minimum noise that can appear at the output of any amplifier is:
             $N_{out\left({\rm {minimum}}\right)}=FGkTB$
       The output noise of a perfect amplifier would be (F = 1):
             $N_{out\left({\rm {perfect}}\right)}=GkTB$
       The difference between these two values is the noised created (added) by the amplifier itself:
             $N_{out\left({\rm {added}}\right)}=N_{out\left({\rm {minimum}}\right)}-N_{out\left({\rm {perfect}}\right)}=FGkTB-GkTB=\left({F-1}\right)GkTB$
       This is the additional (created) noise, appearing at the output.
The total noise out of the amplifier is then given by:
             $N_{total}=N_{out\left({\rm {perfect}}\right)}+N_{out\left({\rm {added}}\right)}=GkTB+\left({F-1}\right)GkTB$
If a second amplifier were added in series, the total output noise would consist the first stage noise amplified by the second stage gain, plus the additional noise of the second amplifier:
             $N_{total}=G_{1}G_{2}kTB+\left({F_{1}-1}\right)G_{1}G_{2}kTB+\left({F_{2}-1}\right)G_{2}kTB$
       If we divide both sides of this expression by the common term: $G_{1}G_{2}kTB$
       we obtain:
             ${\frac {N_{total}}{G_{1}G_{2}kTB}}={\frac {G_{1}G_{2}kTB+\left({F_{1}-1}\right)G_{1}G_{2}kTB+\left({F_{2}-1}\right)G_{2}kTB}{G_{1}G_{2}kTB}}$
       Recall: $F={\frac {N_{out}}{GN_{in}}}={\frac {N_{total}}{G_{1}G_{2}kTB}}$
       Then: $F_{overall}=F_{1}+{\frac {F_{2}-1}{G_{1}}}$
This process can be extended to include more amplifiers in cascade to arrive at:
    Friiss' Formula
             $F=F_{1}+{\frac {F_{2}-1}{G_{1}}}+{\frac {F_{3}-1}{G_{1}G_{2}}}+$
This equation shows that the overall system noise figure is largely determined by the noise figure of the first stage in a cascade since the noise contribution of any stage is divided by the gains of the preceding stages. This is why the 1st stage in any communication system should be an LNA (low noise amplifier).

## Receiver Sensitivity

In a given bandwidth, W, we can show that the noise power N equals:
    $N=F(kT_{0})W$
From N, we can show that the sensitivity of the receiver is equal to
    $SNR\times N$