# 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.

:   

    :   ![](noise_source.gif "fig:noise_source.gif")

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.

:   

    :   ![](noise_circuit.gif "fig:noise_circuit.gif")

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:

$$e_n  = \sqrt {4RP}  = \sqrt {4RkTB}$$

:   

    :   ![](matched_thermal_noise.gif "fig:matched_thermal_noise.gif")

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:

:   

    :   ![](vector_addition.gif "fig:vector_addition.gif")

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

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:   

    :   *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)

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:   

    :   **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.

:   

    :   ![](g.gif "fig:g.gif")

### 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: $G = \frac{{S_{out} }}{{S_{in} }}$

:   

    :   ![](g2.gif "fig:g2.gif")

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:   

    :   Therefore the output signal power is: $S_{out}  = GS_{in}$

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:   

    :   and the noise factor (ratio) can be rewritten as:
        $F = \frac{{S_{in} }}{{N_{in} }} \times \frac{{N_{out} }}{{GS_{in} }} = \frac{{N_{out} }}{{GN_{in} }}$

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:   

    :   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:
        $G_1 G_2 kTB$

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:   

    :   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:
        $F = \frac{{N_{out} }}{{GN_{in} }} = \frac{{N_{total} }}{{G_1 G_2 kTB}}$

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:   

    :   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

$$SNR \times N$$

## Cascaded Systems
