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Noise

The noise level in an electronic system is typically measured as an electrical power P in Watts or dBm, a root mean square (RMS) voltage (identical to the noise standard deviation) in volts, dBµV or a mean squared error (MSE) in volts squared. Noise may also be characterized by its probability distribution and noise spectral density N0(f) in watts per hertz.


Noise Power  amath P = k*T*B endamath  where  amath k = 1.3806504 * 10^-23 J/K endamath  (Boltzmann Constant). T is the temperature in [K] (Kelvin) and B is the Bandwidth in [Hz].


amath P = 10*log_10 (P/P_0) endamath  in [dBm] and P₀= 1 mW


Noise Voltage  amath U = sqrt(P*R) endamath  or  amath U=sqrt(k*T*B*R) endamath  where R = 50 Ω. (usually)


Reference Temperature  amath T = 273.15 K + theta_(a m b) endamath  usually T = 300 K.


Noise Figure   may be given as a linear factor or in dB. It is a measure of degradation of the signal-to-noise ratio (SNR), caused by one or more components in a signal chain.

F = SNR_input / SNR_output   whereas SNR_inputamath = S_(i n)/N_(i n) endamath   and SNR_outputamath = S_(o u t)/N_(o u t) endamath

F_dB amath= 10*log_10 (F) endamath


SNR_input > SNR_output as the 'device' adds noise, decribed by the Noise Figure :-(

⇒ You can compare receivers with the noise-figure. When using the 'sensitivity' a comparison is only possible if this is done at the same bandwidth.


Multi-Stage-Systems

When using multi-stage-systems the noie figures add up using the 'de Friis-Formula'. calculator

F_total = F₁ + amath (F_2 -1 )/ G_1 + (F_3 -1 )/ (G_1 G_2) + ...endamath

This formula lets you know, that :

• The Noise Figure of the first device is decisive for the overall Noise Figure

• The Noise Figures of the following stages loose influence with the Gain of the first stage.


Example : FM-Receiver

We have a Receiver with F = 2 dB operated at an Antenna. Between Receiver and Antenna we have an Coaxial cable of 25 m with an Attenuation of 7 dB/100 m at 100 MHz. This means we have an attenuation of 1.75 dB which is equal to F = 1.75 dB. F = 1.49 ( F = 10^(F_dB/10)). This gives an overall Noise Figure of :

F_total = F₁ + amath (F_2 -1 )/ G_1 = 1.49 + (1.58 - 1)/(1/1.49) = 2.37 endamath ≡ 3.75 dB

The Attenuation of the Coaxial cable worsens the overall Noise Figure.

From a friend, we hear that adding an amplifier will improve the situation. Unfortunately we forgot to ask, where to put it. Amplifier : Gain 20dB, F = 3 dB.

If we put it between Antenna and Coaxial cable, this results in F = 3.03 dB
If we put it between Coaxial cable and Receiver, this results in F = 4.76 dB

Aha !


Noisy Resistors

Resistors are noisy. This noise depends on temperature. Therefore it is called 'thermal noise'. It further depends on bandwidth. (See formula above). Using T = 300 K and B = 1 Hz we get :

P_therm = - 174 (dBm/Hz)


Noise Figure Measurement

In order to measure the noise figure, we need a signal generator, a DUT (device under test) and a spectrum analyser. (Or equivalent equipment.)

• Terminate the input of the DUT with an adequate terminator (usually 50 Ω).
• Connect the spectrum analyser at the output of the DUT.
• Select a useful bandwidth and measure the Noise power at the frequency of interest.

As the terminator is matched to the input of the DUT, it will deliver half of the power to the DUT. (power matching). The DUT will amplify that noise power and add its own noise.

P_term = amath Gain * 1/2 * F * 2 * k * T * B endamath

Now connect the signal generator and set the power level therefore, that the measurement at
the spectrum analyser increases by 3 dB.

P_term + 3 dB = amath 1/2 * F * Gain * 2 * k * T * B   +   P_generator * Gain endamath

rearranging for F yields in :

amath F = P_G / (k * T * B) endamath    where   P_G = P_Generator


Hints :

As the Noise power depends on the Bandwidth, it is wise to note the Power and the measurement-bandwidth.

Decreasing the Bandwidth by a factor of 10 will reduce the noise level by 10 dB. And vice versa.



Credits :

The Visualisation uses ASCIIMathML.js

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