# Understanding Noise Figure

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Understanding Noise Figure
Iulian Rosu, YO3DAC / VA3IUL, http://www.qsl.net/va3iul
One of the most frequently discussed forms of noise is known as Thermal Noise. Thermal noise is a random fluctuation in voltage caused by the random motion of charge carriers in any conducting medium at a temperature above absolute zero (K=273 + °Celsius). This cannot exist at absolute zero because charge carriers cannot move at absolute zero. As the name implies, the amount of the thermal noise is to imagine a simple resistor at a temperature above absolute zero. If we use a very sensitive oscilloscope probe across the resistor, we can see a very small AC noise being generated by the resistor.
• The RMS voltage is proportional to the temperature of the resistor and how resistive it is.
Larger resistances and higher temperatures generate more noise.
The thermal noise phenomenon was discovered (or anticipated) by Schottky in 1928 and first measured and evaluated by Johnson in the same year.
Shortly after its discovery, Nyquist used a thermodynamic argument to show that the opencircuit rms thermal noise voltage Vn across a resistor is given by:
Vn = 4kTRB
where:
k = Boltzmann constant (1.38 x 10-23 Joules/Kelvin) T = Temperature in Kelvin (K= 273+°Celsius) (Kelvin is not referred to or typeset as a degree) R = Resistance in Ohms B = Bandwidth in Hz in which the noise is observed (RMS voltage measured across the resistor is also function of the bandwidth in which the measurement is made).
As an example, a 100 kΩ resistor in 1MHz bandwidth will add noise to the circuit as follows:
Vn = (4*1.38*10-23*300*100*103*1*106) ½ = 40.7 μV RMS
• Low impedances are desirable in low noise circuits. • The thermal noise voltage is dependent only on the resistive component and is
independent of any reactance in the circuit. Reactances (capacitive or inductive) are due to magnetic and electric fields where electron fluctuations are non-existent. • In a series resonant circuit, the noise source is the series loss resistance. • The noise voltage across a parallel resonant circuit is Q times the noise emf (Q is the tuned-circuit quality factor or figure of merit). • In order to compare the noise of different sources we may convert the measurement bandwidth (1MHz in our case) to 1Hz bandwidth (the lowest bandwidth denominator). • Thermal noise is present in all circuit elements containing resistance. • A carbon composition resistor generates the same amount of thermal noise as a metal film resistor of the same value. The noise is independent of the composition of the resistance.

• For a fixed temperature, the thermal noise voltage in a circuit can be reduced by minimizing the resistance and the bandwidth. Further reduction can only be obtained by operating the circuit at lower temperatures.
Noise Bandwidth, B, is defined as the equivalent rectangular pass-band that passes the same amount of noise power as is passed in the usable receiver bandwidth and that has the same peak in-band gain as the actual device has. It is the same as the integral of the gain of the device over the usable frequency bandwidth.
• Typically, bandwidth B is approximately equal to the 3dB bandwidth. • In a receiver, for best sensitivity, B should be no greater than required bandwidth. In RF applications, we usually deal with circuits having matched input and output impedances, and therefore we are more concerned with the power available from a device than the voltage. In this case, it is common to express the noise of a device in terms of the available noise power.
The maximum power transfer theorem predicts that the maximum noise power is transferred from a thermal noise source when the load impedance presents a conjugate match to the source impedance. Available noise power PN which can theoretically be transferred under such conditions is given:
PN = kTB
The factor of 4R has cancelled out so the available noise power does not depend upon the value of the Resistance. This is significant because it means that the available noise power of any resistor (or any noise source), if measured over the same bandwidth B, can be represented by a resistor at temperature T.
• Thus, every noise source has an Equivalent Noise Temperature.
Thermal noise power (in dBW) is defined as:
PN(dBW) = 10 log(kTB)
Where PN in dBW is the Noise Power at the output of the thermal noise source, k is Boltzmann’s constant 1.38 x 10-23(J/K), T is temperature (Kelvin), and B is the bandwidth (Hz). At room temperature (17°C/290K), in a 1 Hz Bandwidth we can calculate the power:
PN(dBW) = 10 log(1.38 x 10-23 x 290 x 1) = -204 (dBW) P(dBW) = 10 log(P(W) / 1W)
Power in dBm takes its reference as 1mW, and results the relation:
0 dBW = 1 W = 30 dBm
Therefore, we can calculate the thermal noise power in dBm at (17°C/290K) in a 1 Hz BW:
Thermal Noise Power = -204 + 30 = -174 dBm/Hz
• Noise Power of -174 dBm/Hz is the reference for any noise power calculation when designing RF systems working at room temperature.

• Relative to the bandwidth, we can use the reference level of -174 dBm/Hz and simply multiply it by the actual bandwidth of the radio channel.
Ex. 1: Let’s calculate the thermal noise floor of the 200 kHz channel bandwidth used by GSM. We just calculate the thermal noise in 200 kHz bandwidth:
kTB for GSM (200 kHz) = -174 dBm/Hz + 10 log (200.000 Hz) = -121dBm The -121dBm is therefore the absolute lowest noise power we get in a 200 kHz GSM channel.
Ex. 2: An SSB receiver has a bandwidth of 2.4 kHz, which makes the thermal noise floor to be: kTB for SSB (2.4 kHz) = -174 + 10 log (2400) = -140dBm
• It does not matter if the RF system operates on 100 MHz or at 2450 MHz, the noise power / Hz will be the same, if the radio channel bandwidth is the same.
• With constant bandwidth, the thermal noise power vs temperature has linear characteristic with Slope = kB. This is an important characteristic that is used for accurate noise measurements.
• Temperatures correspond to power levels. When the temperature of a resistor is doubled, the output power from it is doubled (the voltage is proportional to the square root of the temperature).
• Powers from uncorrelated sources are additive, so noise temperatures are additive. • In addition to thermal noise, amplifiers (or other devices with semiconductors) also
contribute to the total system noise.
Noise Figure and Noise Factor
In 1942 Dwight O. North introduced the term Noise Factor (F), which referred to absolute receiver sensitivity, which is a numeric referred to field strength. That field strength, in a planepolarized wave passing an antenna, necessary to produce a signal power at the detector of a receiver, equal to the total noise power (from all sources) at the same point.
To characterize the receiver alone, Harald T. Friis introduced in 1944 the Noise Figure (NF) concept which characterized the degradation in Signal to Noise Ratio (SNR) by the receiver. The concept of Noise Figure allows the sensitivity of any amplifier to be compared to an ideal (lossless and noiseless) amplifier which has the same bandwidth and input termination.
• Noise Figure (NF) is a measure of how much a device (such an amplifier) degrades the Signal to Noise ratio (SNR).
SNR_input[linear] = Input_Signal[Watt] / Input_Noise[Watt] SNR_input[dB] = Input_Signal[dB] - Input_Noise[dB]
SNR_output[linear] = Output_Signal[Watt] / Output_Noise[Watt] SNR_output[dB] = Output_Signal[dB] - Output_Noise[dB]
• Noise Factor (linear not dB) of a receiver is the ratio of the SNR at its input to the ratio of the SNR at its output.
NoiseFactor_F(linear) = SNR_input[linear] / SNR_output[linear] NoiseFactor_F[dB] = SNR_input[dB] - SNR_output[dB] NoiseFigure_NF[dB] = 10*LOG (NoiseFactor_F(linear))

• Note that SNR at the output will always be smaller than the SNR at the input, due to the fact that circuits always add to the noise in a system.
• The Noise Factor, at a specified input frequency, is defined as the ratio of the total Noise Power per unit bandwidth available at the output port when noise temperature of the input termination is standard (290K) to that portion of engendered at the input frequency by the input termination.
• Noise Factor varies with the bias conditions, with frequency, with temperature, and with source resistance. All of these should be defined when specifying Noise Factor.
The concept of Noise Factor has three major limitations: 1. Increasing the source resistance may decrease the Noise Factor while increasing the total noise in the circuit. 2. If purely reactive source is used, Noise Factor is meaningless since the source noise is zero, making the noise factor infinite. 3. When the device noise is only a small percentage of the source thermal noise (as with some very low noise transistors), the Noise Factor requires taking the ratio of two almost equal numbers. This can produce inaccurate results.
• A direct comparison of two Noise Factors is only meaningful if both are measured at the same source resistance.
• Knowing the Noise Factor for one value of source resistance does not allow the calculation of the noise factor at other values of resistance. This is because both the source noise and device noise vary as the source resistance is changed.
The maximum Noise Figure of the receiver system, when is given the required Receiver Sensitivity and the required Receiver Bandwidth, is:
As can be seen from the equation above, narrow Bandwidth and smaller SNR will relax the required receiver Noise Figure requirements.
• When designing circuits for use with extremely weak signals, noise is an important consideration. The noise contribution of each device in the signal path must be low enough that it will not significantly degrade the Signal to Noise Ratio.
• Noise in an RF system can be generated from external sources, or the system itself. • The Noise level of a system sets the lower limit on the magnitude of a signal that can be
detected in the presence of the noise. So, to achieve the best performance you need to have a minimum residual noise level. • Noise Figure is used to describe the noise contribution of a device. An ideal amplifier would have no noise of its own, but would simply amplify what went in to it.
o For example, a 10dB amplifier would amplify the Signal (and the Noise) at its input by 10dB. Therefore, although the noise floor at the output of the amplifier would be 10dB higher than at the input.

• An “ideal noiseless" amplifier would not change the Signal to Noise ratio (SNR). • A "real life" amplifier will amplify not only the noise at its input, but will contribute its own
noise to signal. This reduces the Signal to Noise ratio at the output of the amplifier. • So, the "real life" amplifier has two major internal components: an "ideal noiseless"
amplifier and a noise source. The noise source adds noise to any signal what enters to the amplifier and then the ideal amplifier amplifies the whole thing by an amount equal to its gain, with no noise contribution of its own.
o For example, a 10dB attenuator placed at the input of an amplifier will increase the total Noise Figure of the system with 10dB.
• An attenuator placed at the input of a system will increase the total Noise Figure with the same amount of its attenuation.
• An attenuator placed at the input of a receiver would not affect the SNR if the level at its output is inside of the input dynamic range of the receiver.
o As an example, let's assume that we have an amplifier at room temperature with 10dB of gain which has only a matched resistor at its input and one matched resistor at its output. - The noise at the input of the amplifier must be -174dBm/Hz. - If the amplifier is known to have a 3dB NF, the internal noise source adds an equal noise to the input noise before amplification. - Then 10dB of gain increases the noise by 10dB. - Therefore, the noise at the output of the amplifier is 13dB higher than at the input, or (-174dBm/Hz + 10dB gain +3dB NF) = -161dBm/Hz.
• Another way to visualize the process is to think in terms of kTB instead of dBm. - If the amplifier has a matched input resistance and its noise power is -174dBm/Hz, then the input resistor is supplying 1kTB of noise to the amplifier. - If the amplifier is known to have a noise figure of 3dB, that tells us that the internal noise source will double the noise before amplification. - Therefore, the internal noise source must supply an additional 1kTB of noise, to yield 2kTB, or twice the noise power the noise source is contributing.
• The noise contribution of the amplifier's noise source is fixed and does not change with input signal.
• Therefore, when more noise is present at the amplifier input, the contribution of the internal noise source is less significant in comparison.
• When the noise into an amplifier is higher than kTB (-174dBm/Hz), the amplifier 's Noise Figure plays a smaller role in the amplifier's noise contribution.
• The Noise Figure (NF) of a device is only calculated with the input noise level at kTB. • The Noise Temperature at the output of an amplifier Tout (in Kelvin) is the sum of the
Noise Temperature of the source Tsource and the Noise Temperature of the amplifier itself Tampl multiplied by the Power Gain of the amplifier.
Tout (K) = G(linear) * ( Tampl (K) + Tsource (K) )

Tout = Noise Temperature at amplifier output in Kelvin. G = Power Gain in linear scale not in dB. Tampl = Noise Temperature of amplifier. Tsource = Noise Temperature of source.
The same formula is valid for attenuators:
Tout = Gatt * ( Tatt + Tsource )
• The Noise Figure of an attenuator is the same as the attenuation in dB. • The Noise Figure of an attenuator preceding an amplifier is the Noise Figure of the
amplifier plus the attenuation of the attenuator in dB.
- For above example both amplifiers have 10dB gain and NF=3dB. - The signal goes in at -40dBm with a noise floor at kTB (-174dBm/Hz). - We can calculate that the signal at the output of the first amplifier is -30dBm and the noise is: (-174dBm/Hz input noise) + (10dB of gain) + (3dB NF) = -161dBm/Hz. - Let see how many kTBs are entering in the second amplifier: (-161dBm/Hz) is 13dB greater than kTB (-174dBm). - 13dB is a power ratio of 20x. So, the noise floor at the second amplifier is 20 times kTB or 20kTB. - Next calculate how many kTBs are added by the noise source of the second amplifier (in this case, 1kTB because the NF=3dB). - Finally calculate increase in noise floor at the 2nd amplifier as a ratio and convert to dB. - Ratio of (input noise floor) + (added noise) to (input noise floor) is: (20kTB+1kTB) / (20kTB) = 21/20 - In dB = 10LOG (21/20) = 0.21dB - Therefore, the 2nd amplifier only increases the noise floor by 0.21dB even though it has a noise figure of 3dB, because the noise floor at its input is significantly higher than kTB. - The first amplifier degrades the signal to noise ratio by 3dB, while the second amplifier degrades it only by 0.21dB. The total Noise Figure of the two amplifiers is 3.21dB.
• When amplifiers are cascaded together in order to amplify very weak signals, it is generally the first amplifier in the chain which will have the greatest influence upon the signal to noise ratio because the noise floor is lowest at that point in the chain.
• The first amplifier in a chain should have low noise figure (noise factor) and high gain.
Determining the total Noise Figure of a chain of amplifiers (or other devices):

NFactor_total = NFact1 + (NFact2-1)/G1 + (NFact3-1)/(G1*G2) + (NFact3-1)/(G1*G2*….Gn-1)

where:

NFactor = Noise factor of each stage (Linear not in dB). Noise Figure[dB] = 10*LOG (NFactor) G = Gain of each stage as a ratio, not dB (for example 4x, not 6dB)

• The first amplifier in a chain has the most significant effect on the total noise figure than any other amplifier in the chain.
• The lower noise figure amplifier should go first in a line of amplifiers (assuming all else is equal).
• For example, we have two amplifiers with equal gain, but with different noise figures. Assume 10dB gain in each amplifier. One amp is NF = 3dB and the other 6dB. When the 3dB NF amplifier is first in cascade, the total NF is 3.62dB. When the 6dB amplifier is first, the total NF is 6.3dB.
• This also applies to gain. If two amplifiers have the same Noise Figure but different gains, the higher gain amplifier should precede the lower gain amplifier to achieve the best overall Noise Figure.
• The overall Noise Factor of an infinite number of identical cascaded amplifiers is: NFactor_total = 1 + M with: M = (F-1) / [1 - (1/G)] where: F is Noise_Factor(linear) of each stage, and G is Gain(linear) of each stage.
NoiseFigure_total[dB] = 10*LOG (NFactor_total(linear))
• At the input of the amplifier the thermal noise power vs temperature has a Slope = kB. • At the output of the amplifier the thermal noise power vs temperature has a Slope = kBG. • The output noise power for absolute temperature of zero is the added noise power, Na,
generated within the amplifier.
Thermal Noise Power [Watt] vs Temperature [Kelvin]

Noise Figure of a Device
• The Noise Figure of a device is the degradation in the Signal to Noise Ratio (SNR) as a signal pass through the device.
The equation of the Noise Figure of a device adopted by the IEEE (Institute of Electrical and Electronics Engineers) is:
Device NF[dB] = 10*LOG [(Na + k*To*B*G) / (k*To*B*G)]
Na=Added Noise Power [W], To =290K, B=Bandwidth[Hz], G=Gain[Linear not dB] , k=Boltzmann ct. Therefore, the Noise Figure of a device is the ratio of the total Noise Power at the output, to
that portion of the Noise Power at the output due to noise at the input when the input source temperature is 290 Kelvin.
• The Noise Figure of a device is independent of the signal level as long as the device is linear (output power vs. input power).
Equivalent Input Noise Temperature
• Every noise source has an equivalent noise temperature. Initially used in Satellite Receivers, the Equivalent Input Noise Temperature (Te), is used to describe the noise performance of a device rather than the Noise Figure. • Te is the Equivalent Input Noise Temperature of source impedance into a perfect
(noiseless) device that would produce the same Added Noise Power (Na). • Te is mostly used as a system parameter, and is defined as:
Te(K) = Na(dBW) / (k*G(linear)*B(Hz))
the relation of Te with the Noise Factor (F) is:
Te(K) = To(K) * (F-1) F = 1 + (Te(K) / To(K))
For example, a device with Noise Figure NF=0.5dB (Noise Factor, F=1.122), at To = 290K, would have an Equivalent Input Noise Temperature, Te = 35.4K
• The use of Equivalent Input Noise Temperature concept is more meaningful and convenient, even Noise Figure and Noise Temperature are related, since them both measure the same thing.
• The concept is based to the fact that the Noise Power is directly proportional to temperature, and at 0K (absolute zero) there is no noise.
For example, if heating the resistor to 400K and this produce the same noise as that generated by the receiver; we could say that the receiver Equivalent Input Noise Temperature is 400K.
• This number is independent of the receiver bandwidth, and we can use it to compare receivers of different bandwidths.

Noise Temperature [Kelvin] vs Noise Figure [dB]
Resulting Noise Temperature referred to the input (Teq) of cascaded stages is given by:
Teq = T1 + (T2/G1) + [T3/ (G1*G2)] + …..
Noise Temperature (in Kelvin) of each component in the cascade is: T(1,2,3…) = To (F(1,2,3…) – 1) Power gain (Linear not dB) of each component in the cascade is: G1,2,3… The Noise Figure of the cascade is: NF[dB] = 10*LOG [1+ (Teq / To)]
Example: Consider the case of a three-stage amplifier, each stage with 13dB gain and 60K noise temperature.
The numerical gain (linear not dB) of each stage is G = 10(13/10) = 20 The noise temperature of the combined stages is:
Teq = 60 + (60/20) + 60/(20*20) = 60 + 3 + 0.15 = 63.15K The Noise Figure of the three-stage cascade amplifier at room temperature is:
NF[dB] = 10*LOG [1+(63.15 / 290)] = 0.85 dB
Antenna Noise Temperature
The performance of a telecommunication system depends on the Signal to Noise ratio (SNR) at the receiver’s input. The electronic circuitry of the RF front-end (transmission lines, amplifiers, mixers, etc.) has a significant contribution to the system noise. However, the antenna itself is sometimes a significant source of noise, too.

• In Satellite and Space Receiving Systems, the noise level coming from the antenna can be very low, limited by the ground noise (due to the side-lobe radiation of the antenna), and by the background sky temperature (with values often below 100K). In these situations, small changes in the Noise Figure of the receiving system may result in much more change of the Signal to Noise Ratio (SNR).
A receiving antenna exhibits noise at its terminals from two sources: 1. The thermal noise generated in its ohmic resistance (usually negligible). 2. The noise received from external sources. Any object with temperature greater than 0K (-273°C) radiates noise energy. The received noise is represented as though it were thermal noise generated in a fictitious resistance equal to the radiation resistance of the antenna, at a temperature Ta that would account for the noise actually measured. Antenna temperature Ta is a parameter that describes how much noise an antenna produces in a given environment. Antenna noise temperature is not the physical temperature of the antenna but rather an expression of the available noise power at the antenna terminals. However, an antenna does not have an intrinsic "antenna temperature" associated with it; rather the antenna temperature depends on its gain pattern and the thermal environment that it is placed in. Antenna temperature is sometimes referred to as noise temperature of the antenna.
Example: Suppose that an antenna with 200-ohm impedance exhibits an rms noise voltage Vn = 0.1uV at its terminals, when measured in a bandwidth B = 104Hz. Vn2 = 4kTaRB Ta = Vn2 / 4kRB = (10-14) / (4*1.38*10-23*200*104) = 90.6K Thus, the noise at the antenna terminals is equivalent to that of a 200-ohm resistor at temperature of 90.6K.
In a satellite receive system the noise temperature of the antenna Ta referred to the receiver input may be defined by the antenna feedline loss L and the sky temperature Tsky:
Ta = [290*(L-1)+Tsky] / L
where Ta and Tsky are in Kelvin and the feedline loss L is linear, as a power ratio (not dB).
Example: In a satellite communication system, if the feedline of the receive antenna has a loss L of 0.4dB (1.10 linear) and the sky (in the direction where the antenna is pointed) have a noise of 15K, then the noise temperature of the antenna Ta referred to the receiver input is 39K.
• Antenna noise temperature vs antenna pointed direction: - When an antenna is pointed to the night-sky, its noise temperature is very low, with temperatures between 3K to 5K at frequencies 1GHz to 10GHz. - The higher the elevation angle, the lower the night-sky temperature because of the lower physical temperature of the atmosphere toward zenith. - The sky noise depends on the frequency and also depends by the time of the day. 