Noise temperature

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A temperature (in degrees Kelvin) assigned to an electrical component such that the noise power delivered by the noisy component to a noiseless matched resistor is given by

P R L = k T s B n

where:

$k$ = Boltzmann’s Constant (1.38 ∙ 10-23 J/K)
$T s$ = noise temperature (degrees K)
$B n$ = noise bandwidth (Hz)

Engineers often model noisy components as an ideal component and a noisy resistor in series (see Figure 1)

Figure 1: Ideal Component (Vsys) and Noisy Resistor (Rsys) Driving a Load Resistor

The temperature of the source resistor is often assumed to be room temperature. $T 0$ is used when the noise temperature is room temperature.

room temperature = $T 0$ = 290 K = 17°C = 62°F (McClaning2002)

1 Applications
2 Noise Factor and Noise Figure
3 Noise Temperature of a Cascade
4 Measuring Noise Temperature
5 References

[edit] Applications

A communications system is typically made up of a transmitter, a communications channel, and a receiver. The communications channel may consist of any one or a combination of many different physical media (air, coaxial cable, printed wiring board traces…). The important thing to note is that no matter what physical media the channel consists of, the transmitted signal will be randomly corrupted by a number of different processes. The most common form of signal degradation is called additive noise. (Proakis2005)

The additive noise in a receiving system can be of thermal origin (thermal noise) or can be from other noise-generating processes. Most of these other processes generate noise whose spectrum and probability distributions are similar to thermal noise. Because of these similarities, the contributions of all noise sources can be lumped together and regarded as thermal noise. The noise power generated by all these sources ( $P n$ ) can be described by assigning to the noise a noise temperature ( $T n$ ) defined in Equation 2.(Skolnik1990)

$T n = P n / (k B n)$ Equation 2: Noise Temperature

In a wireless communications receiver, $T n$ would equal the sum of two noise temperatures:

$T n$ = ( $T a n t$ + $T s y s$ )

$T a n t$ is the antenna noise temperature and determines the noise power seen at the output of the antenna. The physical temperature of the antenna has no affect on $T a n t$ . $T s y s$ is the noise temperature of the receiver circuitry and is representative of the noise generated by the non-ideal components inside the receiver.

[edit] Noise Factor and Noise Figure

An important application of noise temperature is its use in the determination of a component’s noise factor. The noise factor quantifies the noise power that the component adds to the system when its input noise temperature is $T 0$ .

$F = \frac{T_{0} + T_{sys}}{T_{0}}$ Equation 3: Noise Factor

The noise factor (a linear term) can be converted to noise figure (in decibels) using Equation 4.

$F_{dB} = 10 \ \log (F)$ Equation 4: Noise Figure

[edit] Noise Temperature of a Cascade

If there are multiple noisy components in cascade, the noise temperature of the cascade will be limited by the noise temperature of the first component in the cascade. The gains of the components at the beginning of the cascade have a large effect on the contribution of later stages to the overall cascade noise temperature. The cascade noise temperature can be found using Equation 5. (McClaning2002)

$T_{cas} = T_{1} + \frac{T_{2}}{G_{p,1}} + \frac{T_{3}}{G_{p,1} G_{p,2}} + \dots + \frac{T_{n}}{G_{p,1} G_{p,2} G_{p,3} \dots G_{p,n-1}}$ Equation 5: Cascade Noise Temperature

where

$T c a s$ = cascade noise temperature
$T 1$ = noise temperature of the first component in the cascade
$T 2$ = noise temperature of the second component in the cascade
$T 3$ = noise temperature of the third component in the cascade
$T n$ = noise temperature of the nth component in the cascade
$G p,1$ = linear gain of the first component in the cascade
$G p,2$ = linear gain of the second component in the cascade
$G p,3$ = linear gain of the third component in the cascade
$G p, n - 1$ = linear gain of the (n-1) component in the cascade

[edit] Measuring Noise Temperature

The direct measurement of a component’s noise temperature is a difficult process. Suppose that the noise temperature of a low noise amplifier (LNA) is measured by connecting a noise source to the LNA with a piece of transmission line. By observing Equation 5, it can be seen that the noise temperature of the transmission line ( $T 1$ ) has the potential of being the largest contributor to the output measurement (especially when you consider that LNA’s can have noise temperatures of only a few Kelvins). To accurately measure the noise temperature of the LNA the noise from the input coaxial cable needs to be accurately known. ^[1] This is difficult because poor surface finishes and reflections in the transmission line make actual noise temperature values higher than those predicted by theoretical analysis. (Schuster1962)

Similar problems arise when trying to measure the noise temperature of an antenna. Since the noise temperature is heavily dependant on the orientation of the antenna, the direction that the antenna was pointed during the test needs to be specified. In receiving systems, the system noise temperature will have three main contributors, the antenna ( $T A$ ), the transmission line ( $T L$ ), and the receiver circuitry ( $T R$ ). The antenna noise temperature is considered to be the most difficult to measure because the measurement must be made in the field on an open system. One technique for measuring antenna noise temperature involves using cryogenically cooled loads to calibrate a noise figure meter before measuring the antenna. This provides a direct reference comparison at a noise temperature in the range of very low antenna noise temperatures, so that little extrapolation of the collected data is required.(Schuster1962)

[edit] References

*McClaning, Kevin, and Tom Vito. Radio Receiver Design. Atlanta, GA: Noble Publishing Corporation, 2000. ISBN 1-884932-07-X.
*Proakis, John G., and Masoud Salehi. Fundamentals of Communication Systems. Upper Saddle River, New Jersey: Prentice Hall, 2005. ISBN 0-13-147135-X.
*Skolnik, Merrill I., Radar Handbook (2nd Edition). McGraw-Hill, 1990. ISBN 978-0-07-057913-2
*Hu, Robert. “Analysis of the Input Noise Contribution in the Noise Temperature Measurements.” Microwave and Wireless Components Letters, IEEE. Volume 15, Issue 3. pp 141-143. March 2005. 23 April 2008 <http://ieeexplore.ieee.org>.
Schuster, D., C. Stelzried, and G. Levy. “The Determination of Noise Temperatures of Large Parabaloidal Antennas.” Antennas and propagation, IEEE Transactions on. Volume 10, Issue 3. pp 286-291. May 1962. 23 April 2008 <http://ieeexplore.ieee.org>.