Noise figure

The noise figure , sometimes noise factor called, is in the Telecommunications a measure for the noise of a linear two-port network . In this context, a two-port can represent an amplifier stage, for example . The noise figure only applies under the defined conditions and cannot be transferred directly to a real circuit. The noise figure includes the specification of the frequency for which this applies and was determined. A value of 500  MHz is common, since the 1 / f noise is negligible at this frequency .

General

Signal (S) and noise power (N) at the input and output of a two-port (hatched)

The matched to the input impedance of the two-port network glittering resistance is on a noise temperature of 290  K . This temperature value, which corresponds approximately to room temperature , is chosen arbitrarily and designates the standard noise figure . ${\displaystyle T_{0}}$

At the input, the two-port signal power and a noise power are fed, the ratio of which represents the signal-to-noise ratio  (SNR) of the input: ${\displaystyle S_{1}}$ ${\displaystyle N_{1}}$

${\displaystyle \mathrm {SNR} _{\mathrm {ein} }={\frac {S_{1}}{N_{1}}}}$

At its output, the two-port then emits signal power and noise power to the impedance . With an ideally assumed, noise-free two-port, the SNR of the output is ${\displaystyle S_{2}}$ ${\displaystyle N_{2}}$ ${\displaystyle Z_{L}}$

${\displaystyle \mathrm {SNR} _{\mathrm {aus} }={\frac {S_{2}}{N_{2}}}}$

equal to the SNR of the input . ${\displaystyle S_{1}/N_{1}}$

In real two-port devices, such as an electronic amplifier with a gain factor of  G , the amplifier has internal noise sources that are not correlated with the generator , which means that the signal-to-noise ratio at the output is always lower than the signal-to-noise ratio at the input:

${\displaystyle {\frac {S_{2}}{N_{2}}}\leq {\frac {S_{1}}{N_{1}}}}$

The challenge of an amplifier in this context is to add as little intrinsic noise to the signal as possible, so that the useful signal  S at the output is above the noise level of the subsequent processing stages despite the deterioration in the signal-to-noise ratio.

definition

The noise figure  F is given by the ratio:

${\displaystyle F={\frac {\mathrm {SNR} _{\mathrm {ein} }}{\mathrm {SNR} _{\mathrm {aus} }}}={\frac {1}{G}}\cdot {\frac {N_{2}}{N_{1}}}}$

with the amplification factor  G of the amplifier, for which normally applies. However, if there is attenuation, as is the case with a cable , for example , then is ${\displaystyle G>1.}$ ${\displaystyle G<1.}$

The noise figure is often given logarithmically in decibels (dB) as the noise figure :

${\displaystyle F_{\mathrm {dB} }=10\cdot \mathrm {log} {\frac {\mathrm {SNR} _{\mathrm {ein} }}{\mathrm {SNR} _{\mathrm {aus} }}}}$

Since the variables generally depend on the frequency , a sufficiently small bandwidth is selected for the practical determination of the noise figure within the framework of the noise measurement , within which all variables are approximately constant over the frequency. The noise figure thus becomes a function of the frequency, which is then also referred to as the spectral noise figure  F (f).

Linear two-port

It is also possible to describe the noise figure using the additional noise power generated in the linear two-port . The noise power on the output side is made up of the increased noise power supplied on the input side and the noise power generated in the two-port : ${\displaystyle N_{v}}$ ${\displaystyle N_{2}}$ ${\displaystyle G}$ ${\displaystyle N_{1}}$ ${\displaystyle N_{v}}$

${\displaystyle N_{2}=GN_{1}+N_{v}}$

With this, the noise figure of the linear two-port can be represented:

${\displaystyle F={\frac {GN_{1}+N_{v}}{GN_{1}}}=1+{\frac {N_{v}}{GN_{1}}}}$

with the noise figure additionally introduced by the two-port : ${\displaystyle F_{v}}$

${\displaystyle \Rightarrow F_{v}:=F-1={\frac {N_{v}}{GN_{1}}}}$

With ideal, noise-free two-ports,

${\displaystyle N_{v}=F_{v}=0.}$

As a result, the noise figure for the ideal, noise-free linear two-port (independent of frequency) is:

${\displaystyle \Rightarrow F=1.}$

cascade

If several two-ports are connected in series as a cascade - this is the case, for example, when amplifiers are lined up along a longer line - the noise figure  F _{g of} a cascade with n  two-ports can be generalized to:

${\displaystyle {F_{g}=1+(F_{1}-1)+{\frac {F_{2}-1}{G_{1}}}+{\frac {F_{3}-1}{G_{1}\cdot G_{2}}}+{\frac {F_{4}-1}{G_{1}\cdot G_{2}\cdot G_{3}}}+\cdots +{\frac {F_{n}-1}{G_{1}G_{2}G_{3}\cdots G_{n-1}}}=1+\sum _{k=1}^{n}{\frac {(F_{k}-1)}{\prod _{i=0}^{k-1}G_{i}}}\quad {\text{mit }}G_{0}=1}}$

This expanded form of the noise figure is also known as the Friis formula .

Noise temperature

The noise figure of a two-port can also be expressed with the help of the noise temperature  T _e :

${\displaystyle F=1+{\frac {T_{\mathrm {e} }}{T_{0}}}}$

Here, T ₀ is the reference temperature which is set for the standard noise figure of 290K.

An ideal, noise-free amplifier has a noise temperature of T _e = 0 K, which corresponds to a noise figure of 1. A real amplifier that is at a noise temperature of T _e = 290 K , for example , has a noise figure of 2, which means that the SNR at the amplifier output deteriorates by 3 dB. In particular for input amplifiers and to achieve a high SNR, it is therefore necessary to keep the noise temperature of the amplifier as low as possible.

Non-linear two-port

Non-linear two-ports can change the spectra of useful power and noise power at the two-port input in such a way that filter measures can result in noise figures below 1 in favorable cases. A typical example is a demodulator for frequency-modulated useful signals, which produces an improved signal-to-noise ratio at the demodulator output for signal-to-noise ratios at the input above a threshold value.

Optical amplifier

The noise figure describes the decrease in the signal-to-noise ratio of a coherent optical signal when it passes through an optical amplifier. For this purpose, the signal to noise ratios of the electrical current are considered, which an ideal photodetector with the quantum efficiency  1 would deliver in front of or behind the optical amplifier. The electrical powers involved in the S / N ratios are therefore proportional to the square of the corresponding optical powers.

Although the input signal is assumed to be ideal, its power is not completely constant due to the quantum nature of the photons , but varies due to the shot noise .

In addition to the noise already contained in the input signal and amplified in the optical amplifier, there are additional noise components that arise in the amplifier. The mixed product of signal and superluminescence (ASE: amplified spontaneous emission ) usually dominates . If you neglect the other noise components, you get the noise figure for the optical amplifier ( EDFA )

${\displaystyle F={\frac {\mathrm {n} _{\mathrm {ASE} }}{G\cdot h\cdot f}}+{\frac {1}{G}}}$

With

• ${\displaystyle \mathrm {n} _{\mathrm {ASE} }}$ Power density of the ASE noise in W / Hz
• G gain factor
• h Planck's quantum of action
• f Frequency of the optical input signal in Hz

A different formula applies to Raman amplifiers , since amplification and attenuation take place simultaneously along the fiber .

Individual evidence

1. ↑ HW König: The noise figure of linear two-port and amplifier tubes . Conference proceedings frequency , 1955, p. 3-11 .

literature

• Rudolf Müller: Noise . 2nd Edition. tape 15 . Springer Verlag, 1989, ISBN 3-540-51145-8 .
• Curt Rint : Handbook for Radio Frequency and Electrical Technicians. 12th edition. Hüthig and Pflaum Verlag GmbH, 1979, ISBN 3-8101-0044-7 .
• Jürgen Detlefsen, Uwe Siart: Basics of high frequency technology . 2nd Edition. Oldenbourg Verlag, Munich Vienna 2006, ISBN 3-486-57866-9 .
• Anders Bjarklev: Optical Fiber Amplifiers: Design and System Applications . Artech House, Norwood 1993, ISBN 0-89006-659-0 .