2.2 Noise Temperature.
All real systems suffer from the effects of random noise. This noise is an important factor in determining how difficult it will be to detect and measure a given signal in a given amount of time. We therefore need to know the noise level of a given system if we want to decide if it is good enough to use for a particular purpose.
Noise arises in various way, but the most common mechanisms produce:-
- Thermal Noise produced by the random, thermally produced, motions of atoms and electrons in any materials.
- Shot Noise produced by the ‘particle-like’ behaviour of electrons and photons.
Shot Noise is only avoidable if we reduce all the current and photon flux levels to zero — not a very helpful requirement if we're trying to carry an electromagnetic signal from place to place or detect a photon flux! In theory, thermal noise can be reduced to a very low level by cooling a system to a temperature near absolute zero. For the sake of simplicity we will therefore assume that the thermal noise is small enough to be ignored and concentrate our attention on the effects of shot noise. This situation is often referred to in textbooks and scientific papers as being “Shot Noise Limited” or “Photon Noise Limited”. These two descriptions are equivalent in this situation since we assume that the charge carriers we observe are ‘created’ by the detected photons.
Consider a ‘steady’ electric current, I. This consists of a flow of electrons, each carrying a charge, q. On average, the number of charges moving past some place in a circuit during a time, t, will be
where the angle brackets indicate that we're talking about an average value.
In practice the electrons don't pass though a system in a perfectly regular uniform way. The rate at which they flow past tends to vary unpredictably from moment to moment. As a result, a nominally steady current — if observed carefully enough — appears to fluctuate randomly from moment to moment in an unpredictable way. This fluctuation can be described as a Noise Current value which is added to the nominal steady level. By measurement or theoretical analysis it is possible to show that the observed mean squared value of this fluctuation has the value
where B is the bandwidth of the system used to observe the fluctuations. Note that the size of the noise current depends upon the actual current. The higher the actual current level is, the more charges it contains, and the more ‘lumpy’ it appears to be.
Consider now the photoconductive mixer system shown in figure 1.3. Ignoring all noise mechanisms except shot noise we can say this this provides an output signal to noise ratio of
From part 1 we can say that
In practice we can usually expect the LO power level to be much higher than the signal level. This means that 
so, for practical purposes, we can assume that current through the detector is essentially 
. We can therefore rewrite equation 2.11 as
Now we know from before that
Hence
As we'd expect, the signal to noise ratio depends upon the signal power. This result shows that it also depends upon the signal frequency. This is because we're considering shot noise. Each of the electrons in the detector current corresponds to a detected photon. The higher the signal frequency, the higher the individual photon energy, and the smaller the number of photons per second which correspond to a given input power level. As a result, higher signal frequencies are quantised into bigger ‘lumps’ making the shot noise more obvious!
Consider now the situations illustrated in figure 2.1. In each case the signal entering the heterodyne system comes from a Thermal Source of temperature, T. The receiver doesn't know what it is ‘looking at’, it only knows that a given power spectrum is being supplied as an input signal. We can assume that the receiver has an IF filter of bandwidth, B, and that it is a Single Sideband system — i.e. an rf filter only allows it to respond to signals on one side of the LO frequency. It therefore only sees a spectral bandwidth, B, from the source. The spectrum of a thermal Black Body source peaks at a wavelength, 
, such that 
mm°K. For signals in the Rayleigh-Jeans part of the spectrum — i.e. for frequencies 
— the signal power reaching the mixer from a source of temperature, T, will be
(N.B. This result assumes we are only detecting a single mode.)
Combining this with 2.15 we get
In practice the performance of a heterodyne system can be defined in terms of a suitable Noise Temperature value, 
, chosen so that
In effect, 
is the temperature of a thermal source which would provide a signal power equal to the noise power level. Combining 2.17 and 2.18 we find that when the only noise generating process is shot noise the heterodyne receiver's noise temperature value will be
This represents the lowest noise level we could ever hope to attain since we've ignored every possible noise generating mechanism except shot noise. Using a less perfect system we can expect a higher noise temperature.
Now 
is a measure of the noise level produced by our mixer system. It is therefore interesting to note that — since k and h are universal constants — once the signal frequency, f, has been chosen the best possible noise performance it determined by the only by the mixer's quantum efficiency. The maximum possible quantum efficiency will normally be 
— i.e. one electron per photon. (Although this isn't true for avalanche devices the avalanche multiplication process itself generates a large amount of noise. This reduces the attainable S/N value, so we should avoid avalanche devices when seeking the best possible performance.) As a result we find that the noise temperature of any single sideband superheterodyne receiver will always be
where 
is in Kelvins and f is in GHz.
Minimum Possible Noise Temperature versus Frequency
| f
| 100 MHz
| 100 GHz
| 10 THz
| 1000 THz
|
| 3 metres
| 3 mm
| 30 µm
| 3000 Å
|
| 4·8×10-3 °K
| 4·8 °K
| 480 °K
| 48,000 °K
|
From the equation and table we can see that the ultimate noise level of a heterodyne system is very low for signal frequencies below tens of terahertz (THz). At higher frequencies the larger photon energy makes the noise level higher. It should however be remembered that this noise is a result of the signal quantisation, hence it affects any system used to detect visible & near-visible signal powers. Furthermore, whatever the signal frequency, the heterodyne technique is valuable because of its ability to transfer or convert information about a spectrum down to lower frequencies which are more easily processed. At high frequencies single photons may have enough energy that we can begin to detect and analyse them individually – this leads to Photon Counting techniques which we won’t consider here.
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