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Up to now we've looked at the statistical properties of noise in terms of its overall rms level and probability density function. This isn't the only way to quantify noise. Figure 3.3 shows an alternative which is often more convenient.

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As in figure 3.1 we're looking at the Johnson noise produced by a resistor. In this case the voltage fluctuations are amplified and passed through a band-pass filter to an rms voltmeter. The filter only allows through frequencies in some range, . The filter is said to pass a bandwidth, . is the input resistance of the amplifier. Note that this diagram uses a common conventional ‘trick’ of pretending that the noise generated in the resistor is actually coming from an invisible random voltage generator, , connected in series with an ‘ideal’ (i.e. noise-free) resistor. If we build a system like this we find that the rms fluctuations seen by the meter imply that the (imaginary) noise generator produces an average voltage-squared

equation

where: k is Boltzmann's Constant (=1·38 × 10 Ws/K); T is the resistor's temperature in Kelvin; R is it's resistance in Ohms; and B is the bandwidth (in Hz) over which the noise voltage is observed. (Note that, as with the earlier statements about Normal Distribution, etc, this result is not being proved, but given as a matter of experimental fact.) In practice, the amplifier and all the other items in the circuit will also generate some noise. For now, however, we will assume that the amount of noise produced by R is large enough to swamp any other sources of random fluctuations. Applying Ohm's law to figure 3.3 we can say that the current entering the amplifier (i.e. flowing through ) must be

equation

The corresponding voltage seen at the amp's input (across ) will be

equation

hence the mean noise power entering the amplifier will be

equation

For a given resistor, R, we can maximise this by arranging that when we obtain the Maximum Available Noise Power,

equation

This represents the highest thermal noise power we can get to enter the amplifier's input terminals from the resistor. To achieve this we have to match (i.e. equalise) the source and amplifier input resistances. From this result we can see that the maximum available noise power does not depend upon the value of the resistor whose noise output we are examining.

The Noise Power Spectral Density (NPSD) at any frequency is defined as the noise power in a 1 Hz bandwidth at that frequency. Putting into eqn 3.15 we can see that Johnson noise has a maximum available NPSD of just — i.e. it only depends upon the absolute temperature and the value of Boltzmann's constant. This means that Johnson noise has an NPSD which doesn't depend upon the fluctuation frequency. The same result is true of shot noise and many other forms of noise. Noise which has this character is said to be White since we the see the same power level in a fixed bandwidth at every frequency.

Strictly speaking, no power spectrum can be truly white over an infinite frequency range. This is because the total power, integrated over the whole frequency range, would be infinite! (Except, of course, for the trivial example of a NPSD of zero.) In any real situation, the noise generating processes will be subject to some inherent mechanism which produces a finite noise bandwidth. In practice, most systems we devise to observe noise fluctuations will only be able to respond to a range of frequencies which is much smaller than the actual bandwidth of the noise being generated. This in itself will limit any measured value for the total noise power. Hence for most purposes we can consider thermal and shot noise as ‘white’ over any frequency range of interest. However the NPSD does fall away at extremely high frequencies, and this ensures that the total noise power is always finite.

It is also worth noting that electronic noise levels are often quoted in units of Volts per root Hertz or Amps per root Hertz. In practice, because noise levels are — or should be! — low, the actual units may be nV/ or pA/. These figures are sometimes referred to as the NPSD. This is because most measurement instruments are normally calibrated in terms of a voltage or current. For white noise we can expect the total noise level to be proportional to the measurement bandwidth. The ‘odd’ units of NPSD's quoted per root Hertz serve as a reminder that — since power volts (or current) — a noise level specified as an rms voltage or current will increase with the square root of the measurement bandwidth.


Other sorts of noise
A wide variety of physical processes produce noise. Some of these are similar to Johnson and shot noise in producing a flat noise spectrum. In other cases the noise level produced can be strongly frequency dependent. Here we will only briefly consider the most common form of frequency-dependent noise: noise. Unlike Johnson or shot noise which depend upon simple physical parameters (the temperature and current level respectively) noise is strongly dependent upon the details of the particular system. In fact the term ' noise' covers a number of noise generating processes, some of which are poorly understood. For this form of noise the NPSD, , varies with frequency approximately as

equation

where the value of the index, n, is typically around 1 but varies from case to case over the range, .

As well as being widespread in electronic devices, random variations with a spectrum appear in processes as diverse as the traffic flow in and out of Tokyo and the radio emissions from distant galaxies! In recent years the subject of noise has taken on a new interest as it appears that some ‘Chaotic’ systems may produce this form of unpredictable fluctuations.


Summary
This section has shown how random noise arises from the quantised behaviour of the real world. Two types of noise — Johnson Noise and Shot Noise — were described in detail and their nature shows that they are, in practice, essentially unavoidable. You should now know that noise can only be predicted or quantified on a statistical basis because its precise voltage/current at any future instant is unpredictable. That its magnitude is quantified in terms of averaged rms voltages/currents or mean power levels. The concepts of the Maximum Available Noise Power and Noise Power Spectral Density were introduced and we saw that Johnson Noise (and also Shot Noise) have a uniform NPSD — i.e. they have a White power spectrum. Other forms of noise can show different noise spectra, most commonly a ‘1/f ’ pattern.


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University of St. Andrews, St Andrews, Fife KY16 9SS, Scotland.