### What is a noise calculation ?

It is important to know the internal noise level in sensitive circuits for different reasons.  The first one is of course that the performance of the circuit can be limited by the internal noise level.  The second reason is that if one knows the theoretical internal noise level, then a measurement can determine whether there are other noise sources that have been overlooked (such as pick up noise).  If these uncontrolled noise sources are present, then this presents a vulnerability of the product, even if in laboratory conditions, it is still functioning properly.  This lack of control over the quality of the product can be problematic and is to be avoided.

But what exactly does it mean, "knowing the internal noise level" ?  Ideally, it would mean a complete description of the ensemble of the statistical processes that describe the noise for every current and voltage in the circuit.  However, most of the time, it only means one or two things

1. the standard deviation of the momentary voltage or current noise on specific voltages and currents (called RMS value)
2. the time correlation between the voltage or current noise value at time t1 and time t2.

This is a very limited description of the noise ensemble, but for internal noise, it is, most of the time, sufficient.  The instantaneous standard deviation of the noise value gives an idea of the error on an instantaneous voltage or current value, if that value represents some information ; the time correlation, as we will see, is intimately related to the frequency spectrum of the noise, which will be transformed under linear filtering systems.  In fact, the instantaneous standard deviation is included in the time correlation as a special case: its square is the time correlation of the current or voltage noise when the two time instants are identical.

The (second order) time correlation function of the noise on any current or voltage can be calculated if

1. one can model all the contributing noise sources as voltage or current sources in a linear network, and if
2. one knows the (second order) time correlation functions of these sources, as well as
3. an assumption of statistical independence (or a given cross correlation function) of these sources.

This is very often the case when the noise is of small amplitude, and the network behaviour can be approximated as a linear(ized) network.  The fundamental noise sources, and most technological noise sources have a description which is based on a given second order time correlation function.   As the origins of these sources are  the local entropy or a local quantum phenomenon in a given component, of course these sources are statistically independent for different components.  Within a single component, one has to be more careful about the assumption of statistical independence of different internal sources of course.

As the only description we give is a standard deviation (and a mean equal to zero by definition of noise), the maximum entropy principle tells us that we should assume a Gaussian distribution for the noise current or voltage at a given instant in time.   Fundamental noise moreover has really a Gaussian distribution, even if one does study its probability distribution more in detail.

The network solution to a noise problem can be done with standard simulation software, such as any variant of SPICE.  So in principle a noise calculation is done with a network simulator.  But there is a big caveat: for many active components, the noise model is not correct !   If the noise model is wrong, then of course the result of the calculation is wrong.  This is why it is always necessary to make a simplified hand calculation of the noise before trusting a simulation result.  The reason why simulation models do often not contain a correct noise description is not that their publishers are incompetent, but rather that the purpose of the model is not a noise calculation, but a functional simulation.  As such, many circuits are not modelled down to the physical components, but have an equivalent transfer function that describes sufficiently well the behaviour of the component to be reliable for a functional simulation.  An equivalent transfer function is much faster than a detailed simulation of every physical component in the circuit.  But an equivalent transfer function does not describe in any way the noise sources correctly.  So one has to be very careful with SPICE-based noise calculations if one uses externally supplied models.  Entrop-x can help with this task.

### Correlation, RMS value and spectral noise density

Consider a voltage of a node in a network as a function of time: V(t).  By definition of noise, V(t) is the result of the sum of two contributions: the "average" expected signal (the ideal behaviour and the systematic error on it), V0(t), and the "fluctuations", the noise, v(t).

V(t) = V0(t) + v(t)

v(t) is a stochastic process, the noise of which statistical description we're after.

We said before that the only statistical description we are interested in, is the second order time correlation function:

C(t1,t2) = 1/T ∫ v(t1 - t) v(t2 - t) dt    where T is the long time over which the integral is taken, and tends to infinity

An assumption one makes at this point is time invariance.  That is, if we shift time over an arbitrary time s, the statistical properties of the noise don't change:

C(t1,t2) = C(t1 - s,t2 - s)

As such, the correlation function only depends on the time difference of t1 and t2.  The correlation function is nothing else but the average (over time or over the ensemble, which is assumed the same) of the product of the noise value at a certain moment in time, and the noise value at a moment, s time later.

C(s) = 1/T ∫ v(s + t) v(t) dt

The special case (vRMS)2 = C(0) = 1/T ∫  (v(t))2 dt  is nothing else but the expectation value of the local square of the noise: the RMS value (squared).

The Wiener-Khinchin theorem gives the relationship between the correlation function C and the power spectral density:

S(f) = ∫ C(t) e-i 2 π f t dt

The power spectral density S(f) and the correlation function C(t) are simply a Fourier transform pair (with normalisations given above).

The relationship with the RMS value is the following:

(vRMS)2 = C(0) = ∫ S(f) df

The fundamental reason why the spectral density is usually the quantity to be preferred is the following property:

If a noise signal (voltage or current, it doesn't matter, but we will take a voltage as an example) v1(t) with a power spectral density S1(f) is applied at the input of a linear system with a transfer function H(ω), then the output of that linear system, v2(t) will have a power spectral density

S2(f) = | H(2π f) |2 S1(f)

It is this property which allows us to propagate a noise source through any (linear) network.

One still needs one other property: if two statistically independent sources result in two noise contributions v1(t) and v2(t) on the same signal (say, the voltage signal on a node) by the superposition principle, then the spectral density of the resulting noise will be:

S(f) = S1(f) + S2(f)

Noise calculations in linear networks (or linearised networks) are based upon these two properties: each noise source is considered first individually in the network, and its effect on currents and voltages elsewhere is found by using the transfer function (transfer function, trans-impedance function or trans-conductance function) to find its contribution there.

Next the spectral densities of all independent contributions to the same voltage or current are added, according to the second rule.   This gives the resulting spectral densities of all the current and voltage noises in the circuit.

However, there is a big caveat.  One cannot use these resulting voltage and current noise spectral densities as independent sources in a next calculation.  Indeed, all of them contain contributions from the same sources, and are hence correlated.  This problem shows up if one has sub circuits which have already been modelled and have given equivalent noise sources: these equivalent noise sources may very well be correlated.  One can still perform a calculation, but one should use the correlation matrix between these sources instead of a simple sum of spectral densities.

### Units

The spectral density of a voltage noise has as units: V2 s  or V2 / Hz

The spectral density of a current noise has as units: A2 s  or  A2 / Hz

The time-correlation function of a voltage noise has as units: V2

The time-correlation function of a current noise has as units: A2

Finally, the RMS value of a voltage noise has as units: V and the RMS value of a current noise has as units: A.

In some cases, people prefer to work with the square root of S(f).  In that case, the units for the spectral densities are V over square root of Hz, and A over square root of Hz, respectively.

### Fundamental noise sources

The two principal fundamental noise sources are thermal (Johnson) resistor noise, and quantum-mechanical shot noise.  These two noise sources are white noise, that is, their spectral noise density is a constant, independent of frequency.  As such, their expression, which expresses S(f), will not contain f.

The series voltage noise of any resistor at absolute temperature T has the following spectral density:   SR(f) = 4 kB T R ; where kB is Boltzmann's constant.

This can alternatively be modelled as a parallel current noise to the same resistor:  SR(f) = 4 kB T / R (this time, SR is the spectral noise density of a current).

The shot noise is a parallel current noise source to any PN junction through which a DC current IDC is flowing.  It has a spectral noise density SS(f) = 2 q IDC, where q is the elementary electron charge of 1.6 10-19 C.

One notices that these fundamental noise sources are not depending on technological parameters, which is what makes them pretty universal and predictable.  When the noise in a circuit is dominated by fundamental noise sources, then technology is not a limit for the performance of the circuit.  If one needs to do better, then only a new circuit idea can improve the situation and no technical advance can improve upon the situation.   If the limit of a certain circuit was due to fundamental noise in the 1950ies, then this limit still holds for the same approach, no matter the technological improvements.

### Technological noise sources

Technological noise sources need to be modelled according to data from the manufacturer or from measurements ; there is no way to find them out from first principles.  Most technological noise sources do have a frequency dependence, and most of them increase at low frequencies.  An approximate 1/f behaviour of S(f) is often the case.

### Noise, reduced to the input

When one considers a linear electronic circuit with an input signal and an output signal, it can be interesting to see what noise source, added to the input signal, would produce the same noise at the output, than the total noise result of all the internal noise sources in the circuit.

In other words, the noise, reduced to the input, is that noise at the input, that would produce exactly the same noise at the output of a noiseless circuit (which is physically not possible) that has the same transfer function as the circuit under study.

If So(f) is the actual noise spectral density at the output, and H(ω) is the transfer function from the input signal to that output, then the noise, reduced at the input Si(f) is given by:

Si(f) = So(f) / | H(2π f) |2

The noise, reduced to the input is interesting to see the actual "pollution" by a circuit.  For instance, an amplifier A that amplifies an input signal 10 times more, might have a 30 times higher noise spectral density at the output than an amplifier B.  Nevertheless, amplifier A pollutes less the input signal than amplifier B.   Indeed, for the same "pollution", one would expect amplifier A to have a 100 times higher noise spectral density.  There's more information concerning the input signal at the output of amplifier A than there is in the output signal of amplifier B.