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Noise and Noise Parameters
Introduction
The subject of noise is vast and volumes have been devoted to it. Therefore, it is
important that we restrict our coverage of the subject and concentrate on the more
essential basics.
The goal of this article is to present a clear picture of important measurement
parameters that relate to random noise. This article will concentrate to the study of
random noise of the types generated by resistors, diodes, active components etc. This
is essential for gaining good understanding of noise measures such as “Noise Factor”.
We will concentrate only on the properties of random noise and its measurement. Only
the amplitude characteristics and measures of noise will be considered in this article.
We will not consider noise types such as impulsive interference and unwanted CW
radiation from signal sources, nor will we cover the processing of noise by electronic
circuits, such as the way in which an FM system improves Signal/Noise Ratio.
The subject of RF carrier random Phase Jitter and its related parameters will be
covered in Frequency Stability and Frequency Stability Parameters, later in this series
of articles.
The Nature of Random Noise
In simplistic terms, random noise in an electronic circuit is any voltage or current whose
instantaneous value is unpredictable. And yet, some characteristics of random noise
can be predicted to some degree.
Consider pure thermal noise as generated by an ideal resistor. By looking at the longterm statistical behavior of this noise, we find that there are some predictable
characteristics.
For instance, the long-term mean value is small compared to the largest instantaneous
value and with time it is asymptotic to zero. This is a predictable characteristic of pure
thermal noise.
The long-term r.m.s. value of the noise is asymptotic to a finite value. This is another
predictable characteristic of pure thermal noise.
Now consider a symmetrical band of possible instantaneous noise values that is
centered on the zero long-term mean value and has a width equal to twice the longterm r.m.s. value.
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Counter-intuitively, in the long-term, the instantaneous value of the noise will lay in this
band for 68% of the time. Similarly, for a band twice this wide the value will be in this
band for 95% of the time. The value for three times is 99.5%. This predictable statistical
characteristic of pure thermal noise is called a Gaussian distribution, and noise of this
type is called “Gaussian noise”.
Pure thermal noise has no correlation with itself. If we continuously compare the
waveform for the noise with that at a fixed earlier time, then there will be no common
characteristics. We have already said that random noise is unpredictable. However, we
have also said that random noise has predictable characteristics. In fact, random
(unpredictable) noise can have correlation with itself. This form of correlation is known
as “auto-correlation” and is yet another characteristic of pure thermal noise is that it has
no correlation with itself.
HINT: If the RF engineer has problems with understanding auto-correlation, then the
correlation of a sinusoidal signal with itself is well worth considering.
Yet another predictable characteristic of pure thermal noise exists. We must look to the
frequency domain representation of the noise. Clearly, random noise is not periodic.
Another way to look at this is to consider that its period is infinite. This is tantamount to
saying that the Fourier components of the noise are contiguous: the frequency
spectrum is continuous.
Furthermore, this spectrum is uniform to frequencies of around 10,000GHz, after which
it falls off as predicted by quantum theory. If the long-term r.m.s. noise value is
measured by an instrument with a fixed bandwidth, then the same value would be
obtained at all frequencies in this band. Noise with this characteristic is called “white
noise” by analogy with white light, which has a “flat” response for optical frequencies.
Given this analogy, pure thermal noise is called “Gaussian white noise”.
In summary, we have described five predictable characteristic of pure thermal noise
whose instantaneous value is itself unpredictable.
Up to this point we have only considered pure thermal noise which, from an electronic
system viewpoint, can be considered to have an infinite frequency spectrum. Real
systems have finite bandwidths and therefore, we must consider what happens when
we pass Gaussian white noise through such systems. The type of noise that appears
at the output of these system is called “Colored noise”.
What happens when we pass Gaussian white noise through a low-pass filter?
The output noise will be colored, that is, it will be band-limited. However, it is surprising
to discover that the output noise is correlated. The action of the filter is to impose
correlation on uncorrelated noise. This fact is easier to understand when we consider
what happens when we pass Gaussian white noise through a bandpass filter.
If the bandpass filter has a center frequency of f and a bandwidth of 1Hz, then we would
expect to see a sinusoidal signal of frequency f at the output of the filter. In reality the
amplitude and phase of the signal change randomly at a rate determined by the filter
bandwidth. For very small bandwidths, the rate of change will be small and, over a
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relatively short period, the filter output approximates to a pure sine wave. A pure
sinusoidal signal has correlation with itself: the smaller the filter bandwidth, the greater
the correlation. The amplitude of the filter output varies with what is known as a
“Rayleigh distribution”.
Noise Bandwidth and Noise Density
Consider again pure thermal noise as generated by an ideal resistor.
As the frequency spectrum for pure thermal noise is “flat“, it is to be expected that the
long-term noise power observed in a given frequency band is proportional to
bandwidth. The term for bandwidth, in this context, is “noise bandwidth” and is that
provided by a hypothetical filter with infinitely steep response transitions between the
pass-band and stop-bands. The subject of “Noise Power” discussed in greater depth
in Thermal (Johnson) Noise.
The term noise bandwidth can be applied not only to white noise but, also, to any form
of colored noise. For colored noise, the power will not be proportional to bandwidth.
There is a defined noise bandwidth for a real circuit such as a bandpass filter, even
though is does not have infinitely steep transitions. The noise bandwidth of a bandpass
filter is the bandwidth of an idealized hypothetical filter with identical effect on white
noise power reduction.
The noise power in a bandwidth of just 1Hz is specifically known as “Noise Power
Spectral Density”. This is a very common measure of random noise characteristics.
Some Examples of Random Noise
Thermal noise (also known as “Johnson noise”, after an early researcher) has already
been mentioned and results from the motion of electrons within a conductor. The
degree of motion and hence, noise magnitude is dependent upon absolute
temperature. Pure thermal noise is independent of frequency and has a Gaussian
distribution.
Shot Noise results from the random passage of individual charge carriers across a
potential barrier. It was first observed in thermionic devices but is also present when
current flows across a semiconductor junction. Pure shot noise is independent of
frequency and has a Gaussian distribution.
Partition noise is peculiar to thermionic devices and results from random fluctuations
in the way in which current is divided between the various electrodes. The physical
mechanisms for thermal, shot and partition noises are thought to be fully understood.
When a DC current is passed through a resistor or semiconductor, noise in excess of
that which is expected results. For a resistor we would expect to observe just thermal
noise. For a semiconductor we would expect a combination thermal noise and shot
noise, both of which are independent of frequency. Excess or “flicker noise” differs from
thermal, shot and partition noise in that it is inversely proportional to frequency. For this
reason it is sometimes referred to as “pink noise”, where the light analogy is again
used, and it applies to any noise that has a falling frequency spectrum, irrespective of
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its origins. The mechanism for excess noise is still not fully understood. The magnitude
of excess noise in a resistor or semiconductor depends on the DC current. For
resistors it also depends on the component type, being very small for some metal film
resistors. For semiconductors, the excess noise is dependent upon device material,
type and design. It can even vary between devices that are nominally identical.
Thermal (Johnson) Noise
A knowledge of thermal (or “Johnson”) noise is fundamental to the study of noise
measurement and to the design of low noise systems and circuits.
Thermal noise is the result of random electron motion in conductors. The noise from
an ideal resistor is fully predictable. It is also predictable for good quality resistors that
do not suffer from excess noise. It is even predictable for any resistor type over the
normal range of frequencies experienced in RF engineering.
Because of its predictability, pure thermal noise is used as a reference standard for
noise measurement.
Real circuits and device models abound with resistors and, therefore, a knowledge of
thermal noise is of paramount importance in circuit design.
Research has found that the r.m.s. noise voltage VN across a resistor of value R, due
to electron thermal activity alone, is given by:
where
k = Boltzmann’s constant = 1.380662 x 10-23 JK-1
T = Absolute Temperature = 298.15K for 25°C
B = Bandwidth in Hz
We now invoke Thévenin’s (pronounced “tay-venin’s”) theorem to explain this
phenomenon.
Thévenin’s theorem states that the equivalent circuit for any two-terminal linear system
consists of a voltage generator in series with a resistor. The output of the generator is
equal to the open-circuit voltage of the system. The value of the resistor is equal to the
resistance between the two terminals.
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If we apply Thévenin’s theorem to the noisy resistor, we obtain the equivalent circuit
shown in Figure 1. It is important to note that the series resistor is noiseless. If noise
was present in this resistor, then it would have been included twice.
Figure 1. Noise Model for a Resistor
The above model can be used as a model for resistors in a circuit or device model. The
noiseless resistor is “seen” by all other signal/noise voltages and currents.
What happens when we connect a noiseless resistive load to a noisy resistor?
Applying the noiseless model from Figure 1, we can carry out a simple analysis. The
equivalent circuit for resistor and load is shown in Figure 2.
Figure 2. Noise Model with Noiseless Resistive Load
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The noise power delivered to the load is given by:
What happens if we make RL equal to R? This is the condition for obtaining maximum
available noise power PNAV from the resistor. For this condition we have:
At first sight, this expression might seem amazing—the maximum available noise
power PNAV is independent of resistor value R. However, it will be realized that the
noise power is a result of electron mobility, which is only dependent upon absolute
temperature and not on resistor physical size or electrical value. Note that the noise
voltage for a resistor is, however, dependent on resistor value R.
The maximum available noise power kTB is a very convenient reference standard for
noise measurement. Note that when considering noise power spectral density, that is
noise power in a 1Hz bandwidth, the expression for maximum available noise power
simplifies to kT.
Signal/Noise Ratio
How do we compare the noise performances of two electronic systems?
A low frequency analogue engineer’s answer to this question would probably be “by
comparing both Signal/Noise Ratios”. Signal/Noise Ratio is normally written as “S/N”
ratio but, strictly speaking, should be (S+N)/N ratio. However, N is always small so the
two ratios are nearly equal.
The low frequency analogue engineer’s answer is fine as long as the rest of the world
uses the same signal levels when making comparisons between systems. Even RF
engineers use S/N ratio to check the behavior of an individual system but not for
comparing two systems.
Noise Factor and Noise Figure
Consider a two-port linear system. This can be a system where the input and output
frequencies are the same, such as an amplifier. It can also be a system where the input
and output frequencies are different, such as a mixer. However, linearity, over the
signal levels of interest, is a prime requirement.
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To get around the fundamental problems associated with a single S/N ratio, we will
compare the S/N ratios at the input and output.
Thus we can define a hypothetical noise measurement factor (NMF) as:
where
SI = input signal
SO = output signal
NI = input noise
NO = output noise
Here, the signals and noises are not defined, but all have the same dimensions.
The RF engineer may have already seen that the definition of NMF is independent of
signal level. For different levels of SI, SO will change pro-rata if the system is truly
linear. To put this another way, the system gain G = SO/SI is a constant over the signal
levels of interest. Thus we have:
where
G = gain
As SI and SO are not defined, so then is the gain G.
Thus, we have defined a hypothetical noise measurement factor NMF that is
independent of signal levels. It is described in terms of NI, NO, and G, none of which
have been defined.
The definition of NMF forms the basis for the definitions of noise factor and Noise
Figure. We will proceed to the full definitions of “noise factor” and “noise figure”, that
includes those systems having any input impedance.
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Consider the two-port system shown in Figure 3. Here, both the source and load
resistors are considered to be noiseless.
Figure 3. A Two-Port System Pertaining to Noise Factor and Noise Figure
To covert our definition of NMF into something more meaningful, we must define the
quantities SI, SO, NI and NO. We can do this in terms of voltage, current or power as
long as the dimensions for each are the same. However, it turns out best if we work
solely with power.
The RF engineer will, probably, have already realized that the dimensions for the
signals and noises should be power. After all, noise magnitude can only be defined in
terms of r.m.s. voltage/current and, hence, power. Thus we have:
where
PSI = input signal power
PSO = output signal power
PNI = input noise power
PNO = output noise power
GP = power gain
We will now introduce the full definitions for all the parameters.
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Starting with the output of the system, we can simply define PSO and PNO as follows:
where
VSO and VNO are r.m.s. quantities
PSO and PNO are just the signal and noise powers in the load resistance.
In the context of this discussion we will use the concept of “maximum available source
power” to define input signal level. This leads to the definition of “Transducer Power
Gain”, GPT.
The concept of maximum available source power and transducer power gain (GPT)
must be used for both signal and noise. This is convenient as we have already defined
maximum available noise power PNAV for a resistor.
By defining PSAV and PNAV as the maximum available signal and noise powers at the
input and renaming NMF as “Noise Factor F”, we have:
where
PSAV = maximum available signal power
PSO = signal power in load resistance
PNAV = maximum available noise power
PNO = noise power in load resistance
GPT = transducer power gain
What interpretation can we place on this expression?
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The quantity PNO/GPT is the output noise power referred to the input using transducer
power gain. Noise factor is the ratio of the input referred output noise power to the
available input noise power.
This is a very precise and solid definition of a noise defining parameter. The value of
PNAV is the defined quantity “kTB”, which is independent of resistor value. For our final
definition of F, we have:
For this definition to be sound, PNO must also be defined in the bandwidth B. Also, note
that for a linear system, PNO is proportional to the Absolute Temperature T and,
therefore, is mutually canceling with PNAV. This assumes that the complete system and
source are at the same temperature.
HINT: Strictly speaking, the system in Figure 3 should include a noise source for the
output load resistor. This was deliberately omitted for the sake of clarity. However, the
effect of the omission is normally negligible as the output noise PNO is, in most cases,
much larger than the noise generated by the resistor.
Demonstrative Exercise
To bring home the writers point that the definition of noise factor must apply to systems
having any input impedance, the following simple exercise is suggested.
What is the noise factor of an amplifier having a real input resistance of magnitude R0
and no other internal noise sources?
It should be pointed out that the answer is not unity.
Noise Figure {NF) is simply noise factor expressed in dBs. Thus we have:
NF = 10 log F
Noise Factors for Cascaded Systems
Consider two cascaded systems.
When considering noise relative to the overall signal, the noise contribution from the
second system will be small if the gain of the first is large. This because an increase in
gain increases the signal plus noise from the first system, so that the noise from the
second system is a relatively smaller part of the overall signal. The overall noise factor
of two cascaded systems is dependent not only on the individual noise factors for each
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system but also on the gain of the first system. This argument can be extended to any
number of systems. For most well designed systems, the overall noise factor is
arranged to be close to that of the first system.
The expression for the overall noise factor (F) of a cascaded system is given here
without proof.:
where
FN = noise factor of nth system
GN = Transducer Power Gain of nth system
Equivalent Noise Temperature and Equivalent Noise
Resistance
The RF engineer must be made aware of two commonly used noise parameters.
Consider the available noise power from a system such as an antenna. The actual
noise power from the antenna PA will be higher than expected from a straight
calculation of thermal noise from a resistive source at absolute temperature (T).
One way to express this increased noise is by assuming a hypothetical temperature for
the source. Thus we have:
where
TE = Equivalent Noise Temperature
PA = actual available noise power from the antenna
The concept of “Equivalent Noise Temperature” can be extended to other noisy
systems including those containing non-thermal noise sources.
Equivalent Noise Resistance is another technique used to express the total noise in a
system.
Equivalent noise resistance (RE) of a noise generator is the value of resistance that
gives the same noise voltage as the generator at the standard temperature. This
definition can also be expressed in terms of noise current.
Equivalent noise resistances can be introduced into a circuit for noise representational
purposes. However, they must be introduced in such a way that they do not affect other
signals and noises.
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In Conclusion...
In this article we have discussed concepts of noise parameters in general and random
noise in an RF system in particular. The series continues with a discussion of phase,
group, and envelop delay.
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