Noise Figure Overview of Noise Measurement Methods Introduction

Noise Figure Overview of Noise Measurement Methods  Introduction
 Noise Figure
Overview of Noise Measurement Methods
Introduction
Noise, or more specifically the voltage and current fluctuations caused by the random motion of
charged particles, exists in all electronic systems. An understanding of noise and how it
propagates through a system is a particular concern in RF and microwave receivers that must
extract information from extremely small signals. Noise added by circuit elements can conceal
or obscure low-level signals, adding impairments to voice or video reception, uncertainty to bit
detection in digital systems and cause radar errors.
Measuring the noise contributions of circuit elements, in the form of noise factor or noise
figure is an important task for RF and microwave engineers. This paper, along with its
associated appendices presents an overview of noise measurement methods, with detailed
emphasis on the Y-factor method and its associated measurement uncertainties.
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Noise Figure
Overview of Noise Measurement Methods
Contents
Noise Measurements .......................................................................................... 4
Noise Factor, Noise Figure and Noise Temperature ........................................ 5
Active Devices ........................................................................................................................ 5
Passive Devices .......................................................................................................... 8
Noise Figure of Cascaded Stages ..................................................................... 9
Effective Noise Temperature of Cascaded Stages .................................................... 10
Noise Figure Measurements ............................................................................ 10
Y-factor Method ......................................................................................................... 10
Cold Source or Network Analyzer Method ................................................................ 12
Signal Generator (Twice Power) Method .................................................................. 13
Direct Noise Measurement Method ........................................................................... 13
Noise Figure Measurements in Frequency Converters ................................. 14
Frequency Converters with Image Rejection............................................................. 14
Image-reject Filter Included in the Measurement ...................................................... 16
Image-reject Filter Excluded in the Measurement ..................................................... 18
The Nature of Random Noise ........................................................................... 22
Thermal Noise ........................................................................................................... 22
Power Spectral Density of Thermal Noise ................................................................. 22
Shot Noise................................................................................................................. 23
/ Noise .................................................................................................................. 24
Noise Power Spectral Density Graph ........................................................................ 25
Noise in Electronic Components ..................................................................... 26
Resistors ................................................................................................................... 26
Capacitors ................................................................................................................. 26
Inductors ................................................................................................................... 27
Active Devices ........................................................................................................... 27
Conclusion ......................................................................................................... 28
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Noise Figure
Overview of Noise Measurement Methods
Appendices
Appendix 1: Noise of a Resistor, Capacitor and Inductor............................. 29
Resistors ................................................................................................................... 29
Capacitors ................................................................................................................. 30
Piezoelectric Effects in Capacitors ...................................................................................... 32
Inductors ................................................................................................................................. 32
Appendix 2: Shot Noise .................................................................................... 33
Computing Shot Noise .............................................................................................. 33
Derivation Assuming a Poisson Distribution ........................................................................ 33
Appendix 3: Bandwidth Equivalent of Averaging .......................................... 35
Appendix 4: Avalanche Diode Noise Sources ................................................ 36
Avalanche Breakdown............................................................................................... 36
Noise Mechanism ...................................................................................................... 37
A Practical Noise Source........................................................................................... 37
Appendix 5: Error Analysis of the Y-factor Method ....................................... 38
Y-factor Measurements ............................................................................................. 38
Noise Factor of a Measurement Receiver ........................................................................... 38
Noise Factor of the DUT Cascaded with the Measurement Receiver ................................. 38
Uncertainty in Noise Factor Measurements .............................................................. 39
Sensitivities to Measurement Errors .................................................................................... 44
Computing the Sensitivities for Device Gain ............................................................. 51
Sensitivity of
to Power Measurement Errors............................................................. 51
Computing the Gain Sensitivities to the Power Measurement Errors .................................. 52
Summary of Errors .................................................................................................... 57
Errors in Measuring Noise Factor ........................................................................................ 57
Errors in Measuring Device Gain ......................................................................................... 57
Statistical Distribution of Measurement Errors .......................................................... 59
ENR ..................................................................................................................................... 59
Receiver Gain Error ............................................................................................................. 59
Mismatch Errors .................................................................................................................. 60
Coverage Factor ....................................................................................................... 61
Example of Contributions to Noise Figure Measurement .......................................... 62
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Noise Figure
Overview of Noise Measurement Methods Noise Measurements
The noise contribution from circuit elements is usually defined in terms of noise figure, noise
factor or noise temperature. These are terms that quantify the amount of noise that a circuit
element adds to a signal. They can be measured directly using available test equipment as
well as modeled using both system and circuit simulation SW.
Figure 1. Example of an amplifier with signal, thermal noise and additive noise
Consider the amplifier1 shown schematically in Figure 1. Its intended job is to amplify the signal
presented at its input and deliver it to the load. The thermal noise that is present at the input is
amplified along with the input signal. The amplifier also contributes additional noise. The load
receives a composite signal made up of the sum of the amplified input signal, the amplified
thermal noise and the additional noise contributed by the amplifier. Noise figure, noise factor
and noise temperature are figures of merit used to quantify the noise added by a circuit element,
the amplifier in this case.
1
This paper assigns all added noise to the output of the amplifier. Other derivations exist where the added noise is modeled at the DUT input. 4
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Noise Figure
Overview of Noise Measurement Methods
Noise Factor, Noise Figure and Noise Temperature
Noise factor is defined as the signal-to-noise ratio at the input divided by the signal-to-noise
ratio at the output. Noise factor is always greater than unity as long as the measurement
bandwidth is the same for the input and output.
.
Equation 1
Noise figure is Noise Factor expressed in dB
| |.
Equation 2
The definitions for noise figure and noise factor are valid for any electrical network, including
frequency converting networks that contain mixers and IF amplifiers (up-converters or downconverters).
Active Devices
If we consider an electrical network such as amplifier or frequency converter with input signal
we have
, voltage gain
and additive noise referred to the output of
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is the noise present at the input to the system. The parenthetical
is used to
where
indicate that these are frequency-dependent quantities. For simplicity, we will drop this
functional notation in the remainder of the paper unless it is needed for clarity.
| |
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,
Equation 3
An important case exists when the noise at the input is thermal noise, .
has a flat
|
. refers to Boltzmann’s
power spectral density with power level of |
constant, to the absolute temperature in degrees Kelvin and to the system bandwidth
expressed in Hertz.
at 300 Kelvin has a value of 4.14X10-21 W or -174 dBm when
measured in a one Hz bandwidth.
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Noise Figure
Overview of Noise Measurement Methods
Similarly, the signal-to noise ratio at the output is given by
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,
Equation 4
where is the voltage gain of the device under test (DUT) and
the DUT. The noise factor can be computed by taking the ratio.
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is the noise voltage added by
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Equation 5
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It is often more practical to use power gain instead of voltage gain. Let the power gain of the
system be
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In the great majority of cases, the thermal noise and the additive noise are uncorrelated.
Equation 5 becomes
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Equation 6
In the case where the input noise is thermal noise or
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in the above equation refers to a standard temperature, usually 290K.
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Equation 7
Noise Figure
Overview of Noise Measurement Methods
Noise factor and noise figure are an indication of the excess noise (beyond the system thermal
noise) contributed by a functional block in a system.
Effective noise temperature refers to the temperature that a matched input resistance would
require to exhibit the same added noise.
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Equation 8
Effective noise temperature can be related to noise factor by
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Equation 9
.
Equation 10
is the reference temperature, usually 290K. Figure 2. Noise Temperature vs. Noise Figure shows a
graph of Noise Temperature versus Noise Figure. A noiseless device has a noise temperature
of absolute zero or 0 K, while a 4 dB noise figure is equivalent to a noise temperature of
approximately 430 K.
Noise Temperature VS Noise Figure
Reference Temp= 290K
Noise Temperature (K)
1000
100
10
1
0
1
2
3
Noise Figure (dB)
Figure 2. Noise Temperature vs. Noise Figure
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Noise Figure
Overview of Noise Measurement Methods
Passive Devices
Passive devices, those composed only of resistive or reactive elements, have a power gain less
than or equal to unity and contribute no additive noise beyond thermal noise. The noise power
at the output when the input is terminated is always
. Applying Equation 5 and Equation 6
we have
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Equation 11
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The above equation states that the noise figure of a passive device is the reciprocal of its power
gain. A 3 dB attenuator (
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) for example would have a 3 dB noise figure.
Noise Figure
Overview of Noise Measurement Methods
Noise Figure of Cascaded Stages
Consider a two-port network consisting of two stages. The first stage has thermal noise present
at its input. This thermal noise is amplified by the first stage gain and has any additive noise
produced by the first stage added to it. The noise at the output of the first stage is then
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Equation 12
The second stage has the output of the first stage presented to it. The second stage amplifies
the input and contributes additional noise.
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Equation 13
The principle illustrated above can be extended for multiple stages.
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…
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…
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⋯
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Equation 14
The noise factor is the ratio of the SNR at the input to the SNR at the output. For a given input
signal, the ratio for a cascade of k stages is
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…
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…
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Equation 15
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…
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⋯
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⋯
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…
Equation 16
Applying Equation 6 to Equation 16 yields the noise figure calculation for a system consisting of k
cascaded stages. Consider K stages in a system. The kth stage has power gain and noise
factor . Both the signal and the noise from previous stages arrive at the input of the kth stage.
The contribution of the kth stage is reduced by the gain preceding it. Noise Figure calculation for
the cascade of K stages can be found from
⋯
⋯
…
…
.
Equation 17
.Equation 17is often called the Friis formula2
for cascaded stages. It is named after Danish-American electrical engineer Harald T. Friis.
Friis, H.T., Noise Figures of Radio Receivers, Proc. Of the IRE, July, 1944, pp 419-422.
2
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Overview of Noise Measurement Methods
Effective Noise Temperature of Cascaded Stages
The same equation can be manipulated to give the effective noise temperature for cascaded
stages. If we replace the noise factors of each stage by their effective noise temperature we get
1
1
⋯
⋯
…
…
.
Equation 18
Noise Figure Measurements
Y-factor Method
The Y-factor method uses a noise source that can be switched off and on. It is based on
two power measurements, each performed with the same port impedances3 and the same
measurement bandwidth. The Noise source has a specified amount of excess noise. This is
specified as the Excess Noise Ratio or ENR. ENR is the ratio of noise from the source to the
system thermal noise or kTB, often expressed in dB.
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Equation 19
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1
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1
Equation 20
Making a noise figure measurement using the Y-factor method involves the use of a switchable
noise source and four power measurements. The first two measurements are used to
characterize the noise behavior of the receiver used to make measurements. , is the power
, is the power
measured by the measuring receiver with the noise source in the OFF state.
measured by the measuring receiver with the noise source in the ON state. The device under
test (DUT) is inserted between the noise source and the receiver for the next two power
and
are the power measurements made at the DUT output with the
measurements.
noise source turned OFF and ON respectively.
There are then three steps in making the measurement. The first, often called the calibration
step, is to measure the noise figure of the RF receiver used to make the power measurements.
The second step is to make a noise figure measurement on the cascaded receiver and DUT.
The next step is to de-embed the two measurements.
Let the receiver noise figure be
.
Equation 21
3
Noise sources with a port impedance that changes between the “ON” and “OFF” states contribute additional errors to the noise figure measurement. 10
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Noise Figure
Overview of Noise Measurement Methods
The noise figure for the cascade of DUT and receiver has a
Equation 22
The power gain of the DUT is measured by taking the ratio
Equation 23
From the cascaded noise figure equation we have
Equation 24
1
1
1
1
1
1
Equation 25
Substituting the power ratios for the Y-factors, we get
1
Equation 26
Equation 26 expresses the noise figure of the device under test in
terms of the four power measurements in the Y-factor method. This method relying on a series
of power measurements is ideally suited to low-level measurement receivers. It has been
implemented in modern spectrum analyzers as a cost-effective method of making noise figure
measurements.
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Noise Figure
Overview of Noise Measurement Methods
Cold Source or Network Analyzer Method
The cold source method essentially measures the noise power at the output of a device with an
input that is at the reference temperature (cold). It depends on very accurate knowledge of the
device gain. Network Analyzers can measure gain with extreme accuracy, making them ideal for
this method. Like the Y-factor Method, the cold source method requires a calibration step to
determine the measurement receiver’s noise figure. This is done with the use of a calibrated
noise source and a method similar to what is described in Equation 21.
The gain of the device under test is then measured as a function of frequency using the usual
network analyzer methodology. A power measurement is then made as function of frequency
be the noise added by
with the cold source connected to the device under test. If we let
be the noise added by the receiver then the measured power is
the device under test and
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1
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Equation 27
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Equation 28
Some network analyzers4 offer a noise figure measurement option that includes low noise
preamplifiers in their receivers, calibrated noise sources and the software to make
measurements. The Network Analyzer’s ability to make accurate transmission and reflection
measurements means that complete characterization of devices can be made that include
Noise Figure and S-parameters, making Network Analyzer measurements ideal for inclusion
in software-based system models.
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Agilent Application Note: High Accuracy Noise Figure Measurements Using the PNA‐X Series Network Analyzer. 12
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Noise Figure
Overview of Noise Measurement Methods
Signal Generator (Twice Power) Method
Measuring devices with a high noise figure presents a problem for the popular Y-factor method.
The Y-factor approaches unity as the noise figure approaches the source ENR. This affects the
accuracy of the Y-factor measurement. The twice-power method uses a signal generator and a
measuring receiver with an accurately known noise BW such as a spectrum analyzer. The input
to the device under test is terminated with a load at approximately the reference temperature
(usually 290K). A signal generator is then connected to the device under test until the measured
power is exactly 3 dB or twice the power measured with the input terminated. At this point the
sinusoidal power is exactly the same as the noise power and the noise factor can be calculated.
Knowledge of receiver bandwidth is critical but of knowledge of device gain is not needed. The
noise factor of the cascaded DUT and receiver can be computed from
Equation 29
The noise factor for the DUT can be dis-embedded using the formula for cascaded noise figure
in Equation 24.
Direct Noise Measurement Method
Devices with high noise figure can be measured with directly with a spectrum analyzer or other
receivers with accurately known bandwidths as long as the gain is known. The input to device
under test is terminated in a source that is near the reference temperature (290K). The noise
power at its output is measured and noise factor can be computed from
Equation 30
Knowledge of receiver bandwidth is required, as is knowledge of device gain. The noise factor of
the cascaded DUT and receiver can be computed from the formula for cascaded noise figure in
Equation 24.
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Noise Figure
Overview of Noise Measurement Methods
Noise Figure Measurements in Frequency Converters
The super-heterodyne receiver is at the core of most RF communications systems in use today.
The super-heterodyne receiver’s ability to provide high gain and frequency selectivity by
performing key filtering and amplification functions at a fixed intermediate frequency (IF) makes it
the architecture of choice for sensitive receivers ranging from AM radios to the receivers
tracking deep space probes.
The core of the super-heterodyne radio receiver is a frequency conversion (mixing), sometimes
done in several stages that transfers the information contained spectrum surrounding the carrier
to an IF. The noise performance of frequency converters is therefore a key aspect of receiver
design.
Measuring the noise figure, noise factor or noise temperature of receivers and frequency
converters is similar to the methodology described above for elements that operate at a single
frequency, with the exception that the band of frequencies at the input is not the same as the
band of frequencies at the output. Care must also be taken to account for the dual sideband
nature of mixers and the multiple conversions in samplers and other harmonic mixing devices.
Frequency Converters with Image Rejection
Figure 3. Typical frequency conversion stage.
Typical frequency converters used in RF receivers incorporate an image-reject filter prior to the
mixer. The image-reject filter is necessary because of the dual sideband nature of sinusoidal
multiplication.
1
1
ω
ω
ω
ω
2
2
The sum terms above are rejected by the IF band-pass filter, leaving two possibilities for signals
to appear within the IF filter.
cos ω
ω
cos ω ω ω
The above equation has two possible solutions.
ω
ω
ω , and
ω
ω
ω .
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The image-reject filter is designed to pass only the upper or only the lower sideband, but not both
as shown in Figure 4.
Figure 4. An image-reject Filter selects one of the two possible signal sidebands.
Noise can be represented as a broadband random process with frequency content across all
frequencies. Thermal noise is generated in all resistive devices. Shot noise is generated because
of the granular nature of electric currents where discrete electrons flow. There are noise
generators at every stage in the system shown in Figure 3. Noise components generated prior to
the image-reject that lie outside the filter pass-band will be removed by the filter. Both sidebands
of noise components generated after the filter will be converted by the mixer. Passive filters are
mostly reactive and therefore exhibit no thermal noise. One can then assume that all noise
added by the circuit elements in a mixing stage comes from elements located after filtering.
Noise from both sidebands is therefore present in the IF, even though the signal from the
rejected sideband is not.
Figure 5. Both upper and lower noise sidebands are converted.
The additional noise from the rejected sideband is indistinguishable from any noise that is
added by the system and is generally included in the noise figure measurement of receivers.
There are, however, some cases that require special attention.
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Noise Figure
Overview of Noise Measurement Methods
Image-reject Filter is Included in the Measurement
Figure 6. Measuring Noise Figure in a frequency converter that includes an image-reject filter.
Consider a noise figure measurement made in a frequency converter system as shown in
Figure 6. The image-reject filter ahead of the mixer allows the only the excess noise from the
noise source that falls in the desired sideband (1.05 GHz to 1.25 GHz) to enter the mixer, while
thermal noise from both sidebands enters the mixer. Measurements in such a system are
straightforward with the exception that the effect of the frequency converter must be considered.
The noise figure measurement is done in two parts as detailed in Equation 20 – Equation 23.
The first part measures the Y-factor of the measuring receiver. This is done at the frequency
band that exists after the mixer and IF filter. The frequency converter provides a frequency
translation so that the Y-factor measurement for the combined DUT and measuring receiver is
done at the band of frequencies that exists before the mixer. A spectrum analyzer performing
the noise power measurements needs to be tuned so that it covers the 100 MHz to 200 MHz If in
its span. The X-axis of a noise-figure VS frequency plot needs to show the equivalent RF
frequency, covering 1.1 GHz to 1.2 GHz.
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Noise Figure
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The noise figure measurement for a frequency converter can be examined by modifying
Equation 20 – Equation 24 to include the different bands. If we denote frequencies belonging to
the band that precedes the mixer as ω , and frequencies from the band that exists after the
mixer asω , then the noise factor of the receiver, done at the after-mixer band is
.
Equation 31
The noise figure for the cascade of DUT and receiver is measured at the mixer-input band is
Equation 32
The conversion gain of the DUT expressed as a power ratio is measured by taking the ratio
Equation 33
The subscript 1-2 in Equation 27 is used to denote that the gain in question is the frequency
conversion gain or the ratio of the power at
to the power at .
From the cascaded noise figure equation we have
Equation 34
Equation 35
It is important to note that there are two different values of ENR used in Equation 35. The
ENR of the noise source at the IF (mixer output frequency) is used when measuring the
receiver alone. The ENR of the noise source at the mixer input frequency is used when the
frequency converter is in the measurement.
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Overview of Noise Measurement Methods
Image-reject Filter is Excluded in the Measurement
Modular systems often place image-reject filters in separate modules from mixers and IF filters.
It is often useful to make noise figure measurements in a frequency converter that is separated
from its intended image-reject filter. Measurements in such a system will include noise
components from both the upper and lower sidebands as shown in Figure 7. The measurements
must be adjusted to account for the differences in the operating environment, which includes the
image-reject filter, and the test environment which does not.
Figure 7shows an example of a measurement where the image-reject filter is not included. The
noise source contributes noise at both the upper and lower sidebands, both of which are
converted to the same band at IF.
Figure 7. Noise Figure Measurement in a Frequency Converter where the Image-reject Filter is not included in
the measurement.
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Overview of Noise Measurement Methods
The Y-factor measurement equation for measuring the receiver is unchanged from Equation 18.
The Y-factor measurement equation for the double sideband converter must be modified to
account for the contribution from both sidebands. If ENR ω and ENR ω are the excess
and G ω
are
noise ratio at the upper sideband and lower sideband respectively, G ω
the upper and lower sideband conversion gains (or losses) expressed as power ratios and
|N ω | is noise added by the frequency converter stage referred to the post-mixer frequency,
then
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Equation 36
A double-sideband conversion gain for the frequency converter can be measured by taking the
ratio
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1 Equation 37
A useful special case of Equation 37 is when the noise source ENR and mixer conversion gain
are both constant with frequency.
=ENR
Equation 37 then simplifies to
2
In this case, the double-sideband gain is twice the gain of the single sideband case.
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Overview of Noise Measurement Methods
The Y-factor for the receiver is measured at the band of frequencies that exists after the mixer.
.
Equation 38
The Y-factor of the cascaded frequency converter and receiver can be measured. An analysis of
this measurement needs to include both the upper and lower sidebands. Let the subscript U denote
the upper sideband ENR and gain and the subscript L denote the lower sideband.
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From Equation 31,
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1
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In general, noise figure can be computed using Equation 19, repeated here for convenience.
.
We must consider three possible values for ENR when measuring a double-sideband mixer.
ENR ω and ENR ω refer to the excess noise ratio at the lower and upper sidebands at the
mixer inputs. ENR ω refers to the excess noise ratio at the band that exists after mixing. Let us
define ENR ω as an effective double-sideband ENR.
Equation 39
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Noise Figure
Overview of Noise Measurement Methods
Let
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Equation 40
From the equation for cascaded noise factor
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Equation 41
For the case where the conversion gains is flat across the upper and lower sidebands
and the ENR of the noise source is flat over frequency
, Equation 35 simplifies to
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Equation 41
It should be noted that the noise factor for the case where the image-reject filter is included in the
measurement is
1
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Equation 42
One can extend Equation 42 to the lower sideband. For the case where the upper and lower sideband
conversion gains are equal and the ENR is flat.
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Equation 43
Noise Figure
Overview of Noise Measurement Methods
The Nature of Random Noise
Thermal Noise
Everything in the universe is in motion. Even objects that are apparently still have random
vibrations of their molecules. This random vibration is felt by the human senses as heat.
Temperature is, in fact, a measure of the average kinetic energy of these random vibrations.
The same can be said about the molecules in insulators, conductors and semiconductors,
charge carriers (electrons and holes) and all physical structures from which electronic devices
are built. The power from thermal noise is given by
〈
〉
.
Equation 44
The brackets indicate that this is a statistical quantity expressed as an average. The variables
in
. Equation 44 are:
= Noise power expressed in watts,
Absolute temperature in Kelvin,
Bandwidth in Hertz, and
1.3806488
10
Boltzmann’s constant.
The thermal noise emanating from electronic components manifests itself as fluctuations in
voltage and in current. The statistical distributions of voltage and current are nearly Gaussian
when observed within a limited bandwidth.
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Overview of Noise Measurement Methods
Power Spectral Density of Thermal Noise
Thermal noise in an ideal resistor is approximately white of most of the frequencies used by RF
and microwave engineers. A deviation from this flat frequency distribution exists at very high
frequencies, when the quantum nature of electromagnetic waves becomes dominant. The noise
power can be obtained from5
Equation 45
Equation 45 includes the photon energy,
, where h is Planck’s constant. In the great majority
of cases, the operating frequencies used by RF and microwave engineers are such that
≪1
and Equation 46 simplifies to the familiar equation showing a flat power spectral density for
. The exceptions occur at very low noise temperatures and very high
thermal noise,
frequencies.
Figure 8 shows the quantum effects that color the power spectral density of noise. Thermal noise
can be considered to have a flat power spectral density out to 100 GHz or more before the
quantum effects become dominant.
Themal Noise Power (dBm)
1.E+06
‐170
Thermal Noise Including Quantum Effects.
1.E+08
1.E+10
1.E+12
‐175
‐180
‐185
‐190
3K
77K
200K
‐195
Frequency (Hz)
Figure 8. Thermal noise including quantum effects.
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Kerr. A.R and Nanda, J. P. 42 23
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1.E+14
Noise Figure
Overview of Noise Measurement Methods
Shot Noise
Engineers usually consider electrical current to be a continuous quantity. In reality it is composed
of discrete electrons, each with a fixed charge. Current is quantized. Rather than a continuous
flow, current is composed of the effects of individual electrons travelling from the source to the
load in a circuit. The electrons arrive with a uniform distribution over time. The effect of this
variation in time of arrival is called shot noise or corpuscular noise. Appendix 2 derives the
equation for the power spectral density of shot noise. The noise power is proportional to current.
The effective noise current approximates a Gaussian distribution with an RMS value or standard
deviation given by
2
.
/ Noise
Random fluctuations with a power spectra density that varies approximately as 1/ have been
observed in many processes ranging from the flooding patterns of the Nile River to the firing of
human brain neurons. Some noise phenomena also exhibit a deviation from white noise at very
low frequencies. This so called 1/ noise or pink noise has a power spectra density that
approximates a curve that inversely proportional to frequency. It can be dominant at low
frequencies but drops below the flat thermal noise at frequencies ranging a few Hz to a few kHz,
depending on the devices in question.6
6
Johnson, J. B. (1925) 24
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Noise Figure
Overview of Noise Measurement Methods
Noise Power Spectral Density Graph
The following graph illustrates the power spectral density of noise from electronic components
starting at a frequency of 1 Hz and extending to 1014 Hz for at a temperature of 290 K. Low
frequencies display the 1/f effects, which are dominant until their power spectral density drops
below that of thermal noise. Very high frequencies include quantum effects. The power spectral
density can be considered flat for most of the frequencies over which electronic devices operate.
Thermal Noise + Quantum and 1/f Effects ‐162
1/f Region
Themal Noise Power (dBm)
‐164
‐166
Quantum Region
‐168
‐170
‐172
290K
‐174
‐176
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
1.E+09
1.E+10
Frequency (Hz)
Figure 9. An illustration of noise power spectral density.
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1.E+11
1.E+12
1.E+13
1.E+14
Noise Figure
Overview of Noise Measurement Methods
Noise in Electronic Components
All physical matter undergoes molecular vibrations as a result of thermal energy. Temperature is
in fact a measure of the average kinetic energy of the moving molecules. Electronic devices are
no exception. All electronic devices contribute noise at some level due to the vibration of
molecules.
Resistors
Resistors contribute thermal noise caused by the random fluctuations of their internal molecules.
It is often useful to consider the noise contribution of a resistor in terms of its equivalent noise
voltage of its equivalent noise current. These values can be derived from knowledge of kits, the
thermal energy of particles, also known as the thermal noise floor. The derivation for the
equivalent noise voltage caused by thermal noise in a resistor is shown in Appendix 1. And is
given by
〈
〉
The equivalent thermal noise current is given by
〈 〉
The noise power of resistors has a flat power spectral density and depends only on temperature
and resistor value.
Capacitors
Ideal capacitors, like all reactive elements do not exhibit thermal noise. Capacitors, however, are
made with imperfect conductors and dielectrics and therefore have an associated resistance.
The noise voltage from a capacitor can be derived from the noise voltage of a circuit containing a
parallel combination of a resistor and a capacitor. Letting the resistor rise to infinity gives us the
thermal noise power emanating from the capacitor alone as shown in Appendix 1.
.
An interesting result occurs when we let the effective parallel resistor go to infinity. The total noise
power integrated over all frequencies is only dependent on temperature and on the capacitance
value.
.
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Noise Figure
Overview of Noise Measurement Methods
Inductors
Ideal inductors, like all reactive elements, do not exhibit thermal noise. Real inductive
components, however, have the losses in their windings and in their magnetic cores. These
losses can be modeled as an equivalent series resistor, which can be show to contribute noise.
Appendix 1 shows a derivation of the noise in inductors. It is only dependent on the effective
series resistance of the inductor and is given by
4
4
.
Active Devices
Active devices can have many noise contributors. Each of its resistive elements contributes
thermal noise. Bias currents contribute shot noise. The internal reactance of all circuit elements
shapes the power spectral distribution of noise. It is often useful to model all noise sources in a
circuit element in terms of an equivalent noise voltage and current as shown in the Op-Amp in
Figure 10.
Figure 10. Op-Amp with equivalent Noise voltage and current.
The real op-amp is modeled as an ideal noiseless op-amp with the addition of noise voltages and
noise currents at its input. The noise contributions can also be modeled with equivalent noise
sources at the amplifier output. The noise voltage and noise current represent the aggregate
contributions of all the circuit elements inside the op-amp. Resistive elements contribute thermal.
Bias currents contribute shot noise. Reactive elements can shape the noise power spectral
distribution. Transistors, RF amplifiers and all active components can be modeled in this fashion.
The actual values of the equivalent noise sources can be ascertained with measurements or with
careful modeling of the internal circuitry of the amplifier.
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Noise Figure
Overview of Noise Measurement Methods
Conclusion
Noise exists in all electronic systems. An understanding of noise and how to appropriately
measure, model, and account for its effects in a system is an important concern in RF and
microwave receivers that must extract information from extremely small signals. Noise added by
circuit elements can conceal or obscure low-level signals, adding impairments to the signals
being received.
Measuring the noise contributions of circuit elements, in the form of noise factor or noise figure
is an important tasks for RF and microwave engineers. This paper, along with its associated
appendices presents an overview of noise measurement methods, with detailed emphasis on the
Y-factor method and its associated measurement uncertainties.
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Noise Figure
Overview of Noise Measurement Methods Appendix 1: Noise of a Resistor, Capacitor and Inductor
Resistors
Consider a broadband measurement system with characteristic impedance
a matched source at its input. We wish to compute the noise contributed by the source
impedance.
. It has
The source is matched. The source impedance is equal to the conjugate of the system
characteristic impedance.
∗
.
The system noise can be modeled as a noise voltage,
voltage in the receiver.
.
2
The power received is equal to KTB
∗
4
The equivalent noise voltage of a resistor is
The equivalent noise current is
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. This noise voltage generates a noise
Noise Figure
Overview of Noise Measurement Methods
Capacitors
The noise voltage for a capacitor can be derived by substituting the impedance for a parallel RC
circuit for R in the above equation and then considering resistor values that approach infinity.
A parallel RC circuit has impedance given by
.
We now that noise power is caused by the resistive portion of Z. Substituting the impedance into
the equation for noise voltage yields
1
4
4
1
2
4
1
2
2
1
1
2 .
2
Taking the integral over all frequencies and replacing
for ∆ , we get
.
The value of the definite integral is
.
The equation above gives us
4
2
2
.
This is an interesting result in that it tells us that the noise voltage of a capacitor is only
dependent on the temperature and the capacitor value.
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It is useful to note how much noise power a capacitor will deliver to a resistive load.
≫ .
circuit below represents the effective parallel resistance of the capacitor.
≫
If
in the
, the voltage delivered by the capacitor to a resistive load is
1
1
1
1
The power delivered to that same resistive load is
1
1
If ≫
1
, as is the case in the majority of application , then the power delivered to the load is
given by
1
1
If
≪
then
The equations above tell us why the thermal noise from capacitors is usually negligible. , the
effective parallel resistance of capacitors approaches infinity. The noise power delivered to a
load is much smaller than
.
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Overview of Noise Measurement Methods
Piezoelectric Effects in Capacitors
Capacitors can also have additional noise caused by the microphonic effect. 7 Thermal energy
and mechanical vibrations both cause random vibrations in the dielectric. These vibrations are
transformed into noise voltage and current due to the relative motion between the capacitive
plates as well as the piezoelectric nature of many modern ceramic dielectrics. The piezoelectric
effect is highly dependent on the dialectic material and on the capacitor construction. Capacitor
manufacturer have the best information about the piezoelectric properties of their products and
should be consulted for more specific information.
Inductors
The noise voltage from an inductor can be found from exploring the series combination of an
inductor and a resistor.
| |
|
|
4
.
4
The power delivered by an inductor to a load is
|
|
4
4
This is a low pass function.
If
≪
≪
and
4
4
The above equation indicates the noise from inductors is always much less than
7
(Nelson & Davidson, 2002) 32
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.
Noise Figure
Overview of Noise Measurement Methods Appendix 2: Shot Noise
While it is often appropriate to consider electrical current as a continuous quantity, it must be
kept in mind that it is composed of discrete electrons, each with a fixed charge. Current is
quantized. Rather than measuring a continuous flow, the measurement of current is composed
of sensing the effects of individual electrons arriving at a detector with a uniform distribution over
time. The effect of this variation in time of arrival is called shot noise or corpuscular noise.
Computing Shot Noise
Derivation assuming a Poisson Distribution
A measurement of electric current can be considered to be the product of the number of
of each
electrons, , that arrive at a load per second of time, , multiplied by the charge,
electron.
,
Where
refers to the charge on an electron.
1.60217646 × 10-19 coulombs
The time of arrival of the electrons follows a Poisson distribution. The Poisson distributions
describes the behavior of events that occur with a constant probity over time. It states that the
probability that there are occurrences of an event given the expected value of that event is
given by
;
!
.
It can be shown that the Poisson distribution has a mean value of
and a variance of .
Let be the average current. The mean or average value of the number of electron arrivals per
.
second is
The variance in the number of arrivals during an interval of time T is . The standard deviation of
the number of arrivals per second is
.
The standard deviation of current is then
.
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The equivalent low-pass filter for an observation time of T seconds is (see appendix 3).
1
2
, where
2
determines the low-pass frequency of an equivalent filter.
2
2
The equation above states that the variance or power of the current fluctuations are proportional
to bandwidth and that shot noise has a flat power spectral density.
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Noise Figure
Overview of Noise Measurement Methods Appendix 3: Bandwidth Equivalent of Averaging
Consider a zero-mean white Gaussian random process
variance .
) has a Fourier transform
.
lim
1
|
→
|
|
) that is band limited to
and has a
|
The random process is white between the integration limits.
|
|
|
2
|
The flat power spectral density between the integration limits can be computed.
|
|
2
We now need to extend the above equations for the case where the time integration limits are
finite. The finite integration limits are equivalent to multiplying
by a rectangular function with
unity value in the interval
. This is the equivalent of convolving the frequency domain
function with a sinc function.
1
1
1
|
|
|
|
|
|
|
|
|
|
∗ 2
2
2
We can now compute the equivalent low-pass filter to the averaging process
2
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Noise Figure
Overview of Noise Measurement Methods Appendix 4: Avalanche Diode Noise Sources
Avalanche diodes are often used as the source of excess noise. Diodes when biased into the
avalanche breakdown region produce significant amounts of excess noise with good stability, high
bandwidth, and good statistical properties. These noise sources are often used as the stimulus for
noise figure measurements. They are also used as convenient sources of wideband random signals.
Avalanche Breakdown
Avalanche breakdown occurs due to the relatively high potentials that can exist across a revers
biased junction. An electron can accelerate to very high speeds across this potential. A fast electron,
when it strikes the semiconductor crystal lattice, can dislodge an electron from atoms in the crystal,
creating a hole-electron pair. The free electron can in turn be accelerated, generating another holeelectron pair when it strikes the crystal lattice. This process can go on, effectively multiplying the
current that flows.
It is useful to think of each electron that flows as having a multiplication factor, M.
.
in the above equation is an exponent that varies between 3 and 6 depending on the semiconductor
characteristics,
is the diode breakdown voltage and
is the actual voltage across the diode.
Consider a diode whose I-V characteristics are given by the familiar diode equation
1
When the diode is reverse biased with the application of a large negative voltage, the current is given
by
1
Reverse breakdown happens when the reverse bias current is multiplied
The net effect is that a large amount of current can flow, creating the familiar diode breakdown curve
shown in Figure 4.1
Diode Current VS Voltage
0.2
0.15
0.1
0.05
‐12
‐10
‐8
‐6
‐4
0
‐2 ‐0.05 0
‐0.1
‐0.15
‐0.2
Figure 4.1. Diode with a 10 V breakdown.
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2
Noise Figure
Overview of Noise Measurement Methods Noise Mechanism
Like shot noise, current comes in discrete quantities. Unlike shot noise, the size of the discrete
quantity is much larger than one electron, since each electron that crosses the potential barrier
has liberated several others to flow. Each electron that flows is very quickly followed by a large
number others, creating a larger granularity to the current flow.
A useful insight is obtained by thinking of the current flow as composed of discrete quantities that
are much larger than one electron. We can then modify the equation for shot noise and apply a
multiplier, , to the magnitude of each discrete charge. This multiplier accounts for the fact that
each time an electron travels, it brings many others with it.
2
A Practical Noise Source
One of the useful properties of noise diodes is that their noise power is relatively independent of
temperature, depending only on the level of current, which is easily controlled.
Figure 4.2. Noise source that can be switched from a cold to a hot state.
Figure 4.1 shows an example of a noise source that can be used for Y-factor noise
measurements. A 28V control signal is used to switch the diode from cold to hot state. The
Diode produces no excess noise when the control signal is at 0 V. Switching the control signal to
28V produces excess noise. The resistive attenuator is used to set the correct Excess Noise
Ratio (ENR) as well as to improve the source match presented to the device under test.
The presence of an attenuator is the reason that sources with high ENR typically have worse
reflection coefficients. Lower values of ENR with very good match can be achieved by
combining a noise source with a high value of ENR and resistive attenuator.
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Appendix 5: Error Analysis of the Y-factor Method
Y-factor Measurements
The Y-factor is given by making two power measurements. One measurement is made with an
input excess noise source turned on, the other with the noise source turned off.
, where
are power measurements made with the noise source on and off
respectively.
The noise factor of a device can be computed from
Noise Factor of a Measurement Receiver
The noise factor of the test receiver is made by making two power measurements with the noise
source connected directly to the receiver.
, where
is a power measurements
made with the noise source in the off or cold state directly connected to the measuring receiver,
and
is the same power measurement made with the noise source turned on.
Noise Factor of the DUT Cascaded with the Measurement Receiver
The noise factor of the DUT can only be measured as the cascade of the DUT and the
measuring device.
, where
is a power measurements made at the DUT output with the
excess noise source in the off or cold state connected to the DUT input, and
power measurement made with the noise source turned on .
is the same
The cascaded noise factor can be used to compute the DUT noise factor using the Friis8 formula.
The power gain of the device under test can be computed from the power measurements using
The DUT noise factor can be computed using the power measurements made above
The subscript in ENR
refers to the value of ENR that is supplied with the ENR source. It is
used to differentiate from the actual value of ENR that is a contributor to the measurements ofP
andP .
8
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Overview of Noise Measurement Methods
Uncertainty in Noise Factor Measurements
The uncertainty in measuring the Noise factor of the DUT can be computed from the
uncertainties in making the four power measurements.
Sources of Errors in Power Measurements
The noise factor of a device under test is computed from four power measurements as well as
prior knowledge of the Excess Noise Ratio Errors of a noise source. Errors can come from all 5
terms in the equation.
,
Errors in measuring ,
∆
and
∆
∆
∆
Errors in measuring
can be thought of as errors in measuring an incident power.
∆
∆
and
∆
∆
∆
∆
∆
∆
∆
∆
Where:
is the measuring receiver gain error. This absolute error includes calibration error,
∆
frequency response, temperature dependence, etc. This error is common to all power
measurements used in measuring noise figure and gain, and experiences cancellation in ratio
measurements such as the ones used in determining noise figure and gain. The receiver gain
error is common to all measurement and is a function of frequency.
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Noise Figure
Overview of Noise Measurement Methods
There are two cases to consider:
1. Cases where all power measurements are done at the same frequency have a
common gain error.
2. Frequency response can cause the gain error to be different when measurements
are made at different frequencies. This may be the case with frequency converters.
∆ is the linearity of the receiver over its dynamic range. This includes the linearity of any
amplifiers in the signal path as well as quantization errors of internal ADCs.
refers to the amount of noise power added by the receiver. It will also cause variations in
the power measurement result caused by receiver noise. The variance in the power
measurement caused by noise can be reduced to an arbitrarily small value by averaging and
can be assumed to be zero for large numbers of averages.
refers to the noise added by the device under test. The mean value of this additive error
is accounted for in the power measurements outlined in this pepper. Averaging can make the
variations arbitrarily small.
∆ is the error caused by the mismatch in the impedances of the noise source and the
receiver. The mismatch error between the noise source and the receiver is given by
∆
|
|
is the error caused by the mismatch in the impedances of the noise source and device
∆ under test.
∆
|
|
is the error caused by the mismatch in the impedances of the DUT and the receiver.
∆ The mismatch error between the DUT and the receiver is given by
∆
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Overview of Noise Measurement Methods
There are three cases to consider:
1. The noise source reflection coefficient is the same in both the hot and cold
state. This is often the case with noise diodes with a low value of ENR. The
error is correlated between measurements of P1 and P2 when this is the case. The
mismatch error is also correlated between measurements of P3 and P4 when the
noise source reflection coefficient does not change between the cold and hot state.
2. The noise source reflection coefficient changes between the cold and hot state
and the DUT has enough isolation so that its output reflection coefficient is
independent of the source connected to its input. The error is uncorrelated
between P1 and P2 but correlated between P3 and p4.
3. The noise source reflection coefficient changes between the cold and hot state
and the DUT has low isolation so that its output reflection coefficient changes
with changes in the source connected to its input. All mismatch errors are not
correlated and should be treated as independent.
Computing the Sensitivities for
1 The sensitivities of the noise factor measurement to errors in each of the power measurements
can be computed by taking partial derivatives of the noise factor with respect to each of the four
power measurements.
Noise Figure Sensitivity to Errors in Measuring
Noise Figure Sensitivity to Errors in Measuring
1
1
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1
Noise Figure
Overview of Noise Measurement Methods
Noise Figure Sensitivity to Errors in Measuring
1
1
1
1
Noise Figure Sensitivity to Errors in Measuring
1 1 G
1
1
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1
Noise Figure
Overview of Noise Measurement Methods
Sensitivity Table
Table 5-1 lists the sensitivity of the noise factor measurement to errors in each of the four power
measurements.
1 Sensitivity
Equation
1
1
1
1
1
Table 5-1. Sensitivities to power measurement errors.
We must now find the various sources of power measurement errors and compute how they
propagate to the measurement of a device’s noise factor.
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Overview of Noise Measurement Methods
Sensitivities to Measurement Errors
We can now determine the sensitivity of the noise factor measurement to each of the sources of error.
Sensitivity to Measuring Instrument Noise Power Subtraction Uncertainty
The equation for noise factor is
1 The measurements of noise factor and gain depend on the ability to accurately subtract power
levels made at relatively low levels. The quantity
is effectively a measure of the measuring
instrument’s noise floor.
is a measure of the power from the noise source, typically 5 to 15 dB
higher than the kTB.
abd are power measurements made with the DUT in place and the
noise source turned off and on respectively.
Most measuring instruments have limitations on how accurately two power measurements can
be subtracted from each other. These include:
1.
2.
3.
4.
Linearity
ADC Quantization floor
Quantization errors
Spurious signals
The receiver has a minimum difference between two powers that it can accurately resolve then.
.
Let this power subtraction uncertainty be equal to
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Overview of Noise Measurement Methods
The sensitivity of the noise figure measurement to the minimum power is then
1 1 1
1
1
1
1
1
1
1
1
1
Let M be the power subtraction floor expressed as a ratio to KTB. We can also assume that the
ENR specification error is small.
,
1
1
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1
Noise Figure
Overview of Noise Measurement Methods
Sensitivity to ENR Accuracy Errors
ENR accuracy is defined as the error between the specified value of ENR and the actual value.
1
∆
The equation for noise factor then becomes
1 ∆
.
The sensitivity to ENR errors is compute from the partial derivatives,
∆
∆
∆
∆
∆
.
We can see from the formulas for the four power measurements that they have a dependence on the
actual value of ENR, but not on the error in specifying it.
∆
∆
∆
∆
0
∆
3
1
4
3
We can now expand the components of the four power measurements.
kTB
ENRkTB
∆
kTB
kTB
1
∆
1
∆
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Overview of Noise Measurement Methods
Sensitivity to Receiver Gain Errors
For the dependence on the receiver gain we must recognize that the gain is common to all four power
measurements. There is cancellation as the gain used in measuring
subtracts from the gain used
in measuring , and so forth. This means that any receiver gain (measurement accuracy) that is
common to all power measurements does not affect the measurement of noise figure.
We must recognize, however, that the four power measurements are not made simultaneously and
there may be seconds or even minutes between the power measurements. The measurements may
not be at the same frequency in the case of frequency converters. Minute changes in gain as the four
power measurements are made can affect noise figure measurements. These changes can be:

Gain drift over time

Gain drift as the environmental temperature changes

Frequency response if
Let ∆
and
are made at different frequencies than
∆
1
∆
∆
1
1
1
1
1
1
∆
1
∆
1
1
1
1
1
∆
1
1
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1
1
1 1 1
1 1
2
1
√2∆
1 2
1
1
∆
The noise added by the device under test can be expressed as
Substitution yields:
∆
.
be any change in gain that can happen between during the four measurements.
∆
∆
and
1
1
.
Noise Figure
Overview of Noise Measurement Methods
Sensitivity to Linearity Errors
Linearity error is common to all four power measurements, but is not correlated. Its contributions
need to be added in an RSS fashion.
∆
∆
kTB
1 kTBG
∆
1
∆
∆
∆
kTB
1 kTBG
1
1
1
A derivation similar to the one above for gain variations yields the following
∆
√2∆
1
1 Sensitivity to Mismatch Errors
The sensitivity of the four power measurements to mismatches that exist during their
measurement can be computed from the formulas for each of the four power quantities,
reproduced here for convenience.
∆
∆
∆
∆
∆
∆
∆
∆
∆
∆
∆
∆
∆
∆
The sensitivities of the power measurements to each of the mismatch errors are:
∆
∆
∆
∆
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Overview of Noise Measurement Methods
The sensitivity of the noise factor computation to mismatch errors can be computed from:
∆
∆
∆
∆
∆
∆
∆
1
∆
The mismatch error involved in each power measurement is:
∆
1
∆
1
1
1
1
1
2
∆ =1
∆
2
1
1
1
2
1
2
1
2
2
The contributions from each of the mismatches can now be computed using a derivation similar
to the one done for gain and linearity.
∆
∆
∆
∆
∆
∆
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∆
∆
∆
∆
1
∆
∆
2
∆
2
1
∆
∆
∆
1
∆
1
1
1
1 1
2
1
1 2
2
2
Noise Figure
Overview of Noise Measurement Methods
Case 1: The noise source reflection coefficient does not change between the hot and
cold state
If the noise source does not change its reflection coefficient between its hot and cold state, then
the errors in measuring p1 and P2 are correlated, the errors in measuring p3 and p4 are
correlated but the remaining errors from the two pairs of measurements are not correlated to
each other.
∆
∆
∆
∆
∆
∆
∆
There is complete cancellation of the first term and partial cancellation of the other two.
∆
∆
∆
∆
∆
Case 2: The noise source reflection coefficient changes between its cold and hot state
and the DUT has enough isolation so that its output reflection coefficient is independent
of the noise source state.
In this case, power measurement errors due to mismatches at the source are uncorrelated and
the contributions of the mismatch errors at the output are correlated.
∆
∆
∆
∆
∆
∆
∆
There is no cancellation of the errors in measuring p1 and P2, but there is partial cancellation in
measuring the errors in P3 and P4
Case 3: The noise source reflection coefficient changes between its cold and hot state
and the DUT poor isolation so that its output reflection coefficient depends on the noise
source state.
In this case, power measurement errors due to mismatches at the source are uncorrelated and
the contributions of the mismatch errors at the output are also uncorrelated.
∆
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∆
∆
∆
∆
∆
∆
Noise Figure
Overview of Noise Measurement Methods
Computing the Sensitivities for Device Gain
Sensitivity of
to power measurement errors
∆
∆
∆
∆
∆
+
Gain Sensitivity Table
Table 5-2 lists the sensitivity of the gain computation to errors in each of the four power
measurements.
Sensitivity
In Terms of Power
In Terms of ENR and
Table 5-2. Sensitivities to power measurement errors.
We must now examine the various sources of power measurement errors and compute how they
propagate to the measurement of a device gain.
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Computing the Gain Sensitivities to the Power Measurement Errors
Gain Sensitivity to the Noise Power Subtraction Uncertainty
The measurements of noise factor and gain depend on the ability to accurately subtract power
is effectively a measure of the measuring
levels made at relatively low levels. The quantity
instrument’s noise floor.
is a measure of the power from the noise source, typically 5 to 15 dB
abd are power measurements made with the DUT in place and the
higher than kTB.
noise source turned off and on respectively.
Most measuring instruments have limitations on how accurately two power measurements can
be subtracted from each other. These include:
5. Linearity
6. ADC Quantization floor
7. Quantization errors
8. Spurious signals
Let
1 Device gain sensitivity to ENR Error
∆
∆
∆
∆
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∆
∆
∆
∆
0
∆
∆
Noise Figure
Overview of Noise Measurement Methods
Device Gain Sensitivity to Receiver Gain Errors
Gain errors are common to all power measurements, errors add algebraically and cancel. Any
variation in gain that happens during the time that the power measurements are being made will
show up as an error.
Let ∆
made.
Denote any gain variations that happen during the time the four measurements are
∆
∆
1
∆
∆
1
∆
∆
1
1 kTB
1
1 kTB
The noise added by the device under test can be expressed as
Substitution yields:
∆
1
1
∆
∆
1
√2∆
1
1
1
2
1
1
1
1
Errors due to gain inaccuracy of the measuring receiver that are common to all four power
measurements and do not affect the determination of device gain. Any changes to the gain of
the receiver as the four power measurements are being made, however, will affect the
measurement.
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Device Gain Sensitivity to Receiver Linearity Errors
Errors due to receiver linearity are not correlated since the power levels will differ. Linearity
errors must be treated as independent.
1
∆
1
1
1 kTB
1
1 kTB
A derivation similar to the one above for gain yields:
∆
√2∆
1
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1
1
Noise Figure
Overview of Noise Measurement Methods
Sensitivity to mismatch errors
A derivation using the same methodology as that used for gain variations yields:
∆
∆
∆
∆
∆
∆
∆
∆
∆
∆
∆
∆
∆
∆
∆
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∆
∆
∆
∆
∆
∆
2
2
1
∆
∆
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1
∆
∆
1 kTB
∆
1
1
∆
2
1 kTB
1
∆
∆
∆
∆
1
1
1
2
2
2
1
1
2
1
1
2
1
1
∆
∆
2
1
2
2
Noise Figure
Overview of Noise Measurement Methods
Case 1: The noise source reflection coefficient does not change between the hot and cold
state
∆
∆
∆
∆
∆
∆
∆
In this case the mismatch error for P1 and P2 are the same, as are the mismatch errors for
P3 and P4 are the same. There is partial cancellation of the effects as reflected by the
above equation.
Case 2: The noise source reflection coefficient changes between its cold and hot state and
the DUT has enough isolation for its output reflection coefficient to be independent from the
input.
∆
∆
∆
∆
∆
∆
∆
There is still partial cancellation of the mismatch errors at the DUT output.
Case 3: The noise source reflection coefficient changes between its cold and hot
state and the DUT has low solation so its output reflection coefficient depends on the
input.
In this case, all power measurement errors due to mismatches are uncorrelated and
∆
56
∆
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∆
∆
∆
∆
∆
Noise Figure
Overview of Noise Measurement Methods
Summary of Errors
Errors in Measuring Noise Factor
The noise factor of a device can be measured by taking four power measurements.
1 Normalized Errors in Measuring Noise Factor
Error Component
Power
subtraction
uncertainty
∆
ENR
inaccuracy
∆
1
∆
Receiver
Linearity
∆
Mismatch
between the
noise
source and
the DUT
input
Mismatch
between the
DUT output
and the
receiver
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∆
2
√2∆
1
2
√2 ∆
1
Noise
Source
Off
∆
Noise
source
On
∆
Noise
Source
Off
2
1
2
1
∆
Noise
source
On
∆
Noise
Source
Off
∆
Noise
source
On
∆
1
1
2
1
1
2
1
1
1
Table 5-2. Errors in measuring DUT noise factor.
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1
1
Receiver
Gain
Inaccuracy
Mismatch
between the
noise
source and
the receiver
1
1
2
2
1
1
Noise Figure
Overview of Noise Measurement Methods
Errors in Measuring Device Gain
The gain a device is computed from four power measurements.
Normalized Errors in Measuring Gain
Error Component
Power
Subtraction
uncertainty
(SUB DANL
Floor)
∆
ENR
inaccuracy
∆
Receiver
Gain
Inaccuracy
∆
Receiver
Linearity
∆
Mismatch
between the
noise source
and the
receiver
Mismatch
between the
noise source
and the DUT
input
Mismatch
between the
DUT output
and the
receiver
√2∆
1
√2∆
1
Noise
Source
Off
∆
Noise
source
On
∆
Noise
Source
Off
1
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2
1
2
1
∆
2
1
Noise
source
On
∆
Noise
Source
Off
∆
Noise
source
On
∆
2
0
1
1
2
1
2
1
Table 5-3. Errors in measuring DUT gain.
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ENR 1
ENR 1
1
1
2
2
1
1
2
1
1
Noise Figure
Overview of Noise Measurement Methods Statistical Distribution of Measurement Errors
The statistical distribution of measurement is a complex topic, a treatment of which is beyond the
scope of this paper. A summary of the error distributions encountered in the measurement of
noise fugure and gain is included for clarity and completeness.
ENR
The error in determine the ENR of a noise source is the difference between the ENR specified by
the noise source manufacturer and the actual value. This error has many influencing factors,
including measurement frequency temperature, current, noise source aging, etc. Manufacturers
typically specify noise sources have excess noise ratio within ± 1dB from a specified value. A
good assumption is that the ± 1dB spread covers 95% of units with uniform distributions (σ=0.5
dB).
Receiver Gain Error
This is the error with which a receiver can measure a power level coming from a prefect (Usually 50Ω) Ohm source. It is typically includes a reference error and a frequency response
error. This error can also be further improved by making a local calibration using a power meter
and a signal source. Receiver gain errors can be assumed to be normally distributed. If a 95
percentile number is given, it can be assumed to be equal to 2 σ.
Receiver Linearity
Linearity was formerly a big error, especially in spectrum analyzers that used analog circuits for
their log amplifiers. Today’s receivers use D-A converters and DSP. The linearity is much better
but can still be significant. Sources of linearity errors include digitizer quantization errors,
intermodulation and spurious.
Receiver linearity at high input levels is usually limited by its spurious-free dynamic range
(SFDR). Noise figure measurements, however, often measure power levels that are close to the
instrument’s noise floor ( One of the four measurements, P1 is indeed a measure of the
instrument’s noise floor.) Linearity takes on a different character at these low levels.
1. Linearity affects a measuring instrument’s ability to accurately determine the difference
between two power levels. It is often lumped into the instrument uncertainty for gain and
for noise figure measurements.
2. While it is customary to examine the linearity of electrical devices in terms of their 2nd, 3rd,
4th … order intercept points, devices with Analog to digital converters have some lowlevel quantization errors limit linearity at the low levels that are used in noise figure
measurements.
3. Tektronix Spectrum analyzers use a variety of techniques to drop the quantization errors
to levels below the instrument’s noise floor. A good upper bound for the linearity errors:
a. Expressed as a power ratio: ∆
b. Expressed in dB: ∆
The linearity errors can be assumed to be normally distributed.
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Noise Figure
Overview of Noise Measurement Methods
Mismatch Errors
Mismatch errors are of the form
∆
1
1
1
2
and
are vector quantities that have a normally distributed magnitude and an phase that is
uniformly distributed over 0 to 2 radians.
It can be shown that this type of error has distribution that is U-shaped as is shown in Figure 5.1.
It can also be shown that this kind of distribution has a standard deviation of
, and
√2
that 95% of all occurrences will lie within 2√2
of the mean.
Mismatch Distribution for two reflection coefficent gaussian magnitude and uniform pahse
100
90
80
70
60
50
40
30
20
10
0
0.9
0.95
1
1.05
1.1
Figure 5-1. Histogram of mismatch for two reflecting coefficients with Gaussian magnitude and uniform phase
(ρmean=.2, σρ=0.02).
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Noise Figure
Overview of Noise Measurement Methods
Coverage Factor
In general, the value of the coverage factor k is chosen on the basis of the desired level of
confidence to be associated with the interval defined by U = kuc. Typically, k is in the range 2 to
3. When the normal distribution applies and uc is a reliable estimate of the standard deviation of
y, U = 2 uc (i.e., k = 2) defines an interval having a level of confidence of approximately 95 %,
and U = 3 uc (i.e., k = 3) defines an interval having a level of confidence greater than 99 %.
A common approach for errors with distributions that are not normal is to use a coverage factor
that allows a similar level of confidence.
Distribution
Standard Deviation
Normal
Noise
Constrained distributions: -a<x<a
Rectangular (Uniform)
Frequency counter LSB
√3
Triangular
Results from the combination of two uniform
distributions
√6
Cosine
√3
1
U-shaped
Table 5-4. Some typical error distributions
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Distribution of X-values that result from uniformly
distributed phase
Probability of the values of sinusoids
√2
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6
Noise Figure
Overview of Noise Measurement Methods
Example of Contributions to Noise Figure Measurement
The following is an illustrative example of the sources of error in a noise figure measurement.
The table details parameters for the DUT, noise source and the measuring receiver.
Device Under Test
Noise Source
Measuring Instrument
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DUT NF
1
dB
DUT Gain
20
dB
DUT Input Match
1.5
VSWR
DUT Output Match
1.5
VSWR
ENR
15
dB
ENR Uncertainty
0.15
dB
Noise Source match
1.15
VSWR
Noise Figure
10
dB
Instrument Uncertainty for NF
.02
dB
Instrument Uncertainty for Gain
.07
dB
SUB DANL Floor
13
dB
Input Match
1.45
VSWR
Noise Figure
Overview of Noise Measurement Methods
The uncertainty in the noise source ENR and the mismatch errors are the largest contributor to
uncertainty in noise figure. Mismatch errors are the largest contributors to gain uncertainty.
Contributions to uncertainty in NF and gain
Error Contributions
NF Error (dB)
Gain Error (dB)
ENR
0.153
0.000
Gain
0.030
0.030
Linearity
0.105
0.104
Noise subt. Uncertainty
0.003
0.018
Mismatches
0.030
0.454
Total
0.189
0.465
8/2014 © Tektronix All rights reserved.
37W-30477-0
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