noise tutorial part i ~ noise concepts...noise tutorial i ~ noise concepts see last page for...

See last page for document information

Noise TutorialPart I ~ Noise Concepts

Whitham D. ReeveAnchorage, Alaska USA

Noise Tutorial I ~ Noise Concepts


Abstract: With the exception of some solar radio bursts, the extraterrestrial emissions received on Earth’s surface are veryweak. Noise places a limit on the minimum detection capabilities of a radio telescope and may mask or corrupt these weakemissions. An understanding of noise and its measurement will help observers minimize its effects. This paper is a tutorialand includes six parts.

Table of Contents Page

Part I ~ Noise Concepts

1-1 ~ Introduction 1-1

1-2 ~ Basic noise sources 1-1

1-3 ~ Noise amplitude 1-6

1-4 ~ References 1-10

Part II ~ Additional Noise Concepts

2-1 ~ Noise spectrum

2-2 ~ Noise bandwidth

2-3 ~ Noise temperature

2-4 ~ Noise power

2-5 ~ Combinations of noisy resistors

2-6 ~ References

Part III ~ Attenuator and Amplifier Noise

3-1 ~ Attenuation effects on noise temperature

3-2 ~ Amplifier noise

3-3 ~ Cascaded amplifiers

3-4 ~ References

Part IV ~ Noise Factor

4-1 ~ Noise factor and noise figure

4-2 ~ Noise factor of cascaded devices

4-3 ~ References

Part V ~ Noise Measurements Concepts

5-1 ~ General considerations for noise factor measurements

5-2 ~ Noise factor measurements with the Y-factor method

5-3 ~ References

Part VI ~ Noise Measurements with a Spectrum Analyzer

6-1 ~ Noise factor measurements with a spectrum analyzer

6-2 ~ References


See last page for document revision information ~ File: Reeve_Noise_1.doc, Page 1-1


1-1. Introduction

Noise is a random electrical waveform that can obscure the desired radio emissions and signals. The

random nature of a noise voltage waveform means 1) it contains no predictable periodic frequency

components and 2) its amplitude at any future time is unpredictable. Unlike periodic signals, such as

manmade broadcast transmissions that consist of one or more discrete spectral lines corresponding to

their frequency components, random noise has a spectrum that is a continuous function of frequency

and contains no discrete line components (figure 1-1).

Fig. 1-1 ~ Noise and periodic signal spectrums. Left: The noise spectrum is produced by a 20 MHz random noisegenerator and displayed on an oscilloscope. The noise extends continuously four divisions (5 MHz/div) from theleft scale and drops off due to the lowpass filter in the generator. Right: The periodic signal is an unmodulated 15MHz carrier wave (CW) indicated as a single spectral line located three divisions from the left scale.

1-2. Basic noise sources

An important source of noise is the constant agitation of matter at molecular and atomic levels. The

molecules vibrate about their position in solids or collide with each other in gases. These are called

thermal agitations because they are related to temperature. A resistor or copper wire has conduction

electrons that are free to wander randomly throughout the material volume. These electrons as well as

positive ions also present in the material are uniformly distributed, and the entire structure is electrically

neutral. However, because of the random motions, there are statistical fluctuations away from the

neutral state. A very large number of charges are involved and, occasionally, the charge distributions will

not be uniform and a voltage difference will appear across the conductor terminals. This random voltage

is erratic and unpredictable and is called resistor thermal noise, or just thermal noise (also called

Johnson noise).

Thermal noise is proportional to the temperature of the material and the value of its resistance. It is

present at the resistor terminals whether or not it is connected to anything. Thermal resistor noise is

zero for a perfect conductor and any conductor at a temperature of absolute zero (0 kelvin). In noise



analysis and in this paper, all temperatures are in kelvin (K). Temperatures in kelvin are related to

temperatures in celsius by

273.15

273.15

T K T C

T C T K

(the conversion constant is rounded to 273 in ordinary work) (1-1)

Another type of noise is caused by the flow of current across semiconductor junctions in diodes and

transistors. The charge carriers, electrons or holes, enter the junction from one side, drift or are

accelerated across the junction, and are collected on the other side. The average current across the

junction determines the average time interval between two successive carriers that enter the junction.

However, there are random fluctuations in the movement, giving rise to a type of noise called shot

noise. Shot noise also is caused by the random electron emissions from a heated surface, such as the

filament in a vacuum tube or other thermionic device. Resistors and other electronic components in

radio telescopes are called internal noise sources.

Noise sources external to telescopes also are very important. They consist of sky noise, manmade noise

and test and measurement noise. The first two are generalized in a set of charts that extend from 0.1 Hz

to 100 GHz (figure 1-2). These charts are provided for reference as we discuss noise technical

parameters.

Sky noise due to electrical storms, solar and geomagnetic storms, cosmic radio noise and

atmospheric absorption. When these external noise sources interfere with the desired

emissions, they are called electromagnetic interference (EMI), radio frequency interference (RFI)

or just interference. On the other hand, we may be actually investigating certain types of sky

noise, including radio emissions from the Sun and Jupiter, neutral hydrogen emissions or cosmic

microwave background (CMB) radiation, in which case it obviously is not considered

interference. Some of these emissions are due to

synchrotron radiation (see sidebar) from very large

numbers of charged particles in outer space. When

large numbers of independent sources are combined

the resulting emissions have random and very noise-

like characteristics. Also, during their propagation these emissions are randomly modulated by

inter-stellar plasma and the Sun’s plasma (solar wind). Other sources of sky noise may mask the

specific emissions being investigated. For example, lightning can mimic certain types of Jupiter

emissions and the CMB may be strong enough to mask interesting emissions from specific

extraterrestrial sources in the same direction. For our purpose, sky noise is due to natural

phenomena.

Manmade noise due to ignition systems, electric motors and relays, arcing and corona

discharges in electrical power systems and transmission lines, and consumer electrical and

electronic equipment such as LED lighting, televisions, radio receivers and transmitters,

microwave ovens, dimmer switches, ac adapter power supplies, coffee grinders and personal

computers. These noise sources may be close by or far away and can be from terrestrial or

satellite transmitters. Except at locations dedicated for the purpose, radio astronomers have

Synchrotron radiation is electromagneticradiation generated by charged particlesmoving at speeds approaching the speedof light (relativistic speeds) in spiral orbitsaround magnetic field lines.



little or no control over many manmade noise sources and must find ways to minimize their

effects or contend with them.

Test and measurement noise from calibrated noise sources (figure 1-3). These use

semiconductor diodes or special thermionic vacuum tubes that are built into or temporarily

connected to the radio telescope receiver or antenna systems for calibration purposes and are

under control of the radio telescope operator.

Micropulsations

Minimum expectedvalue of atmospheric

noise

Maximum expected valueof atmospheric noise

10lo

g10(F

)(d

B)

Frequency (Hz)

Effect

ive

Nois

eT

em

pera

ture

(K)

300

280

260

240

220

200

180

160

140

120

290 x1030

290 x1028

290 x1026

290 x1024

290 x1022

290 x1020

290 x1018

290 x1016

290 x1014

290 x1012

0.1 1 10 100 1k 10k

(a) 0.1 Hz to 10 kHz: There is very little seasonal, diurnal or geographic variation. The variation in the range of 5 to 10 kHz is due to thevariability of the Earth-ionosphere waveguide mode cutoff and the resulting propagation of atmospheric and other noise at those frequencies

Atmospheric noiseValue exceeded 0.5% of time

Atmospheric noiseValue exceeded

99.5% of time

Manmade noiseQuiet receiving site

Galactic noise

Median business areamanmade noise

Solid line – Minimum expected noise level

10k 100k 1M 10M 100M

10lo

g1

0(F

)(d

B)

180

160

140

120

100

80

60

40

20

0

Effect

ive

Nois

eT

em

pera

ture

(K)

290 x1018

290 x1016

290 x1014

290 x1012

290 x1010

290 x108

290 x106

290 x104

290 x102

290

Frequency (Hz)



(b) 10 kHz to 100 MHz: The atmospheric noise values takes into account all times of day, seasons and the entire Earth’s surface. The curve forgalactic noise above about 5 MHz is greatly affected by the ionospheric absorption that can extend to above 20 MHz. For directional antennasin the HF range (3 - 30 MHz), studies of atmospheric noise show that there can be ±5 dB variation around the indicated noise factor dependingon direction, frequency and geographical location

10lo

g1

0(F

)(d

B)

40

30

20

10

0

–10

–20

–30

–40100M 1G 10G 100G

Frequency (Hz)

Effect

ive

Nois

eT

em

pera

ture

(K)

290 x104

290 x103

290 x102

2900

290

29.0

2.9

0.29

0.029

Galactic noise

Median business areamanmade noise

Galactic noiseToward galactic center withinfinitely narrow beamwidth

Quiet Sun1/2 beamwidth directed at Sun

Sky noise due to oxygen and water vaporVery narrow beamwidth, 0 elevation angle

Sky noise due to oxygen and water vaporVery narrow beamwidth, 90 elevation angle

Black body radiationCosmic background at 2.7 K

Solid line – Minimum expected noise level

(c) 100 MHz to 100 GHz: The minimum galactic noise from a narrow-beam antenna pointed at the galactic pole is 3 dB below the solid curveshown. The sky noise due to oxygen and water vapor absorption is quite variable above about 1 GHz.

Fig. 1-2 ~ Sky and manmade noise across the radio bands from 0.1 Hz to 100 GHz. Except as noted, the curves arefor omni-directional antennas. The left scale on all plots is in terms of noise factor (in dB) and the right scale is interms of noise temperature (in kelvin). (Plots adapted from: [ITU-R P372.8], used with permission)

Fig. 1-3 ~ Calibrated noise sources. Left-to-right: Hewlett-Packard 346D (ENR 21 dB), Renz RQ6 with 20 dBattenuator (ENR 55 dB without attenuator), RF Associates RF-2050S (Noise temperature switchable 43 thousand to1.4 million kelvin), RF Design RFD2305 with 10 dB attenuator (ENR 5 dB with 10 dB attenuator). ENR (excess noiseratio) is discussed in Part V.



All attempts to receive extraterrestrial emissions involve noise. Most extraterrestrial radio sources are

physically very large (but small in angular size when viewed from Earth) and their radiation involves a

very large number of statistically independent radio sources – atoms, electrons, ions and molecules. The

resulting emissions are noise-like and in many situations are indistinguishable from noise due to

manmade radio frequency interference. When these emissions arrive at a radio telescope antenna they

are accompanied by interfering noise, which may be stronger than the desired emissions. Also, each part

of the receiving system and each receiver stage introduce additional interfering noise.

We can describe random noise only in terms of its average properties; an instantaneous measurement

means little. Therefore, noise measurements are made on an average basis over some finite

measurement time. The two most important properties of noise are amplitude (voltage or power) and

spectrum (frequency distribution). Although we cannot predict the noise voltage amplitude at any

instant, we can predict the amplitude range.

We often are interested in the noise power, which is proportional to the square of the noise voltage.

Because an important property of noise is its spectrum, we cannot properly discuss noise power of a

device without knowing the associated bandwidth. In radio discussions it is convenient to normalize

noise power to a bandwidth of 1 Hz, for example, watts per hertz or milliwatts per hertz. This

representation is called noise power density. We will discuss these units of measure and their

logarithmic equivalents later.

1-3. Noise amplitude

Because noise is a random process, it is analyzed on a statistical basis. The noise amplitude prediction

probability is given by a function called the amplitude probability density distribution p(v). When

multiplied by a small voltage increment dv, the amplitude probability density distribution function gives

the probability that, at any given instant, the voltage lies in the interval v to v + dv. The amplitude

density distribution can take on many forms but, as will be seen, only one form is of most interest to us.

Since the noise voltage, v, must exist at some value, the total probability over an infinite range is 1

(100%). In other words, the sum of all the individual products of p(v) and dv throughout the range of

p(v) will be

-( ) 1p v dv

(1-2)

We also are interested in determining the probability that the noise voltage amplitude will fall below

some given value. The sum of the amplitude density distribution over part of the voltage amplitude

range is called the cumulative amplitude probability distribution P(v) and is defined as

-

( )v

P v p v dv

(1-3)

P(v) also is called the cumulative probability and is the probability that a noise voltage lies below the

voltage v at any given time. It is the area under the p(v) curve between the extreme negative value (–∞)



and v. P(v) varies from 0 to 1. When P(v) is 0, there is 0% chance the voltage is below v, and when P(v) is

0.5 there is a 50% chance the voltage is below v, and so on. We cannot say what the values will be at any

instant, only that there is a calculated probability it will be below that value.

Similarly, we are interested in the probability that the noise voltage amplitude will fall within a given

range, say v1 and v2. In this case,

1 2

- -( ) ( )

v v

P v p v dv p v dv

(1-4)

When we analyze noise and its effects, we must make some assumptions. One is that the statistical

averages of the noise, such as its amplitude distribution and spectrum, do not change over time (the

noise is said to be stationary, a property analogous to oscillator stability). Also, we assume the

probability density of the noise is Gaussian; that is the noise conforms to a particular mathematical

function called the Gaussian, or normal, distribution. The normal distribution describes the probability

that a particular amplitude will be exceeded (or not exceeded) in any given measurement period. When

plotted the normal distribution yields the familiar bell-shaped curve (Fig. 1-3). Other distribution

functions exist, but it has been determined both theoretically and experimentally that thermal and shot

noises and many other conditions we encounter in radio astronomy are Gaussian, so we will

concentrate on the Gaussian distribution. It should be noted that many types of interfering noise are not

Gaussian. They have strong periodic components and thus do not meet the definition of noise for our

purposes. Manmade radio frequency interference is an example of non-Gaussian noise.

p(v

)

Fig. 1-3 ~ Gaussian distribution (bell-shaped or normal distribution) curve. Almost 99% of the area under this curvelies between -3σ and +3σ.

For the Gaussian amplitude density distribution

2

221

( )2

v

p v e

(1-5)

and for the Gaussian amplitude distribution



1( ) 1

2 2

vP v erf

(1-6)

where

σ root-mean-square (rms) noise voltage, also called

standard deviation

erf error function

There is no theoretical upper amplitude limit that noise could reach, but very high amplitudes have very

low probability, and real electronic systems have practical amplitude limits. If a large number of

instantaneous amplitude measurements are made on a noise source, it will be found that most of

measurements fall within a range around a certain value, the average value, and only a few have higher

or lower values.

Some representative values for the Gaussian amplitude density distribution from Eq. (1-5) and

amplitude distribution from Eq. (1-6) are given in table 1-1. The two functions are plotted (Fig. 1-4).

From the table, if the noise rms voltage (σ) is, say, 150 mV, then the amplitude distribution function,

P(v), shows that 84% of all noise voltage values will be less than 1σ (150 mV) and 99.9% of all values will

be less than 3σ (450 mV). Also, the probability of the voltage falling between –150 mV (–1σ) and +150

mV (+1σ) is 84.13% – 15.87% = 68.26% and between –450 mV (–3σ) and +450 mV (+3σ) is 99.865% –

1.350% = 98.52%. It should be noted that we assume the noise has no dc component (this is explicitly

true when the noise is capacitor coupled). Thus, the average value of the noise voltage over any

practical time period is zero.

Table 1-1 ~ Gaussian amplitude distribution functionsNote: p(v) is normalized to the rms value (σ)

v p(v) P(v)-5σ 0.000 001 487/σ 0.000 000 287 -4σ 0.000 133 8/σ 0.000 031 67 -3σ 0.004 432/σ 0.001 350 -2σ 0.053 99/σ 0.022 75 -1σ 0.241 97/σ 0.158 65 0σ 0.398 94/σ 0.500 00 +1σ 0.241 97/σ 0.841 34 +2σ 0.053 99/σ 0.977 25 +3σ 0.004 432/σ 0.998 650 +4σ 0.000 133 8/σ 0.999 968 33 +5σ 0.000 001 487/σ 0.999 999 713

The error function is a special function thatfrequently occurs in the mathematics ofprobability and statistics and is determined byintegrating the Gaussian amplitude densitydistribution function.



Fig. 1-4 ~ Gaussian amplitude probability density function (left scale and blue curve) and voltage cumulativeamplitude distribution function (right scale and red curve) plotted together. The cross (+) indicates an examplewhere the cumulative probability (red curve) and area under the amplitude density curve (blue shaded area) are89.97%.

If we capture an oscilloscope display of Gaussian noise, it can be compared to the normal distribution

curve (Fig. 1-5). Examination will show there are only a few instances when the noise voltage exceeds

some high threshold, say 4σ, and many more instances when it exceeds a much lower threshold, say 1σ.

In Part II we will discuss additional noise concepts including spectrum and temperature.

Fig. 1-5 ~ Oscilloscope display of the output from a random noise generator. The generator has a bandwidth of 20MHz and its output was set to 150 mVrms (σ = 150 mVrms). The scope vertical gain was set to 200 mV/division andthe time base to 2.5 μs/division. A few noise peaks exceeded 700 mV (±4.7σ)



1-4. References

[ITU-R P-372.9] Recommendation ITU-R P-372.9, Radio Noise, International TelecommunicationsUnion, Radio Communications Sector, 2007



Document informationAuthor: Whitham D. ReeveCopyright: © 2014 W. ReeveRevision: 0.0 (Adapted from original expanded work, 19 Jun 2014)

0.1 (Updated TOC and references, 7 Jul 2014)0.2 (Minor corrections and minor edits, 18 Jul 2014)

Word count:2739

Size: 3515392


Noise TutorialPart II ~ Additional Noise Concepts


Noise Tutorial II ~ Additional Noise Concepts





1-1 Introduction

1-2 Basic noise sources

1-3 Noise amplitude

1-4 References


2-1 Noise spectrum 2-1

2-2 Noise bandwidth 2-1

2-3 Noise temperature 2-2

2-4 Noise power 2-4

2-5 Combinations of noisy resistors 2-7

2-6 References 2-12


3-1 Attenuation effects on noise temperature

3-2 Amplifier noise

3-3 Cascaded amplifiers

3-4 References


4-1 Noise factor and noise figure

4-2 Noise factor of cascaded devices

4-3 References


5-1 General considerations for noise factor measurements

5-2 Noise factor measurements with the Y-factor method

5-3 References


6-1 Noise factor measurements with a spectrum analyzer

6-2 References




2-1. Noise spectrum

As mentioned in Part I, the spectrum of random noise contains no periodic frequency components and is

a continuous function of frequency. The spectral intensity of the noise describes its frequency content

and is given in units of voltage squared per unit bandwidth. When this is divided by the resistance across

which the voltage is measured, it is equal to the power dissipated in the resistance per unit bandwidth

and is called power spectral density (PSD, also called spectral power density). The most common unit for

SPD is watts/hertz (W/Hz). Noise spectrum often is measured as a voltage, rather than power, in a given

bandwidth. In this case, we define the voltage spectrum of the noise as numerically equal to the square

root of the spectral intensity in units of voltage per square root of bandwidth. Since this is an unwieldy

unit, it frequently is shortened to just volts per hertz (but it really is V/ Hz or volts per root hertz).

Noise that has a constant power spectral density over the range of frequencies in which we are

interested is called white noise. The term white noise is analogous to white light, which is a

superposition of all visible spectral components. White noise does not mean it contains equal

amplitudes at all frequencies because the total noise power would then be infinite. Instead, it means the

spectrum is flat over the range of interest, for example, the audio frequency range or receiver

intermediate frequency bandwidth.

2-2. Noise bandwidth

Noise passes through various filters in any receiving system. The antenna, transmission lines, receiver

input circuits, amplifiers, detectors and output circuits all have bandwidth limits. Filtering generally

makes non-Gaussian noise more Gaussian. The concept of noise bandwidth applies to filters.

When white noise is presented to a receiver, some of the noise power will be rejected by the input filter

and dissipated as heat or reflected back to the source. The amount of noise allowed to pass is

determined by the filter’s noise bandwidth. The noise bandwidth is the bandwidth of an ideal filter with

a rectangular response that passes the same noise power as the real filter. The noise bandwidth is

obtained by finding the area under the actual curve of the filter response with respect to frequency and

converting it to an equivalent flat-top rectangular pass band with the same area. The height of the

equivalent rectangular pass band is made equal to the maximum response of the actual filter curve.

If the mathematical function describing the actual filter response curve is known, the area can be found

by integration from to in frequency ( 2sf to 2sf for a digital filter sampled at sf rate).

Alternately, the area may be found graphically by plotting the response function or results of

measurements. The response curve must be plotted in linear amplitude units and not logarithmic units

such as dB and with linear frequency and not log frequency. For graphical analysis, plotting the

frequency response to the –60 dB bandwidth values usually is sufficient.



For measured filter data, such as s-parameters, the measurements must be made with linear frequency

and not log frequency and a sufficiently small frequency step size must be used. If the measurement

results are in dB, they must be converted to linear ratios. For example, the s21 transmission parameters

of a filter often are measured with a vector network analyzer as complex voltages but displayed or

stored in dB (figure 2-1). For plotting purposes the measurements are converted to voltage magnitudes

by

21( )

2021 10s dB

s

Fig. 2-1 ~ Bandpass filter with –3 dB response from 20 to 90 MHz. The response is measured with a vector networkanalyzer setup for s-parameters and linear frequency sweep from 100 kHz to 180 MHz with 600 kHz step size (300measurement points). The measurements are saved as a Touchstone *.s2p file for later processing. The noisebandwidth is the area under the filter response curve and is equivalent to the rectangular response shown with athick red line.

The magnitude of s21 is a voltage so2

21s is power normalized to 1 ohm impedance. The normalized

power in each frequency increment of the filter response curve is

221step stepArea s f

where

stepf frequency step size

stepArea area for each frequency step

This is repeated across the entire frequency range and the results added,

2

1

21steps

Total stepArea s f

The bandwidth of a rectangular filter with equivalent noise bandwidth is then found by normalizing the

total area to the maximum or peak value of the filter response, as in

221

Totaln

AreaB

Peak s Hz

Since the data in *.s2p files are ASCII text, they may be imported and analyzed in a spreadsheet program

or in a program like Matlab; for example, see [Layne].



The noise bandwidth of a filter always is higher than its 3 dB signal bandwidth. The example of a

bandpass filter was given above. A simple low-pass RC filter has a noise bandwidth 1.57 times its 3 dB

bandwidth [Taub]. In a multi-stage tuned filter, the noise bandwidth factor is close to 1.05 to 1.1 giving a

noise bandwidth only slightly more than its 3 dB bandwidth. Examples involving noise bandwidth are

given in the following sections.

2-3. Noise temperature

Consider an ordinary resistor at temperature T (figure 2-1). As discussed in Part I, a resistor develops a

thermal noise voltage. The corresponding noise power may be derived from the Rayleigh-Jeans and

Planck radiation laws (for example, see [Nyquist]). The open-circuit noise voltage vn across the resistor

has a mean-square value in a frequency band Bn given by

2 4n nv k T R B (2-1)

where

vn open-circuit rms voltage (Vrms) (note that the function of time (t) is implied)

k Boltzmann constant ( 231.38 10 J/K)T temperature (K)R resistance (ohms)Bn noise bandwidth (Hz)

R Vn

T

Fig. 2-1 ~ Resistor with resistance R at temperature T produces an open-circuit noise voltage Vn.

The Boltzmann constant k is a fundamental constant of physics occurring in nearly every statistical

formulation of both classical and quantum physics. Its dimensions are energy (joules) per degree of

absolute temperature (kelvin). The physical significance of k is that it indicates the amount of energy

(heat) in a substance corresponding to the random thermal motions of its molecules.

Taking the square root of both sides of Eq. (2-1),

4n nv k T R B Vrms (2-2)

Since 2nv is proportional to power, Eq. (2-1) and (2-2) indicate that noise power is proportional to

bandwidth and noise voltage is proportional to the square root of the bandwidth; for either case, the

narrower the bandwidth the lower the noise power or voltage (figure 2-2).



(a) 15 MHz (b) 10.7 MHz

(c) 5 MHz (d) 2.5 MHz

Fig. ~ 2-2 - Bandwidth limited noise voltages. The oscilloscope screenshots of voltage measurements on the outputof various low-pass filters connected to a random noise generator whose frequency response is flat from 0 to 20MHz and whose rms output voltage is set to the same value for all filters. All plots are shown at 20 mV/div verticalscale and 5 µs/div horizontal time scale giving full-scale spans of ±100 mV and 50 µs.

These relationships indicate that, to minimize the noise voltage (or noise power) in a receiving system,

the bandwidth should be no higher than necessary. A system with excessively high bandwidth will

detect too much noise and will not be as sensitive to weak signals as a system whose bandwidth is closer

to the emissions of interest. On the other hand, the system bandwidth should not be too narrow or else

too little of the desired emissions energy passes through the filter and it may not be detected.

A noisy resistor can be modeled as a series combination of a gaussian noise voltage generator and an

ideal noiseless resistor (figure 2-3).

R

R

Noiseless

Noisy

T vn

Noise voltage



Fig. 2-3 ~ Noisy resistor model

When the resistor is placed under load, there will be some voltage drop across the source resistance R

and some voltage drop across the load resistance RL (figure 2-4). According to Kirchhoff’s voltage law,

the sum of the voltage drop across each resistor must equal the source voltage, or n R LV V V .

R

Noiseless

RL VL

VR

vn

Noise voltage

Fig. 2-4 ~ Noisy resistor model under load

Assuming both resistors are at temperature T, the maximum power delivered to an external load

resistance RL by the resistor R can be found by noting that maximum power transfer occurs when the

load resistance equals the source resistance, in which case the voltage divides equally across the source

and load, or

2

4 4

2 2 2

nR L

n L nn

vv v

k T R B k T R Bv

(2-3)

2-4. Noise power

The average noise power Navg delivered to the load is

2 4

4 4n L n

avg nL L

v k T R BN k T B

R R

(2-4)

The units for the average power are J J

K Hz J Hz WK s

Eq. (2-4) indicates the average noise power is independent of the actual resistance values when the

source and load are matched. The matching of source and load resistances in real receiving systems

allows maximum transfer of the desired signal power from the source to the load, but it also allows the

maximum transfer of noise power.

Example 2-1:

(a) The temperature used in many investigations is 290 K (17 C). Find the open-circuit voltage of a

50 ohm resistor at this temperature in a bandwidth of 1 MHz



(b) Find the open-circuit voltage of the same resistor at the same temperature in a 1 Hz bandwidth

Solutions: Using Eq. (2-1)

(a) 23 6 64 4 1.38 10 290 50 1 10 0.895 10n nv k T R B Vrms = 0.895 μVrms

(b) 23 94 4 1.38 10 290 50 1 0.895 10n nv k T R B Vrms = 0.895 nVrms

Comment: The ratio of bandwidths in (a) and (b) is 106, or 1,000,000. The square root of 106 is 103, so it

should be no surprise that by decreasing the bandwidth by a factor of 1/106 from 1 MHz to 1 Hz, the

noise decreased by a factor of 1/103 from 0.9 V to 0.9 nV. These voltages are so small they are not

measurable without considerable amplification.

Example 2-2:

Determine the average noise power delivered by a 50 ohm resistor at a temperature of 50,000 K to an

ideal (noiseless) 50 ohm resistor in a 100 kHz bandwidth.

Solution:

Since the source and load are matched, we can use Eq. (2-4), or23 141.38 10 50,000 100,000 6.94 10avg nN k T B W = 0.0694 pW

Because the average thermal noise power is proportional to absolute temperature, it is common for

noise power to be described in terms of noise temperature even though the noise may originate from

something other than a hot resistance. For example, the noise could originate from the cosmic

microwave background (CMB), a local interference source or a solar radio burst but we can refer to it in

terms of its noise temperature.

In radio work a reference temperature T0 = 290 K is used (see

sidebar). Note the subscript for T is the number zero, or

naught, and not the letter o. As a point of reference, the

thermal noise power per hertz bandwidth (noise power

density) available from a resistor at 290 K is

23 210 1.38 10 290 1 4.0 10avg nN k T B W/Hz =

94.0 10 pW/Hz

It often is convenient to calculate power levels in terms of

deciBels with reference to 1 watt (dBW). For a temperature of 290 K the average power is

21, 10 0 1010 log 10 log 4.0 10 204avg dBW nN k T B dBW in a 1 Hz bandwidth.

For a 1 mW reference (dBm), add 30 dB to the previous calculation, or

The reference temperature 290 K is nearEarth’s average surface temperature (289 K),but that is only part of the reason for itschoice. 290 K was chosen to make the value ofkT easy to handle in computations [Friis44,45].Some people incorrectly claim it is based onroom temperature but 290 K is a cold room.

Between the 1940s and 1950s the “standard”reference temperature varied from 288.44 to300 K, depending on the author. The currentvalue was adopted by the Institute of RadioEngineers (IRE) in 1952.



, 204 30 174avg dBmN dBm in a 1 Hz bandwidth.

These noise powers are written –204 dBW/Hz and –174 dBm/Hz. However, the practice of mixing

logarithmic units (dBm) with linear units (Hz) is somewhat risky unless it is clearly understood that

multiplication by the bandwidth does not yield the total power. The powers must first be converted

back to linear units (W/Hz or mW/Hz) before the multiplication. The conversion is

/

1010dBm HzN

avgN mW/Hz (2-5)

For example, the quantity –174 dBm/Hz converted to linear units of mW/Hz is

/ 1741810 1010 10 4.0 10

dBm HzN

avgN

mW/Hz

The total power in a 1 MHz bandwidth for the same noise temperature is

18 18 6 124.0 10 4.0 10 1 10 4.0 10Total nN B mW in 1 MHz bandwidth

and in decibels is

1210 log 4.0 10 114TotalN dBm

The total noise power calculation in dB can be written in a more general way as

,10 log 10 logTotal avg n avg dBm nN P B N B dBm (2-6)

It may be concluded that for a system at a noise temperature of 290 K, the noise floor is –174 dBm (or –

204 dBW) in a 1 Hz bandwidth. Any noise-like emission below this power level, adjusted for bandwidth,

cannot be distinguished from ordinary thermal noise.

2-5. Combinations of noisy resistors

We will now consider a number of noisy resistors and their combined effects. The illustration shows four

noisy resistors, R1 through R4, connected in parallel with a hypothetical noiseless load resistor RL (figure

2-5). We will calculate the noise voltage across RL caused by the four noisy resistors. The concept can be

easily extended to any number of noisy resistors.



Fig. 2-5 ~ Combining noise sources. The noisy resistors (upper) arereplaced by their equivalent noise models (lower).

Since noise voltages are uncorrelated (that is, the voltage at any instant is independent of the voltage at

any other instant), we cannot simply add the individual voltages. Instead, we must calculate the noise

power associated with each source and then add the powers together. To make the calculations we

follow these steps:

1) Using Eq. (2-1) calculate the open-circuit noise voltage of each source based on its resistanceand temperature

2) Short out all the noise voltage sources but one3) Using ordinary circuit analysis, find the voltage across RL due to the one active source4) Calculate the power dissipated in RL due to the one active source5) Repeat steps 2) through 4) for each noise source in succession.6) Add all the noise powers together on a linear basis to find the total noise power dissipated in

RL

7) Calculate the total noise voltage by solving

2Total n

TotalL

vN

R (2-7)

for vTotal-n, or

Total n Total Lv N R (2-8)

Example 2-3:

Find the noise voltage across an ideal load resistor RL for the following resistor values and temperatures

based on 100 kHz bandwidth:

RL: 50 ohms; R1: 100 ohms at 300 K; R2: 1,000 ohms at 600 K; R3: 2,000 ohms at 1200 K; R4: 3,000

ohms at 2400 K

Solution:



Step 1: Referring to the circuit values, calculate the open circuit noise voltage for each resistor from Eq.

(2-1)

R1 → 23 3 71 4 4 1.38 10 300 100 100 10 4.07 10nv k T R B Vrms

R2 → 23 3 62 4 1.38 10 600 1000 100 10 1.82 10v Vrms

R3 → -23 3 -63 4 1.38 10 1200 2000 100 10 3.64 10v Vrms

R4 → 23 3 64 4 1.38 10 2400 3000 100 10 6.30 10v Vrms

Step 2: Short out v2 through v4 (figure 2-9). The only active noise voltage source is v1.

Fig. 2-9 – Example noise resistor network with v2 through v4 shorted out

Step 3: Solve for the voltage across RL. The circuit can be reduced to a simple voltage divider circuit in

which RL is in parallel with R2, R3 and R4 (figure 2-6). Using Kirchhoff’s voltage law, the voltage VL-1 across

RL due to v1 is (the symbol indicates a calculation involving parallel resistors)

7 711 1

1 2 3 4

1004.07 10 2.79 10

100 45.80( )L

L

Rv v

R R R R R

Vrms

Fig. 2-6 – Equivalent voltage divider circuit. VL-1 is the noise voltage across the load resistor due to R1

Step 4: Calculate the noise power dissipated in RL due to the one active noise voltage source. For the

first noise voltage source

2

72151

1

2.79 101.56 10

50L

LL

vN

R

W

Step 5: Repeat steps 2) through 4) for each noise source



5.2): v2 is shorted out

5.3): 6 622 2

2 1 3 4

1,0001.82 10 1.76 10

1000 32.43( )L

L

Rv v

R R R R R

Vrms

5.4):

262

1422

1.76 106.22 10

50L

LL

vN

R

W


5.3): 6 633 3

3 1 2 4

20003.64 10 3.58 10

2000 31.91( )L

L

Rv v

R R R R R

Vrms

5.4):

262

1333

3.58 102.57 10

50L

LL

vN

R

W


5.3): 6 644 4

4 1 2 3

30006.30 10 6.23 10

3000 31.75( )L

L

Rv v

R R R R R

Vrms

5.4):

262

1344

6.23 107.77 10

50L

LL

vN

R

W

Step 6: Add the noise powers to find the total noise power dissipated in RL,

15 14 13 13 121 2 3 4 1.56 10 6.22 10 2.57 10 7.77 10 1.10 10Total L L L LN N N N N W

Step 7: Calculate the noise voltage across RL,

12 64 1.10 10 50 7.41 10Total Total Lv N R

Vrms

Where m resistors are connected in series (figure 2-7), the total resistance RT is simply the sum of the

individual resistances, or

1 2 ...T mR R R R (2-9)

Fig. 2-7 – Series resistors



Again, the noise voltages are uncorrelated so we determine the mean-square noise voltages for the

series combination. The total mean-square noise voltage is

2 2 2 21 2 ...Total mv v v v (2-10)

If all resistors in series are the same temperature, that is, T1 = T2 = T3 . . . = Tm = T, it follows that

21 24 4 ... 4Total n n n mv k T B R k T B R k T B R (2-11)

or

21 24 ( ... ) 4Total n m n Tv k T B R R R k T B R (2-12)

As expected, the available noise power NTotal is

21 2

1 2 1 2

4 ...

4 ... 4 ...n mTotal

Total nm m

k T B R R RvN k T B

R R R R R R

(2-13)

If the resistors are at different temperatures, that is, T1 ≠ T2 ≠ T3 . . . ≠ Tm,

21 1 2 24 4 ... 4Total n n m n mv k T B R k T B R k T B R (2-14)

and the total available noise power is

21 1 2 2

1 2 1 2

4 4 ... 4

4 ... 4 ...Total n n m n m

Totalm m

v k T B R k T B R k T B RN

R R R R R R

(2-15)

or, equivalently,

1 1 2 2

1 1 2 21 2

......

...m m n

Total n m mm T

T R T R T R k BN k B T R T R T R

R R R R

(2-16)

We can now determine an equivalent temperature Tequiv for the series resistor combination, or

1 1 2 2

1 2

...

...m m

equivm

T R T R T RT

R R R

(2-17)

The foregoing expression effectively weights each temperature according to the corresponding resitance

with larger resistances having more weight. The individual contribution of each resistance to the total

equivalent noise temperature, expressed as a fraction α, is



1 1 2 2 ...equiv m mT T T T (2-18)

where

1 2 ...m

mm

R

R R R

(2-19)

Finally, if a matched noise source is connected to the series resistances, the total power available from

that source would be absorbed by the resistances and each resistance would absorb a fraction of the

available power according to

1 2 ...m m

mTotal m

N R

N R R R

(2-20)

Example 2-4:

For the series resistor combination (figure 2-8), determine the effective noise temperature, fractional

contribution α of each resistor, and total available noise power under the following conditions. All

resistances are in ohms and the bandwidth is 6 kHz:

a) T1 = T2 = T3 = T4 = 300 K

b) T1 = 300 K, T2 = 600 K, T3 = 1,200 K, T4 = 2,400 K

Fig. 2-8 – Example series resistor combination

Solution:

a) Since all resistors are the same temperature, T1 = T2 = T3 = T4 = 300 K, the equivalent noise

temperature T = 300 K. To find the fractional contribution α1 of R1 use Eq. (2-19)

1

11 2 3 4

1000.0164

100 1000 2000 3000

R

R R R R

Similarly, for R2, R3, and R4

2

21 2 3 4

10000.1639

100 1000 2000 3000

R

R R R R

3

31 2 3 4

20000.3279

100 1000 2000 3000

R

R R R R

4

41 2 3 4

30000.4918

100 1000 2000 3000

R

R R R R

Note that α1 + α2 + α3 + α4 = 1.0000. The total available power is found from Eq. (2-4)



23 3 171.38 10 300 6 10 2.48 10Total nN k T B W

b) For the case where each resistor is at a different temperature, the equivalent noise temperature is

determined from Eq. (2-17)

1 1 2 2

1 2

... 300 100 600 1000 1200 2000 2400 30001677

... 100 1000 2000 3000k m

equivm

T R T R T RT

R R R

K

The fractional contribution of each resistor is the same as found in a). The total available power is

23 3 161.38 10 1677 6 10 1.39 10Total eq nN k T B W

Another way to find the total equivalent noise temperature is by noting that

1 1 2 2 3 3 4Total eq n nN k T B k T T T T B

Substituting the fractional contributions found in a),

23 3 161.38 10 0.0164 100 0.1639 600 0.3279 1200 0.4918 2400 6 10 1.39 10TotalN W

as expected.

In Part III we derive the noise performance of attenuators and amplifiers.

2-6. References

[Friis-44] Friis, H., Noise Figure of Radio Receivers, pg 419, Proceedings of the IRE, July 1944[Friis-45] Friis, H., Discussion on Noise Figure of Radio Receivers, Proceedings of the IRE, February 1945[Layne] Layne, D., Receiver Sensitivity and Equivalent Noise Bandwidth, High Frequency Electronics,

June, 2014[Nyquist] Nyquist, H., Thermal Agitation of Electric Charge in Conductors, Physical Review, Vol. 32, July

1928[Taub] Taub, H. and Schilling, D., Principles of Communications Systems, 2nd Ed., McGraw-Hill Book

Company, 1986




0.1 (Updated TOC and references, 7 Jul 2014)0.2 (Minor corrections and added filter noise bandwidth calculations, 19 Jul 2014)

Word count:3269

Size: 3339264


Noise TutorialPart III ~ Attenuator and Amplifier Noise


Noise Tutorial III ~ Attenuator and Amplifier Noise





1-1 Introduction


1-3 Noise amplitude

1-4 References


2-1 Noise spectrum

2-2 Noise bandwidth

2-3 Noise temperature

2-4 Noise power

2-5 Combinations of noisy resistors

2-6 References


3-1 Attenuation effects on noise temperature 3-1

3-2 Amplifier noise 3-5

3-3 Cascaded amplifiers 3-8

3-4 References 3-11




4-3 References




5-3 References



6-2 References




3-1. Attenuation effects on noise temperature

All practical transmission lines (coaxial cables, waveguides, open wire) and their associated connectors

introduce loss between their input and output. Poor cable installation practices and improper use and

installation of connectors introduce additional losses. Problems such as these can be especially apparent

at VHF and above. A transmission line or attenuator that is matched at both its input and output can be

characterized as a 2-port network (figure 3-1).

Noisy AttenuatorTemperature TA

Loss LA

vs

R0

R0

TA

TA

vn

VAvA

Fig. 3-1 ~ 2-port circuit used to represent an attenuator. The input is from an external noise voltagesource vn and both input and output terminations are matched. The noise voltage produced by theattenuator is represented by VA. All components are at the same physical temperature TA

The attenuator has loss LA, which is defined as the linear ratio of the attenuator input power to its

output power (LA ≥ 1). It should be noted this definition is equivalent to a positive logarithmic power

ratio in dB. For example, a linear power ratio of 3.0 is equivalent to +4.81 dB. Loss in dB usually is spoken

as a positive value (for example, “The cable loss is 4.81 dB.”). One must be careful to use the correct sign

in calculations. It also is necessary to be careful to not confuse loss with amplifier gain in dB, which also

is spoken as a positive value. When cable or attenuator loss is given as a positive dB value, the loss as a

linear power ratio of input to output is

,

1010

A dBL

AL

(3-1)

where LA,dB is the loss in dB.

For this analysis, all components are at physical temperature TA. We will determine the effective noise

temperature Teff of the attenuator on the basis of thermal equilibrium and will take into account the

equivalent noise power of the voltage source as well as the noise power contributed by the attenuator.

The equivalent noise power from the source at the attenuator input is

s A nN k T B (3-2)

The noise power flowing from the attenuator to the load consists of two components, the noise power

of the source Ns reduced by the attenuator and the noise power NA contributed by the attenuator itself

(figure 3-2). Therefore,



1out s A

A

N N NL

(3-3)

Note that, because of the definition of the attenuator loss LA as the ratio of input to output power, we

must invert it to obtain the fraction of Ns that appears at the attenuator output.

Noiseless AttenuatorTemperature TA

Loss LA

vs

R0

R0

TA

TAvA

vn

VA-in

Fig. 3-2 ~ Attenuator with its internal noise voltage source VA referred to its input andredesignated VA-in. The noise voltage vn of the source is connected to the attenuator input.

For thermal equilibrium, the noise power flowing back to the attenuator from the load must equal Nout.

The total noise power of the load is

L A nN k T B (3-4)

Combining Eq. (3-3) and (3-4),

1A n s A

A

k T B N NL

(3-5)

Substituting Ns from Eq. (3-2)

1

A n A n AA

k T B k T B NL

(3-6)

Solving for NA

11 AA A n A n A n

A A

LN k T B k T B k T B

L L

(3-7)

Let Teff represent the effective noise power of the attenuator referred to its input. This is reduced by the

attenuator, so the attenuator noise power at its output is

1A eff n

A

N k T BL

(3-8)

Combining Eq. (3-7) and (3-8) gives



11 Aeff n A n

A A

Lk T B k T B

L L

(3-9)

Cancelling terms and rearranging,

1eff A AT T L (3-10)

Example 3-1:

Determine the effective noise temperature of an attenuator with losses a) 1.0, b) 1.25 and c) 10.0 (all

linear power ratios). The attenuator physical temperature is 300 K.

Solution:

a) 1 300 1.0 1 0eff A AT T L K

b) 300 1.25 1 75effT K

c) 300 10 1 2700effT K

Comment: The transmission line in a) is lossless and, as expected, its effective noise temperature is 0 K;

that is, it adds no noise to the system. In b) the loss is equivalent to about 1 dB and the noise power

added by the transmission line has a temperature of 75 K. In c) the loss is equivalent to 10 dB and the

added noise power has a temperature of 2700 K, a factor of 36 times higher than b). With large

attenuation values, the effective temperature can be quite high. However, the resulting noise power is

reduced by the attenuator and the noise temperature at its output never rises above the attenuator

physical temperature. However, it is the effective noise temperature that determines the attenuator

noise factor as described later.

All devices, including filters, attenuators, amplifiers, and transmission lines, have an effective noise

temperature and can be evaluated as previously described. For example, consider an antenna that

receives emissions with total noise temperature TAnt and is connected to a transmission line with

physical temperature TL and loss LL. Using the concepts discussed above for an attenuator, the

transmission line effective noise temperature Teff

1eff L LT T L (3-11)

The total noise power from the antenna and transmission line at the output of the transmission line is

1out Ant n eff n

L

N k T B k T BL

(3-12)

Substituting the effective noise temperature Teff from Eq. (3-11) and simplifying



11out Ant n L L n

L

N k T B L k T BL

(3-13)

In terms of total noise temperature TTotal the total noise power is

11 LTotal n Ant n L n

L L

Lk T B k T B k T B

L L

(3-14)

By cancelling terms, the total effective noise temperature of the antenna including the transmission line

contribution is

11 LTotal Ant L

L L

LT T T

L L

(3-15)

The noise temperature of the received emissions is reduced by the factor1

LLand the transmission line

itself increases the noise temperature as much as TL depending on its loss.

3-2. Amplifier noise

Noise is generated within the active and passive components and power supplies of any practical

amplifier. Therefore, the amplifier itself adds noise to the noise power applied to the amplifier input

from an external source. The input noise is amplified and the noise at the output of each stage of a

multi-stage amplifier is amplified again by the stages following it. Any noise in the early stages of a

multi-stage amplifier experiences considerably more amplification than later stages. This leads to the

interesting conclusion, discussed in more detail later, that the early stages of an amplifying system

contribute the most to a receiver’s noise performance.

The noise performance of an amplifier can be analyzed by connecting its input to an external noise

source with noise power NS (figure 3-3). The source noise power can be represented by a noise

temperature TS. TS is not necessarily a physical temperature but accounts for the noise power available

from the source. The source can be an antenna, signal generator, noise generator or a previous amplifier

stage. The amplifier itself contributes its internal noise power Namp to its output. The external noise

source is amplified according to the amplifier gain G; therefore, the total output noise power is

out S AmpN G N N (3-16)

It is convenient to refer the amplifier noise to its input. For this situation, both the noise source NS and

amplifier noise now designated NAmp-in are amplified, and the total output noise power is

out S Amp inN G N N (3-17)



Fig. 3-3 ~ Amplifier input noise. Upper: An external noise source is connected to the amplifier input, where it isamplified and combined with the noise generated within the amplifier. Lower: The internal amplifier noise can bereferred to the amplifier input, where it is combined with the external noise and both are then amplified by anoiseless amplifier.

Assume the amplifier has an input resistance RAmp that is matched to the noise source resistance RS. The

many individual noise sources inside the amplifier are represented by an equivalent noise temperature

TAmp, which is a measure of the noise added to the input by the amplifier. The amplifier has power gain

G. The output of the amplifier is connected to an ideal (noiseless) load resistance RL (figure 3-4).

RS

RL

RAmp

TS TAmp

Input Output

G

Fig. 3-4 ~ Amplifier with gain G connected to a noiseless load resistor RL

First, we consider only the noise available from the two sources, RS and RAmp. We can replace the two

resistors with their noise models (figure 3-5). To determine the noise contribution from these resistors,

we have to make two calculations, one associated with RS and the other associated with RAmp. We

determine the noise power contributed by the first resistor by shorting out the noise voltage source

associated with the second resistor, and then repeat the process for the second resistor. The analysis is

simplified by the fact that RS and RAmp are equal (matched), ensuring that the noise voltage divides

equally across the two resistors.



Fig. 3-5 ~ Noise model of the amplifier input circuit

The noise power available from RS with vAmp shorted is

S S nN k T B (3-18)

This power is dissipated in both RS and RAmp because both have the same voltage and same resistance.

The noise power available from RAmp with vS shorted is

Amp in Amp in nN k T B (3-19)

Again, this power is dissipated in both RS and RAmp. The total power dissipated by RAmp and RS is

Total in S Amp in S n Amp in nN N N k T B k T B (3-20)

Rearranging terms

Total in n S Amp inN k B T T (3-21)

The total effective temperature at the amplifier input is found from

n Total n S Amp ink B T k B T T (3-22)

Cancelling terms

Total S Amp inT T T (3-23)

Eq. (3-23) provides an important result: In a matched system the total amplifier noise temperature

depends only on the sum of the source and amplifier noise temperatures. At the amplifier output, the

noise power is increased by the amplifier gain, G, or

Total Out n S Amp inN G k B T T (3-24)

and it follows that



Total Out S Amp inT G T T (3-25)

The amplifier gain shifts the reference point for the input and amplifier noise to the amplifier output.

We can plot the relationship between the output noise power and the input noise temperature to show

how the amplifier adds noise to the system (figure 3-6). As the input noise temperature is reduced to

zero, the output noise decreases but does not reach zero. The output noise power at the vertical axis

intercept represents the noise contribution of the amplifier, NAmp. A low noise amplifier will intercept

the vertical axis at a lower point than an ordinary or noisier amplifier.O

utp

ut

No

ise

Po

we

r

Input Noise Temperature

TCold THot

NCold

NHot

N Amp

Fig. 3-6 ~ Output noise power as a function of input noise temperature. Two arbitrary input temperature points areshown, TCold and THot with corresponding output noise powers Nold and NHot. The straight plot line has a

slope nG k B and intersects the vertical axis where the input noise temperature TS = 0 K. Even when there is no

input noise from an external source, the amplifier adds noise and the output noise power is N Amp.

3-3. Cascaded amplifiers

Each amplifier in a cascade adds noise (figure 3-7). It was found previously that the total noise power at

the output of an amplifier is determined by the noise power at its input plus the amplifier noise

referenced to its input, both increased by the amplifier gain. Therefore, at the output of the first

amplifier and input of the second amplifier,

1 1 1Out n SN G k B T T (3-26)

where

T1 noise temperature of amplifier 1 referenced to input

G1 gain of amplifier 1



Fig. 3-7 ~ Cascade of m amplifiers, each with respective noise temperatures T1, T2, . . . Tm and gains G1, G2, . . . Gm

Assuming matched conditions, the output of the second amplifier consists of the noise power at its

input from the first amplifier plus the noise power added by the second amplifier, or

2 2 2 1 1Out n n SN G k B T G k B T T (3-27)

Rearranging

2 2 2 1 1Out n SN k B G T G T T

2 2 2 1 2 1Out n SN k B G T G G T T (3-28)

This process can be carried through as many stages of amplification as necessary; for the mth stage

3 3 2 3 2 1 2 3 1...Out m n m m m m m SN k B G T G G T G G G T G G G G T T (3-29)

We can obtain the cascade’s equivalent input noise power by dividing the output power by the total

cascade gain, or

1 2 3

3 3 2 3 2 1 2 3 1

1 2 3

...

Out min

m

m m m m m S

in nm

NN

G G G G

G T G G T G G G T G G G G T TN k B

G G G G

(3-30)

Simplifying and rearranging

321

1 1 2 1 2 3 1

min n S

m

TT TN k B T T

G G G G G G G

(3-31)

If we let Tequiv be the equivalent input noise temperature of the cascaded amplifiers as a whole and note

that in n equivN k B T , then



321

1 1 2 1 2 3 1

mequiv S

m

TT TT T T

G G G G G G G

(3-32)

If we are concerned only with the total equivalent noise power of the cascaded amplifiers, not including

the input noise power due to the source, TS can be dropped from the equation, giving

321

1 1 2 1 2 3 1

mCascade

m

TT TT T

G G G G G G G

(3-33)

Examination of this expression leads to an interesting conclusion if the individual amplifier gains are

high: The first stage noise temperature (T1) can be an important controlling factor in receiver system

noise performance and subsequent stages have considerably less importance. The second stage noise

temperature is reduced by the factor 11 G and the third stage by the factor 1 21 ( )G G . For example, if

the power gain of each stage is 50 (17 dB), then the noise temperature of the second stage is reduced by

the factor 0.02 and the third stage by the factor 0.0004.

Example 3-3:

Find the equivalent noise temperature of cascaded amplifiers shown (figure 3-8) for the conditions

given.

TS

G1

T1

NOut-1 G2

T2

NOut-2

(a) Two amplifiers in cascade: G1 = 60 dB, G2 = 20 dB, T1 = 300 K, and T2 = 20,000 K

TS

NOut-2 G3

T3

NOut-3G1

T1

NOut-1 G2

T2

(b) Three amplifiers in cascade: G1 = 30 dB, G2 = 40 dB, G3 = 50 dB and T1 = 50,000 K, T2 = 20,000 K and T3 = 30,000K

Fig. 3-8 ~ Equivalent noise temperature of cascaded amplifiers



Solutions:

(a) First, convert the gains to linear ratios,

1 60610 10

1 10 10 10

G dB

G and 2 20

210 102 10 10 10

G dB

G

Using Eq. (2-59)

21 6

1

20,000300 300.02

10Cascade

TT T

G K

(b) Converting as above,

30310

1 10 10G ,40

4102 10 10G and

50510

3 10 10G

and

321 3 3 4

1 1 2

20,000 30,00050,000 50,020.003

10 10 10Cascade

TTT T

G G G

K

Comment: In both examples, the equivalent temperature is largely determined by the noise

temperature of the first stage indicating that the noise performances of subsequent stages have little

influence on overall amplifier noise performance.

3-4. References

There are no references in this part.




0.1 (Updated TOC and references, 7 Jul 2014)

Word count:2360

Size: 3216384


Noise TutorialPart IV ~ Noise Factor


Noise Tutorial IV ~ Noise Factor





1-1 Introduction


1-3 Noise amplitude

1-4 References


2-1 Noise spectrum

2-2 Noise bandwidth


2-4 Noise power


2-6 References



3-2 Amplifier noise


3-4 References


4-1 Noise factor and noise figure 4-1

4-2 Noise factor of cascaded devices 4-7

4-3 References 4-11




5-3 References



6-2 References




4-1. Noise factor and noise figure

Noise factor and noise figure indicates the noisiness of a radio frequency device by comparing it to a

reference noise source. The IEEE Standard Dictionary of Electrical and Electronics Terms [IEEE100] does

not distinguish between the terms noise factor and noise figure and defines them in the same way;

however, noise factor commonly is expressed as a linear power ratio and noise figure as a logarithmic

power ratio (dB). This discrepancy causes confusion and will be discussed in more detail later. In this

paper only noise factor is used and it always is made clear if it is a linear or logarithmic power ratio.

One definition of noise factor NF relates the input signal-to-noise ratio (SNR) of a device to its output

SNR, or

in in

out out

S NNF

S N (4-1)

where

Sin signal power at device input

Sout signal power at device output

Nin total noise power available at device input

Nout total noise power available at device output (including noise from the device itself)

All practical devices degrade the SNR, so the output SNR always will be lower (worse) than the input SNR

(figure 4-1).

Inpu

tP

ow

er

Level(

mW

)

Outp

utP

ow

er

Leve

l(m

W)

Fig. 4-1 ~ Amplifier signal and noise levels. Both the input signal and external noise are multiplied by the amplifierpower gain; however, the amplifier adds even more noise at the output, thus lowering the output SNR.

Eq. (4-1) can be rearranged in terms of the device gain



in in

in in Dev

in in

in in Dev in

S NNF

G S G N N

S NNF

G S G N N

(4-2)

where

NDev additional noise due to the device itself, referred to its output ( Dev InG N )

NDev-in additional noise due to the device itself, referred to its input

Eq. (4-2) provides equivalent expressions for the device noise. One uses a noisy device with its internal

noise referred to the output (NDev), and the other uses the combination of a noiseless device with its

internal noise referred to its input (NDev-in), where NDev = GNDev-in. Eq. (4-2) reduces to

1in Dev in Dev in

in in

N N NNF

N N

(4-3)

Alternately,

1in Dev Dev

in in

G N N NNF

G N G N

(4-4)

Eq. (4-3) can be rearranged

( 1)Dev in inN NF N (4-5)

Similarly, for Eq. (4-4)

( 1)Dev inN G NF N (4-6)

Eq. (4-3) shows that the noise factor expresses the noisiness of a device relative to an input noise source

Nin. Therefore, with a noise reference at the device input, the noise factor will represent a measure of

how much noisier the device is than the reference. Using a standard noise reference temperature allows

direct comparison of different devices. As pointed out in Part I, the reference temperature is defined as

T0 = 290 K. For this temperature, the reference noise power per unit bandwidth is

23 210 0 1.38 10 290 4.00 10N k T W/Hz

We can define an effective noise temperature Teff such that the noise power produced by a device

referred to its input is

Dev in eff nN k T B (4-7)



The input noise power to the device from an external source at the reference noise temperature T0 is

0in nN k T B (4-8)

Therefore, from Eq. (4-5),

01eff n nk T B NF k T B (4-9)

Cancelling terms gives

01effT NF T (4-10)

Solving for the noise factor gives

0

0 0

1eff effT T T

NFT T

(4-11)

We can use these results to also find the noise factor of a device that introduces loss, such as an

attenuator or lossy transmission line. First, we substitute the standard noise reference temperature T0

for TA in Eq. (2-17) and then combine it with Eq. (4-10), or

0 01 1eff A AT NF T L T (4-12)

where NFA is the attenuator noise factor.

Simplifying,

1 1A ANF L (4-13)

and solving for the noise factor of the attenuator,

A ANF L (4-14)

where LA is defined as the ratio input power/output power and LA ≥ 1. This expression directly links the

attenuator or transmission line loss to its noise factor. Examples are shown later.

It is important to note that noise factor is not an absolute measure of noise and that NF ≥ 1. For an ideal

device (a device that does not contribute any noise) NF = 1.

As previously mentioned, the definitions of noise factor and noise figure are the same. Both can be

expressed equivalently as linear or logarithmic power ratios. As a logarithmic ratio in dB



10 logdBNF NF (4-15)

However, some important literature (for example, the often cited and excellent application note [Agilent

57-1]) makes a distinction and expresses noise factor as a linear ratio and noise figure as a logarithmic

ratio, or

Noise factor = NF

Noise figure = 10 logdBNF NF

To add even more confusion, some literature uses the letter F and shows noise factor in lower-case (f)

and noise figure in upper-case (F) (for example, see [ITU-R P-372.9]). Unfortunately, lower-case (f) is

used to represent frequency in electrical engineering literature. In this article we use NF to represent

both noise factor and noise figure and always make it clear when a logarithmic ratio in dB is used.

Example 4-1:

Express the following noise factors in dB: (a) 1.0; (b) 1.15, (c) 3.0; (d) 10.0

Solution: Using Eq. (4-15)

(a) 10 log 1.0 0dBNF dB

(b) 10 log 1.15 0.607dBNF dB

(c) 10 log 3.0 4.771dBNF dB

(d) 10 log 10.0 10dBNF dB

Comment: In (a), the noise factor of 1.0 corresponds to an ideal noiseless device.

Example 4-2:

Find the noise factor and effective noise temperature of an amplifier for the conditions given (figure 4-

2).



Nin = 1 mW

Nout = GNin + NDev

Noise-less

Nout = G(Nin + NDev-in)

G = 100

NDev = 100 mW

G = 100

Nout = 200 mW

Nout = 200 mW

Nin = 1 mW

NDev-in = 1 mW

Fig. 4-2 ~ Noise factor of an amplifier. Two equivalent representations are shown, one with the internal noisereferred to the output (upper) and the other with the internal noise referred to the input

Solution:

For the upper drawing use Eq. (4-4), or

100 1 1002

100 1in Dev

in

G N NNF

G N

For the lower drawing use Eq. (4-3), or

1 12

1in Dev in

in

N NNF

N

In dB, the noise factor of the amplifier in both situations is 10 log 10 log 2 3dBNF NF dB. To find

the amplifier effective temperature use Eq. (4-10), 01 1 290 (2 1) 290 290effT NF T NF K.

Example 4-3:

Find the noise factor and effective noise temperature of the attenuator for the conditions given (figure

4-3).

Nin = 1 mW

No = LA Nin + NDev

Nout = LA (Nin + NDev-in)

LA = 5.0

NDev = 0.04 mW

LA = 5.0

No = 0.24 mW

Nout = 0.24 mW

Nin = 1 mW

NDev-in = 0.2 mW

Noiseless



Fig. 4-3 – Noise factor of an attenuator. Two representations are shown as for the amplifiers in the previousexample.

Solution: For either the upper or lower drawing

5.0ANF L

In dB, the noise factor of the attenuator from Eq. (4-14) is

10 log 10 log 5 7dBNF NF dB

To find the effective temperature use Eq. (4-10)

1 290 (5 1) 290 1160effT NF K

4-2. Noise factor of cascaded amplifiers

In Part III we solved for the equivalent input noise temperature of an amplifier cascade. In this section

we will develop the equivalent (or composite) noise factor for an amplifier cascade. Consider two

amplifiers with gains G1 and G2 and noise factors NF1 and NF2 (figure 4-4).

Nin

G1

NF1

NOut-1G2

NF2

NOut-2

Nout

Fig. 4-4 ~ Two amplifiers in cascade. A noise source with noise power Nin is connected to the input of the first

stage.

The output noise power of the cascade due only to the external source at the input is

1 2out ext inN G G N (4-16)

From Eq. (4-6) , the noise power at the output of the first stage due to its internal noise is

1 1 1 1Out inN G NF N (4-17)

The corresponding noise at the output of the second stage due to the first stage internal noise is

2 1 2 1 1 2 1 1Out Out inN G N G G NF N (4-18)

and the output noise of the second stage due to its own internal noise is



2 2 2 1Out inN G NF N (4-19)

The total noise at the output of the cascade is the sum of Eq. (4-16), (4-18) and (4-19), or

2 1 2

1 2 2 2 1 2 11 1

Out out ext Out Out

Out in in in

N N N N

N G G N G NF N G G NF N

(4-20)

Substituting Dev in DevG N N into Eq. (4-4) and rearranging,

In Dev In

In

G N NNF

G N

(4-21)

The cascade can be considered a single device, and all internally generated noise referred to its input

is Cascade in Dev inN N . Also, 1 2CascadeG G G G . Therefore,

Cascade in Cascade inCascade

Cascade in

G N NNF

G N

(4-22)

Since the external noise and internal noise referred to the input are increased by the cascade gain,

Out Cascade In Cascade InN G N N (4-23)

then Eq. (4-22) reduces to

1 2

Out OutCascade

Cascade in in

N NNF

G N G G N

(4-24)

Dividing both sides of Eq. (4-20) by 1 2 inG G N and combining with Eq. (4-24)

2 21 1

1 1

( 1) ( 1)1 ( 1)Cascade

NF NFNF NF NF

G G

(4-25)

This concept can be extended to any number of stages. For m stages

321

1 1 2 1 2 1

( 1)( 1) ( 1)mCascade

m

NFNF NFNF NF

G G G G G G

(4-26)

It is seen that the first stage has the most effect on the cascade noise factor. This result should not be

surprising given the previous noise temperature analysis of an amplifier cascade. As a cross-check, we

can derive the equivalent noise temperature of the cascade from Eq. (4-26) by combining it with Eq. (4-

11) and letting Tm = Teff, or



0

1 mm

TNF

T and

0

1 CascadeCascade

TNF

T (4-27)

Therefore,

32

0 0 01

0 0 1 1 2 1 2 1

1 1 1 1 1 1

1 1

m

CascadeCascade

m

TT T

T T TT TNF

T T G G G G G G

(4-28)

Cancelling terms

321

1 1 2 1 2 1

mCascade

m

TT TT T

G G G G G G

(4-29)

which is identical to Eq. (2-40). As mentioned in Part II, we note that that1

AA

GL

(where GA and LA are

both linear power ratios) so these concepts also apply to attenuation as shown in the following

examples.

Example 4-4:

Find the noise factor of a cascade consisting of a lossy transmission line and an amplifier for the

conditions shown (figure 4-5).

TS

G2

NF2

GCascade

FCascadeG1

NF1

Parameter (dB) Transmission Line Amplifier

Gain or Loss G1 = 10 dB loss G2 = 20 dB gain

NF NF1 = 10 dB NF2 = 6 dB

Fig. 4-5 – Cascade consisting of an attenuator followed by an amplifier

Solution:

First convert the gains and noise factors to linear ratios

Parameter (ratio) Transmission Line Amplifier

Gain or Loss G1 = 1/L1 = 0.1 G2 = 100



NF NF1 = 10 NF2 = 3.981

From equation (4-23),

21

1

( 1) (3.981 1)10.0 39.81

0.1Cascade

NFNF NF

G

, 10 log 39.81 16Cascade dBNF dB

Comment: In this example, the composite noise factor is simply the sum of the transmission line loss and

the noise factor of the amplifier, both in dB. Note that the noise factor does not depend on the amplifier

gain. The composite noise factor could be improved dB for dB by using lower loss coaxial cable or by

shortening it (moving the amplifier closer to the source). This explains why low noise amplifiers are

located close to antennas (the source) and low loss connections and cable always are used in front of

the low noise amplifier.

Example 4-5:

Find the noise factor of the cascade in the previous example except reverse the order of connection

(figure 4-6).

TS

G1

NF1

GCascade

FCascade

G2

NF2

Parameter (dB) Amplifier TransmissionLine

Gain or Loss G1 = 20 dB gain G2 = 10 dB loss

NF NF1 = 6 dB NF2 = 10 dB

Fig. 4-6 – Cascade consisting of an amplifier followed by an attenuator

Solution:

First convert the gains and noise factors to linear ratios

Parameter (ratio) Amplifier Transmission Line

Gain or Loss G1 = 100 G2 = 1/L2 = 0.1

NF NF1 = 3.981 NF2 = 10

From equation (4-25),



21

1

( 1) (10 1)3.981 4.071

100Cascade

NFNF NF

G

, 10 log 4.071 6.1Cascade dBNF dB

Comment: The composite noise factor is dominated by the noise factor of the amplifier and the

transmission line has only a small effect. If the amplifier had higher gain, say 30 dB or more, the

transmission line noise factor would have negligible effect and the composite noise factor would be the

same as the amplifier noise factor.

4-3. References

[Agilent 57-1] Fundamentals of RF and Microwave Noise Figure Measurement, Application Note 57-1, Document No. 5952-8255E, Agilent Technologies, Inc., 2010

[IEEE100] IEEE Std 100-1992, IEEE Standard Dictionary of Electrical and Electronics Terms,Institute of Electrical and Electronics Engineers, 1992

[ITU-R P-372.9] Recommendation ITU-R P-372.9, Radio Noise, International TelecommunicationsUnion, Radio Communications Sector, 2007



Document information

Author: Whitham D. ReeveCopyright: © 2014 W. ReeveRevision: 0.0 (Adapted from expanded original work, 19 Jun 2014)


Word count:1952

Size: 3024896


Noise TutorialPart V ~ Noise Factor Measurements


Noise Tutorial V ~ Noise Factor Measurements





1-1 Introduction


1-3 Noise amplitude

1-4 References


2-1 Noise spectrum

2-2 Noise bandwidth


2-4 Noise power


2-6 References



3-2 Amplifier noise


3-4 References




4-3 References


5-1 General considerations 5-1

5-2 Noise factor measurements with the Y-factor method 5-6

5-3 References 5-8



6-2 References



Part V ~ Noise Factor Measurements

5-1. General considerations

Noise factor is an important measurement for amplifiers used in low noise applications such as radio

telescopes and radar and other radio receivers designed to detect very low signal levels.

Noise factor is determined from noise power measurements. Noise power measurements may be

obtained from a purpose-built noise figure meter (figure 5-1), a spectrum analyzer or even a modern

vector network analyzer. Some receiver systems, for example, the Callisto solar radio spectrometer, can

be used to measure the noise factor of external amplifiers. This part emphasizes using a spectrum

analyzer.

C han 1 C han 2

C han 3 C han 4

Measu re Fo rm atSc ale /

R ef

D ispl ay Av g C al

M ar ker Mar kerSe ar ch

MarkerFu nct ion

St ar t St op Po we r

C ent er Sp an Sw ee p

Sy ste m Loc al Pr es et

Vi deoSa ve/R ecal l Se q

R etu rn

R espon se

Ac ti ve Ch anne l

Po rt 2Po rt 1

En try

0 . - H z

1 2 3 kH z

4 5 6 MHz

7 8 9 G Hz

En tryO ff

<-

/\ \/

St i mu lus I nst ru me nt S ta te R C han nel

Li ne O n /O ff

D iskEj ect

Noise FigureMeter

NoiseSource

Device Under Test

Noise SourcePower

RF

Fig. 5-1 ~ Noise figure meter with noise source and device under test

A calibrated noise source normally is used in noise power measurements. Commercial noise sources

usually provide a flat noise power (or noise temperature) output over the bandwidth being measured.

For example, if measurements are made on a wideband amplifier with a 500 MHz bandwidth in the

frequency range 0.5 to 1.5 GHz, the noise source must cover this range. On the other hand, if a

narrowband amplifier with 15 kHz bandwidth at 20 MHz is to be measured, the noise source only needs

to cover 20 MHz ± 7.5 kHz. Most commercial noise sources have bandwidths above several GHz.

A noise source has two operational states, cold and hot. The cold state is an unpowered (off) state and

the output is 0 nk T B thermal noise. The cold state noise power per Hz bandwidth at reference

temperature T0 is determined from the familiar calculation

23 210 0 1.38 10 290 1 4.002 10Cold nP P k T B W/Hz

In terms of noise power density expressed as dB with respect to 1 W

210 10 log 4.002 10 203.98P dBW/Hz, rounded –204 dBW/Hz



and with respect to the more common 1 mW

0 203.98 30 173.98P dBm/Hz, rounded –174 dBm/Hz

The noise source hot state is the powered (on) state and it provides a known amount of noise in excess

of the cold state noise. Common noise sources use a powering voltage of 28 Vdc (figure 5-2). The excess

noise is expressed as an Excess Noise Ratio, or ENR, and is related to the noise power or noise

temperature above the cold state noise by

0

Hot ColdT TENR

T

(5-1)

where

THot noise temperature when the noise source is in the hot state (powered, on)

TCold noise temperature when the noise source is in the cold state (unpowered, off)

NoiseSource

Power Supply(Typically 28

Vdc)

Cold

Hot

NoiseSource

Power Supply(Typically 28

Vdc)

Cold

Hot

Output

Output

Fig. 5-2 ~ Noise source switching between cold (off) and hot (on) states

ENR normally is given as a logarithmic ratio in dB, or

0

10 log Hot ColddB

T TENR

T

(5-2)

For ordinary measurements TCold = T0 but if the noise source is not at T0, then Eq. (5-1) or (5-2) accounts

for the difference. An undefined situation occurs when THot = TCold in which case

10 log 0dBENR dB; therefore, in all practical measurements, THot > TCold. If 2Hot ColdT T , then

0

210 log 10 log 10 log 1 0Hot Cold Cold Cold

dBCold

T T T TENR

T T

dB

Eq. (5-2) can be rewritten for the most common situation where TCold = T0, or

0

0 0

10 log 10 log 1Hot HotdB

T T TENR

T T

(5-3)



It is seen that ENR is not simply the noise power above the quantity 0 nk T B or the noise temperature

above T0. Even when the noise source is off, it has a noise temperature T0. The hot (on) state noise

temperature may be determined in terms of the ENRdB by solving Eq. (5-3) for THot, or

10 100 0 010 10 1

dB dBENR ENR

HotT T T T

(5-4)

The most common excess noise ratios for commercial noise sources are 5, 6 and 15 dB but much higher

ENRs are available. For example, the Renz RQ6 noise source is especially powerful with an ENR of 55 dB

up to 3 GHz. Noise sources with 5 dB and 15 dB ENR have hot temperatures of

For ENR = 15 dB, 1.5100 010 290 10 290 9460.6

dBENR

HotT T T

K

For ENR = 5 dB, 0.5290 10 290 1207.1HotT K

The hot powers of these noise sources can be calculated by noting that the hot/cold powers are

proportional to the hot/cold temperatures. Therefore,

0

0 0

1Hot HotP P PENR

P P

(5-5)

Solving for PHot gives

0 1HotP P ENR (5-6)

Equivalently,

100 10 1

dBENR

HotP P

(5-7)

For ENR = 15 dB,

15

21 1910 100 10 1 4.002 10 10 1 1.306 10

dBENR

HotP P

W/Hz

The hot power in dBm/Hz is



1910 log 1.306 10 30 158.84HotP dBm/Hz

Similarly, for ENR = 5 dB,

5

21 20104.002 10 10 1 1.666 10HotP

W/Hz

and

2010 log 1.666 10 30 167.78HotP dBm/Hz

As mentioned in Part I, dBW/Hz and dBm/Hz are used for convenience in discussion and are not real

units. One simply cannot multiply the noise powers in dBW/Hz or dBm/Hz by the bandwidth to

determine the total noise power in a wider bandwidth. Instead, the powers must be converted to linear

units (W/Hz or mW/Hz) before the multiplication and then re-converted back to decibel values.

Alternately, the bandwidth can be converted to dB and then added to the noise power in dBW/Hz or

dBm/Hz, as in

/ 10 logHot dBm Hot dBm Hz nP P B (5-8)

Example 5-1: Find the noise power in milliwatts and dBm available from a 15 dB ENR noise source in the

frequency range 250 to 750 MHz. The noise source output is flat over the frequency range 10 MHz to 10

GHz.

Solution: The noise power from this noise source was previously calculated as –158.84 dBm/Hz. The

bandwidth is Bn = 750 – 250 = 500 MHz. Using the first method above, this value is converted to linear

units, multiplied by the bandwidth in Hz and then converted back to dBm, or

, / 158.84

10 6 16 6 810,500 10 10 500 10 1.306 10 500 10 6.53 10

Hot dBm HzP

Hot MHz nP B

mW

In dBm, 8,500 , 10 log 6.53 10 71.85Hot MHz dBmP dBm

Alternately, the bandwidth can be converted to dB and then added to the noise source power, or

6,500 ,dBm / 10 log 158.84 10log 500 10 158.84 86.99 71.85Hot MHz Hot dBm Hz nP P B dBm

In milliwatts,

,500 ,, / 71.85

10 810,500 10 10 6.53 10

Hot MHz dBm HzP

Hot MHzP

mW



If necessary, the ENR of a noise source may be reduced with an external attenuator. The calculation is

the same as shown previously for an attenuator or transmission line, or

,A (1 )Hot Hot A A AT T L L T (5-9)

where

THot,A Hot state temperature of the noise source with the attenuator on its output

LA Attenuator loss as a linear ratio of output power to input power

TA Temperature of the attenuator or transmission line, usually T0

The attenuated ENR is then calculated as before,

, , 0 ,,

0 0 0

10 log 10 log 10 log 1Hot A Cold Hot A Hot AdB A

T T T T TENR

T T T

Using a 15 dB ENR noise source with a 10 dB attenuator (0.1 linear power ratio), the new hot state

temperature is

,A (1 ) 9460.6 0.1 (1 0.1) 290 1207.1Hot Hot A A AT T L L T K

and the new ENR is

,,

0

1207.1 29010 log 10 log 5

290

Hot A ColddB A

T TENR

T

dB

In this example, there would have been no significant error in subtracting the attenuator value from the

ENR (both in dB). Simple subtraction (in dB) is accurate for most practical situations involving typical

noise sources and attenuator values. It should be noted that an attenuator on the output of a noise

source can reduce impedance mismatch error, but error in the attenuation itself directly affects the ENR

used in the noise factor calculations. For example, a

+0.5 dB error in the attenuator value will cause a –0.5

dB error in the ENR (a 10 dB attenuator actually is 10.5

dB and a 5.0 dB ENR noise source actually will be 4.5

dB). It is for this reason that attenuators need to be

accurately measured or precision attenuators be used

with noise sources. There are many sources of error

and uncertainty (see sidebar).

A very high value attenuator connected between a

noise source and device simply provides a noise source

with equal hot and cold temperatures and an

undefined ENR as previously discussed. For example, if

Measurement uncertainty and mismatch error.All measurements are uncertain to some extent,and there are many subtle details that areimportant in accurate noise measurements.Uncertainties are especially important inmeasurements of low noise factors. Forexample, measurement of 0.5 dB noise factorcan easily have more than 0.5 dB uncertaintywhen taking into account connectors, cablesand equipment calibration. Also, an impedancemismatch between the noise source and devicecauses some noise power to be reflected backand unavailable for measurement.Measurement uncertainties and mismatcherrors are dealt with in [Dunsmore].



a 60 dB attenuator at 290 K is applied to a noise source with ENR = 15 dB, calculation to 5 decimal places

gives

,A (1 ) 9460.60521 0.000001 (1 0.000001) 290 290.00917Hot Hot A A AT T L L T K

and

,A (1 ) 290 0.000001 (1 0.000001) 290 290.00000Cold Cold A A AT T L L T K

5-2. Noise factor measurements with Y-factor method

One of several methods used to measure noise factor is called the Y-factor method. It is described in

detail in [Agilent 57-2] and more briefly below. With this method, a pair of hot/cold measurements is

taken and noise factor is then calculated from one of the following equations

10

1

10 log1

1010 log 10 log( 1)

1

dB

dB

ENR

dB dB

ENRNF

Y

ENRNF

Y

NF ENR YY

(5-10)

where

Hot

Cold

PY

P (5-11)

and

PHot noise power measured at the output of the device for the hot state, in suitable linear units

PCold noise power measured at the output of the device for the cold state, in same units as PHot

The noise powers may be measured many ways but a spectrum analyzer is described in detail in the next

section. If the hot and cold noise powers are read from a spectrum analyzer in dBm,

, ,dB Hot dBm Cold dBmY P P (5-12)

and



1010

10

1010 log 10 log 10 1

10 1

dB

dB

dB

ENRY

dB dBYNF ENR

(5-13)

Example 5-2: A noise source with ENRdB = 5.32 dB is used to measure an amplifier with the following

results: PHot = -118.0 dBm and PCold = -121.9 dBm. Find the noise factor.

Solution:

From Eq. (5-11), YdB = –118.0 dBm – (–121.9 dBm) = 3.9 dB

and from Eq. (5-12),3.9

10 1010 log 10 1 5.32 10 log 10 1 3.7dBY

dB dBNF ENR

dB

Where the physical temperature of the noise source is 'ColdT , Eq. (5-10) is modified

'

0

1

1

ColdTENR Y

TNF

Y

(5-14)

Note that Eq. (5-13) reduces to (5-10) when '0ColdT T .

The Y-factor method depends on the linearity of the devices in the measurement chain, so the noise

source ENR should be as low as possible to avoid overdriving them. However, it should not be so low

that the difference between the on and off noise powers is too small to be measured accurately.

There are no simple rules for matching noise source ENR to a device being measured. However, a

general guideline for the Y-factor method is the ENR should be within about 10 dB of the device’s noise

factor. For example, a 5 dB ENR noise source may be used to measure noise factors up to about 15 dB,

and a 15 dB ENR noise source should not be used to measure noise factors below approximately 5 dB or

above 25 dB.

The Y-factor method measures the noise factor of the device on the basis of the noise source

impedance. If the noise source does not match the device input impedance, the measurement will

include mismatch error due to reflections from the device back to the noise source. It is the cold

impedance that is important and the hot impedance less so. Noise sources with lower ENRs are built by

adding an internal high-quality attenuator to a high ENR source, which improves both the cold and hot

impedance match. Most low noise amplifier measurements will be at 50 ohms impedance.

Y-factor method procedure:

1. Connect the calibrated noise source to the device being measured using the highest quality and

lowest loss coaxial cable possible or connect the noise source directly to the device

2. With no power applied to the noise source, measure the noise power at the device output (PCold)



3. Apply power to the noise source and again measure the noise power at the device output (PHot)

4. Calculate Y

5. Correct the noise source ENR for connecting cable loss (if any) between the noise source and the

device and calculate NF

6. Be careful not to mix linear and logarithmic power ratios in the calculations

5-3. References

[Agilent 57-2] Noise Figure Measurement Accuracy – The Y-Factor Method, Application Note 57-2,

Document No. 5952-3706E, Agilent Technologies, Inc. 2013

[Dunsmore] Dunsmore, J., Handbook of Microwave Component Measurements with Advanced

VNA Techniques, John Wiley & Sons, 2012



Document informationAuthor: Whitham D. ReeveCopyright: © 2013, 2014 W. ReeveRevision: 0.0 (Adapted from original expanded work, 19 Jun 2014)


Word count:1977

Size: 2872832


Noise TutorialPart VI ~ Noise Measurements

with a Spectrum Analyzer


Noise Tutorial VI ~ Noise Measurements with a Spectrum Analyzer





1-1 Introduction


1-3 Noise amplitude

1-4 References


2-1 Noise spectrum

2-2 Noise bandwidth


2-4 Noise power


2-6 References



3-2 Amplifier noise


3-4 References




4-3 References




5-3 References


6-1 Noise factor measurements with a spectrum analyzer 6-1

6-2 References 6-10


See last page for document revision information ~ File: Reeve_Noise_6_NFMeasSpecAnalyz.doc, Page 6-1


6-1. Noise measurements with a spectrum analyzer

Most spectrum analyzers can be used to measure the noise factor of active devices (for example, amplifiers and

mixers). Modern analyzers designed to measure digital modulation schemes associated with mobile wireless

systems have provisions to measure noise and noise-like signals (figure 6-1). Spectrum analyzer accuracy may

not be as good as purpose-built noise figure meters but the spectrum analyzer is more than adequate in

ordinary radio work. First, we will discuss spectrum analyzer sensitivity in terms of its noise floor and then go

into actual noise measurements. A low noise floor indicates good sensitivity and is necessary for measuring the

noise factor of an amplifier.

Fig. 6-1 ~ Spectrum analyzers. Left: Agilient N9342C Handheld Spectrum Analyzer (HSA) weighs 3.2 kg. This high-performance instrument has many built-in features that simplify noise measurements including a power spectral densityfunction, noise markers and a built-in low noise preamplifier. However, even with these features, additional gain from anexternal low noise amplifier still is needed to boost the HSA’s sensitivity for noise factor measurements of external devices.Right: Hewlett-Packard 8590A is a high-performance instrument marketed in the mid-1980s as “Portable”. It weighs 13.5kg, and compared to the previous generation of spectrum analyzers it was very portable.

Spectrum analyzer specifications include a parameter called displayed average noise level (DANL), which is the

amplitude of the analyzer’s noise floor over a given frequency range with the input terminated in 50 ohms and

the internal attenuator set to 0 dB. DANL values are normalized to a bandwidth of 1 Hz, so it is necessary to

compensate for the resolution bandwidth (RBW) setting of the analyzer. The change in displayed noise level

Noise is related to the ratio of the old and new RBW by

10log New

Old

RBWNoise

RBW

dB (6-1)

where RBWNew and RBWOld are in the same frequency units, usually Hz. When RBWOld is 1 Hz, equation (6-1) can

be reduced to a bandwidth factor (BF)

10logBF RBW dB (6-2)



For example, if a noise measurement is made in dBm at a resolution bandwidth of 10 kHz, the displayed noise

power would need to be lowered by 40 dB [ 410log 10log 10 40BF RBW dB] for the equivalent noise

in a bandwidth of 1 Hz.

DANL measurements often are at the narrowest RBW setting, but analyzer datasheets usually specify the DANL

measurement conditions. The easiest way to measure a spectrum analyzer’s noise floor is to place a noise

marker at the desired frequency. Modern analyzers internally compensate and display the noise marker value in

dBm/Hz for any RBW setting and also take into account the difference in RBW filter bandwidth compared to an

ideal noise filter.

A noise marker uses an rms (root mean square) detector with averaging performed on a logarithmic scale, called

log power averaging. The log power averaging lowers the displayed noise by 2.51 dB (if root-mean-square, rms,

averaging is used, the 2.51 dB factor is not used the calculation). Some analyzers automatically select the right

settings for a noise marker. However, depending on the measurement, it may be necessary to manually set

some parameters. For example, the input attenuator in most spectrum analyzers has to be manually set to 0 dB

when making DANL measurements. Trace averaging also needs to be set and for noise measurements usually is

at least 40 to 50 sweeps. Trace averaging is used to reduce jitter in the displayed marker value. The sweep times

in older (analog) spectrum analyzers are much longer than modern analyzers so noise measurements with trace

averaging require patience.

For an input noise temperature T0, the DANL in terms of spectrum analyzer noise factor NFSA is given by

DANL dBm/Hz = –174 dBm/Hz + NFSA – 2.51 dB (6-3)

If we are interested in measuring the spectrum analyzer noise factor, solve for NFSA, or

NFSA = DANL dBm/Hz +174 dBm/Hz + 2.51 dB (6-4)

As an example, we will measure the noise factor of two spectrum analyzers, an older (1986) model and a much

newer (2013) HSA. The newer model is equipped with an internal preamplifier that significantly lowers the

analyzer noise factor, so comparative measurements will be made with the preamplifier on and off.

Example 6-1 ~ HP8590A spectrum analyzer: The setup is simple (figure 6-2). The spectrum analyzer display

shows the noise produced by the 50 ohm input termination at T0 combined with the noise produced by the

spectrum analyzer itself (figure 6-3).



Fig. 6-2 ~ Hewlett-Packard 8590A spectrum analyzer terminated in 50 ohms (just right of center) to measure the spectrumanalyzer’s noise floor and noise factor.

Fig. 6-3 ~ Spectrum analyzer display over a 10 MHz frequency band centered on 1 GHz. The reference level is set to –70dBm with a noise marker set to the center frequency (diamond shape partially hidden by the noise spectra at center). Thenoise trace can be seen at a level of approximately -118 dBm with the 1 kHz resolution bandwidth setting. The noise markervalue seen on the upper-right indicates the DANL of –145.84 dBm/Hz at 1 GHz. The vertical scale is 10 dB/division.

The noise factor of this spectrum analyzer at 1 GHz is

NFSA = –145.84 dBm/Hz +174 dBm/Hz + 2.51 dB = –145.84 dBm/Hz + 174 dBm/Hz + 2.51 dB = 30.7 dB (linear

ratio 1166.8)

The noise marker in this example indicates the noise power density and it was only necessary to compensate for

log power averaging by adding 2.51 dB to the difference between the measured level and the theoretical noise



floor of −174 dBm/Hz. An alternate calculation provides comparable results (figure 6-4). First, the measured

noise level in dBm is adjusted for the resolution bandwidth (RBW), in this case 1 kHz, by lowering the measured

level by 10log RBW dB. Next, it is necessary to compensate for the RBW filter’s noise bandwidth. The amount

of compensation depends on the type of filter in the spectrum analyzer and for the 8590A and similar HP analog

spectrum analyzers is 0.52 dB [Agilent 1303, HP 8590A].

Fig. 6-4 ~ Spectrum analyzer is setup the same as the previous example but in this case a normal marker is used. The markervalue seen on the upper-right indicates –117.09 dBm at 1 GHz.

The noise factor for this measurement is

NFSA = –117.09 dBm 310log 10 dB + 0.52 dB +174 dBm/Hz + 2.51 dB = 29.94 dB

which is within 0.8 dB of the previous measurement. This difference can be the result of many small factors and

not necessarily the different marker type.

Example 6-2 ~ N9342C spectrum analyzer (preamplifier off and on): The same physical setup is used in this

example (spectrum analyzer input terminated with 50 ohms). The internal preamplifier is set to off, the trace

captured, preamplifier set to on and the trace captured again (figure 6-5). With the internal preamplifier off, the

noise factor will be that of the spectrum analyzer alone, and with the preamplifier on, the noise factor will be a

composite value that includes the spectrum analyzer and its internal preamplifier.



Fig. 6-5 ~ The spectrum analyzer display has been setup to show the traces in the upper part of the window and markertable below. The green (upper) trace with marker 2 shows the spectrum analyzer noise with the preamplifier turned off.The yellow trace with marker 1 shows the noise with the preamplifier turned on.

The analyzer noise factor with preamplifier turned off is

NFSA = –148.54 dBm/Hz + 174 dBm/Hz + 2.51 dB = 28.0 dB (linear ratio 626.6)

With the preamplifier turned on, the composite noise factor is

NFComposite = –163.21 dBm/Hz + 174 dBm/Hz + 2.51 dB = 13.3 dB (linear ratio 21.38)

These calculations show the internal preamplifier significantly reduces the noise floor and, consequently, the

noise factor. The preamplifier in the N9342C has 25 dB gain (linear ratio of 316.23), providing considerable

amplification without adding a lot of noise to the analyzer.

The noise factor of the preamplifier, by itself, can be determined from the equations for cascaded amplifiers

given in Part III. Solving for the noise factor of the first amplifier in the cascade (in this case, the internal

preamplifier) where NFCascade = NFSys, NF2 = NFSA, NF1 = NFPreamp, and G1 = GPreamp, gives

PrPr

( 1) (626.61 1)21.38 19.40 12.9

316.23SA

eamp Syseamp

NFNF NF

G

dB



In the above measurements, the analyzer’s internal attenuator was set to zero. This was necessary to increase

the sensitivity. An analyzer is designed for unity gain so that it correctly displays the input signal levels. Adding

attenuation or gain changes the transfer function from the input to the display. If the input attenuator is set to

anything but zero, the analyzer must increase its internal gain (usually in its intermediate frequency, IF, stages)

to maintain a correctly displayed level. However, this raises the noise floor an equivalent amount. The concept is

similar for the internal preamplifier except that the spectrum analyzer reduces its internal gain, thus lowering

the noise floor. The noise floor is

, Pr ,dBm dBm A dB eamp dBNoiseFloor DANL L G dB (6-5)

where

LA,dB Attenuator setting in dB

GPreamp,dB Preamplifier gain in dB

It was previously shown that the hot power of a 5 dB ENR noise source is approximately –168 dBm/Hz. The

DANL of the N9342C spectrum analyzer with the preamplifier turned on and attenuator set to zero is about –163

dBm/Hz. This is 5 dB above the noise source hot power, and there is little chance the noise source output will be

visible on the display. Setting the analyzer attenuator to 10 dB attenuation, increases the difference to 15 dB. It

should be noted that an attenuator is necessary to prevent overloading the spectrum analyzer mixer, but it

usually can be set to zero for very low power measurements.

The above discussion indicates that measuring the noise factor of external devices is slightly more involved than

measuring the spectrum analyzer itself. Even with the internal preamplifier, most spectrum analyzers by

themselves do not have enough sensitivity to measure the noise factor of external devices. It is necessary to use

a good-quality low noise amplifier for additional gain between the device being measured and the spectrum

analyzer (figure 6-6).

G3

F3

G1

F1

G2

F2

NoiseSource

Cold

Hot

Amplifier 1 Amplifier 2SpectrumAnalyzer

28 Vdc)

Fig. 6-6 ~ Two amplifiers are connected between the noise source and the spectrum analyzer. Amplifier 1 is the amplifierbeing measured and Amplifier 2 is used to provide additional gain. For best results, all interconnections should be of thebest quality and lowest loss possible.

If the spectrum analyzer has a low noise preamplifier, a total of about 40 to 50 dB external gain is needed,

including the amplifier being measured. Even more gain may be necessary if the analyzer does not have an

internal low noise preamplifier. It should be remembered that measurements close to the analyzer’s noise floor

are problematic because PHot and PCold measurements are nearly the same, resulting in trying to calculate the

logarithm of a number close to zero. It should be noted that external amplifiers introduce problems of their own

and can increase measurement uncertainty.



Examples follow that use the Y-Factor method to measure the noise factor of two amplifiers at 1 GHz, a Mini-

Circuits ZKL-2 and a Chinese amplifier marketed as a low noise amplifier and designated here as CxLNA.

Attempts to measure the noise factor of one of these devices without the gain of the other failed, that is,

measurement of either amplifier is not possible without the additional gain provided by the other. The ZKL-2 has

a nominal gain of 30 dB and 3.45 dB noise factor and the CxLNA has a nominal gain of 17 dB and 1 dB noise

factor.

Example 6-3 ~ CxLNA as Amplifier 1 and ZKL-2 as Amplifier 2. The noise source ENRdB = 5.32 dB at 1 GHz

(RFD2305). The spectrum analyzer’s internal preamplifier is turned on to increase the analyzer’s sensitivity.

However, to maintain proper internal levels the analyzer’s internal attenuator is set to auto. The Y-factor

method is used in which a noise measurement is made with the noise source off (PCold) and another

measurement with the noise source on (PHot). The measured noise factor is a composite value that includes the

spectrum analyzer, its internal preamplifier and the two external amplifiers (figure 6-7).

Fig. 6-7 ~ Spectrum analyzer with traces and marker table for CxLNA as Amplifier 1 and ZKL-2 as Amplifier 2. For thesemeasurements the analyzer was placed in the power spectral density measurement mode, which sets up the properdetector and averaging protocols. The yellow (lower) trace with marker 1 shows the noise level with the noise source off(cold). The green trace with marker 2 shows the noise with the noise source on (hot). The attenuator was set to Autoresulting in 10 dB of attenuation (setting shown just above the grid to the left of center).

The following data are from the marker table:

PCold,dB = –134.39 dBm/Hz

PHot,dB = –128.68 dBm/Hz



and YdB = PHot,dB – PCold,dB = –128.68 dBm/Hz – (–134.39 dBm/Hz) = 5.71 dB

From Eq. (4-9)

5.71

10 10, 10 log 10 1 5.32 10 log 10 1 0.97

dBY

Composite dB dBNF ENR

dB (1.25 linear ratio)

The composite noise factor includes the combined effects of the spectrum analyzer and the two external

amplifiers. To find the noise factor of Amplifier 1 alone it is necessary to use the calculations for cascaded

amplifiers as before. In this case we assume Amplifier 2 noise factor is 3.45 dB (2.213 linear ratio). It is necessary

to know the gain of the Amplifier 1. A measurement using the spectrum analyzer’s tracking generator gives

17.17 dB (52.12 linear ratio) at 1 GHz. Therefore,

21

1

( 1) (2.213 1)1.25 1.23 0.89

52.12

AmplifierAmplifier Composite

Amplifier

NFNF NF

G

dB

The contribution of the spectrum analyzer is ignored in the calculation. Examination of the composite noise

factor for a cascade of three devices shows that the noise factor of the third device (in this case the spectrum

analyzer) is reduced by the factor 1/G1G2, where G1 and G2 are the power gains of the two external amplifiers.

For this example the reduction factor is about 1/60500, which reduces the spectrum analyzer’s contribution to a

negligible value.

Example 6-4 ~ ZKL-2 as Amplifier 1 and CxLNA as Amplifier 2. As before, noise measurement is made with the

noise source off and another with the noise source on (figure 6-8).



Fig. 6-8 ~ Spectrum analyzer traces and marker table with the ZKL-2 as Amplifier 1 and CxLNA as Amplifier 2. The yellow(lower) trace with marker 1 shows the noise level with the noise source off (cold). The green trace with marker 2 shows thenoise with the noise source on (hot). The attenuator was automatically set to 10 dB.

The following data are from the marker table:

PCold,dB = –131.85 dBm/Hz

PHot,dB = –128.05 dBm/Hz

and YdB = PHot,dB – PCold,dB = –128.05 dBm/Hz – (–131.85 dBm/Hz) = 3.80 dB

From Eq. (4-9)

3.80

10 10, 10 log 10 1 5.32 10 log 10 1 3.92

dBY

Composite dB dBNF ENR

dB (2.467 linear ratio)

The noise factor of Amplifier 1 (ZKL-2) alone is determined as previously described. Amplifier 2 (CxLNA) noise

factor was measured as 0.89 dB (1.23 linear ratio). The measured gain of Amplifier 1 at 1 GHz is 30.66 dB

(1164.13 linear ratio). Therefore,

21

1

( 1) (1.23 1)2.47 2.47 3.92

1164.13

AmplifierAmplifier Composite

Amplifier

NFNF NF

G

dB

The noise factor of the ZKL-2 is found to be the same as the composite noise factor because its high gain (30+

dB) reduces the noise contributions of any down-stream devices. These calculations depend on the noise factor



of Amplifier 2 (CxLNA). However, this noise factor is based on the noise factor of Amplifier 1, which was

obtained as a typical value from its datasheet. It is seen that the measured noise factor of Amplifier 1 (3.92 dB) is

about 0.5 dB higher than the assumed value, potentially leading to an error in the calculation. However, in this

case, the high gain of Amplifier 1 reduces the error to a negligible value. There are other potential sources of

error in noise factor calculations and measurement; for example, see [Agilent 1484].

For comparison with the above spectrum analyzer measurements, the noise factors of the two amplifiers were

separately measured with an HP 8970B noise figure meter and HP 346A noise source (table 6-1).

Table 6-1 ~ Comparison of noise factor measurements of CxLNA and ZKL-2 amplifiers with HP 8970B noise figure meter and

N9342C spectrum analyzer

AmplifierGain(dB)

Noise factor(dB)

Measured with 8970B noise figure meter

CxLNA 17.10 0.85

ZKL-2 30.84 3.64

Measured with N9342C spectrum analyzer

CxLNA 17.17 0.89

ZKL-2 30.66 3.92

6-2. References

[Agilent 1303] Spectrum and Signal Analyzer Measurements and Noise, Application Note 1303, Document No.5966-4008E, Agilent Technologies, 2012

[Agilent 1484] Non-Zero Noise Figure after Calibration, Application Note 1484, Document No. 5989-0270EN,Agilent Technologies, Inc., 2004

[HP 8590A] HP 8590A Portable RF Spectrum Analyzer Installation Manual, Manual P/N 08590-90003,Hewlett-Packard Corp., Jan 1987

Noise Tutorial VI ~ Noise Measurements with a SpectrumAnalyzer


Document informationAuthor: Whitham D. ReeveCopyright: © 2013, 2014, 2016, 2017 W. ReeveRevision: 0.0 (Adapted from original expanded work, 19 Jun 2014)

0.1 (Updated TOC and references, 7 Jul 2014)0.2 (Added rms averaging to log power averaging discussion, 24 Dec 2016)0.3 (Revised RBW discussion, 02 Jan 2017)

Word count:3016

Size: 4297728

noise tutorial part i ~ noise concepts...noise tutorial i ~ noise concepts see last page for...

Documents