Fundamental Noise Concepts
•What
is Noise Figure?
•Why do we need to know the Noise Figure?
•How do we define the terms used ?
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What is Noise Figure ?
Small
Signal
Imperfect
Amplifier
Signal larger
But Noisier
Thermal Agitation of Electrons adds
noise to the signal
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In this presentation, we will be concentrating on the effect of thermal noise
on the RF and microwave communication path.
To improve the Signal to Noise Ratio (SNR) at the receiver's output requires
minimizing all sources of noise. Our focus is on the thermal agitation and
random movement of electrons. In this example, a perfect amplifier would
add no noise, and the signal would be an amplified replica. However, in
practice, this noise is broadband in nature, and can mask the wanted signal.
The noise floor, as seen in a given bandwidth, limits the detection of weak
signals.
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Common Types of Noise
Name
Example
Description
Impulse
Ignition, TVI
Not Random, Noise eliminated by
shielding
Quantizing,
Decoding, etc
Bit Error Rate
(BER)
Digital Systems, Digital/Analog
and Analog/DigitalConvertors.
Often Bit Resolution and/or Bit
Fidelity
Shot
Transistors
Quantized nature of current flow,
Implusive in nature
Thermal
Resistors,
Atmosphere
Thermal Agitation of Electrons act
like a signal
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Other types of noise, that are not included in this discussion are shot noise,
ignition, sparks, or spurious signals.
Noise added at the front end of an RF/MW system greatly influences the
cost of the overall system. Noise reduction allows wider repeater spacing
and/or less transmitter power. We will see an example to show that there is
a high economic return for better receiver noise figure.
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What is Noise Figure ? (cont)
Thermal Load
R + jX
RL - j XL
k = 1.38 x 10 joule/K
T = Temperature (K)
B = Bandwidth (Hz)
Available Noise Power,
Pav = kTB
(Power Delivered to a Conjugate Load),
(i.e. R = RL , X = -XL )
Note: At Standard Temperature T (=290K) : kT = 4 x 10 -21 W/Hz = 174dBm / Hz
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Noise follows the normal power transfer laws. A conjugate match is needed
for an optimum noise power transfer from the source to the load.
This is termed the available Noise Power.
Nyquist arrived at the equation Pav. = kTB Watts. 290 K was formally
adopted as the Standard Temperature for determining Noise Figure.
This gives us the figure of -174 dBm/Hz as the universal Noise Floor at that
temperature.
This natural level is encountered everywhere up to around 400 GHz.
Here an example of this relationship:
Doubling the bandwidth will add 3dB to the available Noise Power.
Also doubling the absolute temperature increases the Noise Power by 3dB
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What is Noise Figure ? (cont) Noise added by Amplifier
Noise Added (Na)
Na
Noise (in) x Gain
[N in G]
Noise (in)
To = 290K
Na
Gain 20dB
NF 10dB
Nin
Np = Na + Nin G
at 290K
Imperfect Amplifier
Degrades Signal to
Noise Ratio
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We are now going to talk about Signal to Noise Ratios, and how we define
Noise Figure.
In this example, we have an amplifier with a Gain G = 20 dB. The left graph
shows the Input Signal and noise versus frequency, while the right shows the
Signal and noise after the 20 dB gain.
If the device is noiseless, then the output Noise Floor will go up by its gain,
i.e. from -100 dBm to -80 dBm.
However, the amplifier is noisy. It adds noise, such that the Noise Floor is at
-70 dBm, and additional 10 dB.
The Signal to Noise Ratio has changed by 10dB. This degradation is the
Noise Figure of the device. We will describe this further using the ratio :
 (S N )in 
Noise Figure = 10 ⋅ log 

(
)
S
N
out 

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Why do we measure Noise Figure?
Example...
Transmitter:
ERP
Path Losses
Rcvr. Ant. Gain
Power to Receiver
+ 55 dBm
- 200 dB
60 dB
-85 dBm
Receiver:
Noise Floor @ 290K
Noise in 100 MHz BW
Receiver N.F.
- 174 dBm/Hz
Link Margin: 4 dB
+ 80 dB
+5 dB
Receiver Sensitivity
Power to Antenna: +40 dBm
Frequency: 12 GHz
Antenna Gain: +15 dB
ERP = +55 dBm
Path Losses: -200 dB
-89 dBm
Choices to increase Margin by 3dB
1. Double transmitter power
2. Increase gain of antennas by 3dB
3. Lower the receiver noise figure by 3dB
Receiver NF: 5 dB
Bandwidth: 100 MHz
Antenna Gain: 60 dB
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Here is an example of why we need to know the Noise Figure of a device.
In this example, we have a satellite that transmits with an Effective Radiated
Power (ERP) of +55 dBm, transmitted through a path loss of 200 dB to an
antenna with gain of 60 dB. The signal power to the receiver is -85dBm. The
receiver sensitivity is calculated here using kTB = -174dBm Hz and the
noise power in a 100 MHz bandwidth is 10 ⋅ log 108 = 80 dB of that in an 1Hz
BW.
( )
The noise figure of the device is assumed to be 5 dB. So the receiver noise
floor is at -89 dBm.
If we wanted to double the link margin, then we could double the
transmitter power. This would cost billions of dollars in terms of increased
payload
and/or higher rated, more expensive components and more challenging
engineering issues. Another way is to increase the gain of the receiver.
This would cost millions in terms of size and mechanical engineering, and
the debates over local environmental issues and planning permissions.
While lowering the Noise Figure of the front end would be a fraction of this,
and is the more attractive economically.
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The Definition of Noise Factor
or Noise Figure in Linear Terms
• Noise Factor is a figure of merit that relates the Signal to Noise ratio of
the output to the Signal to Noise ratio of the input.
• Most basic definition was defined by Friis in the 1940’s.
S
 
 N in
S
 
 N  out
G , Na
S
 
 N in
F=
S
 
 N  out
Reference: H.T. Friis: 1944: Proceedings of the IRE
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Our current understanding of noise figure dates back to 1944 when
Harold Friis defined the figure of merit term called Noise Figure.
Because he expected the Signal to Noise (S/N) power ratio to become
worse going through a system, the term is defined as the Signal to Noise
Ratio of the input divided by the Signal to Noise Ratio of the output. This
would therefore yield a figure of merit that would have a minimum value of
1.
There are now two terms that we define that are standard within the IEEE.
Noise Factor is the linear term (F) for measuring Noise Power.
When we use the logarithmic term we use the name Noise Figure (NF).
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Definition of Noise Figure
S
 
 N  in
S
 
 N  out
N out
Noise
N a Added
G , Na
Gain = G =
S
 
 N in
F=
S
 
 N  out
S out
S in
GN in
N out = N a + GN in
F=
f1
f2
N a + GN in
N out
=
GN in
GN in
 N a + GN in
NF (dB) = 10 log
 GN in
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Noise Factor



Noise Figure
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We can simplify this ratio with a couple of definitions. First, let us look at a
device that has a gain G, and an additive Noise Power of Na. The picture of
the device is that of an amplifier, but here it is used for any device that needs
to be measured. Note that we define the gain simply as the the power ratio of
output signal (Sout) over the input signal (Sin).
Noise as we will discuss is additive in nature. Therefore, the Output Noise
Power is the sum of the amplified Input Noise Power (GNin) and the additive
Noise Power of the device (Na). This expression of noise factor is a very
important equation that will be used throughout this presentation.
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Noise Voltage
Standard Equation for Noise Voltage produced by a Resistor R
e 2 = 4kTBR
k = Boltzman' s Constant = 1.38× 10 −23 Joules/°K
T is absolute temperature (°K)
B is bandwidth (Hz)
e is rms voltage
R is Resistor in Ohms
Reference: JB Johnson/Nyquist: 1928: Bell Labs
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As we said earlier in this presentation, noise limits the processing of weak
signals. A receiver designer recognizes the boundary conditions that limit
the usable dynamic range and Signal to Noise Ratio. The Noise Floor limits
the detection of weak signals, and the distortion products limit the upper
level of detectable power. We need to understand the nature of this noise.
Early research at Bell Labs showed the relationship between the voltage
generated by this random electron movement and its volatility with respect
to the absolute temperature. Refer to Application Note 57-1 (5952-8255)
for a more detailed description. Let’s use this equation here in the following
example:
if B = 2.0 MHz, R = 10.0 kΩ, T = 27.0 °C = 290.0 °K
then
e 2 = 4 ⋅ 1.38 ⋅10 −23
Joules
⋅ 290°K ⋅ 2 ⋅106 Hz ⋅ 1 ⋅10 4 Ω
°K
and
e = 18.2 µVolts rms
The noise level limits the detection of signals that are smaller (weaker) than
this value.
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Noise Power is Linear with Temperature
N in
= N a + GkTs B
G , Na
Nout
Output Noise Power
Z s @ Ts
N out = N a + GN in
Slope = kGB
N2
N1
Na
Ts
T1
T2
Temperature of Source Impedance
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This graph shows the relationship of Noise Power as a function of
temperature. We shall use the convention of Zs being a conjugately
match load on the input and Ts as the system temperature. Given the
input noise and the additive noise of the device we have the output
noise power as Nout = Na + GkTsB. If we plot this equation as a
function of system temperature we can see that we have a linear curve.
The slope of the line is kGB. This relationship becomes very important
when making measurements.
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Definition of Effective Noise Temperature
N out = N a + GN in
∑
Z s @ Ts
Z s @ Te
G , Na = 0
Te = (F − 1)Ts
and
F=
Te + Ts
Ts
Output Noise Power
Nout
= GkTe B + GkTs B
= GkB(Te + Ts )
Slope = kGB
Na
Ts
Te
Temperature of Source Impedance
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Because the output Noise Power is linear we can define a new term. Note
the point represented by Te on the graph. This is called the Effective Noise
Temperature of the Noise Power value Na. This is effectively the temperature
that a source impedance needs to be to give the same level of noise power
out of the device as Na. Te is often interchangeable in many noise
calculations. We will let the additive noise be defined as GkTsB.
Rearranging the equations and using the Noise Factor equation:
F=
N out
G ⋅ N in
we obtain a simple relationship between the Noise Factor and effective
noise temperature. These relationship can be used interchangable.
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Friis or Cascade Equation
• Here we see what contribution a second amplifier has on the overall noise factor.
N in
N out = N a 2 + G1 N a1 + G1G2 N in
G = G1G2
Z s @ Ts
G1 , N a1
G2 , N a 2
N a1 = kTe1 B = (F1 − 1) ⋅ kTs B
Na2
N a1G2
N a1
N in =
kTs B
N in = kTs B
kTs BG1
kTs BG1G2
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N a 2 = kTe 2 B = (F2 − 1) ⋅ kTs B
N
F = out
GN in
F12 = F1 +
F2 − 1
G1
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During H. Friis analysis he showed what happens the overall noise factor
when a second stage is added. The graph above shows what happens to
the noise power as it flows through the system. First, we start with the Input
Noise Power. This Noise Power is multiplied by the Gain of the first stage
and the additive noise of the first stage to get its output noise level.
We then follow this Noise Power and it gets multiplied by the second stage
gain and the additive noise of the second stage is added. This leads to the
output Noise Power equation of:
Nout = Na2 + G1Na1 + G1G2 Nin
We can also recognize that the overall gain is just G1⋅G2 and we can use the
effective noise temperature definition for the two stages. We can then
rearrange the Noise Factor equation and get the cascade equation as
follows:
F = F12 = F1 +
F2 − 1
G1
This equation shows that the major factor affecting the Noise Figure is the
first stage Noise Figure. You can reduce any contribution by making the
Gain of the first stage large.
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Effects of Multiple Stages on Noise Factor
• The cascade equation can be generalized for multiple stages
N in
Z s @ Ts
G1 , N a1
G2 , N a 2
Ftotal = F1 +
Ftotal
GN , N N 2
FN − 1
F2 − 1
+L+
G1
G1G2 K GN −1




N
1
F
−


= F1 + ∑  N i

i=2
 ∏ G j −1 
 j =i

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So far we have looked at the cascade equation with only two two devices in
cascade. We can extend the equation by reapplying the cascade equation
for new components.
( see Application Note 57-1 for a further details on the derivation)
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Demo of Cascade Equation
A
Gain (dB)
6.0
Gain
4.0
Noise Figure (dB) 2.3
Noise Factor
1.7
B
12.0
16.0
3.0
2.0
F = F1 +
F2 − 1
G1
C
20.0
100.0
6.0
4.0
2.0 − 1
4.0 − 1
+
= 1.70 + 0.25 + 0.047 = 1.997
4.0
4.0 × 16.0
NFABC = 10 log(1.997) = 3.00 dB
4.0 − 1
2.0 − 1
= 1.7 +
+
= 1.70 + 0.75 + 0.025 = 2.475
4.0
4.0 ×100.0
NFABC = 10 log(2.475) = 3.93 dB
FABC = 1.7 +
FACB
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Let’s look at an example of how the system noise figure is affected by the
order of the devices. Here we have three amplifiers which have gains of
6,12, and 20 dB. The corresponding Noise Figures are 2.3, 3.0, and 6.0 dB.
Using the cascade equation we can calculate the Noise Figure of 3.00 dB.
(Note that all the calculations are done in linear format and not logs).
Remember that we said if we move the gain forward, that this would reduce
the contribution of any stages that follows. Well let us switch the positions
of amplifiers B and C. We then recalculate the noise figure of 3.93 dB.
This is due to the noise figure of amplifier C being so high. When making a
system we can not just worry about the gain of the first stages but also the
Noise Figure.
This example shows that the cascade equation is an important modeling
tool.
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