Derivation of noise figure equations
Any electrical conductor (i.e., a wire or a resistor) that operates at any
temperature above absolute zero generates a thermal noise voltage.
Ideal (lossless) reactive components contribute no noise. You can model a
real RC or RL network as an ideal capacitor or inductor plus a noisy resistor
that includes the real part of any reactance.
We define the noise voltage within a given bandwidth as:
f2
Vn2  4kT  RdF Equation 1
f2
...in which...
V = Integrated root-mean-square (RMS) voltage between frequencies f1 and f2;
k =
R =
T =
f1,
Boltzmann's Constant (1.374 * 10-23 joules per degree Kelvin);
resistance or resistive component of impedance;
temperature in Kelvin (Kelvin = Celsius + 273.16)
f2 = lower and upper limits of bandwidth of integration.
The integral imposes "brick wall" (i.e., perfectly rectangular) limits on the
noise bandwidth.
When a resistance or resistive component of impedance remains constant over a
frequency range of interest, the noise integral simplifies to:
V  4kTRB Equation 2
Thus, for a 51- resistor operating at 290 K, we calculate the noise voltage
produced in a 1-Hz bandwidth as:
V  4(1.374 1023)  290 51 1
V = 0.902 * 10-9, or approximately 0.9 nV.
To obtain peak-to-peak volts, multiply RMS noise voltage by a factor of 5
(for a less than 1 percent probability that noise spikes will exceed the
calculated peak-to-peak value), or by a factor of 6.6 for 0.1% probability.
Note that random noise is uncorrelated. To calculate the sum of several
independent noise voltages, use an RMS calculation: square the individual
voltages, sum the squares and take the square root of the sum of the squares.
If you connect
there's no net
thermal noise.
R connected in
resistive load
two identical resistors at the same temperature in parallel,
transfer of power because each resistor generates its own
However, if you model the noisy resistor as an ideal resistor
series with a noise-voltage source and connected to a matched
R, you can calculate thermal noise power in watts:
Vn2
4R
Pn 
4kTBR
4R
Pn 
2
Pn  kTB Equation 3
For a resistor at room temperature (T = 290 K), noise power is -174 dBm
(decibels referenced to 1 mW) for a 1-Hz bandwidth, or approximately 4*10-21
W/Hz of bandwidth. Note that noise power doesn't depend on resistance—only
upon the defined bandwidth.
...
A nonideal amplifier of gain G delivers noise along with its desired output.
We define a noise factor F as:
F
SNRinput
SNRoutput
Equation 4
...or...
Si
F
So
Ni
Equation 5
No
...where:
Si
So
Ni
No
=
=
=
=
Input signal power
Output signal power
Input noise power
Output noise power
An ideal amplifier would contribute no noise, and the input and output signal
to noise ratios would be equal, resulting in a noise factor F of unity.
A nonideal amplifier of power gain G amplifies its input signal and input
noise and introduces its own internal random noise power Na, resulting in an
output signal to noise ratio of:
So
G  Si

No Na  G  Ni
Equation 6
Which, when substituted into equation 5 produces...
F  1
Na
G N i
Equation 7
If we assume that Ni originates from only random (e.g., thermal) sources, we
can substitute equation 3 for Ni:
F  1
Na
Equation 8
G k T  B
Likewise, if we can assume that Na also originates from truly random (thermal)
sources, we can define an equivalent noise temperature Teq that corresponds to
the noise power produced by a resistor heated to produce power equivalent to
Na.
k  Teq  B 
Na
G
Equation 9
Substituting equation 9 into equation 8 yields...
F 1 
Teq
T
Equation 10
We can also define noise figure NF in decibels:
NF = 10xlog (F) Equation 11
For situations in which a resistor operates at temperature T other than the
reference temperature To (290 K), it's convenient to refer to "excess noise"
as defined by:
EN 
T  To
To
...or...
EN (db)  10 log(
T  To
)
To
In a cascaded system of N stages, each with a differing noise figure Fn and
gain Gn expressed in dB, the overall noise figure becomes:
F  F1 
F2  1 F3  1
Fn  1


G1
G1  G2
G1  G2   Gn
For More Information
Books:
Fish, Peter J., Electronic Noise and Low Noise Design. McGraw-Hill, Inc., New
York, 1994. ISBN 0-07-021004-7. This excellent theoretical and practical
reference book offers many practical tips for reducing noise in a project's
design and hardware implementation phases.
King, Robert, Electrical Noise. Chapman and Hall Ltd., London, 1966. A volume
of the Modern Electrical Series, this 195-page book offers a readable basic
mathematical introduction to noise and its causes.
Ott, Henry W., Noise Reduction Techniques In Electronic Circuits, John Wiley
& Sons, New York, 1976. ISBN 0-471-65726-3. If you're at all involved with
electrical noise in any phase of design or measurements, you probably already
own a copy of this book.
If you purchase only one noise-related book, buy this one!
Pease, Robert A., Troubleshooting Analog Circuits, Butterworth-Heinemann,
Boston, 1991. ISBN 0-7506-9184-0.
Rheinfelder, William A., Design of Low-Noise Transistor Input Circuits,
Hayden Book Co., 1964. Library of Congress Catalog No. 64-13750. In spite of
its title, this book offers an excellent explanation of noise sources and
measurements.
Friis, H. T.: "Noise Figures of Radio Receivers", Proceedings of the
Institute of Radio Engineers (IRE), volume 32, no. 7, pp. 419-422, July,
1944. (Correction, volume 32, no. 12, p. 729, December, 1944.) A classic
paper in which Friis defines receiver noise figures and the expanded case
(for two-port networks).
Web Site:
See http://escriba.cfn.ist.utl.pt/~cupido/nfsa.html for circuit diagrams, PCB
layouts, and photos that describe how to build a noise figure front end for a
spectrum analyzer.