Instrumentation & Measurement Magazine 24-7 - 60

[23] I. S. Gradshteyn and I. M. Ryshik, Table of Integrals, Series, and
Products, Fourth Ed. Orlando, Florida, USA: Academic Press, 1980.
[24] D. Zwillinger, CRC Standard Mathematical Tables and Formulas,
33rd Ed. Boca Raton, Florida, USA: CRC Press, 2018.
Brett T. Walkenhorst (brett@creydos.com) is Founder and
Chief Engineer of Creydos Research in Marietta, GA, USA
and also serves as an Adjunct Professor at Georgia Tech in the
school of Electrical and Computer Engineering. Previously, he
was employed by NSI-MI Technologies in Suwanee, GA, USA.
His expertise is in signal processing and electromagnetics. He
received his B.S. and M.S. degrees in electrical engineering
from Brigham Young University and his Ph.D. degree in electrical
engineering from Georgia Tech.
Ryan T. Cutshall is currently employed at Raytheon Missiles
and Defense, a division of Raytheon Technologies, Tucson, AZ,
USA where he works in the RF Products Department. His work
is primarily focused on antenna design and measurement. He
received his B.S. degree in engineering mechanics from the University
of Wisconsin-Madison in 2008 and his M.S. degree in
electrical engineering from the University of Arizona in 2013.
Daniel R. Frey is currently a Senior Staff and Systems Engineer
employed at Orbit Advanced Technologies, a division of Microwave
Vision Group, in Warminster, Pennsylvania, USA. His
expertise is in signal processing, radar systems, and antenna
measurement systems. He received his B.S. degree in electrical
engineering from Penn State University and his M.S. degree in
systems engineering from University of Pennsylvania.
Appendix A
In this appendix, we calculate the mean of the logarithm of a
real noise signal to be:

x
E r
Y
 

20log10

X
x
1
2X
20log10 ln 2
ln 10

E   20log10

r
Y
40

2Xln 10 0


x
2X

Using the u-substitution of u 
gral, we find:


2X
x
2
 u e duu2
2
ln x e dx 2 ln 2
00
XX X
00
e du
 XX
(61)
  2 ln u e du

2 ln 2 uu
 

22
60
As in Appendix A, we use the u-substitution of u 
reformulate the integral:
x
2 X
to
ln
x
2 X
x

2
x e dx
2X
to evaluate the inteE
  e dx


 

Yr
2


800
2Xln 10 0
20log10
2


x
2X

ln 
x

2 x e dx
2
2X
2
2
1
2
2X
(68)
x
2
1
10 

We begin by re-arranging the integral to use natural logarithms
and take advantage of the Gaussian distribution's
symmetry about zero.

x

2
e dx
2X
(60)
2
e dx
2 2
X
(59)

In this appendix, we calculate the second moment of the logarithm
of real noise to be:
E   


Yr
2
 2X
20log10
x

  

ln 10
2
e dx
2X
2

22
ln 2  ln 4XX 
   ln 4
 ln 2   
48
1


2
We begin by re-arranging the integral to use natural logarithms
and take advantage of the Gaussian distribution's
symmetry about zero.
400
2
(67)
2
1
x
2
2
The half integral of a zero-mean, Gaussian distribution
with variance of 1/2 is given by:
 

2X
1 11
2
1
2
e
x 
 du   
X
Therefore, we can evaluate the first integral of (61) as follows:


 e u du 
2

2
(63)
Equation 4.333 of [23] offers the solution to the second integral
of (61):

u e du  
 
ln 
2
u2

4
 ln 4

where   0.5772157 is the Euler-Mascheroni constant.
Inserting (63) and (64) into (61), we find that:
x

ln

x e dx XX X
2 2
X
2
X

22
X 
2 ln 2 
1
 

  



ln 2 ln 4
By inserting the results of (65) into (60), we find that the mean
of the logarithm of a real noise signal is given by:
E 
Y
 

40
  ln 2
2
2 22ln 10
20


ln 10
 10
20log
X
ln 10

ln 2

Appendix B




X
10
   1
ln
ln 4


rX
X
2 
1
  ln 4
X
ln 2 
(66)
24 ln 4
2
(65)
(64)
e du 
x2
2
(62)
IEEE Instrumentation & Measurement Magazine
October 2021
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