Instrumentation & Measurement Magazine 24-7 - 55

Based on (13), we can write the uncertainty in dB as:
22 2
    YY Yrr r

Yr
Uk N
2
real magnitude  20log 1
,
10

 


1
2




 

X



 20ke
2
dB.
NN
The second moment of Yr
X
given by:
(24)
E   


Y
2
r  2X
 20log10

   

ln 10
x
2
2
e dx
2X
1 6.5623k
log10
22
ln 2  ln 4XX 
   ln 4
 ln 2   
48
1
The expected value of Yr
Logarithmic Averaging of Real Noise
If the noise is sampled with the full-wave rectifier plus digitizer
along the log-amp path, then we have a case of " real,
logarithmic " averaging. To derive the bias and variance
of this case, we start with the probability density function
(pdf) of real noise Xr
X :
variance of 2
x
fx e
2X
Xr
 
1

2 2
X
.
(25)
The random variable Xr is converted to a dB representation,
indicated by the random variable Yr, by implementing the following
equation:
YX (26)
rr .
 20log10

Using this representation of Yr
and standard deviation of Yr

, we may derive the mean
. To do so, it is not necessary to
derive the probability density function of Yr. Rather, if we can
write Y  fX , then  
The expected value of Yr
 
EEY fX , as described in [20].

is derived in Appendix A and results
in the following expression:

E r
Y
 

20log10

X
x
1
2X
20log10 ln 2 .
ln 10
10 


where   0.5772157 is the Euler-Mascheroni constant.
Comparing this result to the true noise power, we discover
that the noise power bias of a " real, logarithmic " averaging device
is:
BIAS
real
logarithmic
  
 20log   
5.5171 dB.
10 XX
  Yr
 ln 10
 


ln 2 20log10
dB E 10log σ
10
10 X

2
(28)
To determine the uncertainty of a noise power value provided
by a " real, logarithmic " averaging device, we must first
compute the variance of a single measurement. The variance
of Yr
of Yr
October 2021
e dx
2 2
X
(27)
x

2
2


result is shown in (27). The variance of Yr
  Yrr
22
r
and the standard deviation of Yr
, which has a mean value of zero and a
50
 Yr
ln 10


is:
 9.6476 dB.
(32)
The reader will notice that the variance and standard deviation
of the measured noise power in the case of a " real,
logarithmic " device are independent of the noise power level.
This is logical because we are analyzing the signal on a dB
scale. An example of a sinusoidal amplitude-modulated signal
will help illustrate this point. Let us assume a signal level
of 1 V peak amplitude with a 10 % modulation index. The signal's
amplitude will vary from 0.9 to 1.1 V, a total variation of
1.74 dB. If this signal is attenuated by 6 dB, then the peak signal
amplitude will be reduced to 0.5 V amplitude, with an amplitude
variation from 0.45 to 0.55 V. The total variation is still
1.74 dB, and will always be so, regardless of how much the amplitude-modulated
signal is attenuated or amplified.
Similarly, the variation of a noise process when assessed on
a logarithmic scale will always be the same regardless of the
actual power level of the noise. This can be readily observed
on a spectrum analyzer by examining the vertical spread of
the noise floor. For a fixed number of averages, that spread
is always the same regardless of the power level at which the
spread is observed.
If N samples of Yr
are averaged, then the standard deviation
of the reported value of the noise power can be computed as:


real,logarithmic N samples 
50

ln 10 NN

The standard deviation above is given in dB. Unlike the
previous cases where we computed the spread of possible values
converted to dB, we can write the k-σ uncertainty directly
as:
U ,
real logarithmic

9.6476k
N
is calculated from the expected value and second moment
as follows:
dB.
(34)
Magnitude-squared Averaging of Complex Noise
Recall that Xc
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9.6476
dB.
(33)
 
 
EE ,
ln 10
YY 2
2
502
was previously computed and the
, therefore, is given as:

(31)
400
2
(30)
.
2
1
x
2
is derived in Appendix B and is
Var E E .
   
(29)
denotes a complex random process. If this noise
signal is sampled with an analog-to-digital converter (ADC)
55
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