Instrumentation & Measurement Magazine 24-7 - 58

Magnitude-squared averaging
Magnitude averaging
Logarithmic averaging
Table 2 - Summary of the expected value bias for six cases of noise power averaging
Real noise
0 dB
-1.9612 dB
-5.5171 dB
-1.0491 dB
-2.5068 dB
Table 3 - Results of the Monte Carlo bias simulations, sample mean compared to true noise power
Real noise
Complex noise
Magnitude-squared averaging
Magnitude averaging
Logarithmic averaging
0.0002 dB (0.0002)
-1.9610 dB (0.0002)
-5.5168 dB (0.0003)
noise processes. Table 3 summarizes the results. Deviations
from the analytic expressions are given in parentheses.
The uncertainty expressions are approximately of the same
form for all six cases. For four of the cases, we truncated a
Taylor series to simplify the expression, which is valid as the
number of samples we average becomes large. In practice, using
N≥100 will yield very good approximations with these
expressions.
All of the simplified expressions follow the form:
 
U
Ak
N
dB.
(58)
The value of A varies for each of the six cases. Table 4 shows
the values of A for each of the six cases.
A Monte Carlo simulation was run with N = 104
averages
and 106
realizations of noise processes. Table 5 summarizes
the simulation's ensemble of standard deviation values of
the sample mean. The values in the table are normalized such
that they correspond to the values of A given in (58). The actual
uncertainties for a 1-σ deviation are smaller by a factor of
N  100. Deviations from the analytic expressions are given
in parentheses.
-0.0001 dB (-0.0001)
-1.0492 dB (0.0001)
-2.5069 dB (-0.0001)
Notice that the uncertainties are smaller for complex noise
than for real noise. The magnitude-squared and magnitude averaging
have lower uncertainties than logarithmic averaging.
The magnitude-squared method, which is the only unbiased
method, has the lowest uncertainty of the three, and is probably
the best choice if available on the instrument being used to
make the noise power measurement.
Fig. 5 summarizes the 1-σ uncertainty values for the six different
methods for different values of N.
Conclusions
Various RF instruments are available for making noise power
measurements, but the system architecture and processing
methods of these instruments may yield different results. In
this paper, we explored some of the RF instruments that are
commonly used for such measurements and analyzed their
biases and uncertainties. With knowledge of the bias, the
user can correct for it. With knowledge of the uncertainty, the
user can make an informed decision about the number of averages
that are needed to achieve the desired measurement
uncertainty. If given a choice, using the " complex, magnitudesquared "
averaging method is recommended as it will result in
an unbiased result with the smallest measurement uncertainty.
Table 4 - Summary of the A value in equation (58) for six cases of noise
power averaging, characterizing the uncertainty of each method
Real noise
Magnitude-squared averaging
Magnitude averaging
Logarithmic averaging
6.1419
6.5623
9.6476
Complex noise
0 dB
Complex noise
4.3429
4.5403
5.5700
Table 5 - Summary of the results of the Monte Carlo simulation, wherein we compared the normalized
standard deviation of the sample mean to the previously derived analytical results
Real noise
Magnitude-squared averaging
Magnitude averaging
Logarithmic averaging
58
6.1436 (0.0017)
6.5654 (0.0031)
9.6481 (0.0005)
IEEE Instrumentation & Measurement Magazine
Complex noise
4.3406 (-0.0023)
4.5386 (-0.0017)
5.5705 (0.0005)
October 2021

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