Instrumentation & Measurement Magazine 24-7 - 56

and converted to successive samples of Xc
X Re X jIm X I jQ
cc   .
can be written as:
  
c
The variance of I and Q are identical, and are given as:

VAR I VAR Q E I E Q
2

Note that I and Q are distributed as chi-squared distri

2
butions with one degree of freedom, where each underlying
normal distribution has a variance of 2
X 2 .
Thus, the expected value of


E X E I EQ
22 2
     2
   
cX
    
2
    
2.
2
X
X is given as:

2
c
(37)
If the noise power is computed by taking the mean of the
magnitude-squared of the complex signal, then the noise
power would be reported as follows:
P
complex
magnitude squared EXc

  
2
2
X .
(38)
This is an example of a " complex, magnitude-squared "
method of averaging.
Comparing this result to the true noise power, we discover
that the noise power bias of a " complex, magnitude-squared "
averaging device is:
BIAS
complex
magnitude squared


 
P
 

dB 10 log
10log

magnitude squared

10
10 0 dB.
2
2




X
X
Therefore, if averaging is performed using the magnitude-squared
of the complex signal, the noise power can be
computed without bias.
The standard deviation of a single sample of a " complex,
magnitude-squared " device can be computed knowing that
c
2
X is a chi-squared distribution with two degrees of freedom,
where each of the underlying normal distributions have
a variance of 2
X 2 . The standard deviation can be computed
as follows [20]:



complex,magnitude squared

  Var
2
2
   22 .X
2
2
2


X
10
Thus, the standard deviation of a reported value after averaging
N samples using a " complex, magnitude-squared "
device is given as:
  2
X
complex,magnitude squared N samples 

N
56
.
(41)
 
2
Xc

X
2
2

(40)
Var 
2
Pn
(39)
complex
Magnitude Averaging of Complex Noise
If the noise is sampled with a device similar to a swept-tuned
superheterodyne spectrum analyzer with the signal routed
through the linear path (Fig. 3), then the output of the envelope
detector is c
      
22
    

  
2
X
.
2
(36)
10
, as would be done
when employing a real-time spectrum analyzer (Fig. 4) then
the complex signal Xc
(35)
This method achieves the Cramer-Rao Lower Bound
(CRLB) for a standard deviation estimator of complex white
Gaussian noise [13], [22].
Using (13), the uncertainty in dB is given by:

UkN
X
complex magnitude squared  10log 1

,
10
2
 
N
10
10log 1
1

k
1


ke
N
log
10 
4.3429k
N
dB.



2
X




(42)
X . If this device were to report power by first
taking the time-average of the output, and then squaring that
result, this is an example of a device that performs " complex,
magnitude " averaging.
To determine the bias incurred by employing " complex,
magnitude " averaging, we must first consider the distribution
of
X . The magnitude of the signal Xc
c
X IQ  22
c
Thus, the distribution of the magnitude of Xc
is given as:
.
(43)
is a Rayleigh
distribution, otherwise known as a chi distribution with two
degrees of freedom, where each underlying normal distribution
has a variance of 2
X 2, see (4) and (5). The mean of this
distribution is given as [20]
 
EX

2
cx .
 

22 2
X

(44)
Thus, the noise power value reported by a " complex, magnitude "
averaging device is given as follows:

4
P EXc
complex
magnitude
  
2
2
X
.
(45)
Comparing this result to the true noise power, we discover
that the noise power bias of a " complex, magnitude " averaging
device in dB is:
BIAS
complex
magnitude

 
P
 

dB 10log10
magnitude
Pn
 10log   1.0491 dB.
X


X
2 
4



2
Thus, a " complex, magnitude " averaging device may be
used to estimate noise power as long as 1.0491 dB is added to
the reported result. This is in agreement with the results shown
in [2] and [16].
IEEE Instrumentation & Measurement Magazine
October 2021
(46)
complex
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