Instrumentation & Measurement Magazine 24-2 - 81

serve as a standard in noise-source characterization. To further
improve the accuracy of the calibration, vector corrections are
performed, which are very similar to the noise-parameter measurement process.

Admittance Constellation
While there are seemingly many ways of determining noise
parameters, most approaches involve manipulating Ys = 1/Zs
by varying the matching network, such as in [5], [9]-[14]. At
least four Ys are required to determine noise parameters [12],
[14]. When the number of admittances, n, is at least four, using
Ys (Ys = Gs+jBs) in (3) results in a system of equations represented as:
Ax = F	(4)

	

where A is an n×4 matrix calculated from Ys, x is the vector
of unknown noise parameters, and F is a vector of measured
noise factors. The least-squares solution to (4) yields the noise
parameters.
For n = 4, entering the four different Ys (Ys1, Ys2, Ys3, and Ys4)
and their corresponding measured noise factors (F1, F2, F3, and
F4) in (4) produces:

	


1


1


1



1


G s1 

Bs21
Gs1

1
Gs 1

Gs 2 

Bs22
Gs 2

1
Gs 2

Gs 3 

Bs23
Gs 3

1
Gs 3

Gs 4 

Bs24
Gs 4

1
Gs 4

Bs1 

Gs1 
Bs 2   A   F1 
   
Gs2   B   F2 
	(5)


Bs 3   C   F3 
   
Gs 3   D   F4 

Bs 4 

Gs4 

where F=[F1 F2 F3 F4]T contains the measured noise factors for
each Ys, and A, B, C, and D are unknowns that relate to the
noise parameters by [12]:

	

F 
A  4 BC  D 2
 min
Rn  B


4 BC  D 2


G
	(6)
opt

2B

D

Bopt  

2B
 Y G  jB
opt
opt
opt


While the determination of noise parameters via (5) and
(6) appears to be straightforward, attaining these measurements in practice relies on accurately measuring noise powers
that are very low and easily contaminated by interference and
noise generated in the measurement equipment, and that are
subject to the repeatability of the Ys-generating network. At
present, researchers are focusing on determining the optimum
set (aka constellation) of Ys that allow for solutions to (4), while
also enabling the greatest reduction in measurement uncertainty, sensitivity to the repeatability of Ys, and measurement
duration.
April 2021	

All previous research has agreed that one of the Ys admittances should be located near the center of the Smith chart; in
other words, it should be near the characteristic admittance of
the measurement system. Regarding other admittances, [14]
have observed that a uniformly distributed pattern of Ys on the
Smith chart resulted in reasonable noise parameter extraction
for a few Ys. Furthermore, [13] showed that it may be beneficial
to expand the uniformly distributed Ys constellation to a nonuniform coverage of the Smith chart. However, [13] required
additional admittances Ys to complete the measurements.
Unlike many previous methods, which sometimes employ
over 20 such admittances, the results of [14] showed that indeed just four admittances Ys were sufficient for solving the
system of equations and determining the noise parameters. In
attempting to identify the optimum Ys constellation pattern,
[14] investigated the system of equations in (5) and focused on
selecting the admittances Ys that created the largest normalized determinant of A. While physical reasons were not given
for this normalization, the results in Fig. 4 from [14] revealed a
reduction in measurement uncertainty for some combinations
of Ys, and suggested that separating Ys by 120° produces a useful Ys pattern for noise-parameter measurement. In [11], the
authors also focused on using four well-spread admittances,
but the location of these admittances was governed by minimizing the condition number of A. The results of [11] indicated
that rewriting A in its noise-wave representation and scaling
its terms by 1−|Γs|2, where Γs is the reflection coefficient associated with Ys, removed singularities in A and improved
matrix conditioning.
Also recently, the authors of [12] showed that four admittances Ys are sufficient for extracting noise parameters,
provided that the Ys are well selected. In that work, linear
algebra was employed to determine which admittances guaranteed both a diagonally dominant A in (5) and a solution to (5).

Fig. 3. Smith chart showing regions of Ys constellations based on [12].

IEEE Instrumentation & Measurement Magazine	81
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