UNCERTAINTY ANALYSIS FOR NIST NOISEPARAMETER MEASUREMENTS 

James Randa 

NIST Technical Note 1530 
NIST Technical Note 1530 

UNCERTAINTY ANALYSIS FOR NIST NOISEPARAMETER MEASUREMENTS 

James Randa 

Electromagnetics Division 

Electronics and Electical Engineering Laboratory 

National Institute of Standards and Technology 

325 Broadway 

Boulder, CO 80305 

March 2008 

U.S. Department of Commerce 

Carlos M. Gutierrez, Secretary 

National Institute of Standards and Technology 

James M. Turner, Acting Director 
Certain commercial entities, equipment, or materials may be identified in this document in order 

to describe an experimental procedure or concept adequately. Such identification is not intended 

to imply recommendation or endorsement by the National Institute of Standards and Technology, 

nor is it intended to imply that the entities, materials, or equipment are necessarily the best 

available for the purpose. 

National Institute of Standards U.S. Government Printing Office For Sale by the 

and Technology Washington: 2008 Superintendent of Documents 

Technical Note 1530 U.S. Government Printing Office 

Natl. Inst. Stand. Technol. Stop SSOP, 

Tech. Note 1530 Washington, DC 20402-0001

34 pages (March 2008) Phone: (202) 512-1800 

CODEN: NTNOEF Fax: (202) 512-2250 

 Internet: bookstore.gpo.gov 
CONTENTS

Page

1. INTRODUCTION ..................................................................................................... 1 

2. THEORETICAL FRAMEWORK............................................................................. 2 

 2.1 Noise Parameters in the Wave Representation ................................................... 2 

2.2 Relation to IEEE Noise Parameters .................................................................... 4 

 2.3 Physical Bounds.................................................................................................. 6 

3. MEASUREMENT METHOD................................................................................... 6 

4. UNCERTAINTIES I: OFF_WAFER, COAXIAL REFERENCE PLANES ............ 8 

 4.1 Type_A Uncertainties ......................................................................................... 8 

 4.2 Type_B Uncertainties ......................................................................................... 9 

 4.3 Input Uncertainties.............................................................................................. 11 

 4.4 Sample Results.................................................................................................... 13 

5. UNCERTAINTIES II: ON_WAFER MEASUREMENTS....................................... 15 

 5.1 Differences from the Coaxial Case..................................................................... 15 

 5.2 Input Uncertainties for On_Wafer Measurements.............................................. 15 

 5.3 Uncertainties in On_Wafer Measurements......................................................... 17 

 5.4 Occurrence and Treatment of Unphysical and Other "Bad" Results.................. 20 

 5.5 Sample Results.................................................................................................... 22 

6. SUMMARY............................................................................................................... 22 

7. REFERENCES .......................................................................................................... 23 

APPENDIX A: JACOBIAN MATRIX FOR X → IEEE ...................................................... 25 

APPENDIX B: OVERVIEW OF THE FORTRAN PROGRAM ......................................... 28 

 B.1 MAIN ................................................................................................................. 28 

 B.2 Subroutine SET_MEAS ..................................................................................... 29 

 B.3 Subroutine T_MEAS.......................................................................................... 30 

 B.4 Subroutine FULL_FIT ....................................................................................... 30 

 B.5 Subroutine FWD_FIT ........................................................................................ 31 

 B.6 Subroutine FCN.................................................................................................. 31 

 B.7 Subroutines GAUSDEV and RECTDEV .......................................................... 31 

iii
UNCERTAINTY ANALYSIS FOR NIST NOISE-PARAMETER 

MEASUREMENTS

James Randa 

Electromagnetics Division 

National Institute of Standards and Technology 

Boulder, CO 80305 

The uncertainty analysis is presented for NIST measurements of noise parameters 

of amplifiers and transistors, in both connectorized (coaxial) and on-wafer 

environments. We treat both the X-parameters, which are based on the wave 

representation of the noise correlation matrix, and the traditional IEEE noise 

parameters. An overview of the Monte Carlo program used to evaluate the type-B 

uncertainties is included. 

Keywords: amplifier noise, measurement uncertainty, Monte Carlo, noise 

measurement, noise figure, noise parameters, transistor noise, uncertainty 

1. INTRODUCTION

 Some time ago, what is now the Electromagnetics Division of the National Institute of 

Standards and Technology (NIST) developed the capability to measure noise parameters of 

amplifiers [1,2]. Recently, modified methods and analysis have been developed and have been 

applied both to amplifiers [3] and to transistors [4], the latter in an on-wafer environment. For 

such measurements to be meaningful, the results must be accompanied by corresponding 

uncertainties. Many complications arise in the analysis for these measurements, and it is the 

purpose of this paper to document and perhaps clarify the procedures used to estimate the 

standard uncertainties [5] in the measured noise parameters. 

 The contributions to the standard uncertainty are divided into two groups. Generally, 

type-A uncertainties are those that are determined by statistical means, and type-B are all others. 

The standard or combined uncertainty (uc) in a quantity is the root sum of squares (RSS) of the 

type-A and type-B uncertainties, 

. 22 += uuu BAc (1) 

Noise parameters are computed by performing a least-squares fit to an over-determined system 

of equations obtained by measuring the output noise temperature (or power) for each of a 

number of different input terminations connected to the amplifier or transistor under test. The 

type-A uncertainties can be computed from the covariance matrix of the fitted parameters, but 

the type-B uncertainties require more effort. The uncertainties in the underlying or input 

quantities, such as reflection coefficients, measured noise temperatures, etc., are known or can be 

estimated; but the problem of propagating these underlying uncertainties to compute 

1
uncertainties in the output noise parameters does not admit a simple analytical solution. 

Therefore, a Monte Carlo approach is used for the type-B uncertainties. Matters are further 

complicated by the fact that it is convenient to perform our analysis in terms of one set of noise 

parameters (what we call the X parameters), but we need to express our results and uncertainties 

in terms of a different set (the IEEE noise parameters) that is in near-universal use. 

 In the next section we present the theoretical framework used in the analysis. Section 3 

reviews the measurement methods, both on-wafer and off-wafer. The uncertainty analysis for the 

off-wafer, connectorized case is presented in Section 4, followed by the on-wafer case in Section 

5. Section 6 contains a summary. There are two appendices: Appendix A presents the Jacobian 

matrix for the transformation from our X parameters to the IEEE noise parameters, and Appendix 

B gives an overview of the Fortran program used to evaluate the type-B uncertainties. 

2. THEORETICAL FRAMEWORK

2.1 Noise Parameters in the Wave Representation 

 Our general theoretical framework was set out in Reference [6]. It follows the wave 

representation formulated by Wedge and Rutledge [7]. For a linear two-port, such as an amplifier 

or transistor, the amplitudes of the incident and outgoing waves are related by 

 

= + cSab , (2) 

where a and b are two-dimensional vectors whose elements are the incident and outgoing wave 

amplitudes, as indicated in Fig. 1, S is the usual 2×2 scattering matrix, and c is a twodimensional vector whose elements are amplitudes of the waves emerging from the two ports 

even in the absence of any input waves, due to the intrinsic noise of the device. Although this 

paper deals with noise parameters of both transistors and amplifiers, the figures will be drawn 

with only the amplifier symbol, or with a generic box, and we will use DUT (device under test) 

to refer to either an amplifier or a transistor. We use a normalization in which the wave 

amplitudes squared have dimensions of spectral power density, power per frequency interval. 

When we refer to noise temperature, we take it to mean the available noise spectral power 

density divided by Boltzmann’s constant kB. 

 The noise properties of the two-port are characterized by the (intrinsic) noise correlation 

matrix N, defined by 

, *

jiij ≡ ccN (3) 

2
1 2

b1 a1 a2 b2

Figure 1. Reference planes for amplifier noise parameters. 

 

Figure 2. Forward (a) and reverse (b) measurement configurations. 

TG, ΓG

1 2

TG, ΓG

2 1

(a) (b)

ΓG Γ2 ΓG

Γ1

TG, ΓG

1 2

TG, ΓG

2 1

(a) (b)

TG, ΓG

1 2

TG, ΓG

2 1

(a) (b)

ΓG Γ2 ΓG

Γ1

where the brackets indicate time or ensemble average (assumed to be equal). We find it 

convenient to work with temperature variables corresponding to scaled elements of the noise 

correlation matrix. We define 

, ,/ ( ) ./ * 212112

2

2122

2

11 B ≡ B ≡ B ≡ SccXkScXkcXk (4) 

Dividing each c2 by S21 effectively refers all four noise parameters (X1, X2, ReX12, ImX12) to the 

input plane 1. In fact, it can (and will) be seen that X2 is equal to the effective input noise 

temperature for a reflectionless input termination, X2 = Te,0 , sometimes also called the “50 Ω

noise temperature.” X1 is the intrinsic noise emerging from the input port (1); it is closely related 

to the Trev of Wait and Engen [1], 

.

1 2

11

1

⎟

⎠

⎞ ⎜

⎝

⎛ −

=

S

X Trev (5) 

 Two different measurement configurations are used in the NIST noise-parameter 

measurements: the “forward” configuration of Fig. 2(a) and the “reverse” configuration of Fig. 

2(b). The forward configuration is the one usually used for noise-parameter measurements, but 

the reverse configuration can also be used, and it provides a very good determination of the 

parameter X1. The reverse configuration is particularly useful in the on-wafer case for poorly 

matched transistors [8]. The output noise temperature for the two configurations can be written 

in terms of the scattering and noise parameters of the amplifier and the reflection coefficient ΓG

and noise temperature TG of the source or generator. For the forward configuration, the equation 

is 

3
, 1

Re2

1 1

1

1 11

12

21

2

11 2

11

2

2

2

2

21

2

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

⎥

⎦

⎤ ⎢

⎣

⎡

− ++ − +

−

⎟

⎠

⎞ ⎜

⎝

⎛ −

⎟

⎠

⎞ ⎜

⎝

⎛ −

= S

X XX

S

T

S

S

T

G

G

G

G

G

G

G

Γ

Γ

Γ

Γ

Γ

Γ

Γ

 (6) 

and for the reverse configuration it takes the form 

, 1Re2

1 1

1

1

1

22

*

122112

12

2

22

2112

2

22

2 2

12

2

1

1

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

− ++ − +

−

⎟

⎠

⎞ ⎜

⎝

⎛ −

⎟

⎠

⎞ ⎜

⎝

⎛ −

= S

XSS XX

S

SS T

S

S

T

G

G

G

G

G

G

G

Γ

Γ

Γ

Γ

Γ

Γ

Γ

 (7) 

where Γ2 is the reflection coefficient of the generator-amplifier combination at plane 2 in Fig. 

2(a), and Γ1 is the reflection coefficient of the generator-amplifier combination at plane 1 in Fig. 

2(b). In performing the fits, the parameters to be determined are the X’s and the “reduced gain,” 

defined by G0 ≡ |S21|

2

. All other S-parameters, including the phase of S21, are taken from 

measurements on a commercial vector network analyzer (VNA). 

2.2 Relation to IEEE Noise Parameters 

 Although many different sets of noise parameters have been suggested, the set in most 

widespread use is the IEEE set [9]. It is therefore desirable to present any measurement results in 

terms of the IEEE set, in addition to the results for the X’s. Even within the IEEE set, there are 

several variants; we use the one in which the effective input noise temperature Te is given by 

,

11

2 2

2

min

⎟

⎠

⎞ ⎜

⎝

⎛ −+

− +=

opt G

Gopt

e tTT

Γ Γ

ΓΓ

 (8) 

where ΓG is the reflection coefficient of the input termination at plane 1, and the four noise 

parameters are Tmin, t, and the complex Γopt. The parameter t is related to the noise resistance Rn

by t = 4RnT0/Z0, where T0 = 290 K and Z0 is the reference impedance, commonly taken to be 50 

Ω. 

 To make the connection between the X-parameters and the IEEE noise parameters, we 

force eq. (6) into the form 

( ), 2 = + TTGT Geav (9) 

where Gav is the available gain, 

.

1

1

1 2

11

2

2

2

21

S

S

G

G

G

GS

av

Γ

Γ

Γ −

⎟

⎠

⎞ ⎜

⎝

⎛ −

⎟

⎠

⎞ ⎜

⎝

⎛ −

= (10) 

This yields the expression for Te in terms of the X-parameters, 

4
[ ( ) 1Re ],

1

2

1

1

1

12 * 11 2 2 2

2

11

1 2

2

X XS

S

T X GG

G G

G

G

G

e ΓΓ

Γ Γ

Γ

Γ

Γ −

⎟

⎠

⎞ ⎜

⎝

⎛ −

+

⎟

⎠

⎞ ⎜

⎝

⎛ −

− +

⎟

⎠

⎞ ⎜

⎝

⎛ −

= (11) 

which can be compared with eq. (8) to derive the relationship between the X-parameters and the 

IEEE noise parameters. Happily, Wedge and Rutledge [7] have done the bulk of the algebra for 

us, obtaining the relationship between the noise correlation matrix of eq. (3) and the IEEE noise 

parameters. We then need only use the definitions of eq. (4) to obtain 

( ).

1

1

,

1

,

1

1

1

2

11 *

min1112

2

2

min2

2

2

2 11

11min1

opt

opt opt

opt

opt

opt

opt

St

TSX

t

TX

St

STX

Γ

Γ Γ

Γ

Γ

Γ

Γ

+

− −=

+

+=

+

− ⎟ +

⎠

⎞ ⎜

⎝

⎛ = −

 (12) 

The inverse relations are 

[( ) ]

( ) ,

1

Re2

, 4 11

2

1 1Re2 ,

2

12 * 2 11

2

111

2

2

min

2

12 * 2 11

2

1 11

⎟

⎠

⎞ ⎜

⎝

⎛ +

⎥⎦

⎤ ⎢⎣

⎡ − + −

=

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎝

⎛

−−=

+−++=

opt

opt

opt

X XSXSX

T

XSXt XS

Γ

Γ

η

η Γ (13) 

where 

( )

( ) .

1 Re2

12112

12 * 1 11

2

112

XSX

XSXSX

−

⎟ −+⎠

⎞ ⎜

⎝

⎛ +

η = (14) 

Thus, if the S-parameters are known, the IEEE parameters can be calculated from the Xparameters, and vice versa. Additional work is required to transform the uncertainties in the 

noise parameters. That point is addressed below, in Section 4. 

5
2.3 Physical Bounds 

 Basic physics and mathematics place certain constraints on the noise parameters. Some of 

the constraints are obvious in either the IEEE representation or the X representation. For 

example, Tmin, t, X1, and X2 must all be positive, and 2112 2 +≤ XXX . Some are less obvious, 

such as |η| ≥ 2. This last inequality can be proved by using eq. (4) to write the numerator and 

denominator of eq. (14) in terms of the c wave amplitudes. The numerator can be put in the form 

1 Re2 ( ) ,

2

1

21

112

2 *

21

2

12 * 1 11

2

112 c

S

Sc

S

c XSXSX −+ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ⎟ −+ = ⎠

⎞ ⎜

⎝

⎛ + (15) 

whereas the denominator is given by 

. 1

21

112

*

21

2

12112 ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ =− c

S

Sc

S

c XSX (16) 

That |η| ≥ 2 then follows from the Schwarz inequality. 

 In actual measurements, as well as in the simulations in the Monte Carlo program 

discussed below, it is possible to obtain unphysical results that violate these bounds. This is rare 

when measuring well-matched amplifiers, but it is not uncommon when measuring poorly 

matched, very low-noise transistors on a wafer. Small measurement errors can conspire to yield 

fitted results for the noise parameters that are slightly (or even dramatically) outside the physical 

bound rather than slightly within it. One approach to this problem would be to constrain the 

fitting procedure so that the fitting parameters range only over physically allowed values. 

Instead, we choose to perform an unconstrained fit and to test whether the result satisfies the 

physical bounds. This has the advantage of alerting us when the measurement results prefer an 

unphysical solution for the noise parameters, rather than just finding the best physical solution. 

(It is also considerably easier than imposing this set of constraints in the fit.) We will return to 

this point when we discuss the Monte Carlo program below. 

3. MEASUREMENT METHOD

 Basically, the measurement method for either off-wafer (“connectorized”) amplifiers or 

on-wafer transistors (or amplifiers) consists of connecting a series of different, known 

terminations to one of the ports of the device under test and measuring the resulting noise 

temperature at the other port. The noise parameters are then computed by performing a weighted 

least-squares fit using eqs. (6) and (7). Most or all of the measurements are performed in the 

forward configuration of Fig. 2(a), in which the known termination (ΓG,i, TG,i) is connected to the 

input port, and the noise temperature (T2,i) is measured at the output port. In the case of poorly 

matched devices, typically transistors on a wafer, it is advantageous to also perform at least one 

reverse measurement, as in Fig. 2(b). The termination used for the reverse measurement is 

typically an ambient-temperature matched load. In principle, more than one reverse measurement 

6
could be used; we are currently investigating the efficacy of that. For well-matched amplifiers, 

the reverse measurement does not improve the noise-parameter uncertainties [6], and so we do 

not use it in the fit for the noise parameters. In such cases, we often perform a reverse 

measurement and use it as a check, comparing the measured value for T1 to the value predicted 

by eq. (7) with the values of the noise parameters that have been measured. 

 The set of input terminations typically comprises about eight ambient-temperature 

terminations and one well-matched non-ambient noise source (usually hot). The input states are 

discrete reflective or matched loads, connected manually to the appropriate port; we do not use a 

tuner. We are currently developing an automated unit to switch automatically among the 

different input terminations [10]. The ambient-temperature terminations include one matched 

load and several reflective terminations chosen to provide adequate coverage of the complex-Γ

unit circle. Although we have not made a great effort to optimize the choice of input reflection 

coefficients, simulations have shown that we would gain little by using an optimized set, such as 

is used by some commercial systems. The reflection coefficients of the input terminations and 

the S-parameters of the device under test are measured on a VNA. As noted above, the quantity 

G0 ≡ |S21|

2

 is treated as a free parameter in the fit; the fitted value usually agrees very well with 

the value obtained from the VNA measurements. Since the same set of terminations is used 

across the entire range of measurement frequencies, the reflection coefficients of the individual 

terminations shift with frequency. The patterns of the reflection coefficients in the complex plane 

at three representative frequencies in the 8 GHz to 12 GHz band are shown in Fig. 3. 

 

 The noise temperatures of the ambient input terminations are taken to be the noise 

temperature corresponding to 296.15 K, 

( )

,

1 )15.296/( , − =iG khf B ⋅ K

e

hf T (17) 

where TG,i is the noise temperature of termination i, assumed to be one of the ambienttemperature terminations. The noise temperature of the non-ambient input termination is 

measured on the NIST coaxial radiometer NFRad [11], as are all output noise temperatures. 

1

2

3

4

5

6

7 8

12 GHz

1

2

3

4

5

6

7

8

10 GHz

1

2

3

4

5

6

7

8

8 GHz

1

2

3

4

5

6

7 8

12 GHz

1

2

3

4

5

6

7

8

10 GHz

1

2

3

4

5

6

7

8

8 GHz

Figure 3. Reflection coefficients of input terminations relative to the unit circle at three 

frequencies. 

7
 Measurement of the output noise temperature for each input termination i yields a set of 

measured values for T2, which we call T2,i(meas), and possibly one or more measured values for 

the reverse configuration, T1, which we call T1,i(meas). Each measured value is accompanied by 

a standard uncertainty, u(T2,i) or u(T1,i). For each measurement, we also have a corresponding 

expression for the output noise temperature from eq. (6) or (7). The reflection coefficients and Sparameters in these expressions have been measured, and so the only remaining unknowns are 

the noise parameters. We refer to the expressions as T2,i(X) and T1,i(X), where the X is understood 

to include G0, as well as X’s. From measurements of the forward configuration, we obtain a set 

of equations of the form 

),()( ,2 i = ,2 i XTmeasT (18) 

for i = 1,...,Nfwd, where Nfwd is the number of forward measurements. If we include a 

measurement in the reverse configuration, we obtain an additional equation of the form 

).()( ,1 i = ,1 i XTmeasT (19) 

These equations can be used to determine the noise parameters and G0. To do so, we perform a 

weighted, least-squares fit, minimizing the function 

, )(

)()(

)(

)()(

,

2

,1

,1 ,1

,

2

,2

2 ,2 ,2 ∑ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ − ∑ + ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ − =

revi i

i i

fwdi i

i i

Tu

XTmeasT

Tu

XTmeasT

χ (20) 

where the first sum runs over the forward measurements, and the second sum over the reverse 

measurements. For well-matched, packaged amplifiers, we do not typically use the reverse 

measurements in the determination of the noise parameters, and consequently the second sum is 

absent. In that case, the equations are all of the form of eq. (6), which can easily be put into 

linear form by defining the variables Z1 = G0X1, Z2 = G0X2, Z12 = G0X12, Z5 = G0. In terms of the 

Z’s, the set of equations becomes linear. A commercial fitting program is used to minimize χ

2

. If 

only forward measurements are used, a linear fitting program would suffice, but if reverse 

measurements are present, the problem cannot be reduced to a set of linear equations. In practice, 

we use the nonlinear fitting program in all cases, because it is the only one that provides the full 

covariance matrix, which we need in order to compute the uncertainties in the IEEE noise 

parameters. The linear fitting program is used on the (sub)set of forward measurements to obtain 

an initial guess for input to the nonlinear fitting program. The nonlinear fitting program uses a 

modified Levenberg-Marquardt method (see, e.g., [12]) to solve a nonlinear regression model 

using least squares. 

4. UNCERTAINTIES I: OFF-WAFER, COAXIAL REFERENCE PLANES

4.1 Type-A Uncertainties 

 Type-A uncertainties are those that are evaluated by a statistical analysis of a series of 

observations [5]. In our case, where the parameters are determined by a fit to an over-determined 

8
system of equations, the packaged fitting program returns the covariance matrix Vij for the fitting 

parameters, and the type-A uncertainties are given by the square roots of the diagonal elements, 

XVXu )()( = iiiA , (21) 

where Xi represents any of the five fitting parameters (X’s and G0). 

 Since the parameters in most common use are the IEEE parameters, we also need to 

evaluate the uncertainties in them. Because the fit is performed in terms of the X-parameters, the 

fitting program does not return values or a covariance matrix for the IEEE parameters. The IEEE 

parameter values are computed from the X-parameters by use of eqs. (13) and (14), and the 

covariance matrix for the IEEE parameters must be computed from the X-parameter covariance 

matrix by use of the Jacobian matrix for the transformation. If we use Ii to represent one of the 

five IEEE parameters (including G0), then the type-A uncertainties in the IEEE parameters are 

given by 

 

∑ ∂

∂

∂

∂ =

=

′′ = ′′ ′′

5

1,

)( .)(

,)()(

ji

ji

j

j

i

i

ij

iiiA

XV

X

I

X

I IEEEV

IEEEVIu

 (22) 

Calculation of the elements of the Jacobian matrix (∂Ii/∂Xi’) is straightforward but tedious, and 

the results are lengthy and unenlightening. They are relegated to Appendix A. 

4.2 Type-B Uncertainties 

 Type-B uncertainties are those that are evaluated by any means other than the statistical 

analysis of a series of observations. In our noise-parameter measurements, these uncertainties 

arise due to possible errors in the input parameters, which include the amplifier S-parameters, all 

measured reflection coefficients, the temperature of the ambient-temperature terminations, the 

noise temperature of the non-ambient termination(s), and the measured output noise 

temperatures. We know or can estimate all these input uncertainties, but the problem of how they 

propagate into uncertainties in the noise-parameters does not admit a simple analytical solution. 

We therefore adopt a Monte Carlo approach, a good description of which can be found in 

Reference [13]. Our Monte Carlo procedure has been described previously in References [6] and 

[14]. Here we present a somewhat more detailed and updated account. 

 The Monte Carlo uncertainty evaluation is built around the generation of simulated 

results for the measurements that were actually performed. These simulated measurement results 

are analyzed in the same way as are the real data, yielding simulated values for the noise 

parameters. This process is repeated a large number of times, and the distribution of values 

obtained for each noise parameter is used to compute its uncertainty. One full simulation of all 

the measurements that were actually performed will be referred to as a “set” of measurements, 

and the number of simulated measurement sets will be denoted Nsim. We choose Nsim to be large 

enough that the computed uncertainties are approximately independent of Nset. A value of 

Nsim = 10,000 is usually more than sufficient, but for poorly matched transistors, it is 

occasionally necessary to use larger values. In all cases, we use a large enough value that any 

9
further increase in Nsim changes the uncertainties by at most 10 % of their values. In most cases, 

the uncertainties are within a few percent of their asymptotic values. 

 Each simulated measurement is performed by choosing a number randomly from the 

probability distribution for that physical quantity. The distribution is constrained to have an 

average value equal to the true value and a standard deviation equal to the standard uncertainty 

for that quantity. We use the real measurement results as the true values. Normal distributions 

are assumed for all quantities except for the ambient temperature, for which a rectangular 

distribution is used, to better model the effect of a thermostatically controlled laboratory 

temperature. (The program also allows the user to choose a normal distribution for the ambient 

temperature, but we do not usually use this. It makes virtually no difference in practice.) For 

each measurement set, we first simulate measurement of the input parameters; these include the 

amplifier S-parameters, the reflection coefficients of all input terminations, the noise 

temperatures of all terminations, and the output reflection coefficients, Γ2,i in Fig. (2), if they are 

measured. (There is an option of measuring Γ2,i directly or of computing it by cascading ΓG,i with 

the S-parameters.) To simulate the measurement of the output noise temperatures, we must first 

compute the true values from eq. (6) (for the forward configuration) or eq. (7) (for the reverse 

configuration), using the true values for all quantities on the right hand side. 

 To simulate a measurement (of a quantity y, for example) we merely add a random error 

term δy to the true value y0, 

,)( 0 = + δyysimy (23) 

where δy is chosen randomly from the appropriate distribution (normal or rectangular), with 

<δy> = 0 and <δy

2

> = u(y)2, where u(y) is the standard uncertainty in y. In practice, we do this by 

choosing a random number from the standard normal distribution (i.e., mean of zero, variance of 

1) and multiplying by the standard uncertainty. For complex quantities, the real and imaginary 

parts are simulated separately. In principle we could include correlation between errors in the 

real and imaginary parts or work with uncertainties in magnitude and phase, but we have not yet 

done so. 

 Because the input variables include sets of variables that are all measured in the same 

manner, correlated errors can and do occur, and the correlations can cause significant effects in 

the noise-parameter uncertainties [6,15]. We therefore include correlated errors in the 

simulations. The correlations that we include are among all the measured reflection coefficients 

and the S-parameters, among all the output noise temperatures, and among the temperatures of 

the ambient-temperature terminations. If there is more than one nonambient input termination, 

we include correlations between them. The way that correlated errors are simulated is to add an 

additional term to eq. (23). If yi and yj are two physical quantities whose errors are correlated (at 

least in part), we write 

)( ).()(

)( ),()(

0,

0,

coryuncyysimy

coryuncyysimy

j jj j

i ii i

δ δ

δ δ

+= +

= + +

 (24) 

10
The correlated (cor) and uncorrelated (unc) error terms satisfy <δyi(cor)> = <δyi(unc)> = 0, 

<δyi(cor)

2

 + δyi(unc)

2

> = ucor(yi)

2

 + uunc(yi)

2

 = u(yi)

2

, and similarly for yj. The random errors for 

the uncorrelated error terms are chosen independently, whereas the same random number (scaled 

by the appropriate uncertainty in each case) is used for both correlated error terms. The 

correlation coefficient is given by 

( )( )

. )()(

)()(

)()(

)()(

)()(

)()()()(

ji

jcoricor

ji

ji

ji

i i j j

ij

yuyu

yuyu

yuyu

corycory

yuyu

coryuncycoryuncy

= =

+ + =

δδ

δδδ δ

ρ

 (25) 

 Each set of simulated noise-temp measurements is analyzed in the same manner as a set 

of real measurements: we perform a least-squares fit to the measurement results and obtain a set 

of the noise parameters (X1, X2, X12, G0). From these and the simulated measurement results for 

the amplifier’s S-parameters, we compute the set of IEEE noise parameters (Tmin, t, Γopt, G0). 

This is done for each of the Nsim sets of simulated measurements. The average and standard 

deviation of the measured values for each parameter (X and IEEE) are computed. The (type-B) 

uncertainty in a single measurement of a parameter is then computed by combining the standard 

deviation in quadrature with the difference between the average and the true value. This is just 

the root-mean-square error (RMSE) of the sample, 

,)()()()( 2

B true = = −+ yyyVaryRMSEyu (26) 

where y is any of the noise parameters, and Var(y) is the variance of the sample of simulated 

results for y. For the complex quantities X12 and Γopt, statistics are done on the real and imaginary 

parts separately. The fact that y is not equal to the true value, may be unsettling at first, but such 

are the vagaries of nonlinear functions.

4.3 Input Uncertainties 

 The simulations require as input not just the true values of all parameters, but also the 

standard uncertainties in the parameters that are directly measured, i.e., the amplifier Sparameters, the reflection coefficients of all terminations, the noise temperatures of all 

terminations, the output reflection coefficient, Γ2,i in Fig. (2a), if it is measured, and the output 

noise temperatures. These input uncertainties (or the parameters that determine the uncertainties) 

are read into the Monte Carlo program along with the input data, so that they can be changed at 

any time, but we have a standard set of uncertainties that we usually use. This set is subject to 

change in the future, as we improve our knowledge of the uncertainties, and especially of the 

correlated errors. 

 The reflection coefficients ΓG,i and Γ2,i (if it is measured), are measured on a commercial 

vector network analyzer (VNA). Since all reflection coefficients are measured with the same 

calibration on the same VNA, there is significant correlation among the errors of the different 

reflection coefficients. The connector used on the radiometer port is type PC-7, and so we 

11
assume the packaged amplifier also has PC-7 connectors. (If the amplifier has some other 

connector, requiring use of adapters, the uncertainty analysis is similar in form to the on-wafer 

case, which is treated in the next section.) The uncertainties in the reflection-coefficient 

measurements can depend on the magnitude of the reflection coefficient; usually the 

uncertainties are larger for larger reflection coefficients. Our program uses one value for small 

|Γ| (less than or equal to 0.5) and another for large |Γ| (greater than 0.5). In the program, we work 

in terms of ucor and uunc, defined after eq. (24) above, and then compute u and ρ from them. The 

values typically used for small |Γ| are ucor = 0.0025 and uunc = 0.001, corresponding to u(Γ) ≈

0.002693 and ρ ≈ 0.8621. These uncertainties are somewhat larger than the manufacturer’s 

specifications, reflecting our own past experience with such measurements [15]. For large |Γ| we 

use ucor = 0.004 and uunc = 0.001, corresponding to u(Γ) ≈ 0.004123 and ρ ≈ 0.9412. The 

uncorrelated part of the uncertainty, uunc = 0.001, is due primarily to connector 

(non)repeatability. The same u(Γ) is used for both the real and the imaginary parts of each 

reflection coefficient, u(ReΓ) = u(ImΓ) = u(Γ). The small S-parameters (i.e., S11, S22, and S12) 

have their errors treated the same way as the reflection coefficients, including the correlations. 

For S21 we use u(ReS21) = u(ImS21) = 0.01. This value was obtained by an extrapolation of the 

VNA specifications, which do not extend to S21 magnitudes as large as we encounter. The input 

uncertainties in S21 are not very important because the magnitude of S21 is treated as a fitting 

parameter, G0 = |S21|

2

. 

 The ambient temperature in the laboratory is set at 296.15 K and is kept within 0.5 K of 

this value. We therefore use a rectangular distribution extending from 295.65 K to 296.65 K for 

the ambient-temperature terminations. Because several hours intervene between measurements 

on the amplifier with different terminations, there is essentially no correlation between the errors 

in the temperatures of different ambient-temperature terminations. 

 In actual measurements of the output noise temperatures on NFRad [10], the 

measurement software evaluates and writes the uncertainty in the noise-temperature 

measurement. The uncertainty depends on a number of details, such as the reflection coefficient 

of the DUT, the magnitude of the noise temperature being measured, statistics of the multiple 

repeat measurements, and characteristics of the radiometer at the measurement frequency. Rather 

than perform a full evaluation of the uncertainty for each simulated measurement, we use the 

following approximate parameterization for the uncertainty in the noise-temperature 

measurements, 

005.02.0)( ( ), Tu meas = + − TTK ameas (27) 

where Ta is the noise temperature corresponding to the ambient temperature of 296.15. This form 

is adequate provided that the reflection coefficient of the DUT is less than about 0.2. For larger 

reflection coefficients, further refinement is required, as described in Section 5. 

 There is significant correlation among the measurements of the output noise temperatures 

for the different terminations. An examination of the uncertainty analysis that was performed for 

noise-temperature measurements in coaxial lines [10,16] leads to the conclusion that most errors 

are correlated if all the reflection coefficients are measured with the same VNA calibration, as 

they typically are in the amplifier noise-parameter measurements. The major source of 

uncorrelated error is the connector (non)repeatability. The magnitude of the connector error 

12
relative to the correlated errors would lead to a value ρ ≈ 0.98. Because correlated errors in the 

noise-temperature measurements often lead to much smaller uncertainties [6], we are reluctant to 

use such a large correlation coefficient without being very sure of it. Until we have better 

knowledge of these correlations, we will use ρ = 0.64, which corresponds to ucor/uunc = 4/3. This 

may lead to a small overestimate of the uncertainty, but we prefer to err on the side of caution. 

 If two nonambient input terminations are used, there is a correlation between the errors in 

their noise temperatures if they are both measured on the same system, as is the case at NIST. 

We have investigated the case of one hot and one cold input noise source, and we find that if 

both are measured on NFRad, the correlation coefficient is ρ ≈ −0.115. The negative sign occurs 

because the correlated errors have different sign for noise temperatures above and below ambient 

temperature. In principle, there is also a correlation between errors in measuring the output noise 

temperature from the amplifier and in measuring the nonambient input temperature(s), since both 

types of measurements are performed on the same radiometer (NFRad) with the same standards. 

In NIST measurements, this effect is usually small for the coaxial amplifier case because the 

nonambient input termination is usually measured with a different VNA calibration, often many 

months before the noise-parameter measurements. We have therefore ignored this correlation in 

the Monte Carlo program for the type-B uncertainties in the noise parameters for coaxial 

amplifiers. At many industrial laboratories, however, the same diode noise source that is used to 

calibrate the radiometer is also used as the hot input noise source. Furthermore, its noise 

temperature is one of the major sources of uncertainty. In this case, the correlation between 

errors in the hot input noise temperature and the measured output noise temperatures can be 

important. 

4.4 Sample Results 

 We have recently measured the noise parameters of a low-noise amplifier (LNA) from 8 

GHz to 12 GHz. This was done to test an automated variable-termination unit (VTU) [10], which 

can be used to automatically switch among a series of input terminations, greatly reducing the 

time required for the measurements. For these measurements, 11 different input terminations 

were used in the forward configuration, of which one was hot (between 1000 K and 1100 K, 

depending on frequency), one cold (about 100 K to 110 K) and the remainder at room 

temperature (about 296.15 K). In addition, one reverse measurement was done, with a matched 

load on the output of the LNA. 

 Results for the X parameters are shown in Fig. 4. The quantity X2 is the equivalent input 

noise temperature for a matched (reflectionless) input termination, often called the 50 Ω noise 

temperature. There are two sets of uncertainties plotted for X2, to show the effect of an output 

attenuator on the measurements. The actual measurements were performed with a 20 dB 

attenuator on the amplifier’s output (for the forward configuration), in order to keep the output 

noise temperature in the measurement range of NFRad. Consequently, the noise at the 

radiometer/DUT measurement plane is weighted 99 to 1 in favor of the attenuator over the 

amplifier, and the uncertainty suffers significantly. The larger uncertainties on X2 in Fig. 4 

correspond to the actual measurements, and the smaller uncertainties represent what could be 

achieved if no attenuator were needed. 

13
 7 8 9 10 11 12 13

f(GHz)

-50

0

50

100

150

200

X(K)

X1

X2 = Te,0

ReX12

ImX12

Figure 4. Results and uncertainties for X parameters. The smaller set of uncertainties for X2

applies to the case with no output attenuator in the measurements. 

 Another noteworthy point in Fig. 4 is the different sizes of the uncertainties for X1 at 

different frequencies; they are much larger at 10 GHz and 11 GHz than at the other frequencies. 

A measurement of the reverse configuration gives the best determination of X1, and one reverse 

measurement was made at each frequency. However, at some frequencies it is not possible to 

obtain a good least-squares fit when all the measured terminations are included. Presumably this 

is due to one (or occasionally two) bad measurements (which contribute disproportionately to 

χ

2

). In those cases, one (or two, if necessary) terminations are deleted, and at 10 GHz and 11 

GHz it was necessary to exclude the reverse measurement, resulting in larger uncertainties for 

X1. 

 In Fig. 5(a–f) we show the results for G0 and the IEEE noise parameters. Because it is in 

common use, we have also shown the minimum noise figure in dB, which is related to Tmin by 

. As in Fig. 4, the two sets of uncertainties correspond to the 

actual measurements (larger uncertainties) and the case with no output attenuator (smaller 

uncertainties). For the phase of Γopt, Fig. 5(f), we show only the actual uncertainties, since they 

are so small. As for the X parameters, the use of the attenuator causes a major increase in the 

uncertainties. The approximate values for the actual uncertainties are: 0.25 dB for G0, 0.10 dB to 

0.16 dB for Fmin, 10 K to 17 K for Tmin, 6 Ω to 12 Ω for Rn, 0.013 to 0.023 for |Γopt|, and 0.8° to 

1.0° for φopt. If an attenuator had not been necessary, these uncertainties would have been 0.05 

dB for G0, 0.03 dB for Fmin, 2.5 K to 3.5 K for Tmin, 0.08 Ω to 0.16 Ω for Rn, 0.003 to 0.007 for 

|Γopt|, and 0.2° to 0.5° for φopt. 

( min 10 0min dBF = + /1log10)( TT )

14
 Fig. 5(a) Results and uncertainties for G0. Fig. 5(b) Results and uncertainties for Fmin. 

 Fig. 5(c) Results and uncertainties for Tmin. Fig. 5(d) Results and uncertainties for Rn. 

 

Fig. 5(e) Results and uncertainties for|Γopt|. Fig. 5(f) Results and uncertainties forφopt. 

7 8 9 10 11 12 13

f(GHz)

30

30.5

31

31.5

32

32.5

33

33.5

34

34.5

35

G0(dB)

7 8 9 10 11 12 13

f(GHz)

0

0.5

1

1.5

2

Fmin(dB)

7 8 9 10 11 12 13

f(GHz)

110

120

130

140

150

160

170

Tmin(K)

5

6

7

8

9

10

11

12

13

Rn(Ω)

7 8 9 10 11 12 13

f(GHz)

7 8 9 10 11 12 13 7 8 9 10 11 12 1

-180

-150

-120

-90

-60

-30

0

30

60

90

120

150

180

φopt

0

0.05

0.1

0.15

0.2

|Γopt|

3 f(GHz) f(GHz)

15
5. UNCERTAINTIES II: ON-WAFER MEASUREMENTS

5.1 Differences from the Coaxial Case 

 The measurement method for on-wafer noise-parameters is similar in principle to the 

method for measuring coaxial amplifiers, but additional complications arise because the 

measurement planes of interest are on-wafer, as indicated in Fig. 6. The reference planes for the 

DUT noise parameters are planes 1 and 2. The locations of the reference planes are chosen to 

assure single-mode conditions in the calibration process. An expanded view of the on-wafer 

structure for microstrip lines is shown in Fig. 7. We do not attempt to de-embed down to the 

transistor reference planes T1 and T2, as is often done in industry. 

 In order to use eqs. (6), (7), (18), and (19) between planes 1 and 2, we need the reflection 

coefficients, S-parameters, and noise temperatures at those planes. A two-tier calibration is 

performed using a multiline TRL set of calibration standards [17,18] that has been fabricated on 

the wafer. The NIST MultiCal® software is used to perform the two-tier calibration. The 

calibrated VNA is then used to measure the reflection coefficients and transistor S-parameters at 

planes 1 and 2, as in the amplifier case in the preceding section. The output noise temperatures 

are measured by NFRad at plane 2′ and corrected to plane 2, as described in Section 5.3 below. 

Similarly, the non-ambient input noise temperature is measured by NFRad using the 

configuration of Fig. 6, but with the on-wafer device replaced by a through line. Again, NFRad 

measures at plane 2′, and from that we compute the noise temperature at plane 1. 

 The uncertainty analysis for measurement of the noise parameters of on-wafer transistors 

is thus similar to that for measurements on packaged amplifiers, but it presents a few additional 

complications. Because the measurement planes of interest are on the wafer, we must 

characterize and correct for the effects of the probes, which introduces additional uncertainty. 

Also, the uncertainties in VNA and noise measurements are different on a wafer from what they 

are in coaxial lines, and therefore the input uncertainties are different for the on-wafer case. The 

third complication is that the transistor may be very poorly matched, leading to relatively large 

values of the reflection coefficient at its output. This requires that we refine our estimate of the 

uncertainty in measuring the output noise temperature. It also requires us to adopt a prescription 

for handling unphysical results in the simulations. We treat each of these complications in turn in 

the following subsections. 

5.2 Input Uncertainties for On-Wafer Measurements 

 The uncertainty in the noise temperatures of the ambient-temperature terminations is the 

same as in the coaxial case above, and the uncertainties in the output noise temperatures and the 

nonambient input noise temperature are treated in the next subsection. Reflection coefficients 

and S-parameters measured at on-wafer reference planes have larger uncertainties than those 

measured at reference planes in coaxial lines. The input uncertainties that we use for the onwafer reflection coefficients and S-parameters (other than |S21|) are uunc = 0.004 and ucor = 0.003, 

corresponding to u(Γ) = 0.005 and ρ = 0.36. For on-wafer measurements, we (currently) use the 

same uncertainties for large and small |Γ|. 

16
Figure 6. Reference planes for on-wafer measurements. 

Figure 7. Detail of reference planes for transistor in microstrip. 

Radiometer Probe 2 Probe 1

2′

2 1

i

1′ DUT

Γ2′,i

Γ2,i

Γ1,i

Γ1′,i

Radiometer Probe 2 Probe 1

2′

2 1

i

1′ DUT

Radiometer Probe 2 Probe 1

2′

2 1

i

1′ DUT

Γ2′,i

Γ2,i

Γ1,i

Γ1′,i

12 μm back from

center of Thru

P 2

2

12 um

M T2 T1 1

1

12 μm back from

center of Thru

P 2

2

12 um

M T2 T1 1

1

5.3 Uncertainties in On-Wafer Noise Temperatures 

 The measurement planes of interest are on the wafer, as indicated in Fig. 6, whereas the 

noise-temperature measurements are performed by NFRad at plane 2′. We must therefore correct 

for the effect of Probe 2 to get the noise temperatures at plane 2. The S-parameters of the probe 

are determined in the two-tier calibration, as described in Section 5.1. To obtain the noise 

temperature at plane 2 T2,i from the noise temperature at plane 2′ T2′,i , we treat the probe as an 

adapter, characterized by its available-power ratio α2′2,i , 

,

1 1

1

2

,2

2

22,2

2

,2

2

22

,22

⎟

⎠

⎞ ⎜

⎝

⎛ − −

⎟

⎠

⎞ ⎜

⎝

⎛ −

=

′

′

′

i i

i

i

S

S

Γ Γ

Γ

α (28) 

where S2′2 is the S-parameter of probe 2 from plane 2′ to plane 2, S22 is the reflection S-parameter 

of probe 2 at plane 2 (what would normally be called S11), Γ2,i is the reflection coefficient at 

plane 2 (from the transistor), and Γ2′,i is the reflection coefficient at plane 2′ (from the probe). 

17
Knowing α2′2,i , we compute the output noise temperature at plane 2 in terms of T2′,i in the usual 

manner, 

( ) , 1

,22

,2 ,22

,2

i

i ai

i

T T

T ′

′ − − ′ = α

α

 (29) 

where Ta is the noise temperature of the probe, which is assumed to be at ambient temperature. 

 From eq. (29) we see that in order to compute the uncertainty in the on-wafer output 

noise temperature T2,i, we need the uncertainties in Ta, T2′,i , and α2′2,i . The uncertainty in Ta is 

the same as in the case of the coaxial amplifier measurements, treated above. The probe effect, of 

course, was absent from the treatment of the preceding section, and there is also a new 

complication with the noise measurement at the coaxial plane T2′,i. Because the on-wafer 

transistor may have a relatively large value of |S22|, the reflection coefficient at the coaxial 

measurement plane Γ2′,i can also be large. For values of |Γ2′,i | larger than about 0.2, the 

uncertainty in T2′,i increases, and the approximation of eq. (27) is no longer adequate. This occurs 

because as |Γ2′,i | increases, the mismatch correction increases, and a larger mismatch correction 

magnifies any error in the measurement of the reflection coefficient. Examination of the 

radiometer equation for NFRad [11] reveals that (T2′,i − Ta) ∝ (1 − |Γ2′,i |

2

)

−1

. Because eq. (28) 

indicates that α2′2,i ∝ (1 − |Γ2′,i |

2

)

−1

 as well, we can effect some simplification by regrouping eq. 

(29), 

( )

, 1

1

,2

,2,2

ai

i aai

TT

TTTT

+′

′ =

+−=

′

′

α

α (30) 

where we have suppressed the subscripts on α for convenience, and pulled out and cancelled 

factors of (1 − |Γ2′,i |

2

) by defining α ′ ≡ (1 − |Γ2′,i |

2

)α and T2′,i′ ≡ (1 − |Γ2′,i |

2

)× (T2′,i − Ta). The 

advantages of the form in eq. (30) are that it removes the necessity of dealing with important 

correlations between the errors in α and T2′,i, and that we can use eq. (27) for the uncertainty in 

T2′,i′. From the usual rules for propagation of uncertainty [5], the uncertainty in T2,i can be written 

as 

,)()( )( )( 1 )( 22

2

2

,2

,2

2

2 ,2

2

a

ai

i i Tuu

TT

Tu Tu ′ + ′

− + ′

′ = ′ α

α α

 (31) 

where we have used T2′,i′/α ′ = T2,i − Ta in the second term on the right side. 

 As already mentioned, u(T2′,i′) and u(Ta) can be treated as in the coaxial amplifier case 

above. For α ′ we use its definition and eq. (28), along with the propagation of uncertainty, to 

write 

18
,)(

1

2

)( 2 )( ,2

2

2

2

,2

,2

22

2

2

22

2

i

i

i Su u

S

u Γ

Γ

α Γα

α

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎝

⎛

−

′ + ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ′ ′ ≈ ′ ′

 (32) 

where a small term has been discarded. The uncertainty for on-wafer reflection-coefficient 

measurements was discussed in the preceding subsection. The relevant numbers for use in eq. 

(32) are u(|S2′2|) = u(|Γ2,i|) = 0.005. 

 A final complication is that we sometimes need to use an attenuator at the output of probe 

2 (plane 2′) in order to keep the measured noise temperature within the dynamic range of the 

radiometer NFRad. If an output attenuator is present, it can be lumped with the output probe, and 

the combined available power ratio is given by the product of the available power ratios of the 

attenuator and the probe, αtotal = αattαprobe. In forming α′, Γ2′,i is to be taken at the attenuator 

output, between the attenuator and the radiometer. From eq. (31) we can see that using an 

attenuator degrades the uncertainty somewhat due to the smaller value for α′. There is no effect 

on the second and third terms on the right hand side because u

2

(α′) ∝ α′ 2, but the first term 

increases somewhat. 

 Pulling everything together, we use eq. (31) for the uncertainty u(T2,i) in noisetemperature measurements at the on-wafer reference plane 2. We get u(T2′,i′) from eq. (27) and 

u(α′) from eq. (32). Equation (31) applies not only to the measured noise temperatures at the 

DUT output, but also to the measurement of the hot (or cold) input noise temperature. We have 

checked that the uncertainties obtained in this manner are consistent with the uncertainties that 

were confirmed in our earlier work [19,20] on verification and consistency checks for on-wafer 

noise-temperature measurements. 

 There is still the question of correlations among the errors in the measurements of the 

output noise temperatures T2,i and also between the errors in T2,i and those in the noise 

temperature of the hot or cold input termination. Of the three terms on the right side of eq. (31), 

the first term corresponds to the highly correlated errors in coaxial noise-temperature 

measurements (ρ = 0.64), considered in Section 4.3 above. The second term is comparable in 

size, but the errors contributing to it are mostly uncorrelated, because the sign and magnitude of 

α′ vary randomly from one termination to another. The third term also corresponds to 

uncorrelated errors, but it is small relative to the other two terms. These considerations lead us to 

use ρij = 0.36 for the correlation coefficient between errors in measurements of the on-wafer 

noise temperatures T2,i and T2,j. As for the correlation between measurements of the output noise 

temperatures and of the on-wafer noise temperatures of the nonambient input terminations, we 

expect this correlation to be larger than in the coaxial case (Section 4.3) because the on-wafer 

measurements are performed with the same VNA calibration, on the same NFRad port, with not 

too long a time between. However, at present, we neglect this correlation, as we do in the coaxial 

case. We hope to include this effect in the future. 

 

 For perspective, it should be remembered that u(T2,i) in eq. (31) is the uncertainty that is 

used for the simulated output noise-temperature measurement for a given termination i. Each 

output noise temperature in each set of simulated measurements has such an associated 

uncertainty, and these uncertainties are used to determine the weighting factors in the least19
squares fit in eq. (20). That fit then produces the simulated measured noise parameters for that 

set, and there are Nsim such simulated measurement sets. 

5.4 Occurrence and Treatment of Unphysical and Other “Bad” Results 

 The final additional complication with which we must deal in on-wafer simulations is the 

occurrence of unphysical results, those that violate the physical bounds discussed in Section 2.3. 

For the CMOS transistors that we have measured [4,21], it is quite possible to obtain unphysical 

results in the simulated measurements (as well as in real measurements). In a real measurement, 

we would normally discard such results and repeat the measurements. In the uncertainty 

program, we compute two sets of uncertainties, one with the unphysical results discarded, as we 

would usually do in practice, and one with the unphysical results included. One reason for 

evaluating the uncertainties with the unphysical results included is that if real measurements 

produce an unphysical result, we will probably want to evaluate the uncertainties in those results, 

in order to see whether they are consistent with physical results within the measurement 

uncertainties. To evaluate the uncertainties in unphysical real measurement results, we need to 

include the simulated unphysical results in the calculation. 

 We impose two further cuts on the set of simulated results. If the best fit is not good 

enough, we discard that set of simulated measurements, as we would with a set of real 

measurements that did not admit a good enough fit. For this purpose we define “good enough” 

by χ

2

/ν ≤ 1, where ν is the number of degrees of freedom in the fit. In several instances, we have 

also evaluated the uncertainties with a cut of χ

2

/ν ≤ 1.5, and it makes little difference. The other 

cut is on the uncertainty in Γopt. It sometimes happens that an acceptable fit is obtained, but that 

the variance of Γopt is enormous (corresponding to an extremely flat minimum in this variable). A 

real measurement set with this property would be discarded, and we do so in the simulation as 

well. The maximum standard deviation allowed for either the real or imaginary part of Γopt is 1. 

Figure 8. Results and uncertainties for X parameters of a transistor on a wafer. 

0 510

f(GHz)

-100

-50

0

50

100

150

200

X(K)

X1

X2

ReX12

ImX12

20
 Fig. 9(a) Results and uncertainties for G0. Fig. 9(b) Results and uncertainties for 

 Fmin(dB). 

0 510

f(GHz)

0

5

10

15

20

G0(dB)

Noise Result

VNA Result

0 510

-1

-0.5

0

0.5

1

1.5

2

B)

 Fig. 9(c) Results and uncertainties for Tmin. Fig. 9(d) Results and uncertainties for Rn. 

 Fig. 9(e) Results and uncertainties for|Γopt|. Fig. 9(f) Results and uncertainties forφopt. 

0 1 2 3 4 5 6 7 8 9 10 11 12

f(GHz)

-40

-30

-20

-10

0

10

20

30

40

50

60

70

80

90

100

110

120

Tmin(K)

0 1 2 3 4 5 6 7 8 9 10 11 12

f(GHz)

0

5

10

15

20

Rn(Ω)

0 1 2 3 4 5 6 7 8 9 10 11 12

f(GHz)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

|Γopt|

0 1 2 3 4 5 6 7 8 9 10 11 12

f(GHz)

0

40

80

120

160

φopt

15

f(GHz)

Fmin(d 21
5.5 Sample Results 

 The preceding analysis was used for measurements of the noise parameters of an onwafer transistor fabricated in 0.13 μm CMOS technology [4]. The set of input terminations for 

the forward configuration comprised one hot source (around 1100 K) and eight ambienttemperature terminations (296.15 K). One reverse measurement was also made, with a matched 

load connected at plane 1′. In the forward measurements, a 10 dB attenuator was connected 

between probe 2 and the radiometer, to keep the measured noise within the linear range of the 

radiometer. The results for the X parameters are shown in Fig. 8 and those for G0 and the IEEE 

noise parameters in Fig. 9 (a–f). 

 

For G0 we have shown a comparison to the value obtained from the VNA measurements. It is 

reassuring that the independent measurements generally agree within the uncertainties. The 

approximate values for the uncertainties in the on-wafer noise parameters are: 0.22 dB to 0.30 

dB for G0, 0.2 dB to 0.3 dB for Fmin, 15 K to 25 K for Tmin, 0.6 Ω to 0.8 Ω for Rn, 0.02 to 0.07 for 

|Γopt|, and 1° to 6° for φopt. As expected, the uncertainties in the on-wafer case are generally 

somewhat larger than for the connectorized amplifier. As in the amplifier case above, if the 

measurements could be done without an output attenuator, the uncertainties would be smaller. 

6. SUMMARY

 We have set forth the uncertainty analysis for NIST measurements of the noise 

parameters of amplifiers and transistors, where the reference planes can be either in coaxial 

transmission lines or on a wafer. The standard uncertainty is the root sum of the squares of the 

type A and type B uncertainties. The type A uncertainties are determined in the weighted leastsquares fit to the overdetermined system of equations that result from the measurements of the 

output noise temperatures for different configurations and terminations. The type B uncertainties 

are evaluated by a Monte Carlo procedure. The program that is used to perform the Monte Carlo 

computation is described in Appendix B. The uncertainty analysis described has been used for 

both coaxial low-noise amplifiers [3] and very poorly matched on-wafer CMOS transistors 

[4,20]. In the amplifier case [3], the results were subjected to checks and verification that 

confirmed the general validity of the uncertainties, as well as the actual values of the parameters. 

____________________________________

 I am grateful to Dave Walker of NIST for many discussions and for performing the 

measurements to which this analysis is applicable. I also thank Ken Wong of Agilent for 

suggesting/requesting this paper. This work was supported in part by a contract from Agilent 

Corp. 

REFERENCES

[1] D. Wait and G. F. Engen, Application of radiometry to the accurate measurement of 

 amplifier noise, IEEE Trans. Instrum. Meas. 40: 433–437 (1991). 

[2] D. F. Wait and J. Randa, Amplifier noise measurement at NIST, IEEE Trans. 

 Instrum. Meas. 46: 482–485 (1997). 

22
[3] J. Randa and D. Walker, Amplifier noise-parameter measurement checks and 

 verification, Digest, 63rd ARFTG Conf., pp. 41–45 (2004). 

[4] J. Randa and D.K. Walker, On-wafer measurement of transistor noise parameters at 

NIST, IEEE Trans. Instrum. Meas. 56(2): 551–554 (April 2007). 

[5] ISO Guide to the Expression of Uncertainty in Measurement, International 

 Organization for Standardization; Geneva, Switzerland (1993). 

[6] J. Randa, Noise-parameter uncertainties: A Monte Carlo simulation, J. Res. Natl. Inst.

 Stand. Technol. 107: 431–444 (2002). 

[7] S. Wedge and D. Rutledge, Wave techniques for noise modeling and measurement, IEEE 

Trans. Microwave Theory Tech. 40: 2004–2012 (November 1992). 

[8] J. Randa, T. McKay, S. L. Sweeney, D. K. Walker, L. Wagner, D. R. Greenberg, J. Tao, 

and G. A. Rezvani, Reverse noise measurement and use in device characterization, 2006 

IEEE Radio Frequency Integrated Circuits (RFIC) Symposium Digest, pp. 345–348, 

San Francisco (June 2006). 

[9] H. Haus et al., IRE standards on methods of measuring noise in linear twoports 1959, 

Proc. IRE, 48: 60–68 (1960). 

[10] D. Gu, D. K. Walker, and J. Randa, Noise-parameter measurement with automated 

variable terminations, submitted to 2008 Conference on Precision Electromagnetic 

Measurements (CPEM 2008), Broomfield, CO (June 2008). 

[11] C. A. Grosvenor, J. Randa, and R. L. Billinger, Design and testing of NFRad—A new 

noise measurement system, NIST Tech. Note 1518 (March 2000). 

[12] D. Luenberger, Linear and Nonlinear Programming, Ch. 7.8, Second ed. Reading, MA: 

Addison-Wesley (1984). 

[13] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes, 

Chap. 14.5, Cambridge: Cambridge University Press (1986). 

[14] J. Randa and W. Wiatr, Monte Carlo estimation of noise-parameter uncertainties, IEE 

Proceedings—Science, Measurement and Technology, 149: 333–337 (2002). 

[15] J. Randa, Simulations of noise-parameter uncertainties, 2002 IEEE MTT-S 

 International Microwave Symposium Digest, Seattle, WA, pp. 1845–1848 (2002). 

[16] J. Randa, Uncertainties in NIST noise-temperature measurements, 

 NIST Tech. Note 1502 (March 1998). 

[17] R. Marks, A multi-line method of network analyzer calibration, IEEE Trans. 

 Microwave Theory Tech., 39(7): 1205–1215 (July 1991). 

23
[18] D. Williams and R. Marks, Transmission line capacitance measurement, IEEE 

 Microwave Guided Wave Lett., 1(9): 243–245 (September 1991). 

[19] J. Randa, Noise temperature measurements on wafer, NIST Tech. Note 1390 (March 

1997). 

[20] J. Randa, R. L. Billinger, and J. L. Rice, On-wafer measurements of noise temperature, 

IEEE Trans. Instrum. Meas., 48(6): 1259–1269 (December 1999). 

[21] J. Randa, S. L. Sweeney, T. McKay, D. K. Walker, D. R. Greenberg, J. Tao, J. Mendez, 

G. A. Rezvani, and J. J. Pekarik., Interlaboratory comparison of noise-parameter 

measurements on CMOS devices with 0.12 μm gate length, Digest, 66th ARFTG Conf., 

pp. 77-81 (2005). 

24
APPENDIX A: JACOBIAN MATRIX FOR X → IEEE

 In order to compute the type-A uncertainties in the IEEE noise parameters from eq. (22), 

we need the elements of the Jacobian matrix for the transformation from X-parameters to the 

IEEE parameters. It is a straightforward exercise to compute them, and we present them here. 

 The easiest elements are the partial derivatives of t = 4RnT0/Z0, which in terms of the X’s 

is given by 

1 [( ) 1Re2 ]. 12 * 2 11

2

1 11 XSXt +−++= XS (A.1) 

The relevant partial derivatives are 

.Im2

Im ,)Re1(2 Re ,1,1 11

12

11

12

2

11

1 2

S

X

t S

X

t S

X

t

X

t −= ∂

∂

+−=

∂

∂

+=

∂

∂ = ∂

∂ (A.2) 

 We consider Γopt next because it is used in writing Tmin. The expression for Γopt is 

( )

( ) .

1 Re2

, 4 11

2

12112

12 * 1 11

2

112

2

XSX

XSXSX

opt

−

⎟ −+⎠

⎞ ⎜

⎝

⎛ +

=

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎝

⎛

−−=

η

η

η Γ

 (A.3) 

We treat the real and imaginary parts of Γopt separately. The partial derivatives can all be written 

in terms of partial derivatives of η, 

. 4 1

Im4 Im 11

2

Im 1

, 4 1

Re4 Re 11

2

Re 1

2

2

4 2

2

2

4 2

i i i

opt

i i i

opt

X X X

X X X

∂

∂

−

− ∂

∂

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎝

⎛

−−= ∂

∂

∂

∂

−

− ∂

∂

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎝

⎛

−−= ∂

∂

η

η

η

η η

η

Γ

η

η

η

η η

η

Γ

 (A.4) 

25
The derivatives of η are given by 

, Im Im Re Re2

,Im Im ,Re Re

2

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+

∂

∂ = ∂

∂

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂ = ∂∂

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂ = ∂∂

i i i

i ii i

X X X

X XX X

η η η η

η

ηηηη

 (A.5) 

where 

. Im2

Im

, Re2

Re

1 ,

, 1

12112

11

12

12112

11

12

1

11

2

11

2

121121

XSX

jS

X

XSX

S

X

X

SS

X

XSXX

−

+− = ∂

∂

−

+− = ∂

∂

∂

∂ ⎟

⎠

⎞ ⎜

⎝

⎛ −+=

∂

∂

− = ∂

∂

η η

η η

η η η

η

 (A.6) 

 Finally, we treat Tmin, which is given by 

( ) .

1

Re2

2

12 * 2 11

2

111

2

2

min

⎟

⎠

⎞ ⎜

⎝

⎛ +

⎥⎦

⎤ ⎢⎣⎡ − + −

=

opt

X opt XSXSX

T

Γ

Γ

 (A.7) 

It is understood that in eq. (A.7), occurrences of Γopt are just a shorthand way of writing the 

expression on the right side of eq. (A.3). The derivatives of Tmin are given by 

26
( )

( )

( )

( ) ⎪

⎩

⎪

⎨

⎧

⎪

⎭

⎪

⎬

⎫

∂

∂

⎥⎦

⎤ ⎢⎣

⎡ +− − +

+

= ∂

∂

⎪

⎩

⎪

⎨

⎧

⎪

⎭

⎪

⎬

⎫

∂

∂

⎥⎦

⎤ ⎢⎣

⎡ +− − +

+

= ∂

∂

⎪

⎩

⎪

⎨

⎧

⎪

⎭

⎪

⎬

⎫

∂

∂

⎥⎦

⎤ ⎢⎣⎡ ⎟ +− − + ⎠

⎞ ⎜

⎝

⎛ −

+

= ∂

∂

⎪

⎩

⎪

⎨

⎧

⎪

⎭

⎪

⎬

⎫

∂

∂

⎥⎦

⎤ ⎢⎣

⎡ +−− − +

+

= ∂

∂

, ImIm2 Re2

1

1

Im

, ReRe2 Re2

1

1

Re

1 Re2 ,

1

1

Re2 ,

1

1

12

2

min12 * 2 11

2

11111

2

2 12

min

12

2

min12 * 2 11

2

11111

2

2 12

min

2

2

min12 * 2 11

2

111

2

11 2 2

min

1

2

min12 * 2 11

2

111

2

2 1

min

X

TXSXSXS

X

T

X

TXSXSXS

X

T

X

TXSXSXS

X

T

X

TXSXSX

X

T

opt

opt

opt

opt

opt

opt

opt

opt

opt

opt

opt

opt

Γ

Γ

Γ

Γ

Γ

Γ

Γ

Γ

Γ

Γ

Γ

Γ

(A.8) 

where the derivatives of |Γopt|

2

 are given by 

i

opt

opt

i

opt

opt

i

opt

X X ∂X

∂

+

∂

∂ = ∂

∂ Γ

Γ

Γ

Γ

Γ Im

Im2

Re

Re2

2

 (A.9) 

and eqs. (A.4) through (A.6). 

27
APPENDIX B: OVERVIEW OF THE FORTRAN PROGRAM

 The type-B uncertainties in our noise-parameter measurements are evaluated with a 

Fortran Monte Carlo program called NIST_NP_UNC. In this appendix we give an overview of 

the structure of the program, discussing each subprogram in turn. 

B.1 MAIN 

 The main program first initializes a few parameters, including the number of simulations 

to be performed (NSET). It queries the user for the names of the output file (OUTFILE2), to 

which the final results are written, and the input file (INFILE), from which the input data are 

read. It also defines OUTFILE1, naming it TRASH.TXT, to which it will write the noise 

parameters from each of the simulated measurements sets. This file is usually discarded, but it 

enables one to examine some details of the individual measurement sets for diagnostic purposes, 

should the occasion arise. The character variable QUANT(9), where 9 indicates the dimension, 

contains the names of the nine quantities for which the program compiles statistics: X1, X2, 

ReX12, ImX12, G0, Tmin, t, ReΓopt, and ImΓopt. 

 After the initialization, MAIN reads the input data. The first line of data read from the 

input file consists of the values for NMEAS, CHAR1, AMB_DIST, CHI_CUT, ALPH0_AT, 

ALPH0_PR. NMEAS is the number of different measurements in each set, i.e., the number of 

different forward-configuration terminations plus the number of reverse-configuration 

measurements made in the actual measurement set from which the noise parameters were 

determined. CHAR1 is a character variable that is either ‘Y’ if the output reflection coefficients 

are measured directly, or ‘N’ if they are determined by cascading the input reflection coefficient 

with the amplifier or transistor S-parameters. AMB_DIST is a character variable that specifies 

the distribution for errors in the ambient temperature, either ‘GAUS’ for a normal distribution, or 

‘RECT’ for a rectangular distribution. CHI_CUT is the maximum value of χ

2

/ν retained in the 

simulations, as discussed in Section 5.4. ALPH0_AT is the value of |S21|

2

 for the attenuator (if it 

is present; see Section 5.3). The other factors in the available power ratio α of the attenuator are 

computed during the course of the program. If no attenuator is present, a value of 1.0 is used. 

ALPH0_PR is the value of |S21|

2

 for the output probe, probe 2 in Fig. 6. The second line of the 

input file contains the “true” values for the noise parameters and G0, whose variable names are 

X1, X2, X12R, X12I, G, where R and I refer to real and imaginary parts. The third line of the 

input file consists of amplifier or transistor S-parameters, in the order S11R, S11I, S12R, S12I, 

S21R, S21I, S22R, S22I. The program then reads the information for the individual 

measurements in each set, contained in lines 4 through 3+NMEAS of the input file, each line 

corresponding to one of the measurements. Each line contains values for the number of the 

measurement (1 through NMEAS), the configuration (1 for forward, 2 for reverse), the noise 

temperature of the input termination, the real and imaginary parts of the reflection coefficient of 

the input termination, and the real and imaginary parts of the output reflection coefficient (plane 

2 in Fig. 2(a) or 6, or plane 1 in Fig. 2b). If the output reflection coefficients are not measured 

directly (CHAR1 = ‘N’), then the last two entries are absent. After reading the termination 

information, the program computes the true values for the IEEE noise parameters and tests 

whether the true values of both the IEEE and the X parameters satisfy the physical constraints. 

The final input quantities are the input uncertainties, which are read from the last five lines of the 

input file. The first of these final lines contains the correlated and uncorrelated uncertainties in 

28
reflection coefficients, first for small reflection coefficients, then for large reflection coefficients 

(see Sections 4.2 and 5.3). The next line contains the correlated and uncorrelated uncertainties 

for S21 and then the connector repeatability. We usually absorb the connector repeatability into 

the input uncertainties in the reflection coefficients, so that this entry is 0, but the option is there 

to enter it explicitly. The next line contains the correlated and uncorrelated uncertainties for 

noise temperatures near ambient, and the following (next to last) line has the uncertainties for the 

hot input termination, the cold input termination, and the correlation between the two (Section 

4.3 or 5.3). The final line of the input file contains the correlated and uncorrelated uncertainties 

for the measurements of the output noise temperatures (Section 4.3 or 5.3). 

 Once all of the input file has been read, the program assigns the appropriate uncertainty 

to each of the quantities whose measurement will be simulated and initializes all the sums that 

will be computed in the Monte Carlo process. It then enters a loop, each pass through of which 

corresponds to one simulated measurement set. In the loop, the first step is to call the subroutine 

SET_MEAS, which generates a complete simulated set of measurements, and returns the results 

for S-parameters, reflection coefficients, input noise temperatures, and output noise 

temperatures. It next calls the subroutine FULL_FIT, which fits the simulated set of 

measurement results to get the noise parameters, in both the X and the IEEE representations. The 

results are added to the set of running sums that include both good and bad results. The program 

then checks the flags for “bad” results (Section 5.4), and, if the results are not bad, they are 

added to the running sums for good results. Thus, we have two sets of sums and averages, one 

for all results and one for only good results. After the Monte Carlo loop is completed, the rest of 

the main program just computes averages and standard deviations and writes the results to the 

output file. The input data are also written to the output file, to ensure that the results are 

associated with the correct input. 

B.2 Subroutine SET_MEAS 

 The subroutine SET_MEAS simulates a complete set of noise-parameter measurements. 

The true values and uncertainties of the variables to be measured are passed from the main 

program through a common block. The number of different output noise temperatures to be 

measured is NMEAS. The simulated measurement results are labeled by _M affixed to the 

variable, except in the case of the output noise temperatures, which are designated T_OUT. The 

uncertainty corresponding to each simulated T_OUT is also computed and passed back to 

MAIN, for use in the fit of the simulated results. 

 The subroutine begins by generating the value for the complex deviate G_DEV. This one 

complex value is used for the correlated parts of the errors in all the reflection coefficients and Sparameters (except S21) in this set of measurements. The deviates are all generated by the 

function subprogram GAUSDEV, which returns a normally distributed deviate with zero mean 

and unit variance. The program then generates simulated measurements for the S-parameters and 

repeats the process for near-ambient noise temperatures and the reflection coefficients. Finally, 

there is a loop that generates a simulated value for each of the output noise temperatures, along 

with the associated uncertainty for each measurement. Each pass through the loop corresponds to 

one of the measurements. The measurement simulation is performed by calling the subroutine 

T_MEAS, and the associated uncertainty is then computed. 

29
B.3 Subroutine T_MEAS 

 This subroutine generates a single measurement of the output noise temperature from one 

of the ports of the DUT. The input information comprises the true values of the noise 

temperature and reflection coefficient of the source that is connected to the DUT, the true value 

of the reflection coefficient of the DUT-source combination, the configuration (forward or 

reverse), the true values or the S-parameters and noise parameters of the DUT, and the input 

correlated and uncorrelated uncertainties for the measured noise temperatures. These input 

uncertainties are not used directly as the standard deviations in the measured noise temperature, 

but are used to set the scale and correlation coefficient for the standard deviations (Section 4.2). 

The subroutine first computes the true value for the output noise temperature for the appropriate 

configuration (forward or reverse). It then computes the uncertainty that is to be associated with 

that measurement, taking into account the effects of potentially large reflection coefficients, the 

probe, and the output attenuator, if it is present (Section 5.3). A simulated measurement result is 

then generated. The present version of the program allows inclusion of an output attenuator 

either for all the measurements or for none at all. We plan to change this soon, to allow use of an 

attenuator on some, but not all the measurements of the output noise temperature. 

B.4 Subroutine FULL_FIT 

 This subroutine performs a weighted least-squares fit to minimize the function χ

2

 defined 

by eq. (20), thereby determining the noise parameters and G0 of the DUT. This same subroutine 

is used in other applications as well, and consequently it includes some features that are not 

needed in this program. After some initial definitions, it calls the subroutine FWD_FIT, which 

performs a linear fit to all the forward measurements in the set. The results of this linear fit are 

used to generate an initial guess for use in the full nonlinear fit. The nonlinear fit also allows 

computation of the covariance matrix, which is used in other applications of FULL_FIT, but not 

here. If there are no reverse measurements included in the measurement set, the linear fit 

produces the full, correct results. Nevertheless, we perform the nonlinear fit in all cases in this 

subroutine because the covariance matrix is needed in the other applications, and the packaged 

program used for the linear fit does not allow computation of the covariance matrix. An odd 

feature of FULL_FIT is that we do not use the exact results of the linear fit as the initial guess for 

the nonlinear fit, but rather we multiply the linear-fit results by 0.9. The reason for this is that if 

the starting point for the nonlinear fitting program corresponds to the minimum of the function 

being minimized (as would be the case when the measurement set includes no reverse 

measurement), the program does not return the correct covariance matrix. (We have, of course, 

checked that the results of the nonlinear fit are independent of the initial guess and that it gives 

the same results as the linear fit when there are only forward measurements present.) 

 After the nonlinear fit, the subroutine checks that χ

2

/ν ≤ CHI_CUT, computes the IEEE 

noise parameters from the X parameters, and checks that the results satisfy the physical 

constraints. It then computes the Jacobian matrix for the transformation from X parameters to 

IEEE parameters, which in this application is used only to check that the type-A uncertainty in 

Γopt is not too large (Section 5.4). 

30
31

B.5 Subroutine FWD_FIT 

 The subroutine FWD_FIT performs a linear, weighted, least-squares fit to the output 

noise temperatures of a series of forward measurements. The relevant S-parameters, reflection 

coefficients, input noise temperatures, measured output noise temperatures, and uncertainties in 

the measured output noise temperatures are all passed to the program via a common block. The X

parameters and G0 are determined by minimizing the χ

2

 defined in eq. (20). The minimization is 

performed by a packaged program that does a weighted, linear, least-squares fit. 

B.6 Subroutine FCN 

 This subroutine defines the “error” in a given measurement for purposes of the fit, i.e., it 

computes the difference between the measured value of the output noise temperature and the 

value computed from eq. (6) or (7) for the set of noise parameters X and G0. It is used in the 

nonlinear fitting routine, which minimizes the weighted sum of the squares of the errors for all 

the measurements in the set, as given by eq. (20). 

B.7 Subroutines GAUSDEV and RECTDEV 

 These two subroutines are taken from and explained in Chapter 7 of Reference [13]. 

RECTDEV returns a random deviate between 0.0 and 1.0, using a system-supplied randomnumber generator. The point of RECTDEV is that it eliminates correlations in k-space, which 

may be present in the system’s random-number generator [13]. GAUSDEV generates and returns 

a normally distributed deviate with zero mean and unit variance, using RECTDEV as the source 

of uniform deviates.