Título/s: | Multivariate statistics applied to assess measurement uncertainty of complex reflection coefficient |

Autor/es: | Benjamín, M.; Silva, H.; Monasterios, G.; Tempone, N.; Henze, A. |

Institución: | INTI-Electrónica e Informática. Laboratorio Metrología RF & Microondas. Buenos Aires, AR |

Editor: | IEEE |

Palabras clave: | Técnicas estadísticas; Errores; Mediciones; Incertidumbre; Radiofrecuencia |

Idioma: | eng |

Fecha: | 2014 |

Notas: | Fuente: IEEE, de acuerdo a su política de copyright: "Los autores / empleadores pueden reproducir o autorizar a terceros a reproducir la Obra, material extraído literalmente de la Obra o trabajos derivados para el uso personal del autor o para uso de la empresa, siempre que se indique la fuente y el aviso de copyright de IEEE, no deben usarse las copias de ninguna manera que implique el respaldo de IEEE de un producto o servicio de cualquier empleador y las copias en sí no se ofrecen para la venta" |

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Multivariate Statistics Applied to Assess Measurement Uncertainty of Complex Reflection Coefficient M. Benjamı´n, H. Silva, G. Monasterios, N. Tempone, and A. Henze Instituto Nacional de Tecnologı´a industrial (INTI), Electro´nica e Informa´tica, Lab. Metrologı´a RF & Microondas metrologiarf@inti.gob.ar Abstract—In this paper we show an alternative mathematical interpretation of the error propagation law for complex quantities established in supplement 2 of the GUM [1]. We use this interpretation to study VNA’s one port reflection mea- surement which includes several terms that represent complex quantities. We show two different approaches to solve the problem that arises when trying to establish the variance matrix of the sum of two complex quantities. We also give explicit formulae to estimate uncertainty with both approaches. Keywords—GUM, multivariate statistics, RF metrology, VNA measurement, complex quantities, measurement uncertainty. I. THEORY Supplement 2 of the GUM [1] establishes a method for the expression of uncertainty in multiple output measurements. This means determining the variance matrix of the output vector. In the special case where measurand and input quan- tities are complex, an alternative formulation of the error propagation law is presented in [2]. This formulation is a matrix analogy of the well known law in the univariate case. Let f : Cm → C be an analytic function and the complex measurand Y = f(z1, . . . , zm), [2] shows that the variance matrix of Y is var [Y ] = m∑ i=1 m∑ j=1 J(ci)cov [zi, zj ] J(cj)t (1) Where ci = ∂f∂zi are sensitivity coefficients, J(ci) their matrix representation and J(cj)t the transpose of J(cj) J(ci) = ( Re(ci) −Im(ci) Im(ci) Re(ci) ) (2) The covariance matrix of two complex variables is cov [zi, zj ] = ( u[Re(zi),Re(zj)] u[Re(zi),Im(zj)] u[Im(zi),Re(zj)] u[Im(zi),Im(zj)] ) (3) Where u[Re(zi),Re(zj)] is the covariance between Re(zi) and Re(zj). u[Re(zi),Re(zi)] = u2[Re(zi)] is the variance of Re(zi). If z and w are complex variables and c and d are constant complex values, covariance matrices have the following prop- erties var [z] = cov [z, z] (4) cov [z, w] = cov [w, z]t (5) cov [cz, dw] = J(c)cov [z, w] J(d)t (6) cov [z1 + z2, w] = cov [z1, w] + cov [z2, w] (7) var [z + w] = var [z] + var [w] + cov [z, w] + cov [w, z] (8) var [−z] = var [z] (9) Using (4), (6) and (8) in equation (1) the variance matrix of the measurand can be interpreted as var [Y ] = m∑ i=1 m∑ j=1 cov [cizi, cjzj ] = var [ m∑ i=1 cizi ] (10) The complex variables z and w are uncorrelated if cov [z, w] = ( 0 0 0 0 ) (11) From (8) we get that var [z + w] = var [z] + var [w] (12) We say z is “circular” and has a circular variance matrix if var [z] = u2z ( 1 0 0 1 ) (13) Where u2z = u2[Re(z)] = u2[Im(z)]. If c is a constant complex value, var [cz] = |c|2u2z ( 1 0 0 1 ) (14) II. MEASURMENT MODEL The following equation expresses the true reflection coef- ficient Γ of a one port device measured with a VNA. Γ = Γm −D M(Γm −D) + T −R (15) Here Γm is the measured value of Γ. D, M and T are the residual directivity, source-match and reflection tracking respectively. R accounts for external error terms, such as cable flexibility, stability, etc. All terms are complex quantities. The following approximations can be considered: T ≈ 1, M,D,R ≈ 0 and Γm ≈ Γ. After complex differentiation and using the approximate values, ∂Γ ∂D = −1 ∂Γ ∂T = −Γ ∂Γ ∂M = −Γ 2 ∂Γ ∂R = −1 (16) Replacing in (10) and using (9), we have that var [Γ] = var [D + Γ2M + ΓT + Γm +R] (17) III. UNCERTAINTY ANALYSIS All input variables are assumed uncorrelated , and except for Γm they all satisfy (13). The Ripple technique described in [3] is a standard approach to assess var [D] and var [Mef ]. Where Mef , shown in Fig.1, is the “Effective Test Port Match ”. This quantity is similar to M and is assumed to satisfy (13). Guidelines in [3] establish 19
? Fig. 1. Ripple signal flow graph for Mef to use Mef as M , and warns that Mef could be correlated with D. Using (12) and (14) in (17) we get var [Γ] = var [D + Γ2Mef ]+|Γ|2var [T ]+var [Γm]+var [R] (18) The last three terms of the sum do not present any analytical difficulties. For the first one, [3] neither explains how to calculate the covariance needed nor gives a bivariate treatment to the problem. In order to overcome this situation we present two dif- ferent approaches we developed. One that overestimates var [ D + Γ2Mef ] and another where the covariance between Mef and D is calculated. Although the second approach is more rigorous, it makes more assumptions about the model. The first approach is also of general interest beyond the case of the reflection coefficient measurement. It serves as an example of what can be done to establish the variance matrix of the sum of two complex quantities when their covariance matrix is unknown. A. Overestimation of the Variance of the Sum Using (8), the fact that D and Mef satisfy (13), and assuming that u[Re(D), Im(Mef )] = u[Im(D),Re(Mef )] = 0 var [D+Γ2Mef ] = ( u2[Re(D)+Re(Γ2Mef )] 0 0 u2[Im(D)+Im(Γ2Mef )] ) (19) Due to missing information about covariance, diagonal ele- ments can not be calculated. Cauchy-Schwarz inequality for random scalars is used to bound them in order to get a worst case variance matrix as follows var [D+Γ2Mef ] = (uD + |Γ|2uMef )2 ( 1 0 0 1 ) (20) B. Covariance Matrix of D and Mef Solving the flow graph in Fig. 1 yields Mef ≈ M + D + L (21) Where L is the reflection coefficient of the air-line used for the Ripple technique. L is assumed to satisfy (13) and is uncorrelated with all input variables. Using (4), (7) and (21) cov [Mef , D] = cov [M, D] + cov [D, D] + cov [L, D] = var [D] (22) Using (5), (6), (8), (14), (22) and assuming that D is circular, it is possible to obtain the following equality var [ D + Γ2Mef ] =var [D] + |Γ|4var [Mef ] + . . . . . . + (J(Γ2) + J(Γ2)t)var [D] (23) Finally, this can be expressed as var [ D + Γ2Mef ] = ( (1 + 2Re(Γ2))u2D + |Γ|4u2Mef )(1 0 0 1 ) (24) IV. MEASURAND’S VARIANCE MATRIX In this section we establish var [Γ] for both approaches. In order to get a circular variance matrix for the measurand we overestimate var [Γm] with a circular variance matrix. A common practice is to do this with u2Γm = u 2(Re(Γm)) + u2(Im(Γm)) (25) For the first approach, using (20) in equation (18) we get an overestimated variance matrix var [Γ] = u2c1 ( 1 0 0 1 ) (26) where u2c1 = (uD + |Γ|2uMef )2 + |Γ|2u2T + u2Γm + u2R (27) For the second approach, using (24) in equation (18) we get var [Γ] = u2c2 ( 1 0 0 1 ) (28) where u2c2 = (1+2Re(Γ2))u2D+|Γ|4u2Mef+|Γ|2u2T+u2Γm+u2R (29) In order to compare both matrices it is enough to compare uc1 and uc2. V. COMPARISION OF BOTH APPROACHES We show uc1 and uc2 defined in (27) and (29) for three real measurement of low, medium and high reflection coefficients at 18 GHz with a type N connector Γlow Γmed Γhigh |Γ| 0.058 0.559 0.980 The values obtained for uc1 and uc2 were ×10−3 Γlow Γmed Γhigh uc1 7.8 11.9 20.5 uc2 7.8 7.8 12.2 VI. CONCLUSION Formulating the variance matrix of a measurand as in (10) allows a better understanding of the error propagation law. The approach shown in (III-A) is a useful resource when the covariance between two complex variables is unknown. In (V) we showed that overestimation was too big for measurements with medium or high reflection values. Therefore, overestima- tion should be avoided whenever possible, for example using an approach similar to (III-B). Equations (28) and (29) are of special interest for one port re- flection measurements as they establish an explicit expression for the measurand’s variance matrix. REFERENCES [1] BIPM, IEC, IFCC, ISO, IUPAC, UPAP and OIML “Evaluation of Measurement Data - Supplement 2 to the ‘Guide to the Expression of Uncertainty in Measurement’. Extension to any number of output quantities.” JCGM 102:2011. [2] B. D. Hall “On the propagation of uncertainty in complex-valued quantities,” Metrologia, vol 41 pp 173 - 177. 2004 [3] EURAMET cg-12 v2.0. “ Guidelines on the Evaluation of Vector Network Analysers (VNA),” European co-operation for Accreditation, 2011. Ver+/- | |

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