DesignCon 2012
Fast and Optimal Algorithms for Enforcing Reciprocity, Passivity and Causality in S-parameters
Dr. Kaviyesh Doshi, LeCroy Corporation
[email protected], 845-425-2000 Anirudh Sureka, LeCroy Corporation
[email protected], 845-425-2000 Peter J. Pupalaikis, LeCroy Corporation
[email protected], 845-425-2000
Abstract Scattering or S-parameters obtained from direct measurements or through EM wave simulations may violate physical laws due to measurement or numerical errors. If used in simulations, such S-parameters lead to incorrect results and wrong conclusions. This paper focuses on three such properties of S-parameters – causality, passivity and reciprocity. Tests are provided to detect if any of the three conditions are violated. Separate algorithms are provided to make S-parameters causal, passive and reciprocal. The proposed algorithms are proven to be the best in the solution space, and are easy to implement.
Patent Disclosure Portions of the information provided in this paper are the subject of patents applied for.
Author(s) Biography Kaviyesh Doshi received his Ph.D. in Electrical Engineering from University of California, Santa Barbara in June 2008. He joined LeCroy Corporation, a manufacturer of high-performance measurement equipment located in Chestnut Ridge, New York in August 2008, as a Research & Development Engineer. At LeCroy, he has been involved in design and development of SPARQ – a TDR based instrument for measuring Sparameters. He has filed patents in the area of de-embedding and signal processing for time domain network analyzer. Anirudh Sureka received his Masters in Systems Engineering and Masters in Biomedical Imaging in June 2003 from University of Michigan, Ann Arbor. He joined LeCroy Corporation, New York in August 2003, as a Research & Development Engineer. At LeCroy, he has been involved in design and development of various signal processing applications for high end digital oscilloscopes, probes and digitizers. He has worked on developing the DBI technology and SPARQ – a TDR based instrument for measuring Sparameters. He has patents in the area of filter designs, DBI architecture and signal processing for time domain network analyzer." Peter Pupalaikis was born in Boston, Massachusetts in 1964 and received the B.S. degree in electrical engineering from Rutgers University, New Brunswick, New Jersey in 1988. He has worked at LeCroy Corporation for 16 years and currently manages integrated-circuit development and intellectual property as Vice President of Technology Development. Prior to LeCroy he served in the United States Army and as a consultant in embedded systems design. Mr. Pupalaikis holds numerous patents in the area of measurement instrument design and is a member of Tau Beta Pi, Eta Kappa Nu and the IEEE signal processing, communications, and microwave societies.
Introduction High bit rate serial data systems require detailed measurements of the channels through which they must be transmitted. These channels include cables, backplanes, connectors and PC boards. Scattering or S-parameters are the nearest thing to a universal language for expressing the characteristics of such a channel in a variety of applications from simulation to modeling and even for de-embedding and virtual probing. Their central role adds to the necessity of their accurate measurement. Accurate measurement of S-parameters is not a trivial task. There are multiple sources of errors while measuring S-parameters using network analyzers - errors introduced due to imperfect knowledge of the calibration standards, errors introduced due to measurement noise, errors due to uncompensated systematic errors etc. Also it is not always possible to connect the network analyzer directly to the device under test (DUT). This necessitates the use of de-embedding algorithms, which further introduces errors in the measured S-parameters of the DUT. S-parameters obtained using 3D EM wave simulation may also be erroneous either due to numerical errors in solving the partial differential equations or due to model mismatch. These errors in S-parameters may manifest themselves as violation of certain physical laws. Besides leading to incorrect conclusions, such violations present significant difficulties while using such S-parameters in system simulations like generating and analyzing eye patterns, or equalization of the modeled channels etc. This paper focuses on rectifying the violation of three such properties of S-parameters – causality, passivity and reciprocity. Current literature has multiple algorithms to enforce causality, passivity and reciprocity (for example see presentations by (Shlepnev, 2010)). The algorithms proposed in this paper are fast and optimal. They are optimal in the sense that, with minimum perturbation, the measured S-parameters are made to satisfy the passivity criteria. For reciprocity the solution is optimal in least squares sense. The algorithms are fast because they are direct as opposed to iterative. Numerical examples are presented to demonstrate the validity and efficiency of the proposed algorithms. Note that the definitions, algorithms and proofs presented here are valid for the case when the real part of the reference impedance is positive – which is the most typical scenario. For the case when the real part is negative will be a topic of future discussion. Before proceeding to the algorithms some concepts of matrix algebra are reviewed in the next section. These concepts will help in understanding the proposed algorithms.
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Mathematical Preliminaries In this section we define certain properties of matrices that will be useful in understanding the passivity problem and the proposed solution. A short intuitive explanation is also provided for non-trivial concepts. Refer to (Strang, 2006) or (Meyer, 2004) for an in-depth explanation of these concepts. Readers familiar with matrix algebra can skip the following section. Definition 1: Let C n denote the space of complex vectors of length n × 1 . For vector x ∈ C n , || x || 2 is the standard Euclidean norm defined as n
∑| x
|| x || 2 =
i
|2 .
i =1
|| x ||22 can be considered as the total energy in the discrete signal represented by x . Definition 2: Let C m×n denote the space of complex matrices of size m × n . Frobenius norm of a matrix A ∈ C m× n , is similar to the Euclidean norm of a vector and is defined as:
|| A ||F =
n
m
∑∑| A
i, j
|2 ,
i =1 j =1
where Ai , j is the (i, j ) element of the matrix A . || A ||2F can be considered as the total energy in the discrete signal represented by matrix A . Definition 3: For matrix A ∈ C m× n , || A ||2 is the induced 2-norm and is defined as:
|| A ||2 = max || Ax ||2 . || x || 2 =1 x∈C n
[1]
The induced norm of a matrix, also known as the operator norm, signifies the transformation that a vector undergoes when multiplied by the matrix. The above optimization problem states that in the space of unit norm vectors, we want to find a unit norm vector x such that when it is multiplied by the matrix A , the energy in the transformed vector ( || Ax ||2 ) is maximized, i.e. the matrix A “amplifies” the vector x by maximum amount possible (see Figure 1). The induced norm of the matrix is this maximizing amplification factor.
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Figure 1 : Induced Matrix 2-norm in R 3 (from (Meyer)) For example 2 0 = 2, 0 1 2 0 0.4 = 0.6, 0 − 0.6 2 − 0.60 1 = 1.37. 0.90 0.89 2 The optimization problem defined in [ 1 ] has a closed form solution and the induced 2norm of a matrix is given as
|| A ||2 = σ 1 , where σ 1 is the largest singular value of the matrix A. Singular values of a matrix are defined in the following paragraph. Definition 4: Given a matrix A ∈ C n× n , λ (possibly complex) is an eigenvalue of A if Ax = λx ,
for some vector x . The vector x is called the eigenvector corresponding to the eigenvalue λ . One way to define the singular values of a matrix is to use the above definition of eigenvalue. If σ is a singular value of A , then
AH Ax = σ 2 x ,
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and x is the corresponding singular vector. Thus σ 2 is the eigenvalue and x is the eigenvector of AH A . Here AH is the conjugate transpose of the matrix A , i.e. (i, j ) element of matrix AH is given as: ( A H )i , j = ( A* ) j ,i ,
where A* is the complex conjugate of the matrix A . A property of singular value is that it is always positive and real. Refer (Meyer) for further details regarding this property. Definition 5: For each A ∈ C n× n , there exists unitary matrices U ∈ C n× n , V ∈ C n× n and a diagonal matrix Σ ∈ R n× n such that
A = UΣV H . [2]
Equation [ 2 ] is known as the singular value decomposition (SVD) of the matrix A . SVD is a well-known matrix decomposition and more details can be found in (Meyer). Note that for SVD to exist, the matrix A need not be a square matrix, but since the Sparameter matrix is always square, the SVD for square matrices will be discussed. Here the three matrices have special significance. Both U and V are unitary matrices, i.e. UU H = U H U = I and VV H = V H V = I , where I is the identity matrix. This implies that j − th column of a unitary matrix ( j ∈ {1,2,⋯ n} ) is of norm one, i.e. || U *, j ||2 = 1 and that any two columns of the unitary matrix are orthogonal to each other. Same is true for each of the rows of a unitary matrix. For the SVD, columns of V are the eigenvectors of matrix AH A and are also called right singular vectors of A . The columns of U are the eigenvectors of the matrix AAH and are also called the left singular vectors of A . Σ is a diagonal matrix with singular values of A on the diagonal. Also by convention, the singular values are arranged in decreasing order, i.e. 0 ≤ Σ n, n = σ n ≤ Σ n −1, n −1 = σ n −1 ≤ … ≤ Σ1,1 = σ 1 . If the matrix A is of rank r , then only the first r singular values are non-zero. In the following section the passivity enforcement problem is defined and a novel solution using the SVD is proposed. It is further proved that the proposed solution is the optimal solution.
Passivity Enforcement Let S denote an S-parameter matrix at frequency ωk . For an n-port passive device, S is a complex matrix of order n. As described in (Saraswat, Achar, & Nakhla, 2005) for each sampled frequency ωk , passivity criteria requires that the matrix S should satisfy the following property: 4
|| S ||2 ≤ 1 , [3]
where || S || 2 is the induced 2-norm of the matrix S. For a short explanation for why condition described in Equation [ 3 ] holds, consider the following relationship for the s-parameter matrix defined at each frequency point, Sa = b , [4]
where a and b are the power-wave vectors such that a j is the power wave incident on port j and b j is the power wave coming out of port j of the system described by the s2
parameter matrix S. In this case || a || 2 represents the total power input to the system, 2
while || b || 2 represents the total power leaving the system. If the system is passive, then the total power leaving the system cannot exceed the total power entering the system. Mathematically this can be written as:
|| b || 2 ≤|| a || 2 Substituting equation [ 4 ] in the above equation we have:
|| Sa || 2 ≤|| a || 2 or || Sa || 2 ≤ 1, || a || 2 which by definition of induced 2-norm (equation [ 1 ]), represents equation [ 3 ]. In this section an algorithm is described to enforce passivity when the norm criteria (equation [ 3 ]) is violated. It is further showed that the proposed solution is the optimal solution in the class of the defined problem. To enforce passivity, we wish to perturb the matrix S by minimum possible amount, so that the norm criteria is satisfied i.e. at each frequency, find smallest possible matrix ∆S , such that:
|| S − ∆S ||2 ≤ 1 . [5]
Here smallest can be defined in different ways. We want to define smallest in the sense of Frobenius norm and induced 2-norm of the matrix. The passivity enforcement problem defined under two different classes of optimization can be written as: Problem 1: Minimize || ∆S || F such that || S − ∆S ||2 ≤ 1 is satisfied. Problem 2: Minimize || ∆S || 2 such that || S − ∆S ||2 ≤ 1 is satisfied. 5
Problem 1 is the most commonly accepted form of the passivity enforcement problem. An intuitive explanation for the problem is that since ∆S perturbs the individual elements of the S-parameter matrix, minimal || ∆S || F implies that the total energy of the perturbation matrix is as minimal as possible. In problem 2 the minimal || ∆S || 2 signifies that the perturbation is such that it causes minimal change in the “amplification” of the input power-wave vector. The proposed solution is optimal for both classes of problem. The passivity criteria stated in [ 3 ] implies
σ n ≤ σ n −1 ≤ … ≤ σ 1 ≤ 1, where σ j ' s are the singular values of the matrix S (see Definition 3). Suppose that the first k (k ≤ n) singular values violate the passivity criteria. Then
σ j = 1+ | δ j |, j ∈ {1,2,… k} [6]
A novel algorithm to enforce passivity is described below:
1. Perform singular value decomposition (SVD) of the matrix S, i.e. S = UΣV H . 2. Let ∆Σ be a diagonal matrix of order n, with the diagonal entry defined as | δ |, j ∈ {1, 2,… k} ∆Σ j , j = j 0, j > k 3. Now form the matrix ∆S such that
∆S = U∆ΣV H . [7]
Then || ∆S || 2 =| δ 1 | , || ∆S || F =
k
∑| δ
i
| 2 , (See (Meyer) for more details) and
i =1
|| S − ∆S ||2 = 1 . 4. Thus passivity enforced s-parameter matrix is given as: S passive = S − ∆S . [8]
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To prove that this ∆S is the optimal solution in the induced 2-norm sense (problem 2), let B be a matrix such that
|| B || 2
