Estimation of Crosstalk among Multiple Stripline Traces Crossing a Split by Compressed Sensing

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1 Estimation of Crosstalk among Multiple Stripline Traces Crossing a Split by Compressed Sensing Tao Wang, Yiyu Shi, Songping Wu, and Jun Fan Department of Electrical and Computer Engineering, Missouri University of Science and Technology, Rolla, MO 65409, USA {yshi, swhv7, jfan}@mst.edu Abstract In printed circuit board (PCB) designs, it is common to split power/ground planes into different partitions, which leads to more crosstalk among signal traces that route crossing a split. It is of general interest to develop a crosstalk model for various geometric parameters. However, the long time required to simulate the structure with any given set of geometric parameters renders general modelling approaches such as interpolation inefficient. In this paper, we develop an empirical model based upon the compressed sensing technique to characterize the crosstalk among traces as a function of geometric parameters. A good agreement between the empirical model and full-wave simulations is observed for various test examples, with an exceptionally small number of samples. dominant factors that impact the peak crosstalk values. The other trivia geometric parameters that do not affect crosstalk significantly are fixed at values listed in Table I. In order to make a fast estimation of split-related crosstalk for arbitrary geometries, empirical expressions as a function of these dominant parameters are developed based on a simple multivariable interpolation methodology [6]. I. INTRODUCTION Power/ground plane often has different partitions to supply different required voltage levels in a printed circuit board (PCB). The split usually leads to disruption of highspeed signal return path for the traces routing in proximity to the split plane [1]. From the signal integrity point of view, the crosstalk among multiple traces crossing a split usually is nonnegligible even if the separation of the traces is large enough that the direct trace to-trace coupling is negligible. In today s digital world, low-voltage source is preferred to minimize power consumption and thus even a small amount of noise added to the signal may cause system breakdown [2, 3]. Thus, it is desirable to quantify the impact of the split on crosstalk and provide effective design guidelines to mitigate splitinduced crosstalk for signal integrity engineers. The coplanar transmission line model has been widely used to model a microstrip line trace over a split power/ground plane, which effectively represents the characteristics of the discontinuity [2~4], but it is only suitable for microstrip geometries. Full-wave tools are commonly used to analyze the impact of split as well as slotted planes on crosstalk for stripline geometries [5]. However, full-wave simulations are usually time-consuming, and thus are not suitable for fast crosstalk estimations needed for PCB layout screening. Previous research shows that the overall crosstalk among two 10cm traces crossing a split, as displayed in Figure I, can be decomposed into two portions: the direct trace-to-trace coupling and the split-related coupling. The former portion can be evaluated readily by W-element. For the latter portion, the trace-to-trace separation d, the trace-to split-plane distance h, and the signal rise time t r are found to be the (a) Perspective view (b) Top view (c) Cross section view Figure I. Geometry under study The simulation time required to obtain each sample in the multi-variable interpolation method is extremely long, since each sample needs to be calculated by 3D full-wave solver, e.g. Ansoft HFSS and it usually takes almost one day to finish one simulation. Accordingly, the number of samples

2 available to perform interpolation is also limited. As a result, the empirical model lacks of accuracy, with the maximum error of 11.2%. Table I. Geometric parameters settings b [mm] 80 s [mm] 40 t [mm] 1 β [ ] 90 w [µm] 135 ε r 4.35 To tackle this problem, we propose to use compressed sensing technique to build the model. Compressed sensing is a recently developed technique to recover signals with an exceptionally small number of randomly distributed samples. While it has been applied widely in the area of signal processing, its application to the electromagnetic compatibility (EMC) remains unexplored. In this paper, we use the crosstalk estimation for multiple stripline traces crossing a split as a vehicle to demonstrate the power of compressed sensing in the EMC problems where the computational cost to obtain samples are high. The remainder of the paper is organized as follows. Section II gives a brief review of the compressed sensing techniques. Section III adapts the technique to the crosstalk estimation problem. Experimental results are presented in Section IV and concluding remarks are given in Section V. II. OVERVIEW OF COMPRESSED SENSING Compressed sensing is a recently developed technique in the field of signal processing. Its key idea is to use an exceptionally small number of samples to recover a desired signal, under the assumption that the signal has sparse representation in certain basis functions. In this section, we will use a one-dimensional signal to briefly review the technique. Consider a signal () in the t-domain, which can be represented as ()= (1) where (i=1,..., N) are the basis functions in a Hilbert space. are the coefficients and can be calculated as = <, >,=1,, (2) where <> operation is the inner product defined in the space spanned by the basis functions. If the basis functions are chosen properly, many of the coefficients can be zero. Specifically, if the vector ( ) formed by the coefficients has at most k non-zero entries, we call it k-sparse. Under the assumption that we are able to find a set of basis functions to represent f(t) with k-sparse coefficients, compressed sensing enables us to accurately recover with =((/)) samples, when the sampling are random enough to follow certain properties. There are many different ways to choose the basis functions and to reconstruct the signal. In Section III, we will describe one that best fits the crosstalk estimation for multiple stripline traces crossing a split. A more detailed description of the compressed sensing techniques, including these conditions required to apply the technique, is beyond the scope of the paper. Interested readers are referred to [7] [8] for more details. III. APPLICATION TO CROSSTALK ESTIMATION To apply the compressed sensing technique to the problem, we will first need to select a proper set of basis functions such that the crosstalk, as a function of d, h, and t r, has a sparse representation. While there are many possible candidates such as wavelet functions and polynomials, in our experiments we find that discrete cosine functions offer the best sparsity and accuracy. In this section, we use bold to indicate a vector (e.g. f), bold capitalization to indicate a matrix (e.g. A), and subscript to denote the element-wise index (e.g. f i ). Without loss of generosity, we discretize the range of interest for d, h and t r and label them in integers, i.e., d={1, 2,..., P}, h={1, 2,..., Q} and t r = {1, 2,..., R}.As such, the basis functions we selected are gi, j, k ( d, h, tr ) = π (2d 1)( i 1) π (2h 1)( j 1) π (2tr 1)( k 1) cos cos cos 2P 2Q 2R where 1 i P,1 j Q,1 k R. These basis functions need some constant coefficients to normalize, but omitting them does not affect our algorithm. The crosstalk function f(d, h, t r ) can be represented using these basis functions as P Q R r = i, j, k r (3) i= 1 j= 1 k = 1 f ( d, h, t ) α ( i, j, k) g ( d, h, t ) where α ( i, j, k) are the coefficients. (3) might look familiar to some readers, as it is in fact the discrete cosine transform (DCT). Next, we randomly select ( ) samples with geometry parameters (,h, ) (=1,2,,) and measure the corresponding crosstalk f u. As such, we can obtain a set of equations based on these sampling points, i.e., P Q R u = i, j, k u u ru (4) i= 1 j= 1 k = 1 f α ( i, j, k) g ( d, h, t ) It is worthwhile to note here, that in (4), the only unknowns are the coefficients α ( i, j, k). And we can re-cast it in a compact form as = (5) where A is a constant matrix formed by gi, j, k ( du, hu, t ru ). α is a vector formed by α ( i, j, k). And f is a vector formed by f u.

3 If we can get the coefficients by directly solving (5), and insert them back to (3), we will have an analytical expression for the crosstalk estimation. Unfortunately, we will not be able to do so, because the number of equations (M), which is equal to the number of samples available, is much smaller than the number of variables (N=PQR). In other words, (5) is an underdetermined equation. With the assumption that the coefficients α( i, j, k) are sparse, however, we can approximately solve it using an optimization. Specifically, we can solve min (6) subject to = where α 0 is the zero norm (the number of non-zeros in α). The meaning of such an optimization is to minimize the nonzeros in the coefficients subject to the measurement data available. Zero-norm is a nonlinear function, and thus (6) is still very difficult to solve. Accordingly, we resort to an approximate version of (6), by replacing the zero-norm with one-norm, i.e., min (7) subject to = It is well established in literature that the optimal solution (7) is also sparse. It is obvious that the quality of the compressed sensing algorithm depends on how to efficiently solve (7). While many different methods can be used such as the interior point methods [11] and the homotopy method [13], in this paper we choose to use the iteratively-weighted least squares (IRLS) method [12], as in the experiments we find that it leads to the most accurate results, with a minimum runtime. Denote F()=:=. The key idea of IRLS is that if (7) has a solution, then the solution of the weighted least squares problem = F(), (8) coincides with when the weight =. Inspired by such relationship, given a real number () >0 and a current estimation (), the IRLS method solves the following optimization problem to get an updated weight (n+1) () =argmin (),, () (9) where (,,) + + (10) The updated weight is then plugged in to (8) to get an updated x (n+1). () is replaced by a non-increasing update (). The process iterates until () =0. The optimality is clear, as when such condition holds, (8) and (9) will give an optimal solution x () = () (11) This in-turn indicates that the optimal solution of (8) coincides with the optimal solution of (7). Further discussion about the details of IRLS is beyond the scope of this paper. Interested readers are referred to [12] for more details. We will simply outline our implementation in Algorithm 1, where we have defined for a nonincreasing rearrangement () for the absolute values of the elements of z, such that () is the i-th largest element in z. Step 1: Initialize by taking () = (1,1,,1). Set () =1. Step 2: Set () =argmin F(), (), (which is equivalent to (8)), and () =min (), ( () ), where k is some fixed integer. Step 3: Set () =argmin (),, () Step 4: If () =0, stop; return x (n+1) else, go to Step 2. Algorithm 1. The IRLS method. Finally, we would like to present an extra benefit of our method. With the analytical expression (3), we are able to easily calculate the sensitivity information with respect to each geometry parameters accurately. Such information is extremely valuable to guide the design optimization. IV. EXPERIENTIAL RESULTS In order to recover the crosstalk between two traces over a split as a function of trace-to-trace separation d in the range of 0.6~2.9mm, the trace-to split-plane distance h in the range of 180~370µm and the signal rise time t r in the range of 75~255 psec, 21 samples designed at special locations are obtained by full-wave simulations. In full-wave simulations, port 1, as shown in Figure I, is excited by a 2V step source, while the other three ports are terminated by the trace characteristic impedance, which varies for different h. Noise voltage on port 3, namely near-end crosstalk (NEXT) and noise voltage on port 4, known as far-end crosstalk (FEXT) are simulated by Ansoft HFSS. According to previous study [6], split related noise is generated due to inductive coupling from the split to the trace, and therefore split related NEXT and FEXT have the same shape of waveforms but they are out of phase. In reality, it is the biggest noise pulse of the crosstalk that causes signal integrity issues. Thus, only the amplitude absolute values of split related crosstalk are extracted at 21 sampling points to verify the proposed method. A list of these sampling points is illustrated in Table II. We first verify that the basis functions (DCT) we used actually lead to a sparse representation of the crosstalk. For this purpose, we construct a model based on the sampling points in Table II, and calculate the coefficients by solving the one-norm problem (7). The results are depicted in Figure II. This indicates that the problem is indeed suitable for our compressed sensing based technique. To visualize where these non-zero coefficients are in the DCT basis, we fix t r and only plot the coefficients with various d and h in Figure III. As we can see from the figure, these non-zero coefficients are gathered near the low frequency region (close to zero). This reflects that the function is smooth and has little sharp transitions. Table II. Sampling points.

4 i d (mm) h (um) tr (psec) xtalk (mv) better than the results reported in [6] using the same number of samples. i Figure III. Plot of the coefficients α(i, j, 1) Table III. Accuracy verification. d (mm) h (um) tr (psec) xtalk (mv) xtalk' (mv) relative error % % % % % % % % % j count coefficient values Figure II. Histogram of the coefficients α(i, j, k). In addition, we verify the accuracy of our model by running cross-validation on the samples. Specifically, we randomly select 18 of these samples to build the model, estimate the crosstalk of the remaining three samples using the model and compare it with the actual measurements. Such cross-validation α(i, j, 1) is run three times, and the results are shown in Table III. From the table we can see that the model predicts quite close to the actual measurement, with a minimum error of 0.37% and a maximum error of 5.81%. This is significantly Figure IV. Max/mean/min relative error v.s. the number of sampling points used. M is the actual number of samples and N is the total number of coefficients ( M N ). Another interesting thing to see is how the accuracy of our method changes with the number of samples used. We normalize the number of samples against the total number of

5 coefficients (N), as the maximum number of samples required to determine all the coefficients is N (in this case (5) becomes a determined equation). The results are depicted in Figure IV, where N= From the figure we can see that the minimum error stays below 5% even when M/N=0.025%. On the other hand, the mean error and maximum error drop quickly at the beginning, and then saturate after the number of samples increase to 18 (M/N=0.1%). This is an amazingly small number compared with 17760, the total number of coefficients (variables). This also verifies the efficacy of our algorithm. V. CONCLUSIONS The impact of PCB geometric parameters on the splitrelated crosstalk is largely determined by the trace-to-trace separation d, the trace-to split-plane distance h, and the signal rise time t r. A compressed sensing based model is established to predict the amplitude of the split-related crosstalk among traces over a split, as a function of these 3 deterministic variables. The method only required an exceptionally small number of samples, thus significantly reduce the time required to build the model. Different test cases are designed to test the accuracy of the model, and it is found that the maximum error is 5.81%. REFERENCES [1] J. Kim, and J. Kim, Effects on Signal Integrity and Radiated Emission by Split Reference Plane on High-Speed Multilayer Printed Circuit Boards, IEEE Trans. Adv. Packag., vol. 28, no.4, pp , 2005 [2] F. Xiao, Y. Nakada, K. Murano, and Y. Kami, Crosstalk Analysis Model for Traces Crossing Split Ground Plane and Its Reduction by Stitching Capacitor, Electronics & Communications in Japan, vol. 90, no.8, pp , 2007 [3] J. Chen, W. Shi, A. Norman, and P. Ilavarasan, Electrical Impact of High-speed Bus Crossing Plane Split, in Proc. IEEE Symp.EMC., vol. 2, pp , 2002 [4] J. Kim, H. Kim, Y. Jeong, J. Lee, and J. Kim, Slot Transmission Line Model of Interconnections Crossing Split Power/Ground Plane on High-speed Multi-layer Board, in Proc. IEEE Workshop on Signal Propagation on Interconnects, pp , 2002 [5] J. Miller, I. Novak, G. Blando, B. Williams, and R. Dame, Examining the Impact of Split Planes on Signal and Power Integrity, in Proc. DesignCon 2010, February, 2010 [6] S. Wu, M. Herndon, H. Shi, and J. Fan, Crosstalk among multiple stripline traces crossing a split, DesignCon 2011, Santa Clara. [7] Tsaig Y., and Donoho D. L., Extensions of compressed sensing, Signal Processing, Vol. 86, No. 3, , [8] Donoho D. L., Compressed sensing, IEEE Trans. Inf. Theory, Vol. 52, No. 4, , [9] Deanna Needell, Roman Vershynin, Uniform Uncertainty Principle and Signal Recovery via Regularized Orthogonal Matching Pursuit, Foundations of Computational Mathematics Volume 9, Number 3, [10] J. Cand`es, J. Romberg, and T. Tao, Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math., 59(8): , [11] Y. Nesterov and A. Nemirovskii. Interior-point polynomial algorithms in convex programming, volume 13 of SIAM Studies in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, [12] M. I. Daubechies, R. DeVore, M. Fornasier, and C. Gunturk. Iteratively re-weighted least squares minimization for sparse recovery. Comm. Pure Appl. Math., 63(1): 1 38, [13] M. Osborne, B. Presnell, and B. Turlach. On the LASSO and its dual. J. Comput. Graph. Statist., 9(2): , [14] Massimo Fornasier and Holger Rauhut, Compressive Sensing. (Chapter in Part 2 of the "Handbook of Mathematical Methods in Imaging" (O. Scherzer Ed.), Springer, 2011)