VISA: Versatile Impulse Structure Approximation for Time-Domain Linear Macromodeling

VISA: Versatile Impulse Structure Approximation for Time-Domain Linear Macromodeling Chi-Un Lei and Ngai Wong Dept. of Electrical and Electronic Engineering The University of Hong Kong Presented by C.U. Lei ASP-DAC 21

Outline Digital IIR filter approximation technique Eigenvalue-free, initial-guess guess-free, linear macromodeling technique Introduction Macromodeling Process Time Domain Macromodeling Algorithm (VISA) Numerical Results Conclusion

Introduction Accurate interconnect/via/package models esp. in high frequency simulation to capture EM effects Compact & efficient macromodels for simulation

Introduction Import data Data preprocessing Tabulated data of input-output response Electromagnetic data Time / frequency Calculation / measurement Math. model Export netlist Model postprocessing Macromodel requirement: Accurate, causal, stable, passive

Existing algorithms Frequency-domain Vector Fitting (VF) [99] Iterative continuous frequency domain approximation (p input ports, q output ports) ( ) ( ) ( ) ( ) P s p m s ( ) M m uv, uv, uv, uv, ( ) = m Q s = m= q m s f s f s Time-domain Macromodeling Techniques Approximation using (truncated) time-sampled system response Easier to capture e.g. 4SID [99], GPOF [99],, TD-VF(z VF(z) ) [3,8]

Problems Issues and Alternatives Numerical-sensitive calculation Affected by the initial poles, noisy signal Require eigenvalues calculation Unstable pole flipping Not an accurate approx. High (multiport) computation complexity Other techniques has much higher computation complexity (Expensive SVD computation) Digital IIR filter approximation Macromodeling SISO LS [ISCAS 8] VISA [ASPDAC 1] Implication: finite-length discrete response sequences... like FIR filter responses!!!

Outline Introduction Algorithm (VISA) Walsh s s theorem & Complementary signal Formulation Convergence: SM iteration, model order reduction, P-norm approximation Numerical Results Conclusion

Least-Squares Approximation Macromodeling of a time-sampled response = IIR Approx. of FIR filter Problem: Approximation Interpolation problem Input / Output description of an allpass filtering operation Designing an allpass operator 1 L F( z) = f + f1z + + flz 1 N Pz ( ) p + p1z + + pn z 1 N Qz ( ) 1+ qz 1 + + qn z H ( z) = =, N < L Δ ( z) = F( z) H( z) 1 l Δ z = Δ z Δ z * 2norm: ( ) ( ) ( ) 2 2π j z = 1 dz z

Find P(z) Suppose Q(z) Known Walsh s Theorem: z =, z = 1/ α, z = 1/ α,, z = 1/ α * * * 1 1 2 2 N N Pz ( ) p+ pz + + p z = = F( zi ), i =,1,, N Qz ( ) 1 qz q z 1 N i 1 i N i 1 N i + 1 i + + N i Pz ( i ) Δ ( zi) = F( zi) = Δ( z) minimized! 2 Qz ( ) ( N + 1) 1 z Q( z ) Δ ( z) = R( z) where R( z) = r + rz Qz ( ) + + r z ( N + 1) 1 z Q( z ) P( z) Rz ( ) = Fz ( ) Qz ( ) Qz ( ) i Finding z i s can be numerically ill. Smarter way making use of the FIR nature of F(z), we can rewrite Δ(z) as 1 ( L 1) 1 L 1

Equivalent Filtering Problem Complementary Signal: All-pass filter! f L f L-1.. f 1 f 1 L-1 L p N p N-1.. p 1 p 1 N-1 N z -L F(z -1 ) z -N Q(z -1 ) / Q(z) z -N P(z -1 ) 1/ Q(z) z -L + z -(L-1) R(z -1 ) - r L-1 r L-2.. r 1 r 1 L-2 L-1 For the first L time instances rl-1 rl-2.. r1 r= z -N Q(z -1 ) / Q(z) f L f L-1.. f 1 f

Finding Numerator P(z) ( N + 1) 1 z Qz ( ) Pz ( ) Δ ( z) = R( z) = F( z) Qz ( ) Qz ( ) Easily prove that 2 L 1 r 2 i i= = Δ () z P(z) easily found!! For the first time L instances: f L f L-1.. f 1 f z -N Q(z -1 ) / Q(z) 1 L-1 L r L-1 r L-2.. r 1 r - 1 L-2 L-1

Complementary Signal Adopted from [YH62] Original Signal Approximating Signal Allpass Filtering Error Signal (Need to minimize!!!)

Finding Denominator Q(z) For the first L time instances f L f L-1.. f 1 f 1 L-1 L z -N Q(z -1 ) / Q(z) r L-1 r L-2.. r 1 r 1 L-2 L-1 *Find Q(z) such that energy output of the all-pass filter A(z)=z -N Q(z -1 )/Q(z) is concentrated in time L, Recall Define 1 N Q( z) 1 q1z = + + + qn z () Q ( z) = 1 Q ( z) = 1+ q z + q z + + q z ( k) ( k) 1 ( k) 2 ( k) N 1 2 N 1 ( ) ( ) 1 ( ) ( 1) ( k k k N ) 1 2 N = 1+ z q + q z + + q z = 1 + z Q ( z) 1 ( k ) 1

Finding Q(z) by Iterations () Q ( z) = 1 Q ( z) = 1+ q z + q z + + q z ( k) ( k) 1 ( k) 2 ( k) N 1 2 N 1 ( ) ( ) 1 ( ) ( 1) k k k N ( 1 2 N ) = 1+ z q + q z + + q z = 1 + z Q ( z) 1 ( k ) 1 f L f L-1.. f 1 f 1 L-1 L z -N Q(z -1 ) / Q(z) r L-1 r L-2.. r 1 r 1 L-2 L-1 Define A ( k ) N ( k) 1 z Q ( z ) ( z) = ( k 1) Q ( z) N ( k) 1 ( k) z Q ( z ) N ( k) 1 ( k) U ( z) = X( z) = z Q ( z ) X ( z) ( k 1) Q ( z) ( k) ( N 1) ( k) 1 ( k) N ( k) X ( z) z Q1 ( z ) = U ( z) z X ( z)

Finding Q(z) by Iterations ( k) ( N 1) ( k) 1 ( k) N ( k) X ( z) z Q1 ( z ) = U ( z) z X ( z) ( k ) ( k ) x () u () ( k) ( k) ( k ) ( k) x (1) x () q N u (1) ( k ) q N 1 ( k) ( k) = ( k ) x ( N 1) x () x () ( k ) q 1 ( k) ( k) ( k) ( k ) x ( L 1) x ( L N) u ( L 1) x ( L N 1) A q = u + b ( k) ( k) ( k) ( k) ( k) ( k) ( k) Automatically uk ( ): = A q b has minimum norm By iterations, Q (k) (z) converges to Q(z) What s amazing: Q (k) (z) is always stable!

Finding Q(z) by Iterations ( k) ( N 1) ( k) 1 ( k) N ( k) X ( z) z Q1 ( z ) = U ( z) z X ( z) ( k ) ( k ) x () u () ( k) ( k) ( k ) ( k) x (1) x () q N u (1) ( k ) q N 1 ( k) ( k) = ( k ) x ( N 1) x () x () ( k ) q 1 ( k) ( k) ( k) ( k ) x ( L 1) x ( L N) u ( L 1) x ( L N 1) A q = u + b ( k) ( k) ( k) ( k) ( k) ( k) ( k) Automatically uk ( ): = A q b has minimum norm By iterations, Q (k) (z) converges to Q(z) What s amazing: Q (k) (z) is always stable!

Convergence of VISA Calculation of Q(z) ) is a simplification of Steiglitz- McBride (SM) iteration without initial guess P(z) ) is not required for the calculation of Q(z) min = N s k = 1 min N s k = () i 1 ( ) z Q z N k k L Q Q ( i 1 ) ( z ) 1 k ( i 1 ) ( z ) k 2 z 1 ( k ) F z 2 () i 1 L 1 ( ) ( ) z Q z z F z N k k k 2 Ns 1 () i = min Q 2 ( z ) F( z ) P z k = Q ( i 1 ) ( z ) k () i ( ) k k k 2 SM iteration

Model order selection Appropriate model order (N) for efficient and accurate simulation Similar pattern between approximant error and Hankel singular value (HSV) of the impulse response HSV distribution and the ratio of first and last HSV H = eig f f f f f 1 2 L 2 f3 L

P-norm approximation Modify the minimization criteria to suit different macromodeling requirements P-norm approximation User-defined norm approximation Norm-constrained approximation min L 1 n= ( ) ( z ) ( k 1 ) ( z ) ( ) ( ) ( ) Δ = min B q d k k k k n p L 1 1 ( k = min Q ) ( z ) H( z ) P z n= Q n p ( k ) ( ) n n n Applying convex programming for solving p p

MIMO VISA Extension Walsh s s theorem can be applied to MIMO situation Stacking the responses into a column for LS solving ( k) Less computation complexity TD-VF: O((pq+1) (pq+1)n 2 Lpq) Much higher! VISA: O(N 2 Lpq) ( k) B 1,1 d 1,1 ( k) ( k) B1,2 ( k ) d1,2 q = ( k) ( k) B d p, q p, q Single-port response

VISA Features No eigenvalue computations No initial guess required Less initial-pole pole-sensitive calculation Guaranteed stable pole calculation Converge to the near-l 2 optimal solution Low (multi-port) computation complexity P-norm approximation for different modeling requirement Quasi-error bound for model order selection

Outline Introduction Algorithm (VISA) Numerical Results Three-port counterwise RF circulator Benchmark examples Conclusion

Numerical Examples Three-port counterwise RF circulator More accurate : 18% less avgerage R.M.S. error after convergence Faster : >15X faster for convergence and >17X faster to achieve a -4dB accuracy L 2 error 1 (a) L 2 error 1 2 (b) 1 TD-VF MIMO VISA 1-2 2 4 CPU Time 1-2 2 4 CPU Time

Mag. (db) Mag. (db) Mag. (db) Mag. (db) Mag. (db) S(1,1) Mag. (db) -1-2 -3.5 1 Normalized Frequency S(1,3) -5-2.5 1 Normalized Frequency S(2,2) -4.5 1 Normalized Frequency S(3,1) -1-2.5 1 Normalized Frequency S(3,3) -1-2 -3.5 1 Normalized Frequency Mag. (db) Mag. (db) Mag. (db) -1-2 S(1,2).5 1 Normalized Frequency S(2,1) -2-4 -1-2.5 1 Normalized Frequency S(2,3).5 1 Normalized Frequency S(3,2) -2-4.5 1 Normalized Frequency MIMO VISA Original Magnitude Magnitude Magnitude Magnitude Magnitude S(1,1).2 -.2 5 1 Time [ns] S(1,3).1 -.1 5 1 Time [ns] S(2,2).2 -.2 5 1 Time [ns] S(3,1).2 -.2 5 1 Time [ns] S(3,3).2 -.2 5 1 Time [ns] Magnitude Magnitude Magnitude Magnitude.2 S(1,2) -.2 5 1 Time [ns] S(2,1).1 -.1 5 1 Time [ns] S(2,3).2 -.2 5 1 Time [ns] S(3,2).1 -.1 5 1 Time [ns] MIMO VISA Original

Numerical Examples HSV (error bound) computation: 58 sec. CPU time P-norm approximation 3.5% L 2 error, 29.5% L inf error and 24.5% CPU time reduction Four benchmark examples [IG8] 51% less L 2 error, >21 X faster, comparing to TD-VF Magnitude of HSV 1 (a) 1-1 1-2 5 1 15 2 Value Magnitude 1 1-1 1-2 (b) HSV MIMO VISA 2 4 6 Model Order

Conclusion VISA: Linear Macromodeling using time- sampled response Simplified Steiglitz-McBride algorithm No initial-pole assignment No eigenvalues calculation Robust and efficient computation P-norm approximation

Thank you! Thank you! Questions are welcome Authors: C. U. Lei * culei@eee.hku.hk Dr. N. Wong nwong@eee.hku.hk Department of Electrical and Electronic Engineering, The University of Hong Kong

Back-Up Slides

Walsh Theorem Among the set of rational functions H(z)= )=P(z)/Q(z), with prescribed poles α 1, α 2,..., α n that are fixed and located in z < 1, the best approximation in the LS sense to F(z) ) (analytic in z > 1 and continuous in z 1) is the unique function that interpolates to F (z)) in all the points. z =, 1/α 1, 1/α 2,,..., 1/α n,, where * denotes complex conjugate Approximation Interpolation problem Input / Output description of an allpass filtering operation Design of an allpass operator

Complementary Signal Adopted from [YH62] Original Signal Approximating Signal Allpass Filtering Error Signal (Need to minimize!!!)