Simulation und Optimierung analoger Schaltungen Optimization Methods for Circuit Design

Transcription

1 Technische Universität München Department of Electrical Engineering and Information Technology Institute for Electronic Design Automation Simulation und Optimierung analoger Schaltungen Optimization Methods for Circuit Design Compendium H. Graeb

2 Version (WS 08/09 - SS 10) Michael Eick Version (SS 07 - SS 08) Husni Habal Presentation follows: H. Graeb, Analog Design Centering and Sizing, Springer, R. Fletcher, Practical Methods of Optimization, John Wiley & Sons, 2nd Edition, Status: February 1, 2010 Copyright Simulation und Optimierung analoger Schaltungen Optimization Methods for Circuit Design Compendium H. Graeb Technische Universität München Institute for Electronic Design Automation Arcisstr Munich, Germany graeb@tum.de Phone: All rights reserved.

3 Contents 1 Introduction Parameters, performance, simulation Performance specification Minimum, minimization Unconstrained optimization Constrained optimization Classification of optimization problems Classification of constrained optimization problems Structure of an iterative optimization process without constraints with constraints Trust-region approach Optimality conditions Optimality conditions unconstrained optimization Necessary first-order condition for a local minimum of an unconstrained optimization problem Necessary second-order condition for a local minimum of an unconstrained optimization problem Sufficient and necessary conditions for second-order derivative 2 f (x ) to be positive definite Optimality conditions constrained optimization Constrained descent direction r Necessary first-order conditions for a local minimum of a constrained optimization problem Necessary second-order condition for a local minimum of a constrained optimization problem Sensitivity of the optimum with regard to a change in an active constraint Optimization of Analog Circuits i

4 3 Worst-case analysis Task Typical tolerance regions Classical worst-case analysis Task Linear performance model Optimization type max f, specification type f > f L Optimization type min f, specification type f < f U Realistic worst-case analysis Task Optimization type max f, specification type f f L : Optimization type min f, specification type f f U General worst-case analysis Task Optimization type max f, specification type f f L : Optimization type min f, specification type f f U : Summary of discussed worst-case analysis problems Statistical parameter tolerances Univariate Gaussian distribution (normal distribution) Multivariate normal distribution Transformation of statistical distributions Example Optimization of Analog Circuits ii

5 5 Expectation values and their estimators Expectation values Definitions Linear transformation of expectation value Linear transformation of variance Translation law of variances Normalizing a random variable Linear transformation of a normal distribution Estimation of expectation values Expectation value estimator Variance estimator Variance of the expectation value estimator Linear transformation of estimated expectation value Linear transformation of estimated variance Translation law of estimated variance Optimization of Analog Circuits iii

6 6 Yield analysis Task Acceptance function Parametric yield Statistical yield analysis/monte-carlo analysis Variance of yield estimator Estimated variance of yield estimator Importance sampling Geometric yield analysis for linearized performance feature ( realistic geometric yield analysis ) Yield partition Defining worst-case distance β W L as difference from nominal performance to specification bound as multiple of standard deviation σ f of linearized performance feature Yield partition as a function of worst-case distance β W L Worst-case distance β W L defines tolerance region specification type f f U Geometric yield analysis for nonlinear performance feature ( general geometric yield analysis ) Problem formulation Advantages of geometric yield analysis Lagrange function and first-order optimality conditions of problem (210) Lagrange function of problem (211) Second-order optimality condition of problem (210) Worst-case distance Remarks Overall yield Consideration of range parameters Optimization of Analog Circuits iv

7 7 Yield optimization/design centering/nominal design Optimization objectives Derivatives of optimization objectives Problem formulations of analog optimization Unconstrained optimization Univariate unconstrained optimization, line search Wolfe-Powell conditions Backtracking line search Bracketing Sectioning Golden Sectioning Line search by quadratic model Unimodal function Multivariate unconstrained optimization without derivatives Coordinate search Polytope method (Nelder-Mead simplex method) Multivariate unconstrained optimization with derivatives Steepest descent Newton approach Quasi-Newton approach Levenberg-Marquardt approach (Newton direction plus trust region) Least-squares (plus trust-region) approach Conjugate-gradient (CG) approach Optimization of Analog Circuits v

8 9 Constrained optimization problem formulations Quadratic Programming (QP) QP linear equality constraints QP - inequality constraints Example Sequential Quadratic programming (SQP), Lagrange Newton SQP equality constraints Penalty function Sizing rules for analog circuit optimization Single (NMOS) transistor Sizing rules for single transistor that acts as a voltage-controlled current source (VCCS) Transistor pair: current mirror (NMOS) Sizing rules for current mirror Optimization of analog circuits: tasks Analysis, synthesis Sizing Nominal design, tolerance design Optimization without/with constraints A Matrix and vector notations 113 A.1 Vector A.2 Matrix A.3 Addition A.4 Multiplication A.5 Special cases A.6 Determinant of a quadratic matrix A.7 Inverse of a quadratic non-singular matrix A.8 Some properties Optimization of Analog Circuits vi

9 B Abbreviated notations of derivatives using the nabla symbol 119 C Norms 121 D Pseudo-inverse, singular value decomposition (SVD) 123 D.1 Moore-Penrose conditions D.2 Singular value decomposition E Linear equation system, rectangular system matrix with full rank 125 E.1 underdetermined system of equations E.2 overdetermined system of equations E.3 determined system of equations F Partial derivatives of linear, quadratic terms in matrix/vector notation129 G Probability space 131 H Convexity 133 H.1 Convex set K R n H.2 Convex function Optimization of Analog Circuits vii

10 Optimization of Analog Circuits viii

11 1 Introduction 1.1 Parameters, performance, simulation design parameters x d R n xd e.g. transistor widths, capacitances statistical parameters x s R nxs e.g. oxide thickness, threshold voltage range parameters x r R nxr e.g. operational parameters: (circuit) parameters x = [ x T d xt s x T r ] T supply voltage, temperature performance feature f i e.g. gain, bandwidth, slew rate, phase margin, delay, power (circuit) performance f = [ f i ] T R n f (circuit) simulation x f(x) e.g. SPICE A design parameter and a statistical parameter may refer to the same physical parameter. E.g., an actual CMOS transistor width is the sum of a design parameter W k and a statistical parameter W. W k is the specific width of transistor T k while W is a width reduction that varies globally and equally for all the transistors on a die. A design parameter and a statistical parameter may be identical. 1.2 Performance specification performance specification feature (upper or lower limit on a performance): f i f L,i or f i f U,i (1) number of performance specification features: n f n P SF 2n f (2) performance specification: f L,1 f 1 (x) f U,1. f L,nf f nf (x) f U,nf f L f(x) f U (3) Optimization of Analog Circuits 1

12 f (a) strong local minimum weak local minimum global minimum x f (b) x Figure 1. Smooth function (a), i.e. continuous and differentiable at least several times on a closed region of the domain. Non-smooth continuous function (b). 1.3 Minimum, minimization without loss of generality: optimum minimum because: max f min f (4) { minimum, i.e., a result min minimize, i.e., a process (5) min f(x) f (x)! min (6) min f(x) x, f(x ) = f (7) 1.4 Unconstrained optimization f = min f(x) min f min f(x) x x min{f(x)} x = argmin f(x) argmin f argmin f(x) x x argmin{f(x)} (8) Optimization of Analog Circuits 2

13 1.5 Constrained optimization E: set of equality constraints I: set of inequality constraints min f(x) s.t. c i (x) = 0, i E c i (x) 0, i I (9) Alternative formulations where, min x f s.t. x Ω (10) min f (11) x Ω min {f(x) x Ω} (12) { c i (x) = 0, i E Ω = x c i (x) 0, i I } Lagrange function combines objective function and constraints in a single expression L(x, λ) = f(x) λ i c i (x) (13) i E I λ i : Lagrange multiplier associated with constraint i Optimization of Analog Circuits 3

14 1.6 Classification of optimization problems deterministic, stochastic continuous, discrete local, global scalar, vector constrained, unconstrained with or without derivatives The iterative search process is deterministic or random. Optimization variables can take an infinite number of values, e.g., the set of real numbers, or take a finite set of values or states. The objective value at a local optimal point is better than the objective values of other points in its vicinity. The objective value at a global optimal point is better than the objective values of any other point. In a vector optimization problem, multiple objective functions shall be optimized simultaneously (multiplecriteria optimization, MCO). Usually, objectives have to be traded off with each other. A Pareto-optimal point is characterized in that one objective can only be improved at the cost of another. Pareto optimization determines the set of all Pareto-optimal points. Scalar optimization refers to a single objective. A vector optimization problem is scalarized by combining the multiple objectives into a single overall objective, e.g., by a weighted sum, least-squares, or min/max. Besides the objective function that has to be optimized, constraints on the optimization variables may be given as inequalities or equalities. The optimization process may be based on gradients (first derivative) or on gradients and Hessians (second derivative), or it may not require any derivatives of the objective/constraint functions. 1.7 Classification of constrained optimization problems objective function constraint functions linear linear linear programming quadratic linear quadratic programming nonlinear nonlinear nonlinear programming convex linear equality convex programming constraints (local global minimum) concave inequality constraints Optimization of Analog Circuits 4

15 1.8 Structure of an iterative optimization process without constraints Taylor series of a function f about iteration point x (κ) : f(x) = f(x (κ) ) + f ( x (κ)) T ( ) x x (κ) ( x x (κ) ) T 2 f ( x (κ)) (x x (κ)) +... (14) = f (κ) + g (κ)t (x x (κ)) ( x x (κ) ) T H (κ) (x x (κ)) +... (15) f (κ) : value of function f at point x (κ) g (κ) : gradient (first derivative, direction of steepest ascent) at point x (κ) H (κ) : Hessian matrix (second derivative) at point x (κ) Taylor series about search direction r starting from point x (κ) : x(r) = x (κ) + r (16) f(r) = f (κ) + g (κ)t r rt H (κ) r +... (17) Taylor series about step length along search direction r (κ) starting from point x (κ) : x(α) = x (κ) + α r (κ) (18) f(α) = f (κ) + g (κ)t r (κ) α r(κ)t H (κ) r (κ) α (19) = f (κ) + f (α = 0) α f (α = 0) α (20) f (α = 0) : slope of f along direction r (κ) 2 f (α = 0) : curvature of f along r (κ) repeat determine the search direction r (κ) determine the step length α (κ) (line search) x (κ+1) = x (κ) + α (κ) r (κ) κ:= κ + 1 until termination criteria are fulfilled Steepest-descent approach search direction: direction of steepest descent, i.e., r (κ) = g (κ) Optimization of Analog Circuits 5

16 Figure 2. Visual illustration of the steepest-descent approach for Rosenbrock s function f (x 1, x 2 ) = 100(x 2 x 2 1) 2 + (1 x 1 ) 2. A backtracking line search is applied (see Sec , page 67) with an initial x (0) = [ 1.0, 0.8] T and α (0) = 1, α := c 3 α. The search terminates when the Armijo condition is satisfied with c 1 = 0.7, c 3 = 0.6. Optimization of Analog Circuits 6

17 with constraints Constraint functions and objective functions are combined in an unconstrained optimization problem in each iteration step Lagrange formulation penalty function Sequential Quadratic Programming (SQP) Projection on active constraints, i.e. into subspace of an unconstrained optimization problem in each iteration step active-set methods Trust-region approach model of the objective function: f(x) m ( x (κ) + r ) (21) min r m ( x (κ) + r ) s.t. r trust region (22) e.g., r < search direction and step length are computed simultaneously trust region to consider the model accuracy Optimization of Analog Circuits 7

18 Optimization of Analog Circuits 8

19 2 Optimality conditions 2.1 Optimality conditions unconstrained optimization Taylor series of the objective function around the optimum point x : f(x) = f(x ) + f (x ) T (x x ) }{{}}{{} f For x = x + r close to the optimum: g T (x x ) T 2 f (x ) }{{} (x x ) +... (23) H f : value of the function at the optimum x g : gradient at the optimum x H : Hessian matrix at the optimum x f(r) = f + g T r rt H r +... (24) x is optimal there is no descent direction, r, such that f (r) < f. x 2 f ( x (κ)) gradient (κ) x direction steepest descent direction f (x) > f ( x (κ)) f (x) = f ( x (κ)) f (x) < f ( x (κ)) x 1 Figure 3. Descent directions from x (κ) : shaded area. Optimization of Analog Circuits 9

20 2.1.1 Necessary first-order condition for a local minimum of an unconstrained optimization problem x : stationary point descent direction r: f ( x (κ)) T r < 0 steepest descent direction: r = f ( x (κ)) r 0 g T r 0 (25) g = f (x ) = 0 (26) Figure 4. Quadratic functions: (a) minimum at x, (b) maximum at x, (c) saddle point at x, (d) positive semidefinite with multiple minima along trench. Optimization of Analog Circuits 10

21 2.1.2 Necessary second-order condition for a local minimum of an unconstrained optimization problem r 0 rt 2 f (x ) r 0 2 f (x ) is positive semidefinite f has non-negative curvature (27) sufficient: r 0 rt 2 f (x ) r > 0 2 f (x ) is positive definite has positive curvature (28) Figure 5. Contour plots of quadratic functions that are (a),(b) positive or negative definite, (c) indefinite (saddle point), (d) positive or negative semidefinite. Optimization of Analog Circuits 11

22 2.1.3 Sufficient and necessary conditions for second-order derivative 2 f (x ) to be positive definite all eigenvalues > 0 has a Cholesky decomposition: f 2 (x ) = L L T with l ii > 0 f 2 (x ) = L D L T with l ii = 1 and d ii > 0 (29) all pivot elements during gaussian elimination without pivoting > 0 all principal minors are > 0 x 2 unconstrained descent unconstrained c i = const f ( x (κ)) c i ( x (κ) ) x (κ) descent f = const x 1 Figure 6. Dark shaded area: unconstrained directions according to (35), light shaded area: descent directions according to (34), overlap: unconstrained descent directions. When no direction satisfies both (34) and (35) then the cross section is empty and the current point is a local minimum of the function. Optimization of Analog Circuits 12

23 2.2 Optimality conditions constrained optimization Constrained descent direction r descent direction: f ( x (κ)) T r < 0 (30) constrained direction: ( c i x (κ) + r ) ( c ) ( i x (κ) + c ) i x (κ) T r 0 (31) Inactive constraint: i is inactive ( c ) i x (κ) > 0 then each r with r < ɛ satisfies (31), e.g., ( c ) i x (κ) r = c i (x (κ) ) f (x (κ) ) f ( x (κ)) (32) (32) in (31) gives: where, ( c ) [ i x (κ) 1 c ( ) i x (κ) T ( ) ] f x (κ) 0 (33) c i (x (κ) ) f (x (κ) ) 1 c ( ) i x (κ) T ( ) f x (κ) c i (x (κ) ) f (x (κ) ) 1 Active constraint (Fig. 6): i is active c i ( x (κ) ) = 0 then (30) and (31) become: f ( x (κ)) T r < 0 (34) c i ( x (κ) ) T r 0 (35) no constrained descent direction exists: no vector r satisfies both (34) and (35) at x : f (x ) = λ i c i (x ) with λ i 0 (36) no statement about sign of λ i in case of an equality constraint (c i = 0 c i 0 c i 0) Optimization of Analog Circuits 13

24 2.2.2 Necessary first-order conditions for a local minimum of a constrained optimization problem x, f = f (x ), λ, L = L (x, λ ) Karush-Kuhn-Tucker (KKT) conditions L (x ) = 0 (37) c i (x ) = 0 i E (38) c i (x ) 0 i I (39) λ i 0 i I (40) λ i c i (x ) = 0 i E I (41) (37) is analogous to (26) (13) and (37) give: f (x ) λ i c i (x ) = 0 (42) i A(x ) A (x ) is the set of active constraints at x A (x ) = E {i I c i (x ) = 0} (43) (41) is called the complementarity condition, Lagrange multiplier is 0 (inactive constraint) or constraint c i (x ) is 0 (active constraint). from (41) and (13): L = f (44) Optimization of Analog Circuits 14

25 2.2.3 Necessary second-order condition for a local minimum of a constrained optimization problem f (x + r) = L (x + r, λ ) (45) = L (x, λ ) + r T L (x ) + 1 }{{} 2 rt 2 L (x ) r + (46) }{{} 0. =. f + 1 {}}{ 2 rt [ 2 f (x ) λ i 2 c i (x )] r + (47) for each feasible stationary direction r at x, i.e., i A(x ) F r = r r 0 c i (x ) T r 0 i A (x ) \A + c i (x ) T r = 0 i A + = { j A (x ) j E λ j > 0 } (48) necessary: sufficient: r T 2 L (x ) r 0 (49) r F r r T 2 L (x ) r > 0 (50) r F r Optimization of Analog Circuits 15

26 2.2.4 Sensitivity of the optimum with regard to a change in an active constraint perturbation of an active constraint at x by i 0 c i (x) 0 c i (x) i (51) L (x, λ, ) = f (x) i λ i (c i (x) i ) (52) f ( i ) = L ( i ) ( L = x T x + L i λ T λ + L ) i T i x,λ }{{}}{{} 0 T 0 T = L i = λ i x,λ f ( i ) = λ i (53) Lagrange multiplier: sensitivity to change in an active constraint close to x Optimization of Analog Circuits 16

27 3 Worst-case analysis 3.1 Task indexes d, s, r for parameter types x d, x s, x r left out index i for performance feature f i left out Given: tolerance region T of parameters Find: worst-case performance value, f W, that the circuit takes over T and corresponding worst-case parameter vectors, x W optimization specification good bad worst-case performance max f lower bound: f f L f f f W L = f (x W L ) min f upper bound: f f U f f f W U = f (x W U ) 3.2 Typical tolerance regions box: T B = {x x L x x U } { ellipsoid: T E = x β 2 (x) = (x x 0 ) T C 1 (x x 0 ) βw 2 C is symmetric, positive definite }, x 2 x U,2 x 2 β = β W x L,2 T B x 0 T E x L,1 x U,1 x 1 x 1 Figure 7. Tolerance box, tolerance ellipsoid. Optimization of Analog Circuits 17

28 3.3 Classical worst-case analysis indexes d, s, r for parameter types x d, x s, x r left out index i for performance feature f i left out Task Given: hyper-box tolerance region T B = {x x L x x U } (54) linear performance model f (x) = f a + g T (x x a ) (55) Find: worst-case parameter ( vectors ) x W L/U and corresponding worst-case performance values f W L/U = f W L/U xw L/U x 2 x U,2 f = f a g x W U f = fw U = f (xw U) x L,2 x W L x a f = f W L = f (x W L ) T B x L,1 x U,1 x 1 Figure 8. Classical worst-case analysis with tolerance box and linear performance model. Optimization of Analog Circuits 18

29 3.3.2 Linear performance model sensitivity analysis: forward finite-difference approximation: f a = f (x a ) (56) g = f (x a ) (57) f a = f (x a ) (58) f (x a,i ) g i = f (x a + x i e i ) f (x a ) x i (59) e i = [ ] T (60) i-th position f, f (a) f f(x a ) a b f g = b a x a x f, f (b) f f(x a + x) f f(x a ) g = f(xa+ x) f(xa) x x a x a + x x Figure 9. Linear performance model based on gradient (a), based on forward finitedifference approximation of gradient (b). Optimization of Analog Circuits 19

30 3.3.3 Optimization type max f, specification type f > f L min f (x) }{{} min g T x s.t. { x xl x x U x W L, f W L = f (x W L ) (61) specific linear programming problem with analytical solution corresponding Lagrange function: L (x, λ L, λ U ) = g T x λ T L (x x L ) λ T U (x U x) (62) first-order optimality conditions: L (x) = 0 : g λ L + λ U = 0 (63) λ L/U 0 x U x W L 0, x W L x L 0 λ L,j (x W L,j x L,j ) = 0, j = 1,, n x λ U,j (x U,j x W L,j ) = 0, j = 1,, n x { λ T L (x W L x L ) = 0 λ T U (x U x W L ) = 0 a i b i = 0 a T b = a i b i = 0 i i a T b = 0, a i 0, b i 0 a i b i = 0 i (64) second-order optimality condition holds because 2 L (x) = 0 either constraint x L,j or constraint x U,j active, never both, therefore from (63) and (64): either: g j = λ L,j > 0 (65) or: g j = λ U,j < 0 (66) component of the worst-case parameter vector x W L : x L,j, g j > 0 x W L,j = x U,j, g j < 0 undefined, g j = 0 (67) worst-case performance value: f W L = f a + g T (x W L x a ) = f a + j g j (x W L,j x a,j ) (68) Optimization of Analog Circuits 20

31 3.3.4 Optimization type min f, specification type f < f U min f (x) }{{} min g T x s.t. { x xl x x U x W U, f W U = f (x W U ) (69) component of the worst-case parameter vector x W U : x L,j, g j < 0 x W U,j = x U,j, g j > 0 undefined, g j = 0 (70) worst-case performance value: f W U = f a + g T (x W U x a ) = f a + j g j (x W U,j x a,j ) (71) Optimization of Analog Circuits 21

32 3.4 Realistic worst-case analysis indexes d, s, r for parameter types x d, x s, x r left out index i for performance feature f i left out Task Given: ellipsoid tolerance region { } T E = x (x x 0 ) T C 1 (x x 0 ) βw 2 (72) linear performance model f (x) = f a + g T (x x a ) (73) = f 0 + g T (x x 0 ) with f 0 = f a + g T (x 0 x a ) (74) Find: worst-case parameter vectors x W L/U values f W L/U = f ( ) x W L/U and corresponding worst-case performance x 2 x W L f = f W L = f (x W L ) x 0,2 T E x 0 f = f a + g T (x 0 x a ) g f = f W U = f (x W U) x W U x 0,1 x 1 Figure 10. Realistic worst-case analysis with tolerance ellipsoid and linear performance model. Optimization of Analog Circuits 22

33 3.4.2 Optimization type max f, specification type f f L : min f (x) }{{} min g T x s.t. (x x 0 ) T C 1 (x x 0 ) β 2 W x W L, f W L = f (x W L ) (75) specific nonlinear programming problem (linear objective function, quadratic constraint function) with analytical solution corresponding Lagrange function: L (x, λ) = g T x λ ( ) βw 2 (x x 0 ) T C 1 (x x 0 ) (76) first-order optimality conditions: L = 0 : g + 2λ W L C 1 (x W L x 0 ) = 0 (77) due to linear function f, the solution is on the border of T E, i.e., the constraint is active: (x W L x 0 ) T C 1 (x W L x 0 ) = β 2 W (78) λ W L > 0 (79) second-order optimality condition holds because: 2 L (x) = 2λ W L C 1, C 1 is positive definite, and because of (79) (77) gives: x W L x 0 = 1 2λ W L C g (80) substituting (80) into (78) gives: 1 4λ 2 W L g T C g = β 2 W (81) inserting λ W L from (81) into (80) to eliminate λ W L gives a worst-case parameter vector in terms of the performance gradient and tolerance region constants: x W L x 0 = β W g T C g C g = β W σ f C g (82) substituting (82) in (74) gives the corresponding worst-case performance value: f W L = f (x W L ) = f 0 + g T (x W L x 0 ) = f 0 β W g T C g (83) = f 0 β W σ f Gaussian error propagation, linear transformation of a normal distribution Optimization of Analog Circuits 23

34 3.4.3 Optimization type min f, specification type f f U (75) becomes min f (x) }{{} min g T x s.t. (x x 0 ) T C 1 (x x 0 ) β 2 W x W U, f W U = f (x W U ) (84) (82) becomes x W U x 0 = + β W g T C g C g = + β W σ f C g (85) (83) becomes f W U = f (x W U ) = f 0 + g T (x W U x 0 ) = f 0 + β W g T C g (86) = f 0 + β W σ f Optimization of Analog Circuits 24

35 3.5 General worst-case analysis indexes d, s, r for parameter types x d, x s, x r left out index i for performance feature f i left out Task Given: ellipsoid tolerance region: { } T E = x (x x 0 ) T C 1 (x x 0 ) βw 2 (87) general smooth performance f (x) Find: worst-case parameter vectors x W L/U values f W L/U = f ( ) x W L/U and corresponding worst-case performance x 2 f (x W U ) f x W U x 0 f (x W L ) f (W U) = f W U f = f W U = f (x W U ) T E f = f (x 0 ) = f 0 f (W L) = f W L x W L f = f W L = f (x W L ) x 1 Figure 11. General worst-case analysis with tolerance ellipsoid and nonlinear performance function. Optimization of Analog Circuits 25

36 3.5.2 Optimization type max f, specification type f f L : min x f (x) s.t. (x x 0 ) T C 1 (x x 0 ) β 2 W (88) specific nonlinear programming problem (non-linear objective function, one quadratic inequality constraint) with numerical solution, e.g., by Sequential Quadratic Programming (SQP), which yields f (x W L ) assumption: unique solution on border of T E Linearization of objective function f at worst-case point x W L, i.e., after solution of (88): substituting (89) in (88) gives: f (W L) (x) = f W L + f (x W L ) T (x x W L ) (89) min f (W L) (x) }{{} min f(x W L ) T x s.t. (x x 0 ) T C 1 (x x 0 ) β 2 W (90) structure identical to realistic worst worst-case analysis (75), replace g by f (x W L ) worst-case parameter vector: β W x W L x 0 = f(x C f (x W L ) T W L) C f(x W L ) = β W σ f (W L) C f (x W L) (91) worst-case performance value: f (W L) (x 0 ) = f 0 = f W L + f (x W L ) T (x 0 x W L ) f W L = f 0 + f (x W L ) T (x W L x 0 ) (92) substituting (91) in (92) gives the corresponding worst-case performance value: f W L = f 0 β W f (x W L ) T C f (x W L ) = f 0 β W σ f (W L) (93) Optimization of Analog Circuits 26

37 3.5.3 Optimization type min f, specification type f f U : (88) becomes (89) becomes min f (x) s.t. (x x 0 ) T C 1 (x x 0 ) β 2 W (94) f (W U) (x) = f W U + f (x W U ) T (x x W U ) (95) (90) becomes min f (x W U ) T x s.t. (x x 0 ) T C 1 (x x 0 ) β 2 W (96) worst-case parameter vector: β W x W U x 0 = + f(x C f (x W U ) T W U) C f(x W U ) = + β W σ f (W U) C f (x W U) (97) worst-case performance value: f W U = f 0 + β W f (x W U ) T C f (x W U ) = f 0 + β W σ f (W U) (98) Optimization of Analog Circuits 27

38 3.6 Summary of discussed worst-case analysis problems For each performance feature f i there exists a worst-case parameter vector x W L,i and/or x W U,i respectively and a corresponding worst-case performance value f W L,i and/or f W U,i. x W L,i and x W U,i are unique in the classical and realistic worst-case analysis. Several worst-case parameter vectors may exist in the general worst-case analysis. worst case feasible region objective function good for analysis type classical hyper-box linear uniform distribution unknown distribution discrete circuits range parameters realistic ellipsoid linear normal distribution general ellipsoid non-linear IC transistor parameters Worst-case analysis requires design technology (circuit, performance features) and process technology (statistical parameters, parameter distribution). Optimization of Analog Circuits 28

39 4 Statistical parameter tolerances modeling of manufacturing variations through a multivariate continuous distribution function of statistical parameters x s cumulative distribution function (cdf): cdf (x s ) = xs,1 xs,nxs pdf (t) dt (99) (discrete: cumulative relative frequencies) probability density function (pdf): dt = dt 1 dt 2... dt nxs pdf (x s ) = nxs xs,1 xs,nxs cdf (x s ) (100) (discrete: relative frequencies) x s,i denotes the random number value of a random variable X s,i Optimization of Analog Circuits 29

40 4.1 Univariate Gaussian distribution (normal distribution) x s N ( x s,0, σ 2) (101) x s,0 σ 2 σ : mean value : variance : standard deviation probability density function of the univariate normal distribution: pdf N ( xs, x s,0, σ 2) = 1 2π σ e 1 2 xs xs,0 σ 2 (102) pdf N 3σ 2σ σ 0 σ 2σ 3σ x s x s,0 1 cdf N 0.5 3σ 2σ σ 0 σ 2σ 3σ x s x s,0 Figure 12. Probability density function, pdf, and corresponding cdf of a univariate Gaussian distribution. The area of the shaded region under the pdf is the value of the cdf as shown. x s x s,0 3σ 2σ σ 0 σ 2σ 3σ 4σ cdf (x s x s,0 ) 0.1% 2.2% 15.8% 50% 84.1% 97.7% 99.8% 99.99% Optimization of Analog Circuits 30

41 4.2 Multivariate normal distribution x s N (x s,0, C) (103) x s,0 : vector of mean values of statistical parameters x s C: covariance matrix of statistical parameters x s, symmetric, positive definite probability density function of the multivariate normal distribution: pdf N (x s, x s,0, C) = 1 2π n xs det (C) e 1 2 β2 (x s,x s,0,c) (104) β 2 (x s, x s,0, C) = (x x s,0 ) T C 1 (x x s,0 ) (105) C = Σ R Σ (106) 1 ρ σ 1 0 1,2 ρ 1,nxs Σ =..., R = ρ 1,2 1 ρ 2,nxs. 0 σ.... nxs ρ 1,nxs ρ 2,nxs 1 σ1 2 σ 1 ρ 1,2 σ 2 σ 1 ρ 1,nxs σ nxs σ 1 ρ 1,2 σ 2 σ2 2 σ 2 ρ 2,nxs σ nxs C = σ 1 ρ 1,nxs σ nxs σn 2 xs (107) (108) R : correlation matrix of statistical parameters σ k : standard deviation of component x s,k, σ k > 0 σ 2 k : variance of component x s,k σ k ρ k,l σ l : covariance of components x s,k, x s,l ρ k,l : correlation coefficient of components x s,k and x s,l, 1 < ρ k,l < 1 ρ k,l = 0 : uncorrelated and also independent if jointly normal ρ k,l = 1 : strongly correlated components Optimization of Analog Circuits 31

42 x s,2 x s,0,2 x s,0 β 2 (x s ) = const x s,0,1 x s,1 Figure 13. Level sets of a two-dimensional normal pdf (a) (b) (c) x s,2 β 2 (x s) = const x s,2 β 2 (x s) = const x s,2 β 2 (x s) = const x s,0,2 x s,0 x s,0,2 xs,0 x s,0,2 x s,0 x 1 x s,0,1 x s,1 x s,0,1 x s,1 x s,0,1 x s,1 Figure 14. Level sets of a two-dimensional normal pdf with general covariance matrix C, (a), with uncorrelated components, R = I, (b), and uncorrelated components of equal spread, C = σ 2 I, (c). Optimization of Analog Circuits 32

43 x 2 ρ = 0 a 2 σ2 β 2 = const = a 2 a 2 σ 1 x 1 Figure 15. Level set with β 2 = a 2 of a two-dimensional normal pdf for different values of the correlation coefficient, ρ. Optimization of Analog Circuits 33

44 4.3 Transformation of statistical distributions y R ny, z R nz, n y = n z z = z (y), y = y (z) such that the mapping from y to z is smooth and bijective (precisely z = φ (y), y = φ 1 (z)) cdf y (y) = = y z pdf y (y ) dy = z(y) ( ) pdf y (y (z )) y det dz z T pdf z (z ) dz = cdf z (z) (109) y pdf y (y ) dy = z pdf z (z ) dz (110) ( ) pdf z (z) = pdf y (y (z)) y det (111) z T univariate case: pdf z (z) = pdf y (y(z)) y z (112) In the simple univariate case the function pdf z has a domain that is a scaled version of the domain of pdf y. ( ) y determines the scaling factor. In high-order cases, the random z variable space is scaled and rotated with the Jacobian matrix y determining the scaling z T and rotation. pdf y (y) pdf z (z) y 1 y 2 y 3 y z 1 z 2 z 3 z Figure 16. Univariate pdf with random number y is transformed to a new pdf of new random number z = z (y). According to (109) the shaded areas as well as the hatched areas under the curve are equal. Optimization of Analog Circuits 34

45 4.3.1 Example Given: probability density function pdf U (z), here a uniform distribution: pdf U (z) = { 1 for 0 < z < 1 0 otherwise (113) probability density function pdf y (y), y R random number z Find: random number y z y from (109) z y 0 pdf z (z ) }{{} 1 for 0 z 1 dz = pdf y (y ) dy (114) from (113) hence z = y pdf y (y ) dy = cdf y (y) (115) y = cdf 1 y (z) (116) This example details a method to generate sample values of a random variable y with an arbitrary pdf pdf y if sample values are available from a uniform distribution pdf z : insert pdf y (y) in (115) compute cdf y by integration compute inverse cdf 1 y create uniform random number, z, insert into (116) to get sample value, y, according to pdf y (y) Optimization of Analog Circuits 35

46 Optimization of Analog Circuits 36

47 5 Expectation values and their estimators 5.1 Expectation values Definitions h (z): function of a random number z with probability density function pdf (z) Expectation value E {h (z)} = E {h (z)} = pdf(z) + h (z) pdf (z) dz (117) Moment of order κ Mean value (first-order moment) Central moment of order κ m (κ) = E {z κ } (118) m (1) = m = E {z} (119) E {z 1 } m = E {z} =. (120) E {z nz } c (κ) = E {(z m) κ } c (1) = 0 (121) Variance (second-order central moment) σ: standard deviation Covariance c (2) = E { (z m) 2} = σ 2 = V {z} (122) cov {z i, z j } = E {(z i m i ) (z j m j )} (123) Variance/covariance matrix C = V {z} = E {(z } m) (z m) T V {z 1 } cov {z 1, z 2 } cov {z 1, z nz } cov {z 2, z 1 } V {z 2 } cov {z 2, z nz } =..... cov {z nz, z 1 } cov {z nz, z 2 } V {z nz } V {h (z)} = E (124) { (h (z) E {h (z)}) (h (z) E {h (z)}) T } (125) Optimization of Analog Circuits 37

48 5.1.2 Linear transformation of expectation value E {A h (z) + b} = A E {h (z)} + b (126) special cases: E {c} = c, c is a constant E {c h (z)} = c E {h (z)} E {h 1 (z) + h 2 (z)} = E {h 1 (z)} + E {h 2 (z)} Linear transformation of variance special cases: Gaussian error propagation: V {A h (z) + b} = A V {h (z)} A T (127) V { a T h (z) + b } = a T V {h (z)} a V {a h (z) + b} = a 2 V {h(z)} V { a T z + b } = a T C a = i,j a i a j σ i ρ i,j σ j = i j ρ i,j=0 a 2 i σi 2 i Translation law of variances V {h (z)} = E { (h (z) a) (h (z) a) T } (E {h (z)} a) (E {h (z)} a) T special cases: V {h (z)} = E { (h (z) a) 2} (E {h (z)} a) 2 V {h (z)} = E { h (z) h T (z) } E {h (z)} E { h T (z) } V {h (z)} = E {( h 2 (z) )} (E {h (z)}) 2 (128) Normalizing a random variable z = z E {z} V {z} = z m z σ z (129) E {z } = E {z} m z = 0 (130) σ z { V {z } = E (z 0) 2} = E { (z m z ) 2} = 1 (131) σ 2 z Optimization of Analog Circuits 38

49 5.1.6 Linear transformation of a normal distribution x N (x 0, C) f (x) = f a + g T (x x a ) (132) mean value µ f of f: µ f = E { f } = E { f a + g T (x x a ) } = E {f a } + g T (E {x} E {x a }) variance σ 2 f of f: σ 2 f µ f = f a + g T (x 0 x a ) (133) { (f ) 2 } { (g = E µf = E T (x x 0 ) ) } 2 { } = E g T (x x 0 ) (x x 0 ) T g { } = g T E (x x 0 ) (x x 0 ) T g σ 2 f = gt C g (134) Optimization of Analog Circuits 39

50 5.2 Estimation of expectation values Expectation value estimator Ê {h (x)} = ˆm h = 1 n MC n MC µ=1 h ( x (µ)) (135) x (µ) D (pdf (x)), µ = 1,..., n MC sample of the population with n MC sample elements, i.e., sample size n MC sample elements x (µ), µ = 1,, n MC, that are independently and identically distributed, i.e, E { h ( x (µ))} = E {h (x)} = m h (136) V { h ( x (µ))} = V {h (x)}, cov { h ( x (µ)), h ( x (ν))} = 0, µ ν (137) ˆφ (x) = ˆφ ( x (1),..., x (n MC) ) : estimator function of φ (x) (138) Variance estimator ˆV {h (x)} = n 1 MC ( ( ) ) ( ( h x (µ) ˆm ) ) h h x (µ) T ˆm h (139) n MC 1 µ=1 estimator bias: x (µ) D (pdf (x)), µ = 1,..., n MC ˆV {h (x)} = 1 n MC ( ( ) ) ( ( h x (µ) m ) ) h h x (µ) T m h (140) n MC µ=1 } b ˆφ = E {ˆφ (x) φ (x) (141) unbiased estimator: } E {ˆφ (x) = φ (x) b ˆφ = 0 (142) consistent estimator: strongly consistent: { } lim ˆφ P φ < ɛ = 1 (143) n MC ɛ 0 (144) variance of an estimator (quality): { ) ) } T {ˆφ} Q ˆφ = E (ˆφ φ (ˆφ φ = V + b ˆφ b Ṱ (145) φ b ˆφ = 0 : Q ˆφ = V {ˆφ} (146) Optimization of Analog Circuits 40

51 5.2.3 Variance of the expectation value estimator Q ˆmh = V { ˆm h } = V } {Ê {h (x)} = V { 1 n MC n MC µ=1 h ( x (µ))} (127) Q ˆmh = 1 n 2 MC V { nmc µ=1 I nh,n h h (µ) } h (µ) = h ( x (µ)), I nh,n h identity matrix of size n h of h (µ) (127) Q ˆmh = Q ˆmh = 1 n 2 MC 1 n 2 MC V [ ] I nh,n h I nh,n h... I nh,n h [I nh,n h I nh,n h... I nh,n h ] V h (1) h (2). h (n MC) h (1) h (2). h (n MC) I nh,n h I nh,n h. I nh,n h (137) Q ˆmh = = 1 n 2 MC 1 n 2 MC [I nh,n h I nh,n h... I nh,n h ] n MC V {h} V {h} V {h} I nh,n h I nh,n h. I nh,n h Q ˆmh = V { ˆm h } = 1 n MC V {h} (147) replace Q ˆmh by ˆQ ˆmh, V {h} by ˆV {h}, (127) by (150) to obtain the variance estimator of the expected value estimator ˆQ ˆmh = ˆV { ˆm h } = 1 n MC ˆV {h} (148) standard deviation of the mean estimator decreases with 1/ n MC, e.g., 100 times more sample elements for 10 times smaller standard deviation in the expectation value estimator Optimization of Analog Circuits 41

52 5.2.4 Linear transformation of estimated expectation value Ê {A h (z) + b} = A Ê {h (z)} + b (149) Linear transformation of estimated variance ˆV {A h (z) + b} = A ˆV {h (z)} A T (150) Translation law of estimated variance ˆV {h (z)} = n MC { [Ê h (z) h T (z) } n MC 1 Ê {h (z)} Ê { h T (z) }] (151) Optimization of Analog Circuits 42

53 6 Yield analysis 6.1 Task Given: statistical parameters with normal distribution, eventually obtained through transformation performance specification Find: percentage/proportion of circuits that fulfill the specifications statistical parameter distribution (manufacturing process) pdf (x s ) = pdf N (x s ) = 1 2π n xs det (C) e 1 2 β2 (x s) (152) β 2 (x s ) = (x s x s,0 ) T C 1 (x s x s,0 ) (153) performance acceptance region, performance specification (customer) A f = {f f L f f U } (154) solution requires either: non-normal distribution pdf f (f) or: non-linear parameter acceptance region A s = {x s f (x s ) A f } (dashed lines in Fig. 17) x s,2 f 2 β = const A s f U,2 x s,0 f (xs,0 ) A f f L,2 x s,1 f L,1 f U,1 f 1 Figure 17. Optimization of Analog Circuits 43

54 6.1.1 Acceptance function δ (x s ) = { 1, f (xs ) A f 0, f (x s ) / A f = { 1, xs A s circuit functions 0, x s / A s circuit malfunctions (155) Parametric yield Y = = pdf (x s ) dx s (156) A s δ (x s ) pdf (x s ) dx s (157) yield: expected value of the acceptance function = E {δ (x s )} (158) Optimization of Analog Circuits 44

55 6.2 Statistical yield analysis/monte-carlo analysis sample of statistical parameter vectors according to given distribution x (µ) s N (x s,0, C), µ = 1,..., n MC (159) (numerical) circuit simulation of each sample element (simulation of the stochastic manufacturing process on circuit level) x (µ) s f (µ) = f ( ) x (µ) s (160) evaluation of the acceptance function statistical yield estimation δ (µ) = δ ( { ) 1, f (µ) x (µ) A f s = (161) 0, f (µ) / A f Ŷ = Ê {δ (x s)} = 1 n MC δ (µ) (162) n MC µ=1 number of functioning circuits = (163) sample size = n + n MC = #{ + } #{ + } + #{ - } (Fig. 18) (164) x s,2 β = const A s x s,1 Figure 18. Optimization of Analog Circuits 45

56 6.2.1 Variance of yield estimator V {Ŷ } (162) = V V {δ(x s )} = E{δ 2 (x s ) } (147) {Ê {δ (xs )} = 1 V {δ (x s )} = σ 2 n Ŷ MC (165) } (E {δ (x }{{} s )}) = Y (1 Y ) }{{} (166) δ(x s) } {{ } Y Y } {{ } Y Estimated variance of yield estimator ˆV {Ŷ } (162) = ˆV (151) = = } (148) {Ê {δ (xs )} = 1 1 n MC Ŷ n MC n MC ˆV {δ (xs )} (167) n MC 1 [Ê{δ2 (x s )} (Ê }{{}} {δ {{ (x s)} } ( ) 1 Ŷ δ(x s) }{{} Ŷ Ŷ ) 2 }{{} Ŷ 2 ] (168) n MC 1 = ˆσ2 Ŷ (169) Ŷ is binomially distributed: probability that n + of n MC circuits are functioning n MC : Ŷ is normally distributed (central limit theorem) in practice: n + > 4, n MC n + > 4 and n MC > 10 ˆσ 2 Ŷ 0.25 n MC n MC Ŷ 0% 50% 100% Ŷ = 85%: n MC ˆσŶ 11.9% 5.1% 3.6% 1.6% 1.1% Figure 19. Optimization of Analog Circuits 46

57 confidence interval, confidence level P (Y [Ŷ k ζ ˆσŶ, Ŷ + k ζ ˆσŶ ]) = }{{} confidence interval kζ k ζ 1 2π e t2 2 dt } {{ } confidence level (170) e.g., n MC = 1000, Ŷ = 85% ˆσ Ŷ = 1.1%; k ζ = 3: P (Y [81.7%, 88.3%]) = 99.7% given: yield estimator Ŷ, confidence interval Y [Ŷ Y, Ŷ + Y ], confidence level ζ% find: n MC ) ) ζ = cdf N (Ŷ + kζ ˆσŶ cdf N (Ŷ kζ ˆσŶ k ζ (171) Y = k ζ ˆσŶ ( ˆσŶ ) (172) Ŷ 1 Ŷ ˆσ 2 = Ŷ n MC 1 (173) n MC = 1 + ( Ŷ ) 1 Ŷ (k ζ ) 2 Y 2 (174) number n MC for various confidence intervals and confidence levels: Ŷ ± Y ζ 90% k ζ % % % % ± 10% % ± 5% % ± 1% 3,452 4,900 8,462 13, % ± 0.01% 66,352 given: Ŷ! > Y min, significance level α find: n MC null hypothesis H 0, Ŷ < Y min, rejected if all circuits functioning, n + = n MC assuming H 0 holds, the probability of falsely (i.e., Ŷ < Y min) rejecting H 0 is P ( rejection ) test definition = P ( n + = n MC ) (175) binominal distribution = Ŷ n MC n MC > (falsely) < Y n MC min! < α (176) log α log Y min (177) Optimization of Analog Circuits 47

58 number n MC for a minimum yield and significance level: Y min α = 5% α = 1% 95% % % 3,000 4, % 30,000 46, Importance sampling Y = δ (x s ) pdf (x s ) dx s (178) = = E pdfis pdf (x s ) δ (x s ) pdf IS (x s ) pdf IS (x s ) dx s (179) { δ (x s ) pdf (x s ) pdf IS (x s ) } = E {δ(x s ) w(x s )} (180) sample created according to a separate, specific distribution pdf IS E {w(x s )} = pdf IS pdf(x s ) pdf IS (x s ) pdf IS(x s ) dx s = 1 (181) goal: reduction of estimator variance with V {Ê {δ(xs )}} = V {δ(x s)} n MC V {δ(x s )w(x s )} = pdf IS! < Eq. (182) is statisfied for n MC = n IS, if: V {δ(x s )w(x s )} pdf IS n IS δ (x s ) { } = V Ê {δ(x s )} pdfis pdf IS (182) pdf (x s ) pdf IS (x s ) pdf (x s) dx s Y 2 (183) {x pdf IS(x s ) > pdf(x s ) (184) s δ(x s)=1} Optimization of Analog Circuits 48

59 6.3 Geometric yield analysis for linearized performance feature ( realistic geometric yield analysis ) yield in case that A s is a half of R ns defined by a hyperplane linear model for one single performance feature, index i left out: f = f L + g T (x s x s,w L ) (185) e.g. after solution according to (210) or (211), f L = f (x s,w L ) g = f (x s,w L ) (186) specification type: f f L (187) x s,2 β 2 (x s ) = β 2 W L A s,l, f f L g f = f L x s,0 f = f 0 x s,a β 2 (x s,a ) β 2 (x s) = (x s x s,0) T C 1 (x s x s,0) = const x s,1 Figure 20. Optimization of Analog Circuits 49

60 6.3.1 Yield partition Y L = x s A s,l ( ) pdf N (x s ) dx s = pdf f f df (188) f f L linearized performance feature normally distributed (133), (134): f N f L + g T (x s,0 x s,w L ), g T C g }{{}}{{} (189) f 0 =f(x s,0 ) σ 2 f yield written in terms of the pdf of the linearized performance feature: Y L = 2π σf e 1 2 f L 1 f f 0 σ f «2 df (Fig. 21) (190) pdf f Y L f L f 0 f Figure 21. Optimization of Analog Circuits 50

61 6.3.2 Defining worst-case distance β W L as difference from nominal performance to specification bound as multiple of standard deviation σ f of linearized performance feature f 0 f L = g T (x s,0 x s,w L ) (191) f 0 f L = { +βw L σ f, f 0 > f L circuit functions β W L σ f, f 0 < f L circuit malfunctions (192) variable substitution: t = f f 0, df dt = σ f (193) σ f Y L = β W L 1 2π e t2 2 t L = f L f 0 σ f = dt = { βw L, f 0 > f L +β W L, f 0 < f L (194) ±β W L 1 e ( t ) 2 2 dt (t = t, 2π ) dt = 1 dt Yield partition as a function of worst-case distance β W L Y L = ±β W L 1 e t 2 2 dt 2π f 0 > f L : +β W L (195) f 0 < f L : β W L standard normal distribution, statistical tables, exact within given digits, no estimation Optimization of Analog Circuits 51

62 6.3.4 Worst-case distance β W L defines tolerance region f (x s,a ) = f L (185) g T (x s,a x s,w L ) }{{} f (x s ) = f L + g T (x s x s,w L ). = 0 (196) {. }} { g T (x s,a x s,w L ) = f L + g T (x s x s,a ) (197) i.e., x s,w L in (185) can be replaced with any point of level set f = f L because of (196) from Fig. 20: β 2 (x s,a ) = 2 C 1 (x s,a x s,0 ) = { λa g, f 0 > f L +λ a g, f 0 < f L, λ a > 0 (198) substituting (201) in (199) (x s,a x s,0 ) = λ a 2 C g (199) = g T (x s,a x s,0 ) = λ a 2 σ2 f (197) = f L f 0 (192) = β W L σ f (200) λ a 2 = β W L σ f (201) (x s,a x s,0 ) = β W L σ f C g (202) (x s,a x s,0 ) T C 1 (x s,a x s,0 ) = β 2 W L 1 σ 2 f 2 g T C g } {{ } 1 (203) (x s,a x s,0 ) T C (x s,a x s,0 ) = β 2 W L (204) β 2 W L is level parameter ( radius ) of ellipsoid that touches level hyperplane f = f L Optimization of Analog Circuits 52

63 6.3.5 specification type f f U (192) becomes f U f 0 = { +βw U σ f, f 0 < f U circuit functions β W U σ f, f 0 > f U circuit malfunctions (205) (195) becomes Y U = ±β W U 1 e t 2 2 dt 2π f 0 < f U : +β W U (206) f 0 > f U : β W U (197) becomes f (x s ) = f U + g T (x s x s,a ) (207) Optimization of Analog Circuits 53

64 6.4 Geometric yield analysis for nonlinear performance feature ( general geometric yield analysis ) Problem formulation x s,2 β 2 (x s) = (x s x s,0 ) T C 1 (x s x s,0 ) = const x s,0 f x s,w L f (W L) = f L f = f L x s,w L A s,l A s,l x s,1 Figure 22. lower bound, nominally fulfilled / upper bound, nominally violated f (x s,0 ) > f L/U : max x s pdf N (x s ) s.t. f (x s ) f L/U (208) lower bound, nominally violated / upper bound, nominally fulfilled f (x s,0 ) < f L/U : max x s pdf N (x s ) s.t. f (x s ) f L/U (209) statistical parameter vector with highest probability density on the other side of the acceptance region border lower bound, nominally fulfilled / upper bound, nominally violated f (x s,0 ) > f L/U : min x s β 2 (x s ) s.t. f (x s ) f L/U (210) lower bound, nominally violated / upper bound, nominally fulfilled f (x s,0 ) < f L/U : min x s β 2 (x s ) s.t. f (x s ) f L/U (211) statistical parameter vector with smallest distance (weighted according to pdf) to the acceptance region border specific form of a nonlinear programming problem, quadratic objective function, one nonlinear inequality constraint, iterative solution with SQP: worse-case parameter set x s,w L/U worst-case distance β W L/U = β ( ) x s,w L/U performance gradient f ( ) x s,w L/U Optimization of Analog Circuits 54

65 6.4.2 Advantages of geometric yield analysis Y L/U = A s,l/u pdf (x s ) dx s ; the larger the pdf value, the larger the error in Y if border of A s is approximated inaccurately; A s is exact at point x s,w L/U with highest pdf value, A s differs from A s the more, the smaller the pdf value Y L/U accuracy of Y L/U systematic error, depends on the curvature of performance f in x s,w L/U duality principle in minimum-norm problems: minimum distance between point and convex set equal to maximum distance between point to any separating hyperplane case 1: Y (A s) greatest lower bound of Y (A s ) concerning any tangent hyperplane x s,0 A s A s Figure 23. case 2: Y (A s) least upper bound of Y (A s ) concerning any tangent hyperplane x s,0 A s A s Figure 24. in practice, error Y L/U Y L/U 1%... 2% Optimization of Analog Circuits 55

66 6.4.3 Lagrange function and first-order optimality conditions of problem (210) L (x s, λ) = β 2 (x s ) λ (f L f (x s )) (212) L (x s ) = 0 : 2 C 1 (x s,w L x s,0 ) + λ W L f (x s,w L ) = 0 (213) λ W L (f L f (x s,w L )) = 0 (214) βw 2 L = (x s,w L x s,0 ) T C 1 (x s,w L x s,0 ) (215) assumption: λ W L = 0; then f L f (x s,w L ) (i.e., constraint inactive); from (213): x s,w L = x s,0 and f (x s,w L ) = f (x s,0 ) > f L (from (210)), which contradicts assumption, therefore: λ W L > 0 (216) f (x s,w L ) = f L (217) from (213): x s,w L x s,0 = λ W L C f (x s,w L ) 2 (218) substituting (218) in (215): ( ) 2 λw L f (x s,w L) T C f (x s,w L) = βw 2 L (219) 2 λ W L 2 = β W L f (x s,w L ) T C f (x s,w L ) (220) substituting (220) in (218): x s,w L x s,0 = β W L f (x s,w L ) T C f (x s,w L ) C f (x s,w L ) (221) (221) corresponds to (91), worst-case analysis Lagrange function of problem (211) (221) becomes L (x s, λ) = β 2 (x s ) λ (f (x s ) f L ) (222) x s,w L x s,0 = +β W L f (x s,w L ) T C f (x s,w L ) C f (x s,w L ) (223) Optimization of Analog Circuits 56