Stochastic Projection Based Statistical Expression Generator

In the recent years, analysis frameworks of the statistical performances, such as the statisti-cal static timing analysis [90], the statististatisti-cal leakage analysis [6, 7, 76], the statististatisti-cal wave-form/delay analysis of interconnects [60, 91–93] and the statistical power grid analysis [94], have been proposed. The general analysis frameworks of statistical performances analyzers ex-pressed the target performance as the parametric form of the physical parameters up to second order polynomials. The statistical static timing analyzer [90] and interconnect waveform/delay analyzer [60, 91–93] applied the second order Taylor expansion to simulate the target perfor-mance. Due to the variations of delays for gate or interconnect corresponding to the parameters are usually in a controllable range, the Taylor expansion can present reasonable estimations for the delays of gates and interconnects [60, 90–93]. However, as shown in section 3.2.1, leakage currents/powers are sensitive to the variations of the physical parameters, and their variation ranges will be large due to the exponential dependency to the physical parameters. In this situa-tion, since the Taylor expansion presents accurate results under the assumption that the variation of the estimating waveforms is small, it might be not suitable for estimating the performance correlated to the leakage currents/powers. For example, Mi et. al [94] have addressed that the second order Taylor expansion did not present accurate voltage waveform analysis for the power grid considering the leakage currents.

Comparing with the Taylor expansion, the polynomial chaos (PC)) [86] is another frame-work to estimate an output quantity of a system with the sources having statistical fluctuations.

Similar with the Taylor expansion framework, the PC expresses the target quantity as a poly-nomial of the variation sources fluctuating the input sources of the system. The difference between PC and Taylor expansion is that PC generates orthonormal bases corresponding to the probability distribution of the variation sources to represent the output of the system. With the orthonormal property for the representing bases, the PC achieves the minimal mean square er-ror estimation for the target quantity with a specific approximation order of the polynomials.

Therefore, under a specific approximation order, PC can be more accurate than Taylor expan-sion. In other words, under a specific accuracy, Taylor expansion requires higher approximation fficiency of PC is equal to that of

Taylor expansion if the transformed systems for calculating the coefficient corresponding to each orthonormal bases can be solved individually.

Based on the framework of PC, the statistical leakage power analyzers [76,90], the statistical power grid analysis [94] and the author’s previous works on the statistical on-chip thermal anal-ysis [8, 95] have been developed. However, the leakage power models adopted by [76, 90, 95]

do not take the temperature effects into account. As addressed in section 3.2.1, this leads to considerable errors on the leakage power prediction. Although the author’s previous work [8]

has took into account the temperature effect in the leakage power model, as addressed in sec-tion 3.2.1, the leakage power models is still not adequate for the accurate leakage power predic-tion. Since the on-chip temperature is transformed from the on-chip power, our previous works can not provide sufficient accuracy for the statistical thermal estimation. Therefore, although the framework of PC is adopted in this statistical expression generator, we propose an adaptive leakage power modeling technique to deal with the issue of complex leakage power models for advanced technologies.

Polynomial Bases

With the random vector ξ = [ξ1, ξ2, · · · , ξ|ξ|]^T constructed by the KL expansion stated in sec-tion 3.2.2, a set of |ξ|-dimensional Hermite polynomial (HPs) [86] can be constructed to serve as the bases to approximate the on-chip temperature distribution. The HPs of ξ with order r is

Γr(ξi₁, · · · , ξi_r)= (−1)^r ∂^r

∂ξi₁· · ·ξi_r

exp −1 2ξ^Tξ

!

, (3.21)

where each in with 1 ≤ n ≤ r is the index of the selected random variable in ξ. With equa-tion 3.21, the zeroth, first, and second orders of HPs are

Γ0 = 1, (3.22)

Γ1(ξi₁) = ξⁱ1, (3.23)

Γ2(ξi₁, ξi₂) = ξi₁ξi₂ −δi₁i₂, (3.24)

respectively. Here, δi1i2 is the Kronecker delta. The HPs satisfy the following orthogonal prop-erty [86].

EnΦⁱ(ξ)Φj(ξ)o = E nΦ²i(ξ)o δi j, (3.25)

whereΦⁱ(ξ) is the concise expression ofΓ^r(ξi₁, · · · , ξi_r). There is a one-to-one mapping between Φ[·] and Γ[·], and between i and i1· · · ir.

Stochastic Projection Based Electro-Thermal Updating Scheme

With the KL expanded random vector ξ of the channel length L and oxide thickness tox, the on-chip temperature T (r, L, tox) can be approximated as T (r, ξ). Since L and tox are the vari-ation sources of the input source, the leakage powers, of the heat transfer equvari-ations, based on the framework of PC [86], the on-chip temperature T (r, L, tox) can be approximated by the following polynomial expression.

where each Tk(r) = E{T(r, ξ)Φk(ξ)} is the projection temperature profile at any arbitrary po-sition r of die corresponding to Φk(ξ), and NPC is the truncation number that is equal to 1+ P_n^p_=1n!¹ Qn−1

r=0(NKL+ r). Here, p is the order of HPs, and NKL = Ntox+ NL.

Substituting equation (3.26) into equation (3.19) and approximating p(r, L, Tox,Tb) to be p(r, ξ, bT), the residual of equation (3.19) is With a similar procedure, the residual of equation (3.20) can also be obtained. Based on the principle of stochastic Galerkin projection [86], the residuals of the statistical heat transfer equations (3.19)–(3.20) are enforced to be orthogonal to each H-PC, i.e. E {R(r, ξ)Φk(ξ)} = 0 for each k. Therefore, we have the following un-coupled deterministic heat transfer equation for solving each Tk(r).

in equation (3.28) is equal to

Algorithm Stochastic Projection Based Electro-Thermal Updating Scheme Input: Initial average die temperature µⁱⁿⁱ_T obtained by 1-D thermal model Output: The H-PC expression of on-chip temperature bT(r, ξ)

1 Begin

10 Obtain the projected power density profile onto the k-th HP, p_k(r) ← p_d(r)δ0k+ ps,k(r)+ pg,k(r);

† The deterministic thermal simulator mentioned in Chpater 2 is employed to

solve equations (3.28) and (3.29). Note that any deterministic thermal simulators can be used here.

“stdev” means the standard deviation.

Figure 3.6: The electro-thermal updating scheme of the stochastic projection based statistical expression generator.

Here, pd(r) is the dynamic power density profile. En

pg(r, ξ, bT)Φk(ξ)o

and En

ps(r, ξ, bT)Φk(ξ)o are the statistical projected power density profiles onto the k-th H-PC basis for the gate-leakage and subthreshold-leakage power density profiles pg(r, ξ, bT) and ps(r, ξ, bT), respectively. The term δ_0k in both equations (3.29) and (3.30) is from E{Φk(ξ)}= δ0k [86].

Any existing deterministic thermal simulators, such as [5, 51–59] and the GIT thermal sim-ulator mentioned in Chapter 2, can be utilized to obtain each Tk(r) after En

p(r, ξ, bT)Φk(ξ)o has been calculated. Since the subthreshold leakage and gate tunneling leakage power density pro-files are temperature dependent, as shown in Figure 3.6, a developed electro-thermal updating scheme is performed to obtain each Tk(r) in equation (3.26) for expressing bT(r, ξ).

First, the initial bT(r, ξ) is estimated by utilizing the 1-D equivalent thermal circuit with the nominal total on-chip power, and all of the projected coefficient functionsTbk(r)’s are set to be zeros. Then, by executing the projection algorithms presented in the next subsection 3.3.1 with the leakage power models shown in section 3.2.1, the projection power profiles of corresponding

to eachΦ^k(ξ) of the circuit, which are p_s,k(r) and p_s,k(r) shown in Lines 8∼10, can be obtained.

After that, each Tk(r) is solved by using the deterministic thermal simulator mentioned in 2, and the HP expression of bT(r, ξ) is updated by using Line 11. Finally, the MaxStdError is calculated as the maximum absolute error between the standard deviation of bT(r, ξ) and the standard deviation of bTpre(r, ξ). The above computation process is repeated until MaxStdError is less than a given threshold value.

The developed stochastic projection based electro-thermal updating scheme has the follow-ing advantage. Because the deterministic heat transfer equations for solvfollow-ing different Tk(r)’s are un-coupled, each Tk(r) can be solved individually. Moreover, since equations (3.28)-(3.29) corresponding to each Tk(r) have the same thermal conductivity κ, the system handling process of an employed deterministic thermal simulator, such as the LU decomposition of the tridiago-nal matrix [51], the establishment of the multi-grid cycle [54, 55], and the basis construction of the deterministic thermal simulator stated in Chapter 2, can be performed only once for solving all Tk(r)’s. In this work, we employ the deterministic thermal simulator stated in Chapter 2 to solve Tk(r)’s because of its high efficiency for the thermal estimation in early design stages⁵

With the statistical expression shown in equation (3.26), the mean and variance profiles of the statistical on-chip temperature distribution can be approximated as

En

Tb(r, ξ)o

= T0(r), (3.31)

Varn

Tb(r, ξ)o

=

N_PC

X

k=1

T_k²(r)EnΦ²k(ξ)o . (3.32)

Projection Coefficient Calculation of Leakage Power Consumption

In this subsection, for a specific type of gate located at an arbitrary parameter modeling grid, two algorithms are proposed to calculate the projection coefficient of leakage powers corresponding to each HP. As the locations of gates are given, for completing the electro-thermal updating scheme shown in Figure 3.6, the projected power density profiles p_s,k(r) and p_g,k(r) shown in Lines7∼8 of Figure 3.6 can be obtained.

According to the deterministic thermal simulator stated in Chapter 2, the die is divided into a mesh for obtaining Tk(r)’s. Similarly, as mentioned in section 3.2.2, the die is divided into

a mesh for obtaining the explicit forms of the device channel length and oxide thickness at the parameter modeling grids. Under an acceptable accuracy, the required mesh sizes for modeling physical parameters and obtaining Tk(r)’s are different. Therefore, the mesh sizes of parameter modeling and temperature simulation grids is set to be different. Based on the above setting, if the grids of the temperature simulation mesh overlap those of the parameter modeling mesh, the values of Tk(r) are averaged for calculating the projection coefficients of leakage powers in an arbitrary parameter modeling grid.

As mentioned in section 3.2.1, the complex fitting form are required to be adopted for ac-curately modeling the leakage powers. Therefore, we adopt the accurate but complex leakage power models presented in section 3.2.1 and propose two approximating strategies to trace the temperature dependency of the leakage powers. For both the subthreshold and the gate tun-neling leakage powers, in each parameter modeling grid, the HP expression of the temperature distribution by each iteration, bTpre(r shown in Figure 3.6, is substituted in to the leakage power models. Although the temperature distribution is approximated by the second order HPs in the output of this statistical expression generator, for reducing the complexity, the first order HP expression of the temperature is employed to obtain the explicit forms of leakage powers . Be-sides, with the mean temperature profile obtained by each iteration shown in Figure 3.6 being the expansion point, an adaptive Taylor expansion is proposed for simplifying the explicit form of the subthreshold leakage power model.

Projection Coefficient of Gate Tunneling Leakage Power For a specific type of gate in the m-th parameter modeling grid, the proposed gate tunneling leakage power model mentioned in section 3.2.1 can be written as

P_gm(Lm, t_oxm, Tm)= Vdd × a₀e^{a1Lm+a2Tm+a3toxm+a4t2oxm}, (3.33)

where Lm, t_oxmand Tmare the device channel length, the device oxide thickness and the average temperature in the m-th parameter modeling grid, respectively. And ai’s are the fitting constants of the specific type of gate.

Approximating Lmand t_oxmas the KL expansions shown in equations (3.15)–(3.16) and uti-lizing the first order HP expression obtained by each iteration shown in Figure 3.6, the average

Algorithm Gate Tunneling Leakage Power Projection Input: Vdd and ai’s of each gate leakage power model;

µLm, µtoxm and µTm; vectors gLm, gtoxm, hLm and htoxm; Vt_oxm and Λt_oxm which are the eigen-vector matrix and the diagonal eigen-value matrix of Gt_oxm, respectively.

Output: qm[k]= En

Figure 3.7: The evaluating algorithm of projection coefficients of gate tunneling leakage power up to second order of HPs.

Function PROVECs

NPar, cParm, VParm, ΛParm, ϕ, $, ρ, Θ Input: The dimension of related vectors and matrices NPar;

Vector cPar_m ; Matrices VPar_m, and ΛPar_m . Output: scalars ϕ and $; vector ρ; matrix Θ

1 Begin

Figure 3.8: The function that evaluates the related vectors for calculating the leakage powers.

temperature in m-th parameter modeling grid Tm= T^m(ηL, ηt_ox) can be written as

Tbm(ηL, ηt_ox)= µT_m + h^TL_mηL+ h^Tt_oxmηt_ox, (3.34) where µT_m is the mean of average temperature in the m-th parameter modeling grid. hL_m and ht_oxmare the vectors of projection coefficients of Tm(ηL, ηt_ox) corresponding to HPs. Substituting bTm(ηL, ηt_ox) into equation (3.33), for each electro-thermal iteration, the projection coefficient corresponding to the k-th HP of P_gm(Lm, t_oxm, Tm) can be approximated by

According to the derivation given in APPENDIX B and replacing the subscript Par, which is shown in Figure 3.8, with toxm, the evaluating algorithm of equation (3.35) is summarized in Figure 3.7. The function PROVECs shown in Fig 3.8 evaluates related vectors for calculat-ing EnΦ^kPar(ξ) exp(c^T_Par

mηPar+ aη^TParGPar_mηPar)o

. Here, the subscript Par means the physical

Algorithm Subthreshold Leakage Power Projection

Input: µP_sm and βi’s in equation (3.38); vectors gL_m, gt_oxm, hL_m

and ht_oxm; VL_m and ΛL_m which are eigen-vector matrix and diagonal eigen-value matrix of GL_m, respectively. Vt_oxm

and Λt_oxm which are eigen-vector matrix and diagonal eigen-value matrix of Gtoxm, respectively

Output: qm[k]= En

Figure 3.9: Subthreshold leakage power projection algorithm.

parameter L or tox, and m means the m-th grid. ηPar is a standard normal random vector repre-senting the KL expanding random vector of the physical parameter, and a is a constant. VPar_m

is the eigen-vector matrix of GPar_m, and ΛPar_m is the eigen-value matrix of GPar_m multiplied by a. Φk_Par(ξ) is the HPs of ξ up to the second order. kParis the index of HPs.

Although the algorithm in Figure 3.7 only calculates the projection coefficient of the gate tunneling leakage power up to the second order of HPs, it can be easily extended to the higher order of HPs.

Subthreshold Leakage Power Projection As mentioned in section 3.2.1, for a specific type of gate in the m-th parameter modeling grid, the adopted subthreshold leakage power model can be written as

P_sm(Lm, t_oxm, Tm)= Vdd × b₀e^{f˜s(Lm,toxm,Tm)}×e^{b1Lm+b2L2m+b3toxm+b4t2oxm+b5Tm}, (3.36)

where bi’s are the fitting constants, and ˜fs equals to the remaining terms of fs shown in equa-tion (3.2) with excluding the polynomial terms {Lm, L²_m, toxm, t²_oxm, Tm}. By utilizing the Taylor expansion with the expansion point at (µL_m, µt_oxm, µT_m), ˜fs(Lm, toxm, Tm) can be approximated as d0+d1∆L^m+d2∆L²m+d3∆toxm+d4∆t²oxm+d5∆T^m. Here, di’s are µT_m dependent Taylor expansion coefficients. With the approximated ˜fs(Lm, t_oxm, Tm), P_smcan be approximated as

P_sm(Lm, t_oxm, Tm) ≈ µP_sme^{β1∆Lm+β2∆Lm2+β3∆toxm+β4∆toxm2 +β5∆Tm}, (3.37)

where β1 = b1+ d1, β2= b1+ 2b2+ d2, β3 = b3+ d3, β4 = b3+ 2b4+ d4, β5 = b5+ d5, and µP_sm

is equal to Vdd× b₀e^{(b1+d0)µLm+b2µ}

2

Lm+b3µ_toxm+b4µ²_toxm+b5µ_Tm

.

Utilizing the KL expansions of Lm and t_oxm, and the first order HP expression of Tm, we have∆L^m = g^TL_mηL,∆toxm = g^Tt_oxmηt_ox, and∆T^m = h^TL_mηL+ h^Tt_oxmηt_ox. Then, for a specific type of gate located in the m-th parameter modeling grid, the projection coefficients of Psm(Lm, t_oxm, Tm) corresponding to k-th HP basis can be approximated by equation (3.38) for each electro-thermal iteration.

En

P_sm(Lm, t_oxm,Tbm)Φk(ξ)o = µP_smE



e^{cTLmηL+β2ηLTGLmηL}· e^{cTtoxmηtox+β4ηTtoxGtoxmηtox}Φk(ξ)

, (3.38)

where µP_sm = V^ddb0e^{(b1+d0)µL+b2µ2L+b3µtox+b4µ2tox+b5µTm}, ct_oxm = β3gt_oxm + β5ht_oxm, cL_m = β1gL_m + β5hL_m, Gt_oxm = gt_oxmg_t^T

oxm and and GL_m = gL_mg_L^T

m.

According to the derivation given in APPENDIX B, the calculating algorithm of equa-tion (3.38) up to the second order of HPs is summarized in Figure 3.9. Since the expressions of both exponents in equation (3.38) are similar with the expression of the second exponent in equation (3.35), the calculating steps (Lines 6 ∼ 19) in Figure 3.9 are similar with the calculat-ing steps (Lines 5 ∼ 18) in Figure 3.7.

As indicated by [6, 76], the number of parameter modeling grids can be much less than the number of gates while maintaining the acceptable accuracy. Thus, the simulated die is divided

Algorithm Stochastic Projection Based Electro-Thermal Analysis Input: Geometries of the die, spatial correlation models of channel

length and oxide thickness, design information such as .def file, .lef file, .lib file, package structure and leakage power models Output: The H-PC expression of bT(r, ξ).

The mean profile and the variance profile of bT(r, ξ).

1 Begin

2 Set the thermal parameters, and get the initial average die

temperature µⁱⁿⁱ_T by 1-D thermal model described in section 3.2.3;

3 For m ← 1 to Ng

14 Obtain the projected gate-tunneling leakage powers onto the H-PC bases for the n-th gate type in the m-th parameter modeling grid by the algorithm shown in Figure 3.7;

15 Obtain the projected sub-threshold leakage powers onto the H-PC bases for the n-th gate type in the m-th parameter modeling grid by the algorithm shown in Figure 3.9;

16 EndFor

17 EndFor

18 For k ← 0 to NPC

19 Obtain the projected power density profile onto the k-th H-PC basis, pk(r), by using the projected powers calculated

from Lines 14 and 15;

23 Update mean and variance profiles by equations (3.31) and (3.32);

24 MaxStdError ←max

Figure 3.10: Stochastic projection based electro-thermal analysis algorithm. NumGateType in Line13 is the number of gate types given from the industrial library file.

into Ng grids for modeling parameters that is much less than the number of simulated temper-ature grids. Gates locate in the same parameter modeling grid share the same KL expansions of the channel length and oxide thickness; hence, they share the same GL_m and Gt_oxm in the m-th parameter modeling grid. Therefore, the number of eigen-decompositions is Ng rather than the number of gates. In addition, the eigen-functions and eigen-values only depend on the spatial covariance functions of the channel length and oxide thickness; thus, each GL_m and Gt_oxm are known after the spatial covariance functions are given. Generally, the empirical spatial covariance functions of the channel length and the oxide thickness can be extracted before the circuit design [79, 80, 83, 85]. Therefore, the eigen-decomposition of each GL_m and Gt_oxm can be calculated before the thermal simulation. The complete analysis algorithm is presented in Figure 3.10.

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