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**Design of Power Efficient VLSI Arithmetic: Speed and Power Trade-offs**

Vojin G. Oklobdzija, Ram Krishnamurthy Intel AMR / ACSEL Laboratory Intel Corp/ University of California Davis Tutorial Presentation 16th International Symposium on Computer Arithmetic Santiago de Compostela, SPAIN June 18, 2003

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Issues to be addressed How do we compare different topologies for their efficiency ? How do we estimate speed and efficiency of our algorithm ? What criteria's should we use when developing a new algorithm ? How does power enter into this equation ? June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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Additional Issues Determine which topology is the best for given Power or Delay budget Determine which topology can stretch the furthest in terms of speed or power June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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Metric

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**Previously used estimates**

Counting the number of gates (logic levels): not accurate June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Critical path in Motorola's 64-bit CLA**

As opposed to Ripple or Carry-Skip Adders the critical path in the Carry-Lookahead-Adder travels in vertical direction rather than a horizontal one as shown in the previous slide. Therefore the delay of Carry-Lookahead-Adder is not directly proportional to the size of the adder N, but to the number of levels used. Given that the groups and super-groups in the Carry-Lookahead-Adder resemble a tree structure the delay of a Carry-Lookahead-Adder is thus proportional to the log function of the size N. This log dependency makes Carry-Lookahead-Adder one of the theoretically fastest structures for addition. However, it can be argued that the speed efficiency of the Carry-Lookahead-Adder has passed the point of diminishing returns given the fan-in and fan-out dependencies of the logic gates and inadequacy of the delay model based on counting number of gates in the critical path. In reality, Carry-Lookahead-Adder is indeed achieving lesser speed than expected, especially when compared to some techniques that consume less hardware for the implementation. An example of a Carry Lookahead Adder, and a critical path as implemented in Motorola processor is shown in this slide. June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Motorola's 64-bit CLA Modified PG Block**

Intermediate propagate signals Pi:0 are generated to speed-up C3 June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Fan-In and Fan-Out Dependency (Oklobdzija, Barnes: IBM 1985)**

June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Delay Comparison: Variable Block Adder (Oklobdzija, Barnes: IBM 1985)**

Complexity June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Design Objective Design takes time:**

finding results afterward is not of much value There is a disconnect between measures used by computer arithmetic when developing an algorithm and what is obtained after implementation we want to estimate as close to the measured results A simple tool that can evaluate different design trade-off for a given technology is needed Power trade-off is the most important speed and power are tradable June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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Logical Effort Theory “Back of the Envelope” complexity: good for estimating speed Gate delay = linear function of load Slope: logical effort gate driving characteristics Intersect: parasitic gate internal load “Logical Effort” accuracy is not sufficient We needed to extend and refine the method However, that becomes more than “Back of the Envelope” Logical Effort does not account for possible power-delay trade-offs June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Logical Effort Theory Excel –a platform of choice (ARITH-16)**

Simple enough Can provide computation quickly Easy to enter a given design Technology characterization is needed: This needs to be done only once: available for every design afterwards Domino gate = 2 stages of dynamic and static Different driving characteristics of these stages Multi-output gate (carry-look-ahead, Ling/conditional sum) Energy model needs to be included June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Energy Motivation AGUs: performance and peak-current limiters**

*courtesy of Intel Corp. Cache Processor thermal map Temp (oC) Execution core AGU 120oC AGUs: performance and peak-current limiters High activity thermal hotspot Goal: high-performance energy-efficient design June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Kogge-Stone Adder Critical path = PG+5+XOR = 7 gate stages**

31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Carry-merge gates XOR Critical path = PG+5+XOR = 7 gate stages Generate,Propagate fanout of 2,3 Maximum interconnect spans 16b Energy inefficient June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Sparse-tree Adder Architecture**

Generate every 4th carry in parallel Side-path: 4-bit conditional sum generator 73% fewer carry-merge gatesenergy-efficient June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Kogge-Stone adder (8-stage)**

D = 8*(GBH)1/8* *P June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**MXA2 – Architecture & Result**

Multiplexer-based Generate carries using radix-2 (P,G) 4-bit conditional sum selected by carries 4-b cell width = 17m 9-stage critical path Per-stage effort = 3.7 Total effort delay = 33.3 Total parasitic = 22.5 Total delay = 55.8 June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**HC2 – Architecture Generate even carries using radix-2 (P,G)**

Generate odd carries from even carries CMOS adder for sum 1-b cell width 4m 10-stage critical path June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**HC2 – Circuits & Results June 18, 2003**

16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**KS2 – Architecture & Results**

Generate carries using radix-2 (P,G) CMOS adder for sum Similar circuits as HC2 1-b cell width 4m 9-stage critical path June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**KS4 – Architecture Generate carries using redundant radix-4 (P,G)**

Dynamic circuit 1-b cell width 4m 6-stage critical path June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**KS4 – Circuits & Result June 18, 2003**

16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**CLA4 – Architecture Generate carries using radix-4 (P,G,C)**

1-b cell width 4m 15-stage critical path (P,G,C) Network G-Path P-Path June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**CLA4 – Circuits & Result June 18, 2003**

16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**LNG4 – Architecture Generate carries using Ling pseudo-carries**

Conditional sums selected by local & long carries 1-b cell width 5.1m; 9-stage critical path June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**LNG4 – Circuits & Result June 18, 2003**

16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Results from Simulation**

Fairly consistent with logical effort analysis Per-stage delay 1.4 FO4 (static) 0.8 FO4 (dynamic) June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Delay of Representative 64-b Adders**

June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**What happened when Power is considered ?**

June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Energy-Delay Space Energy Emin Dmin Delay speed barrier**

Different Adders Emin power limit Dmin Delay June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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Logical Effort June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Delay in a Logic Gate Delay of a logic gate has two components**

d = f + p Logical effort describes relative ability of gate topology to deliver current (defined to be 1 for an inverter) Electrical effort is the ratio of output to input capacitance parasitic delay effort delay, stage effort electrical effort is also called “fanout” f = gh electrical effort = Cout/Cin logical effort *from Mathew Sanu / D. Harris June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Logical Effort Parameters: Inverter**

Delay g=2.2 (logic effort) d=gh+p p=3.8ps (parasitic delay) Fanout: h =Cin/Cout d = gh + p Delay increases linearly with fanout More complex gates have greater g and p *from Mathew Sanu / D. Harris June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Normalized Logical Effort: Inverter**

*from Mathew Sanu / D. Harris 6 5 4 g = p = d = 1 inverter Normalized delay: d 1 3 gh + p = h+1 effort delay 2 1 parasitic delay 1 2 3 4 5 Fanout: h = Cout/Cin Define delay of unloaded inverter = 1 Define logical effort ‘g’ of inverter = 1 Delay of complex gates can be defined w.r.t d=1 June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Computing Logical Effort**

DEF: Logical effort is the ratio of the input capacitance to the input capacitance of an inverter delivering the same output current Measured from delay vs. fanout plots of simulated gates Or estimated, counting capacitance in units of transistor W *from Mathew Sanu / D. Harris June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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L.E for Adder Gates *from Mathew Sanu / D. Harris Logical effort parameters obtained from simulation for std cells Define logical effort ‘g’ of inverter = 1 Delay of complex gates can be defined w.r.t d=1 June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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Normalized L.E Gate type Logical Eff. (g) Parasitics (Pinv) Inverter 1 Dyn. Nand 0.6 1.34 Dyn. CM 1.62 Dyn. CM-4N 3.71 Static CM 1.48 2.53 Mux 1.68 2.93 XOR 1.69 2.97 Logical effort & parasitic delay normalized to that of inverter *from Mathew Sanu June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Delay of a string of gates**

Delay of a path, D = di = gihi pi gi & pi are constants To minimize path delay, optimal values of hi are to be determined D is minimized when each stage bears the same effort, i.e. gihi = g i+1h i+1 *from Mathew Sanu / D. Harris June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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** Minimizing path delay gi Logical Effort of a string of gates:**

Path Electrical Effort: Branching Effort Path Branching Effort: Path Effort: F=GBH G = Cout(path) H = hi = Cin(path) Con-path + Coff-path Con-path b = bi B = Delay is minimized when each stage bears the same effort: f = gihi = F1/N The minimum delay of an N-stage path is: NF1/N + P *from Mathew Sanu / D. Harris June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Inclusion of Wire Delay into Logical Effort**

June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Wiring Load Wiring in hand analysis Wiring in HSPICE Wire length**

Only lumped capacitance included Wiring in HSPICE Short wire: 1-segment -model RC network Long wire: 4-segment -model RC network Using worst-case wire capacitance Wire length Estimated from most critical 1-bit pitch June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Modeling interconnect cap.**

Include interconnect cap in branching factor Coff-path Coff-path PG CM0 PG CM0 Adder bitpitch Adder bitpitch Cint CM0 CM0 Con-path Con-path Con-path + Coff-path Con-path + Coff-path+Cint Cint b = = 2 b = = 2+ Con-path Con-path Con-path = 2 + I I : % int. cap to gate cap in 1 adder bitpitch June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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Branching g0 g1 g2 g3 Logical Effort assumes the “branching” factor of this circuit to be 2. This is incorrect and can create inaccuracies June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Correction on Branching**

f0 = f1 , f2 = f3 Td1 = (f0 + f1 + parasitics) Td2 = (f2 + f3 + parasitics) Minimum Delay occurs when Td1 = Td2 June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**“Real” Branching Calculation**

Branching only equals 2 when: This explains why we had to resort to Excel ! June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Technology Characterization**

June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Characterization Setup**

Logical Effort Requirements: Equalize input and output transitions. Logical Effort is characterized by varying the h (Cout/Cin) of a gate. By using a variable load of inverters each gate can be characterized over the same range of loads. The Logical Effort of each gate is characterized for each input. Energy is characterized for each output transition of the gate caused by each input transition. i.e. for an inverter: energy is measured for tLH and tHL June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**LE Characterization Setup for Static Gates**

In tLH tHL Average Energy .. Variable Load June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**LE Characterization Setup for Dynamic Gates**

In tHL Energy Variable Load June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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LE Table (Static CMOS) Technology: P/N Ratio = 2 INV = 3.67, pINV = 4.29 Measured on worst-case single-input switching June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Static CMOS Gates: Delay Graphs**

June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Static Gates: Pull-up Delay Graph**

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**LE Table (Dynamic CMOS)**

Technology: Minimum-sized keeper included Measured on all-input switching of worst path June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Dynamic CMOS: Delay Graphs**

June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Dynamic CMOS: Delay Graphs**

June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Energy Calculation June 18, 2003**

16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Energy Calculation 16X Minimal Size Dyn-NAND 8X Minimal Size Dyn-NAND**

June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Energy Calculation June 18, 2003**

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**Energy Calculation June 18, 2003**

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**Energy Calculation NAND-2 June 18, 2003**

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Examples June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**64-Bit Adders Han-Carlson (prefix-2, HC2): Static and Dynamic**

Han-Carlson (prefix-2, HC2-2): Dynamic-Static Kogge-Stone (prefix-2, KS2): Static and Dynamic Kogge-Stone (prefix-2, KS2-2): Dynamic-Static Quaternary-Tree (prefix-2, QT2): Static and Dynamic Included wire delay, tdelay = 0.7RwireCwire Included wire energy, Ew = CwireV2 June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Test Setup 1mm wire Cwire A0 S0 Adder A63 S63 Cwire**

H=(Cin + Cwire)/Cin June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Energy-Delay Estimates**

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**Adders: Energy Dynamic: KS, HC QT KS Static HC Dynamic-Static**

June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Dynamic Static Implementation of Carry-Merge stage**

inverters to be eliminated Regular Domino Implementation Compound-Domino Implementation June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Energy-Delay comparison of 64-bit KS, HC and QT adders**

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**Adders: Critical Path Energy**

QT dynamic-static HC-dynamic KS dynamic HC dynamic-static QT static KS dynamic-static HC-static KS-static June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Intel 32-bit Adder 0.13u 1.2V [VLSI-2002]**

KS KS estimated QT QT Estimated June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Energy-Delay comparison of 32-bit QT and KS adders: estimated vs**

Energy-Delay comparison of 32-bit QT and KS adders: estimated vs. simulation in 0.10mm technology June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Est. Results: All Adders w/o Wires**

June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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**Est. Results: All Adders w/ Wires**

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Conclusion Using realistic measures for comparing various designs leads to better design choices Power is as important as speed Making comparison in Energy-Delay space is necessary: power can always be traded for speed and vice versa Wire effects are significant Leakage currents ? June 18, 2003 16th International Symposium on Computer Arithmetic, Santiago de Compostela, SPAIN

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