242?“?
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`3:L
`I...
`gig-JUNE 1997
`
`*uomr PUBLICATION or THE EEEE cmcun's AND SYSTEMS SOCIETY -
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`THE IEEE COMPUTER SOCIETY @
`THE IEEE soup-STATE cmcurrs some-n!
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`VOLUME 5
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`NUMBER 2
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`@ IEEE TRANSACTIONS ON
`iVERY LARGE SCALE
`INTEGRATION (VLSI) SYSTEMS
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`PAPERS
`
`Pipelines! H—Trees for HighSpeed Clocking of Large Integrated Systems in Presence of Process Variations ............
`......................................................................... M. Nektii G Bots, and Y. Snttrtrin
`
`Scheduling Tests for VLSI Systems Under Power Constraints ............ R. M. Chou. K. K Snlttjn. and l” D. Agmwnl
`Hierarchical Interconnection Structures for Field Programmable Gate Arrays .................. Y.— 1". Lot and P. -T. Wang
`Unifiabic Scheduling and Allocation for Minimizing System Cycle Time ............... S.
`(Ci—Y. Huang and. W. H. Wolf
`VLSI Array Algorithms and Architectures for RSA Modular Multiplication ............ Y.-J. Jermg and W. P. Barrios-mt
`An Architectural Co—Syntheeis Algorithm for Distributed, Embedded Computing Systems ................... W. H. Wot)“
`
`
`TRANSACTIONS BRIEFS
`
`VLSI Compressor Design with Applications to Digital Neural Networks ................... 1). Zhang and M. I. Etmarry
`Diagnosis and Correction of Multiple Logic Design Errors in Digital Circuits ............................................
`....................................................................... P.-l’. (Enter-TIM) Chung and I. N. Hajj
`Design of a 32 b Monolithic lvlicroprocessor Based on GaAs HMBSFET Technology ...................................
`............................................ (I. K. V. Tien, K. Lewis H. J. Great) ”I. Teen. and}. F. McDonald
`
`Correction to “Control—Flow Versus DataFlowBased Scheduling. Combining Both Approaches in an Adaptive
`Scheduling System" .............................................. R. A. Bergamascftt S Raje. I Nair, and}; Trevtit'yrm
`
`CALLS FOR PAPERS
`
`International Conference on Next Decades of High Technologies ..........................................................
`Workshop on Enhancing Successfltiented EQ. and IQ. for Students .....................................................
`International Symposium on Low Power Electronics and Design ..........................................................
`
`244
`245
`246
`
`Petitioner Microsoft Corporation - Ex. 1061, Cover 1
`Petitioner Microsoft Co oration - Ex. 1061, Cover :
`
`

`

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`Petitioner Microsoft Corporation - Ex. 1061, Cover 2
`Petitioner Microsoft Corporation - Ex. 1061, Cov
`
`

`

`IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS
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`FRANCKY Ctt'rrIIooR
`LMEC. Belgium
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`Sum]. R. Das
`Elec. Eng. Dept.
`Univ. of Ottawa, Canada
`WnI-Ctn ano
`NASAIJet Prop, Lab.
`California Inst. of Technol,
`ERIC R. Possum
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`Photnbit. LLC, CA
`GRAHAM HELLESTRAND
`Univ. of New South Wales. Australia
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`NIRM .IHA
`Elec. Eng. Dept.
`Princeton Univ. NJ
`PINAKI anuntoan
`EE and CS Dept.
`Univ. of Michigan. Ann Arbor
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`Elec. & Comput. Eng. Dept.
`Northeastern Univ. MA
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`Univ. of Illinois. Urbana
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`NTT. Japan
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`Kristina Pnant
`Elec. Eng. Dept.
`Univ. of Minnesota, Minneapolis
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`Univ. of Hannover, Germany
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`VIKTOR Pansartnn
`Elec. Eng. Dept.
`Univ. of Southern California
`SARMA B. K. Vnuotruttt
`Elec. & Comput. Eng. Dept.
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`Petitioner Microsoft Corporation - Ex. 1061, Cover 3
`Petitioner Microsoft Corporation - Ex. 1061, Cover 3
`
`

`

`JEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, VOL. 5. NO. 2. JUNE I997
`
`I61
`
`Pipelined H—Trees for High—Speed Clocking of
`Large Integrated Systems in Presence of
`Process Variations
`Mohamed Nekili, Guy Bois, and Yvon Savaria, Member, {85
`
`r
`
`.
`
`Petitioner Microsoft Corporation - Ex. 1061, Cover 4
`Petitioner Microsoft Corporation - EX. 1061, Cover 4
`
`Abstract—This paper addresses the problem of clocking large
`high-speed digital systems, as well as deterministic skew mod-
`eling, a related problem. A conventional method for clocking a
`large digital system is to use a set of metallic lines organized
`as a tree. This method is limited by the bandwidth of the
`clock network. Another limitation of existing solutions is that
`available skew models do not directly take into account process
`variations. in order to provide a reliable skew model, and to
`avoid the frequency limitation, we propose a novel approach
`that distributes the clock with an H-tree, whose branches are
`composed of minimumsized inverters rather than metal. With
`such a structure, we obtain the highest clocking rate achievable
`with a given technology. Indeed, clock rates around 1 GHz
`are possible with a 1.2 pm CMOS technology. From the skew
`modeling standpoint, we derive an analytic expression of the
`skew between two leaves of the H-tme, which we consider to
`be the difference in root-to-leat‘ delay pairs. The skew upper
`bound obtained has an order of complexity which, with respect
`to the H-tree size D,
`is
`the same as the one that may be
`derived from the Fisher and Kung model for both side-to-
`side and neighbor-to-neighbor communications, Le, a Q{D2),
`whereas. the Stciglitz and Kugelrnass probabilistic model predicts
`@(D x \/l.ug ll}. In an H-tree implemented with metallic lines,
`the leaf-to-lcat' skew is obviously bounded by the delay between
`the root and the leaves. However, with the logic based H-lree
`proposed in this paper, we arrive at a nonobvious result, which
`states that the leaf-to-leaf skew grows faster than the root-to-
`leai‘ delay in presence of -a uniform transistor time constant
`gradient. This paper also proposes generalizations of the skew
`model to l) the case of chips in a wafer subject to a smooth,
`but nonuniform gradient and 2} the case of H-tree configurations
`mixing logic and interconnections;
`in this respect,
`this paper
`covers the H-tree configurations based on the combination of
`logic and interconnections.
`
`Index Terms—~H-trec, high—speed clocking, pipelining, process
`variations, skew.
`
`I.
`
`INTRODUCTION
`
`HE evolution of VLSI chips toward larger die sizes and
`
`Tfaster clock speeds makes clock design an increasingly
`important issue. A striking example of what can be accom-
`plished with aggressive clock design is the DEC alpha chip 11],
`
`[995. This
`l994; revised September 22.
`Manuscript received March 24,
`work was supported by the INI (lostitut National de formation on informa-
`tique) in Algiers. CIDA (Canadian Intemational Development Agency), and
`strategic grant and operating grants from the Natural Sciences and Engineering
`Research Council of Canada.
`The authors are with the Department of Electrical Engineering, Ecole
`Polytechnique of Montreal. Station "Centre-Ville," Monti-cal. P.Q. li3C 3A7,
`Canada.
`
`Publisher Item Identifier S 1063-8210(97301949-5.
`
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`des1gned to operate at more than Zogfimmh‘fibeds.
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`clock skew becomes a very significant problem. Available
`literature dealing with skew ['2]—[8], [10], [I ll approaches the
`problem both from deterministic and probabilistic standpoints.
`In the deterministic approaches, Friedmann and Powell [6]
`emphasize the use of a hierarchical clock distribution, while
`others [2], [3], [8], [ll] suggest the length equalization of the
`different paths followed by the clock throughout the circuit.
`Shoji
`[5] suggests an approach that guarantees a synunetry
`beIWeen paths that contribute to propagate “0” and “1." This
`symmetry ensures proper operation despite some types of
`process variations [5]. Except for the work of Fisher and Kong
`[4'], which provides bounds on skew, the other authors do not
`deal with the analytic modeling of system skew.
`In the probabilistic approaches, Kugelmass and Steiglitz [7]
`consider the delay of a clock signal along a given path as a
`sum of delays along path segments, each of these segments
`behaving according to a probabilistic law. Then, by assuming
`independence between these delays, the total delay, as well as
`the skew, can then be described by a normal law. By assuming
`independence and the linearity of delay with line length, their
`approach becomes an oversimplification of the reality. Other
`authors [10] consider the skew as a dispersion in the physical
`parameters of a circuit (e.g., geometrical dimensions) and in
`the process (e.g., sensitivity to temperature).
`The work that is most directly related to that presented in
`this paper is the work of Fisher and Kong [4]. These authors
`have developed two deterministic skew models {the difference
`model and the summation model), from which they determined
`bounds on skew. However, these models do not directly refer
`to a process variation model. The difference modeltends to
`be unrealistically optimistic, whereas, under the summation
`model, Fisher and Kung reached a pessimistic result which
`states that, from a skew standpoint, synchronous systems are
`not feasible with large two—dimensional arrays.
`In order to avoid the frequency limitation when using
`metallic lines, we propose a logic-based H-tree structure
`that provides the highest clocking rate achievable with a
`given technology in Section II. To provide a reliable skew
`model, Section III suggests a model based on delay difi‘crences
`combined with a model of electrical variations in the process
`parameters. Under this model, we derive an analytic expression
`of the skew between any leaf pair, which we consider to
`be the difference in root-to-leaf delay pairs. Even though the
`model of electrical variations described in this paper assumes
`1063~«8210I97$10.00 © 199’? IEEE
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`
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