1. Copyright  1995-1999 SCRA 1 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Scheduling and Assignment for DSP RASSP Education & Facilitation Program Module 22 Version 03.00 Copyright 1995-1999 SCRA All rights reserved. This information is copyrighted by the SCRA, through its Advanced Technology Institute (ATI), and may only be used for non-commercial educational purposes. Any other use of this information without the express written permission of the ATI is prohibited. Certain parts of this work belong to other copyright holders and are used with their permission. All information contained, may be duplicated for non-commercial educational use only provided this copyright notice and the copyright acknowledgements herein are included. No warranty of any kind is provided or implied, nor is any liability accepted regardless of use. The United States Government holds “Unlimited Rights” in all data contained herein under Contract F33615-94-C-1457. Such data may be liberally reproduced and disseminated by the Government, in whole or in part, without restriction except as follows: Certain parts of this work to other copyright holders and are used with their permission; This information contained herein may be duplicated only for non-commercial educational use. Any vehicle, in which part or all of this data is incorporated into, shall carry this notice.

2. Copyright  1995-1999 SCRA 2 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Rapid Prototyping Design Process Scheduling/Assignment SYSTEM DEF. FUNCTION DESIGN HW & SW PART. HW DESIGN SW DESIGN HW FAB SW CODE INTEG. & TEST VIRTUAL PROTOTYPE RASSP DESIGN LIBRARIES AND DATABASE Primarily software Primarily hardware HW & SW CODESIGN

3. Copyright  1995-1999 SCRA 3 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Module Goals l To present a state of the art review of scheduling and assignment methodologies for DSP l Examine and study the impact of these methods on RASSP design environments l Consolidate understanding through examples

4. Copyright  1995-1999 SCRA 4 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Module Outline l Introduction l FSFG as a Representation of DSP Algorithms l Measures of Performance l Retiming l Iteration Period and Iteration Period Bounds l Formal Methods for FSFGs l Perfect-rate FSFGs and Unfolding l Lookahead l Scheduling Approaches l Optimal Scheduling and Assignment Considering communication Delays and Finite Resources l Summary

5. Copyright  1995-1999 SCRA 5 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Module Outline l Introduction l FSFG as a Representation of DSP Algorithms l Measures of Performance l Retiming l Iteration Period and Iteration Period Bounds l Formal Methods for FSFGs l Perfect-rate FSFGs and Unfolding l Lookahead l Scheduling Approaches l Optimal Scheduling and Assignment Considering Communication Delays and Finite Resources l Summary

6. Copyright  1995-1999 SCRA 6 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Introduction l Graphical capture of algorithms via block- oriented environments is equivalent to a fully- specified flow graph representation of the timing l A formal approach to scheduling allows the transformation of the timing graphs to optimize execution times, processor utilization, and sample periods l Most commercial and research tools begin with the flow graph methodology that follows but only recently has communications and I/O been included within the cost constraints

7. Copyright  1995-1999 SCRA 7 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Module Outline l Introduction l FSFG as a Representation of DSP Algorithms l Measures of Performance l Retiming l Iteration Period and Iteration Period Bounds l Formal Methods for FSFGs l Perfect-rate FSFGs and Unfolding l Lookahead l Scheduling Approaches l Optimal Scheduling and Assignment Considering Communication Delays and Finite Resources l Summary

8. Copyright  1995-1999 SCRA 8 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Scheduling and Assignment for DSP l Common representation of DSP algorithms 1. Language-driven executable specification 2. Graphics/flow graph-driven specification l Option 2 is preferred by most optimization algorithms l Goals of optimization m Provide efficient implementation with respect to sampling rate, clock period, latency, area, power, etc., of a given algorithm m Provide modular and scaleable implementation capable of being upgraded in performance and functionality m Include communications cost and cost of memories and other peripherals

9. Copyright  1995-1999 SCRA 9 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Difference Between DSP and General-purpose Schedulers l DSP schedules are optimized over multiple iterations of data (essentially a semi-infinite stream of data); GP schedules are executed only once l Real-time and memory/buffer constraints exist for DSP l DSP uses numerically intensive arithmetic with high data bandwidths (most arithmetic is deterministic) l Library-based synthesis and design procedure is suitable for DSP applications

10. Copyright  1995-1999 SCRA 1010 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP FSFG Used to Represent DSP Functional/Timing l A convolution - A fine-grain depiction l IRST - A coarse-grain depiction h(0)h(1)h(2) x(1) x(0) y(1) y(0) DD x(.) = data input y(.) = data output h(.) = coefficients D = unit delays Velocity Filter Scene Register Scene Segment Clutter Filter Restore Registration Video In D FSFG = Fully Specified Flow Graph (Data/Control Flow Graph)

11. Copyright  1995-1999 SCRA1 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Example: FSFG for an Adaptive Filter Z  x[n] x[n-N+1] h[0] h[1] h[2] h[N-2] h[N-1] d[n] y[n] Feedback d[n] = desired signal e[n] = error signal

12. Copyright  1995-1999 SCRA 1212 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Second Order IIR Filter x[n] 1 2 3 4 5 6 7 8 9 10 11 y[n] Nodes 5, 7, 9, 11 = Multipliers (computational delay) Nodes 1, 3, 4, 8 = Adders (computational delay) Nodes 2, 6, 10 = Data Routers (zero delay)

13. Copyright  1995-1999 SCRA 1313 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Module Outline l Introduction l FSFG as a Representation of DSP Algorithms l Measures of Performance l Retiming l Iteration Period and Iteration Period Bounds l Formal Methods for FSFGs l Perfect-rate FSFGs and Unfolding l Lookahead l Scheduling Approaches l Optimal Scheduling and Assignment Considering Communication Delays and Finite Resources l Summary

14. Copyright  1995-1999 SCRA 1414 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Performance Measures l Sample period = average time between arrival/acceptance of two successive data samples l Latency = the time for the DSP to produce a result y[0] in response to input x[0] l In DSP applications, one usually considers minimization of sample period to be more important than latency DSP x[2] x[1] x[0] y[2] y[1] y[0] Latency Sample Period Input Data Output Data Time

15. Copyright  1995-1999 SCRA 1515 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Fully Specified Flow Graph l FSFG = m V i - set of computational nodes m E i - set of directed edges (implying data precedence) m D i - initial assignment of delays l Goal Transform Improve Performance without changing functionality i = initial f = final

16. Copyright  1995-1999 SCRA 1616 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Clock Period D x[n] 2 units 1 unit clock 2 units y[n] Minimum clock period = 2 + 1 = 3 units D x[n] 2 units clock 1 unit clock 2 units y[n] D Minimum clock period = 2 units Minimum Clock Period = Computational delay of the longest zero-delay path

17. Copyright  1995-1999 SCRA 1717 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Clock Period (Cont.) l Formally where m T(u) = computational latency (in wall-clock time) of node u m p = path of edges m P l = set of paths with zero delays (purely combinatorial) m T c = smallest clock period Goal: Optimize the clock period T C = max T(p), p  P l

18. Copyright  1995-1999 SCRA 1818 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Module Outline l Introduction l FSFG as a Representation of DSP Algorithms l Measures of Performance l Retiming l Iteration Period and Iteration Period Bounds l Formal Methods for FSFGs l Perfect-rate FSFGs and Unfolding l Lookahead l Scheduling Approaches l Optimal Scheduling and Assignment Considering Communication Delays and Finite Resources l Summary

19. Copyright  1995-1999 SCRA 1919 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Cutset Definition l A Cutset consists of the minimal set of edges that divide an FSFG into two disjoint parts  D-D A A B B C C A-A, B-B, D-D are valid cutsets C-C is not a valid cutset Edge 1-2 and edge 4-5 are directed in opposite directions with respect to cutset A-A

20. Copyright  1995-1999 SCRA 2020 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Time Scaling l All delays in an FSFG can be scaled by a positive number “s” without changing the functionality (except the input/output rates) l Scaling is useful (as will be seen) but leads to loss in efficiency D x[2] x[1] x[0] y[2] y[1] y[0] 2D x[2] - x[1] - x[0] y[1] _ y[0] c1 SCALING spacing in the input

21. Copyright  1995-1999 SCRA 2121 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Nodal Transfer vu 3D D 2D D D r(u) = number of delays REMOVED from EACH input arc and moved to each output arc (can be +ve or -ve) vu 2D 4D Edge e r(v) = 1 r(u) = -1 4 = 2 + 1 - (-1) D f (e) = D i (e) + r(v) - r(u) l Same functionality, different timing - RETIMING

22. Copyright  1995-1999 SCRA2 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Legal Retiming l D f (e) >= 0 for all edges l D f (e) >= 1 for a SYSTOLIC FSFG m Systolic FSFGs have minimum clock period, or the highest clock frequency l Total number of delays in a loop remains constant with and without retiming (no loop can have a zero delay)  2D D r(1) = 1  D 2D r(1) = 0 r(2) = 1 r(3) = 0 Retiming

23. Copyright  1995-1999 SCRA 2323 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Delay Transfer Across Cutsets  2D 4D D 2D D A A  3D D A A Delay Transfer l Remove k delays from positive edges and move them to negative edges (k can be positive or negative) l No algorithmic change in behavior

24. Copyright  1995-1999 SCRA 2424 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Results from Retiming l Theorem I: A systolic FSFG can always be realized l Theorem II: If the number of delays in a loop is less than the number or arcs, then time scaling is required for obtaining systolic schedules l Theorem III: The delays in a retimed circuit can be minimized by setting up the following Linear Integer Program for r(v) l Where e is a directed edge from v to u Minimize  r  v r(v) {Fanout(v) - Fanin(v)} subject to r(u) - r(v) <= D i (e) -1

25. Copyright  1995-1999 SCRA 2525 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Module Outline l Introduction l FSFG as a Representation of DSP Algorithms l Measures of Performance l Retiming l Iteration Period and Iteration Period Bounds l Formal Methods for FSFGs l Perfect-rate FSFGs and Unfolding l Lookahead l Scheduling Approaches l Optimal Scheduling and Assignment Considering Communication Delays and Finite Resources l Summary

26. Copyright  1995-1999 SCRA 2626 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Iteration Period and Delays in a Loop l Case 1 m Bound on iteration period = = (2+2+2)/3 = 3 units m Data sample can be accessed every 3 units (on average) l Case 2 m Bound on iteration period = = (2+2+2)/3 = 2 units  x[.] T(1) = 2 T(2) = 2 y[.] 2D T(3) = 2  x[.] T(1) = 2 T(2) = 2 y[.] 2D T(3) = 2 D Case 1 and Case 2 are different algorithms, though!

27. Copyright  1995-1999 SCRA 2727 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Iteration Period Bound l An FSFG is bounded below by the Iteration Period Bound (IPB) l Thus the IPB can be reduced by increasing the delays in the loops or by reducing the computational latency of the loops IPB = max Total computational latency of a loop Total number of delays in the loop all loops Scheduling and assignment efficiency is limited by loops in the DSP flow graph!

28. Copyright  1995-1999 SCRA 2828 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Retiming as a Mechanism to Reduce Area/Conserve Resources x[n] 1 3 4 57 8 911 A A D D 1 3 4 57 8 911 D D D Cycle Multiply Addition 1 5,9 --- 171 2 7,11 4258 3 --- 8,1393 4 --- 34114 Large area, low utilization Too many multiplies in one cycle Initial Schedule Re-timed Schedule [Madisetti95] D © IEEE 1995

29. Copyright  1995-1999 SCRA 2929 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Top-down Methodology for Scheduling DSP Algorithm Dataflow Specifications Transformations Estimation of Performance Scheduling Assignment Allocation Objective Functions Constraints Low PowerSmall Area High Clock Speed Small Latency

30. Copyright  1995-1999 SCRA 3030 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Partial Summary of Results l Flow graph transformations can result in improved efficiency of implementation for the same algorithm l Retiming cannot improve upon the bound, but it can help achieve the bound on the iteration bound l A formal approach is necessary for top down design, optimization, and implementation

31. Copyright  1995-1999 SCRA 3131 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Module Outline l Introduction l FSFG as a Representation of DSP Algorithms l Measures of Performance l Retiming l Iteration Period and Iteration Period Bounds l Formal Methods for FSFGs l Perfect-rate FSFGs and Unfolding l Lookahead l Scheduling Approaches l Optimal Scheduling and Assignment Considering Communication Delays and Finite Resources l Summary

32. Copyright  1995-1999 SCRA 3232 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP FSFG Transformation and Scheduling l Various transformations exist that lead to efficient implementation of flow graphs m Unfolding m Retiming m Lookahead m Loop shrinking l Approach m Step 1: Classify different types of schedules m Step 2: Try to find methods to construct these schedules, if they exist

33. Copyright  1995-1999 SCRA3 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Example  D  C 2D D IPB = 2 P1 P2 D1 A1 D2 A2 D3 A3 C1 B1 C2 B2 C3 B3 P1 P2 D1 A1 B1 C2 D3 A3 B3 C1 D2 A2 B2 C3 Processors P1, P2 0 1 2 3 4 5 6 P1 P2 D1 A1 B1 D2 A2 B2 D3 C1 C2 Time IP = 3 Time IP = 2 a) b) c) a) Non-overlapping static schedule b) Overlapping static schedule c) Cyclostatic schedule

34. Copyright  1995-1999 SCRA 3434 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Questions To Be Answered at This Point l Can every FSFG achieve the iteration period bound? l Can every FSFG achieve the IPB in addition to the processor bound? l Can transformations assist in achieving the IPB? l Are optimal schedules static or cyclostatic? l Can the IPB itself be reduced? l Can transformations reduce the processor bound?

35. Copyright  1995-1999 SCRA 3535 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Module Outline l Introduction l FSFG as a Representation of DSP Algorithms l Measures of Performance l Retiming l Iteration Period and Iteration Period Bounds l Formal Methods for FSFGs l Perfect-rate FSFGs and Unfolding l Lookahead l Scheduling Approaches l Optimal Scheduling and Assignment Considering Communication Delays and Finite Resources l Summary

36. Copyright  1995-1999 SCRA 3636 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP 0 1 2 3 4 5 6 P1 P2 Perfect-rate FSFGs ACDEB D D D 0 1 2 3 4 5 6 P1 P2 D1 C1 A1 D2 C2 A2 E1 B1 E2 B2 Time IP = IPB ACDEB D D A1 D1 C1 A2 D2 C2 B1 E1 B2 E2 Time IP = IPB Optimal static schedules are possible for PERFECT-RATE FSFGs

37. Copyright  1995-1999 SCRA 3737 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Perfect-rate FSFGs (Cont.) Conclusion l Because perfect-rate FSFGs have static rate- optimal schedules, can all FSFGs be transformed to PR-FSFGs? l Answer: Yes, all FSFGs can be transformed to PR-FSFGs m Note communication cost and processor cost is not optimal, only sample period is optimized

38. Copyright  1995-1999 SCRA 3838 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Unfolding l How does one convert FSFGs to perfect-rate FSFGs? l Answer: Through UNFOLDING! (method 1)

39. Copyright  1995-1999 SCRA 3939 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Example of Unfolding ABC D D B1C1A2 D A1B2C2 D Original FSFG Unfolded-by-2 FSFG By unfolding-by-2 we expose two iterations of the DSP algorithm at the same time

40. Copyright  1995-1999 SCRA 4040 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Optimum Unfolding l Unfold the flow graph by the LCM of the number of delays in each loop l Example: Unfold by 6 results in IP = 16 units on 4 processors AB D EDC D D D D A = 20 units B = 5 units C = 10 units D = 10 units E = 2 units IP = 20 units IPB = 16 units Note: Node A = 20 units which is larger than IPB

41. Copyright  1995-1999 SCRA 4141 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Module Outline l Introduction l FSFG as a Representation of DSP Algorithms l Measures of Performance l Retiming l Iteration Period and Iteration Period Bounds l Formal Methods for FSFGs l Perfect-rate FSFGs and Unfolding l Lookahead l Scheduling Approaches l Optimal Scheduling and Assignment Considering Communication Delays and Finite Resources l Summary

42. Copyright  1995-1999 SCRA 4242 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Method 2: Lookahead l Lookahead actually reduces the iteration period bound m Pipelining and unfolding reduce the iteration period, but do not affect the iteration period bound l Design Flow m Step 1: Check if sample period is satisfied by IPB m Step 2: Lookahead can be used to reduce IPB m Step 3: Retiming, unfolding, or pipelining can then be used to reduce the iteration period to that of the IPB

43. Copyright  1995-1999 SCRA 4343 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP What is Lookahead? Statement 1: x[n] = ax[n-1] + u[n] Statement 2: x[n+1] = ax[n] + u[n+1] Statement 2 cannot be executed until x[n] is computed Rewriting the equations again (with a lookahead of two) Statement 1: x[n] = bx[n-2] + au[n] + u[n-1] Statement 2: x[n+1] = a(ax[n-1]+u[n]) + u[n+1] = a 2 x[n-1] + au[n] + u[n-1] where b = a 2 Note that x[n+1] depends on x[n-1] and not on x[n] Therefore, these statements could be executed in parallel

44. Copyright  1995-1999 SCRA4 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP What is Lookahead? (Cont.) Read Ops x y z Multiply x y z Add 2 x y z Add 1 x y z Store State x y z 0 1 2 3 4 5 6 7 8 Read Ops x y Multiply x y Store State x y Add x y First instruction Second Instruction begins after x[n] is ready Original first order Filter Filter with lookahead of 2 Note: Factor sample period due to increased lookahead Time

45. Copyright  1995-1999 SCRA 4545 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP What is Lookahead? (Cont.) l Lookahead reduces the IPB, but at the cost of additional hardware (adders and multipliers) l Lookahead adds delay (or memory) into the feedback loop m E.g., x[n] depends on x[n-2] => Two delays x[n] depends on x[n-1] => One delay l Iteration period bound = m Lookahead increases the denominator l Alternative Total Computational Latency in Loop Total Number of Delays A D A 10D Lookahead by 10

46. Copyright  1995-1999 SCRA 4646 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Flowchart l Step 1: Use Lookahead by M l Step 2: Retime to distribute delays l Step 3: Obtain a static rate-optimal schedule ABC MD ABC D D Flowchart for optimization with Lookahead and Retiming

47. Copyright  1995-1999 SCRA 4747 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Module Outline l Introduction l FSFG as a Representation of DSP Algorithms l Measures of Performance l Retiming l Iteration Period and Iteration Period Bounds l Formal Methods for FSFGs l Perfect-rate FSFGs and Unfolding l Lookahead l Scheduling Approaches l Optimal Scheduling and Assignment Considering Communication Delays and Finite Resources l Summary

48. Copyright  1995-1999 SCRA 4848 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Taxonomy of Scheduling Approaches SchedulingStaticSub-OptimalOptimalHeuristicApproximateDynamic Physically Non-Distrib Physically Distributed Non-CooperativeCooperativeSub-OptimalOptimalHeuristicApproximate

49. Copyright  1995-1999 SCRA 4949 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP FSFG for Second Order IIR Filter I1I1 I2I2 x[n] y[n] 1 3 5 2 4 6 7 8 DDDD d_add = 1 unit time d_mult = 2 unit time Flow Graph Bound Definition Value  j  l d j Iteration Period IPB = max, l  L 3 Bound n l  j  l d j Iteration Period PB = 4 Bound IPB Period Delay Bound D i/o = max  j  p d j - n p IPB, r  P 3

50. Copyright  1995-1999 SCRA 5050 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Second Order Filter- A Cyclostatic Schedule 41234123 Cycling Vector, C 6060 6060 7070 4040 4040 2020 8 -1 1010 3131 3030 5151 5151 Period Matrix, S Iteration Tile Iteration Tile repeated in P x T space 5 5 6 6 7 4 4 2 8 1 3 3 5 5 6 6 7 4 4 2 8 1 3 3 5 5 6 6 7 4 4 2 8 1 3 3 5 5 6 6 7 4 4 2 8 1 3 3 5 5 6 6 7 4 4 2 8 1 3 3 5 5 6 6 7 4 4 2 8 1 3 3 T P

51. Copyright  1995-1999 SCRA 5151 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Comparison of Several Scheduling Algorithms SchedulingIteration Period No. of Processors Throughput Delay Comm. Considered Single Iteration not optimal no Direct Blocking not optimal no Maximum- Spanning tree not optimaloptimalnot optimalno Optimum Unfolding optimizedoptimalnot optimalno Cyclo-Static optimal no CSPP optimal no Generalized PSSIMD optimal yes Scheduling- Range | Fixed rate Max TP optimized fixed optimal optimized not optimal yes DSMP-C1 optimal yes MULTIPROC optimal yes

52. Copyright  1995-1999 SCRA 5252 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Module Outline l Introduction l FSFG as a Representation of DSP Algorithms l Measures of Performance l Retiming l Iteration Period and Iteration Period Bounds l Formal Methods for FSFGs l Perfect-rate FSFGs and Unfolding l Lookahead l Scheduling Approaches l Optimal Scheduling and Assignment Considering Communication Delays and Finite Resources l Summary

53. Copyright  1995-1999 SCRA 5353 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Optimal Scheduling and Assignment l Focus on an optimal scheduling and assignment methodology l Optimize objective function subject to m Constraints

54. Copyright  1995-1999 SCRA 5454 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP IPB Model l Minimize IPB subject to  Precedence constraints: Node v precedes node w (arc v->w) t w -t v > d w -n vw IPB, for e(v,w)  E m Non-negativity constraints: IPB, t i > 0, for i = 1,..., N l For the second order IIR filter, this problem becomes m Minimize IPB m Subject to: q Precedence Constraints t 1 - t 3 > 1 t 2 - t 1 > 1 t 2 - t 4 > 1 t 4 - t 2 + IPB > 2 t 3 - t 2 + 2 IPB > 2 t 8 - t 2 > 1 t 8 - t 7 > 1 t 7 - t 6 > 1 t 7 - t 5 > 1 t 6 - t 2 + IPB > 2 t 5 - t 2 + 2 IPB > 2 q Non-negativity constraints t 1, t 2, t 3, t 4, t 5, t 6, t 7, t 8, IPB > 0

55. Copyright  1995-1999 SCRA5 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Finite Resources Model l Minimize the number of processors P and periodic throughput delay D i/o T  j=0 to U 2 j p j + D i/o l Subject to  Precedence constraints: Node v precedes node w (arc v-> w). t w - t v > d w - n vw IP, for e(v,w)  E where t i =  t=1 to T t x it  Job completion constraints: All nodes must be scheduled  t=1 to T x it = 1, for i = 1,..., N  Processor modulo and non-preemption constraints: At most P processors can be scheduled at a time  i=1 to N  Qn x iq <  j=0 to U 2 j p j, t=1,...,T n = 0,...IPB-1 where Qn = {q | q mod IP = n, for q = t to t+d i -1}  0-1 constraints: x it  {0,1} for i = 1,..., N and t = 1,...,T p j  {0,1} for j = 1,...,u

56. Copyright  1995-1999 SCRA 5656 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Adding Communications 12P-1P 12 P Originating PE j Destination PE k cp jk Originating PE j Destination PE k Edge e i 1P1P 1 2 P 1 2 E Communication paths Decision variable x ejk

57. Copyright  1995-1999 SCRA 5757 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Multi-processor System 3030 1010 5050 6060 2 -1 4040 7 -1 8 -1 Period matrix, S x[n] y[n] 1 3 5 2 4 6 7 8 DDDD d_add = 1 unit time d_mult = 1 unit time 3 4 7 9 2 5 6 8 10 11 Second order IIR filter PE 1 PE 2 PE 3 PE 4 1 22 1 1 1 c jk Communication network

58. Copyright  1995-1999 SCRA 5858 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Second Order IIR Filter Edge Comm. Path 1 1 --> 2 2 1 --> 1 3 2 --> 2 4 2 --> 2 5 2 --> 1 6 2 --> 4 7 2 --> 3 8 4 --> 3 9 3 --> 3 10 2 --> 4 11 3 --> 4 PE Nodes 1 3, 1 2 4, 2 3 6, 7 4 5, 8 3 14 2 5 86 7 PE 1 PE 2 PE 4 PE 3 Edge mapping PE assignmentCommunications

59. Copyright  1995-1999 SCRA 5959 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Multi-processor System 3030 1010 5050 6060 2 -1 4040 7 -1 8 -1 Period matrix, S x[n] y[n] 1 3 5 2 4 6 7 DDDD d_add = 1 unit time d_mult = 1 unit time 3 4 7 9 2 5 6 8 10 11 Second order IIR filter PE 1 PE 2 PE 3 PE 4 1 1 1 1 c jk Communication network

60. Copyright  1995-1999 SCRA 6060 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Randomly Connected Cost (RCC) Model for Second Order IIR Filter Edge Comm. Path 1 1 --> 1 2 2 --> 1 3 2 --> 1 4 1 --> 2 5 1 --> 2 6 1 --> 3 7 1 --> 3 8 3 --> 4 9 3 --> 4 10 1 --> 4 11 4 --> 4 1 23 47 85 6 PE 1 PE 2 PE 4 PE 3 Edge mapping Communications 2 -1 1 0 3030 4040 5050 6060 7 -1 8 -1 2 0 1 1 3 1 4141 5 1 6161 7 0 8080 Comm. that increase the IP RCC Model = Randomly Connected Cost Model

61. Copyright  1995-1999 SCRA 6161 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Scheduled Communications for Second Order IIR Filter PE 1 PE 2 PE 3 PE 4 2->3,4,8.3->1 5->7 2->5,6 Inserted cycle T1A PE 1 PE 2 PE 3 PE 4 Inserted cycle T1B Route 2->8 through PE2 2->8 PE 1 PE 2 PE 3 PE 4 Inserted cycle T1C 2->8 PE 1 PE 2 PE 3 PE 4 4->2 6->7 Inserted cycle after T2

62. Copyright  1995-1999 SCRA 6262 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Including Memory and Registers j k Edge e i R1 R2 M1 M2 +1 +2 x1 x2 1 2 E +x+x RMRM PE 1 Network Decision variable

63. Copyright  1995-1999 SCRA 6363 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Modified FSFG for Second Order IIR Filter 1111 331 10 15 2 12 4 6 14 7 16 12 4 6 14 17 2 9 1 11 3 5 13 1 11 3 5 13 7 8 8 18

64. Copyright  1995-1999 SCRA 6464 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Comparison of Processor Mapping Models Model Min Max Num Proc Network Multiple Reg & Increase IP of Comm Mem Fully connected cost Yes No Given Full No No Fully connected cost Yes Yes Given Full No No capacity Randomly connected Yes No Given Random No No cost Randomly connected Yes Yes Given Random No No cost & capacity Randomly connected Yes Yes Given Random Yes No cost & capacity with Multiple FU Randomly connected Yes Yes Given Random Yes Yes cost & capacity with Multiple FU & Memory & Reg Combined Scheduler Min IP No Given Random No No Combined Scheduler Min IP Yes Given Random Yes Yes with Multiple FU

65. Copyright  1995-1999 SCRA 6565 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Module Outline l Introduction l FSFG as a Representation of DSP Algorithms l Measures of Performance l Retiming l Iteration Period and Iteration Period Bounds l Formal Methods for FSFGs l Perfect-rate FSFGs and Unfolding l Lookahead l Scheduling Approaches l Optimal Scheduling and Assignment Considering Communication Delays and Finite Resources l Summary

66. Copyright  1995-1999 SCRA6 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Review of Approach to Scheduling DSP Algorithm Dataflow Specifications Transformations Estimation of Performance Scheduling Assignment Allocation Objective Functions Constraints Low PowerSmall Area High Clock Speed Small Latency

67. Copyright  1995-1999 SCRA 6767 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP Summary l A formal approach to recent results to the scheduling and assignment of non-ideal systems has been presented l This framework is very powerful and flexible for a variety of computational granularities found in signal processing l RASSP is currently developing a number of constraint-based schedulers

68. Copyright  1995-1999 SCRA 6868 Methodology Reinventing Electronic Design Architecture Infrastructure DARPA Tri-Service RASSP References [IEEE] All referenced IEEE material is used with permission. [Madisetti95] V.Madisetti, “VLSI Digital Signal Processors”, IEEE Press, 1995; © IEEE 1995 [Richards97] Richards, M., Gadient, A., Frank, G., eds. Rapid Prototyping of Application Specific Signal Processors, Kluwer Academic Publishers, Norwell, MA, 1997.