2. Tiny Triplet Finder Jinyuan Wu, Z. Shi Dec. 2003

3. Hits, Hit Data & Triplets Hit data come out of the detector planes in random order. Hit data from 3 planes generated by same particle tracks are organized together to form triplets.

4. Three data items must satisfy the condition: x A + x C = 2 x B. A total of N 3 combinations must be checked (e.g. 5x5x5=125). Three layers of loops if the process is implemented in software. Large silicon resource may be needed without careful planning. Triplet Finding Plane APlane BPlane C

5. Tiny Triplet Finder (Animation) x1 x2 x3 Implement x1 and x3, the plane x2 is for eye- guide only. (1) Fill in hits to x1 and x3 planes. (2) Cycle through x2 hits. Line matched.

6. Tiny Triplet Finder Bit Array/Shifter C Bit Array/Shifter A Bit-wise Logic Block Hash Sorter C Hash Sorter A

7. TTF Operations Phase I: Filling Bit Arrays Note: Flipped Bit Order Physical Planes Bit Array/Shifters For any hit… Fill a corresponding logic cell. x A + x C = 2 x B x A = - x C + constant

8. TTF Operations Phase II: Making Match For any center plane hit… Logically shift the bit array. Perform bit- wise AND in this range. Triplet is found. Physical Planes Bit Array/Shifters

9. TTF Operations Phase II: Making Match (Next Hit) Triplet is found through bit-wise AND. Fake triplet may exist. It will be cut out in the later stages. Physical Planes Loop to next center plane hit… Logically shift the bit array. Bit Array/Shifters

10. TTF Operations Phase II: Making Match (More…) Physical Planes Loop to next center plane hit… Logically shift the bit array. Triplet is found through bit-wise AND. Bit Array/Shifters

11. TTF Operations Phase II: Making Match (and More…) Physical Planes Loop to next center plane hit… Logically shift the bit array. Triplet is found through bit-wise AND. Bit Array/Shifters

12. TTF Operations Phase II: Making Match (Last Hit) Physical Planes Loop to next center plane hit… Logically shift the bit array. Triplet is found through bit-wise AND. Bit Array/Shifters

13. Multiple Hits & Triplets Keep Them All Each bin may be filled with more than one hits. Hits data are kept in hash sorters – allowing multiple hits per bin. There are may be more than one match in each bit-wise AND operation. They are all sent out, one-by-one, to the later stages for fine cut and arbitration processes.

14. Boundary Issues Beyond Just Bit-wise AND When the track hits near the boundary of a bin, simple bit-wise AND may miss the triplet. The bit-wise OR-AND logic will cover the boundary. The logic cells in most of today’s FPGA’s have 4 inputs. So the OR- AND bit-wise logic doesn’t increase any resource usage.

15. DA BitLogic KA XB EN CKC Pop Halt Bit64A Q KS WR EN SR Bit64C Q KS WR EN SR Hash2blk DIQQ K Push Pop WT RePop SR RDY MD Hash2blk DIQQ K Push Pop WT RePop SR RDY MD Cut and Arbitration QT DA EN DB RDYDC Shift Reg. 6 steps DQ4 EN FIFO Interface Hit Feeder A QAD EN KA Push FIFO Interface Hit Feeder C QCD EN KC Push A FIFO Interface Hit Feeder B SAD EN QB Chk SC XB 172 LUT FF 92 LUT FF 192 LUT 111 FF 77 LUT 66 FF 77 LUT 66 FF Tiny Triplet Finder: Block Diagram

16. Schematics

17. Simulation (1) Filling hits on Plane A and C. (2) Looping over hits on Plane B. Pipelined internal operations… Triplets are grouped together. This combination is a fake triplet. The extra combination halts the pipeline for proper operation.

18. Logic Cell Usage Both 64- and 128-bit TTF designs fit $100 FPGA comfortably. A simple 64-bit Hough transform design is shown for scale. A $1200 FPGA is shown for scale.

19. Use 4 bit arrays. There are 3 constraints total. More constraints help to eliminating fake tracks. It is possible to use bit- wise majority logic (such as 3-out-of-4) to accommodate detector inefficiency issues. Pentlet Finding Beyond Just Bit-wise AND Plane APlane CPlane EPlane BPlane D

20. Comparison of Baseline and Tiny Triplet Finder: Data Flow FIFO AM: Long Doublet FIFO Station N-1 bend Station N-1 Non-bend Station N Non-bend FIFO AM: Triplet FIFO AM: Short Doublet FIFO AM: Short Doublet AM: Short Doublet Station N bend Station N+1 bend Station N+1 Non-bend Station N-1 bend Station N-1 Non-bend Station N Non-bend FIFO FTF: Triplet FIFO Hash Sorter: Short Doublet Station N bend Station N+1 bend Station N+1 Non-bend FIFO Hash Sorter: Short Doublet Hash Sorter: Short Doublet Hash Sorter Hash Sorter

21. The End Thanks

22. Hash2blk DIQQ K Push Pop WT RePop SR RDY MD Cut, Arbitration and Combine QT DA EN DT RDYDC Shift Reg. 2 steps DQQ EN FIFO Interface Hit Feeder A’ QAD EN KA Push 77 LUT 66 FF Short Doublet Stage: 1 of 3 Identical Ones Triplet Feeder SAD EN QT Push Station N-1 Non-bend FIFO Hash Sorter: Short Doublet Triplets In Triplets Out EPStnIDHitMap y2y (8+3bits) x1x (8+3bits)  y1(400u) y1y (8+3bits)  x1(400u) y3y (8+3bits) H EPStnIDHitMap y2y (8+3bits) y1y (8+3bits) y3y (8+3bits) H x1y (7bits) x2y (7bits) x3y (7bits) x2y (7bits) x3y (7bits)

23. A3 A2 A1 A0 S0,S1 K(8:0) DI(27:0) DOB(27:0) IdN(7:0) CB8RE CE SR Q CB10E CE Q A0 A1 S0 COMP A EQ B FD DQ DQ EN FD DQ DQ EN RAM 512x18 D A O WE EN DO FD DQ RAM 256x36 D A O WE EN DO FD DQ EN RDY MD PUSH POP WT REPOP DUMPD SR CLK POPQ PUSHQ DIQ(27:0) IdP(7:0)IdQ(7:0) IdK(7:0)IdKQ(7:0) EVeq EV(9:0) EV(8:0) Vertex II Implementation Hash Sorter: Vertex-II Implementation

24. A3 A2 A1 A0 S0,S1 K(7:0) DI(28:0) DOB(28:0) IdN(6:0) CB7RE CE SR Q CB9E CE Q A0 A1 S0 COMP A EQ B FD DQ DQ EN FD DQ DQ EN RAM 256x16 D A O WE EN DO FD DQ RAM 128x36 D A O WE EN DO FD DQ EN RDY MD PUSH POP WT REPOP DUMPD SR CLK POPQ PUSHQ DIQ(28:0) IdP(6:0)IdQ(6:0) IdK(6:0)IdKQ(6:0) EVeq EV(8:0) EV(7:0) Cyclone Implementation Hash Sorter: Cyclone Implementation

25. Bit Array/shifter XILINX Implementation DQ WE A(3:0) DQ WE A(3:0) DQ WE A(3:0) DQ WE A(3:0) Each Bit Array/Shifter has 64 bins. Each bin is a 16-bit RAM (look-up table). The RAM operates like a D Flip-flop. Each xc2v1000 FPGA has 10,240 such RAM’s. DQ WE A(3:0) DQ WE A(3:0) DQ WE A(3:0) DQ WE A(3:0)

26. Bit Array/shifter Filling the Hits Each hit is written into 8 to 16 different RAM’s with different addresses. One clock cycle (not 8 to 16) is used to fill a hit.

27. Bit Array/shifter Shift Reading With given addresses of the RAM’s, hit patterns appear at the outputs of the arrays. Changing addresses shifts the hit patterns. One array shifts in unit of 1 bin, the other in unit of 8 bins. The relative shift of the two patterns covers 128 bins. One clock cycle is used for a reading, regardless the distance of the relative shifting.

28. Bit Array/shifter Comments: Xilinx or Not DQ WE A(3:0) DQ WE A(3:0) DQ WE A(3:0) DQ WE A(3:0) Writing/reading operations need only single clock cycle. Storage and shifter are combined — big resource saving. Maximum 16 clock cycles are needed for resetting.  Non-xilinx implementation: Additional 64x8=512 LE’s may be needed.  But resetting needs only one clock cycle. DQ WE A(3:0) DQ WE A(3:0) DQ WE A(3:0) DQ WE A(3:0)

29. A triplet has two parameters. With two hits in two planes, two parameters can be determined. The 3 rd hit in the 3 rd plane will provide the first constraint. Assume each parameter is sliced into 64 bins. (That’s not too many.) There are 64 x 64 = 4096 “roads”, i.e., possible track configurations. Any hit can belong to 64 possible roads. Two hits in two planes have one common road. Three hits must have one common road to be in a triplet. Challenge on Triplet Finding (1) The Facts in “Road” Language

30. It is possible to use (static) contents addressable memory (CAM) to implement triplet finding. Hit patterns are pre-stored in CAM. When a hit pattern match the pre-stored one, the address represent the road. Each of 3 planes has 64 possible hit locations. (64x64=4096 roads) Assume one road can generate one hit pattern, a CAM array with 64x3=192 input bits, 4096 words is needed. It can be done with 128 small CAM’s (32-bit 32-words). EPC20K1000E has 160. (80%) Oops: boundary issues have not be considered. (More patterns  ) Challenge on Triplet Finding (2) A Possible Implementation Using CAM CAM D( ) A WE Match Plane APlane BPlane C Road CAM D( ) A WE Match CAM D( ) A WE Match

31. Book a 2-D (64x64) histogram. Each bin of the histogram represent a road. Each hit from plane A, C or B increments the counts of the 64 bins in a row, column or diagonal. The bins with count 3 are valid triplet roads. Each histogram bin is a counter plus selection logic. Assume each bin can be implemented with 4 Altera logic elements (LE’s). 4096 bins need 16,384 LE’s, EPC20K1000E has 38,400. (16,384/38,400 = 43%) However: How to find the bins with count 3 and encode them is not trivial. Challenge on Triplet Finding (3) A Possible Implementation Using Hough Transformation

32. Silicon Usage (Estimate) The Tiny Triplet Finder (TTF) silicon usage is acceptable. The hash sorter silicon usage is acceptable. The Baseline, if not implemented automatically, could reduce the size a lot.

33. RAM D A O WE EN D CC8RLE Q L CE SR A0 A1 A2 A3 S CB8SE S1 CE SS Q FD DQ CE RAM D A O WE EN DO NBin DI DOB IdNext FD DQ CE

34. 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 3 2 1 0 7 6 5 4 11 10 9 15 14 13 12 8 0 1 2 30 1 2 30 1 2 30 1 2 30 1 2 30 1 2 3

35. 0 5 6 0 1 2 3 0 1 5 6 0 1 2 3 0 1 2 5 6 0 1 2 3 0 1 2 3 5 6 0 1 2 3 4 0 1 2 3 5 6 0 1 2 3 4 5 0 1 2 3 5 6 0 1 2 3 4 5 6 0 1 2 3 5 6 0 1 2 3 4 5 6 7 0 1 2 3 5 6 0 1 2 3 44 4’ 4444 7 7777 44 777

36. 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 1 2 3 5 6 7 0 1 2 3 4 5 6 7 2 3 5 6 7 0 1 2 3 4 5 6 7 3 5 6 7 0 1 2 3 4 5 6 7 5 6 7 0 1 2 3 5 6 7 5 6 7 0 1 2 3 6 7 5 6 7 0 1 2 3 7 5 6 7 0 1 2 3 4444 4 44