1. EEEB0765 Digital Signal Processing for Embedded Systems 5 Systolic Arrays
Assoc. Prof. Dr. Peerapol Yuvapoositanon
Dept. of Electronic Engineering
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

2. Outline The concept of Systolic arrays
Hands on Simulation with Simulink and S-function
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

3. What Is a Systolic Array?1
A systolic array is an arrangement of processors in an array where data flows synchronously across the array between neighbors, usually with different data flowing in different directions.
1 “Systolic Arrays & Their Applications, J. Bleak”
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

4. Examples of Systolic Arrays
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

5. Example 11 Calculate the force of Planets Conventional method: N2
Systolic method: N
We will need one processing element for each body, lets call them planets.
1 “Systolic Arrays & Their Applications, J. Bleak”
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

6. B2 B3 B1 B4 B6 B5
Six planets in orbit with different masses, and different forces
acting on each other, so are systolic array will look like this.
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

7. Each processor will hold the newtonian force formula:
Fij = Gmimj/d2ij , G being the gravitational constant.
Then load the array with values. B1 B2 B3 B4 B5 B6
Now that the array has the mass and the coordinates of Each body we can begin our computation.
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

8. Fij = Gmimj/d2ij B6, B5, B4, B3, B2, B1 B1m B1c B2m B2c B3m B3c B4m
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

9. Fij = Gmimj/d2ij B6, B5, B4, B3, B2 B1m B1c B2m B2c B3m B3c B4m B4c
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

10. Fij = Gmimj/d2ij B6, B5, B4, B3 B1m B1c B2m B2c B3m B3c B4m B4c B5*B1
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

11. Fij = Gmimj/d2ij B6, B5, B4 B1m B1c B2m B2c B3m B3c B4*B1 B5*B2 B6*B3
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

12. Fij = Gmimj/d2ij B6, B5 B1m B1c B2m B2c B3*B1 B4*B2 B5*B3 B6*B4
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

13. Fij = Gmimj/d2ij B6 B1m B1c B2*B1 B3*B2 B4*B3 B5*B4 B6*B5
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

14. Fij = Gmimj/d2ij Done Done Done Done Done Done
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

15. Example 21 Calculate Matrix Multiplication Conventional method: N3
Systolic method: N
1 “Systolic Arrays & Their Applications, J. Bleak”
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

16. Example for N=2 N3=23=8 2 Mult. 2 Mult. 2 Mult. 2 Mult.
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

17. Finding C[i,j] What is the algorithm to find C[i,j]?
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

18. Matrix Multiplication (Conventional)
For I = 1 to N For J = 1 to N For K = 1 to N C[I,J] = C[I,J] + A[J,K] * B[K,J]; end
This is N3
computation
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

19. C++ Code for Matrix Multiplication
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

20. Systolic Array Matrix Computation
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

21. EEEB0765 DSP fort Embedded Systems http://embedsigproc. wordpress
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

22. EEEB0765 DSP fort Embedded Systems http://embedsigproc. wordpress
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

23. EEEB0765 DSP fort Embedded Systems http://embedsigproc. wordpress
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

24. EEEB0765 DSP fort Embedded Systems http://embedsigproc. wordpress
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

25. EEEB0765 DSP fort Embedded Systems http://embedsigproc. wordpress
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

26. EEEB0765 DSP fort Embedded Systems http://embedsigproc. wordpress
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

27. EEEB0765 DSP fort Embedded Systems http://embedsigproc. wordpress
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

28. * =
Lets try this using a systolic array. 5 3 2 2 5 4 3 2 P1 P2 P3 P4 P5 P6 P7 P8 P9

29. Clock tick: 1 5 3 2 2 5 4 3 2
2 4
3*3 P1 P2 P3 P4 P5 P6 P7 P8 P9 9

30. Clock tick: 2 5 3 2 2 5 3 2
4*2
3*4
3 5
2*3 P1 P2 P3 P4 P5 P6 P7 P8 P9 17 12 6

31. Clock tick: 3 5 3 2
2*3
4*5
3*2 3
5*2
2*4
3*3 P1 P2 P3 P4 P5 P6 P7 P8 P9 23 32 6 16 8 9

32. Clock tick: 4 5
2*2
4*3
3*3
5*5
2*2 5
2*2
3*4 P1 P2 P3 P4 P5 P6 P7 P8 P9 23 36 18 25 33 4 13 12

33. Clock tick: 5
2*5
3*2
5*3
5*3
5*2
3*2 P1 P2 P3 P4 P5 P6 P7 P8 P9 23 36 28 25 39 19 22 6

34. Clock tick: 6
3*5
5*2
2*3 P1 P2 P3 P4 P5 P6 P7 P8 P9 23 36 28 25 39 34 32 12

35. Clock tick: 7
5*5 P1 P2 P3 P4 P5 P6 P7 P8 P9 23 36 28 25 39 34 32 37

36. Same answer! In 2n + 1 time 23 36 28 25 39 34 28 32 37 P1 P2 P3 P4 P5
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

37. Hands-on Simulation Let us do a 3-by-3 Matrix multiplication.
Using Systolic Arrays and Simulink.
The PE in systolic array is a Simulink s-function.

38. Simulink model * Use systolic array to find the multiplication of * =
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

39. EEEB0765 DSP fort Embedded Systems http://embedsigproc. wordpress
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

40. EEEB0765 DSP fort Embedded Systems http://embedsigproc. wordpress
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

41. EEEB0765 DSP fort Embedded Systems http://embedsigproc. wordpress
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

42. Part of “mdlUpdate”
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

43. PE_sfcn_sim.mdl
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

44. A Process Element
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

45. PE is a mask
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

46. PE mask is an S-function
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

47. PE_sfcn.m
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

48. EEEB0765 DSP fort Embedded Systems http://embedsigproc. wordpress
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon

49. EEEB0765 DSP fort Embedded Systems http://embedsigproc. wordpress
EEEB0765 DSP fort Embedded Systems Assoc. Prof. Dr. P.Yuvapoositanon