How to draw systolic array. How matrix Multiplication is done in Systolic array. Time for matrix multiplication.

1. Systolic Array
Matrix Multiplication

2. 3 x 3 Matrix
• A =
𝑎00 𝑎01 𝑎02
𝑎10 𝑎11 𝑎12
𝑎20 𝑎21 𝑎22
• B =
𝑏00 𝑏01 𝑏02
𝑏10 𝑏11 𝑏12
𝑏20 𝑏21 𝑏22
𝑐 =
𝑎00𝑏00 + 𝑎01𝑏10 + 𝑎02𝑏20 𝑎00𝑏01 + 𝑎01𝑏11 + 𝑎02𝑏21 𝑎00𝑏02 + 𝑎01𝑏12 + 𝑎02𝑏22
𝑎10𝑏00 + 𝑎11𝑏10 + 𝑎12𝑏20 𝑎10𝑏01 + 𝑎11𝑏11 + 𝑎12𝑏21 𝑎10𝑏02 + 𝑎11𝑏12 + 𝑎12𝑏22
𝑎20𝑏00 + 𝑎21𝑏10 + 𝑎22𝑏20 𝑎20𝑏01 + 𝑎21𝑏11 + 𝑎22𝑏21 𝑎20𝑏02 + 𝑎21𝑏12 + 𝑎22𝑏22
𝑡𝑖𝑚𝑒 = 3 ∗ 𝑛 − 2
For n x n mesh

3. Clock cycle 00

4. Clock cycle 01
C=a00b00
a00
b00

5. Clock cycle 02
C=a00b00+a01
b10
C=a00b01
C=a10b00
a01
b10
a10
b00
a00
b01

6. Clock cycle 03
C=a00b00+a01
b10+a02b20
C=a00b01+a01
b11
C=a00b02
C=a10b00+a11
b10
C=a10b01
C=a20b00
a02
b20
a11
b10
a01
b11
a00
b02
a10
b01
a20
b00

7. Clock cycle 04
C=a00b00+a01
b10+a02b20
C=a00b01+a01
b11+a02b21
C=a00b02+a01
b12
C=a10b00+a11
b10+a12b20
C=a10b01+a11
b11
C=a10b02
C=a20b00+a21
b10
C=a20b01
a12
b21
a02 a01 a00
b12
a11
b11
a21
b20
b10
b00
a10
b02
a20
b01

8. Clock cycle 05
C=a00b00+a01
b10+a02b20
C=a00b01+a01
b11+a02b21
C=a00b02+a01
b12+a02b22
C=a10b00+a11
b10+a12b20
C=a10b01+a11
b11+a12b21
C=a10b02+a11
b12
C=a20b00+a21
b10+a22b20
C=a20b01+a21
b11
C=a20b02
a12
b21
a02 a01 a00
b22
a11
b11
a22
b20
b10
b00
a10
b02
a20
b01
b12
a21

9. Clock cycle 06
C=a00b00+a01
b10+a02b20
C=a00b01+a01
b11+a02b21
C=a00b02+a01
b12+a02b22
C=a10b00+a11
b10+a12b20
C=a10b01+a11
b11+a12b21
C=a10b02+a11
b12+a12b22
C=a20b00+a21
b10+a22b20
C=a20b01+a21
b11+a22b21
C=a20b02+a21
b12
a02 a01 a00
b22
a22
b20
b10
b00
a12 a11 a10
b12
a21
b21
b11
b01
a20
b02

10. Clock cycle 07
C=a00b00+a01
b10+a02b20
C=a00b01+a01
b11+a02b21
C=a00b02+a01
b12+a02b22
C=a10b00+a11
b10+a12b20
C=a10b01+a11
b11+a12b21
C=a10b02+a11
b12+a12b22
C=a20b00+a21
b10+a22b20
C=a20b01+a21
b11+a22b21
C=a20b02+a21
b12+a22b22
a02 a01 a00
b22
a22
b20
b10
b00
a12 a11 a10
b12
a21
b21
b11
b01
a20
b02