convolution circuits synthesis perkowski. fir-filter like structure b4b3 b2b1 +++ a4000 a4*b4

Convolution circuits synthesis

Perkowski

FIR-filter like structure

b4 b3 b2 b1

++ +

a4 0 0 0

a4*b4

Think what you can do in all possible ways with two vectors of items (numbers)?

• 1. Dot product• 2. Convolution (polynomial multiplication)• 3. Cartesian Product• 4. Kronecker Product• 5. Other?

Think what you can do in all possible ways with two matrices of items (numbers)?

Convolution• Perhaps the most important operation on data.

• Not related to operators that operate on items.

• It is a pattern of moving data and operating on them

• Although first systolic processors were not for convolution, it is the standard and common object of systolic, cellular and parallel design of algorithms and hardware.

• Every image processing project such as Hadamard, Fourier, Hough or other transform includes convolution – like circuit/system design in one way or another.

• This part of design is truly creative.

I have two vectors A=(a1,a2,a3,a4) and B=(b1,b2,b3,b4)

b4 b3 b2 b1

++ +

a4 0 0

a4*b4

a3

a3*b4+a4b3

b4 b3 b2 b1

++ +

a3 a4 0

a4*b4

a2

a3*b4+a4b3 a4*b2+a3*b3+a2*b4

b4 b3 b2 b1

++ +

a2 a3 a4

a4*b4

a1

a3*b4+a4b3 a4*b2+a3*b3+a2*b4

a1*b4+a2*b3+a3*b2+a4*b1

b4 b3 b2 b1

++ +

a1 a2 a3

a4*b4

0

a3*b4+a4b3 a4*b2+a3*b3+a2*b4

a1*b4+a2*b3+a3*b2+a4*b1 a1*b3+a2*b2+a3*b1

We insert Dffs to avoid many levels of logic

b4 b3 b2 b1

++ +

a4a2 a3

a4*b4a4*b3 a4*b2 a4*b1

b4 b3 b2 b1

++ +

a3a1 a2

a4*b4 a4*b3+a3b4 a4*b2+a3b3a4*b1+a3b2 a3b1

b4 b3 b2 b1

++ +

a20 a1

a4*b4 a4*b3+a3b4 a4*b2+a3b3+a2b4 a4*b1+a3b2+a2b3

a3b1+a2b2 a2b1

The disadvantage of this circuit is broadcasting

We insert more Dffs to avoid broadcasting

b4 b3 b2 b1

++ +

a4a2 a3

a4*b40 0 0

0 0 0

b4 b3 b2 b1

++ +

a3a1 a2

a4*b4 a3b4 a4b30

a4 0 0

0

Does not work correctly like this, try something new….

b4 b3 b2 b1

a3a1 a2

a4*b4

a3b4 a4b3

0

a4 0 0

0

a2b4

a1b4

a3b3

a2b3

a1b3

00

0

0

a4b2

a3b2

a2b2

a1b2

0

0

0

a4b1

a3b1

a2b1

First sum

Second sum

FIR-filter like structure, assume two delays

b4 b3 b2 b1

++ +

b4 b3 b2 b1

++ +

b4 b3 b2 b1

++ +

b4 b3 b2 b1

++ +

b4 b3 b2 b1

++ +

b4 b3 b2 b1

++ +

b4 b3 b2 b1

++ +

b4 b3 b2 b1

++ +

b4 b3 b2 b1

++ +

b4 b3 b2 b1

++ +

b4 b3 b2 b1

++ +

b4 b3 b2 b1

++ +

b4 b3 b2 b1

++ +

b4 b3 b2 b1

++ +