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