get boolean expression for decoder if encoder logic is known

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fussylogic

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Hi Friends,

I want to generate decoder Boolean expression if I have a truth table for encoder. What is the procedure? The encoder encodes, input which is 4 bit to 5 bit. How can I used these 5 bits and get the 4 bits back.
Following is the truth table:


A B C D d4 d3 d2 d1 d0
0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 1 0 0
0 0 1 0 0 0 0 0 1
0 0 1 1 0 0 1 0 1

Thanks
Fussylogic
 

I presume you should generate a new truth table, but now considering the input entry (A,B,C,D) as output (d1,d2,d3,d4), and vice versa:

Code: 
d4	d3	d2	d1	d0     A	B	C	D		
0	0	0	0	0      0	0	0	0		
0	0	1	0	0      0	0	0	1		
0	0	0	0	1      0	0	1	0		
0	0	1	0	1      0	0	1	1


+++
 
Let's say your inputs are written as a (4x4) matrix, X:

Code: 
X =

     0     0     0     0
     0     0     0     1
     0     0     1     0
     0     0     1     1
and your outputs are a (4x5) matrix Y:
Code: 
Y =

     0     0     0     0     0
     0     0     1     0     0
     0     0     0     0     1
     0     0     1     0     1
We want to find some (4x5) "encoding" matrix, Z, such that:
Code: 
X*Z = Y
One solution (with minimum Frobenius norm) is to take the Moore-Penrose pseudoinverse of X, so that Z=X+Y (where + denotes pseudoinverse). In this case, we get:
Code: 
Z =

     0     0     0     0     0
     0     0     0     0     0
     0     0     0     0     1
     0     0     1     0     0
So, to encode X, we compute Y = X*Z. To decode Y, we compute X = Y*ZT (where T denotes matrix transpose).
 
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