Announcments

Homework 3 is due on Sept 27th, Monday before class.
Hazards: Ex1
Assume all gates have a propagation delay of 10ns.

Identify the transition when static 1 hazard will happen,

draw the corresponding timing diagram,

and modify the circuit to hazard free circuit.
Hazards: Ex1 Soln
\[ f = A\bar{B} + BC \]
 \(\bar{A}\)  \(A \) 
 \(\bar{B}\)  \(B \)  \(\bar{B}\) 
\(\bar{C}\) 
0  0  0  1 
\(C\) 
0  1  1  1 

Identify adjacent nonoverlapping terms

Identify the variable that is transitioning between them.

Identify the longer path of the circuit with that variable.

For a SOP circuit, the transition is that makes to 0>1 for longer circuit and 1>0 for shorter circuit.
 \(\bar{A}\)  \(A \) 
 \(\bar{B}\)  \(B \)  \(\bar{B}\) 
\(\bar{C}\) 
0  0  0  1 
\(C\) 
0  1  1+ 1  1+1 
Hazards: Ex2
Propagation delay of NOT gate=3 ns, AND/OR gate=5 ns
\[ F = (A + C)(\bA+\bD)(\bB+\bC+D)\]
Multilevel synthesis

SOP and POS are two level circuits

Fanin is the number of inputs to a gate

Twolevel circuits have higher fanin, but smaller propagation delays.

Fanin is typically limited by the technology used.
Approaches Multilevel synthesis

Factorization

Functional decomposition
Factorization Ex1
\[ f = ABC + ABD + \bar{A}\bar{B}C \]
Factorization Ex2
\[f_1 = ABD + CD \]
\[f_2 = AB\bar{D} + C \bar{D} \]
Functional decomposition Ex1
\[ f = \bx_1 x_2 x_3 + x_1 \bx_2 x_3 + \bx_1 \bx_2 x_4 + x_1 x_2 x_4 \]
Functional decomposition Ex1
\[ f = \bx_1 x_2 x_3 + x_1 \bx_2 x_3 + \bx_1 \bx_2 x_4 + x_1 x_2 x_4 \]
  \(\bar{x}_1\)  \(x_1 \) 
  \(\bar{x}_2\)  \(x_2 \)  \(\bar{x}_2\) 
\(\bar{x}_3\)  \(\bar{x}_4\) 
0  0  0  0 
\(x_4\) 
1  0  1  0 
\(x_3\) 
1  1  1  1 
\(\bar{x}_4\) 
0  1  0  1 
Functional decomposition Ex2
\[ f = \sum m(0, 6, 8, 4) \]
  \(\bar{x}_1\)  \(x_1 \) 
  \(\bar{x}_2\)  \(x_2 \)  \(\bar{x}_2\) 
\(\bar{x}_3\)  \(\bar{x}_4\) 
1  0  0  1 
\(x_4\) 
0  0  0  0 
\(x_3\) 
0  0  0  0 
\(\bar{x}_4\) 
0  1  1  0 