Question

B.26 [5] <§B.6> Rewrite the equations on page B-44 for a carry-lookahead logic for a 16-bit adder using a new notation. First, use the names for the CarryIn signals of the individual bits of the adder. That is, use c4, c8, c12, … instead of C1, C2, C7, …. In addition, let Pi,j; mean a propagate signal for bits i to j, and Gi,j; mean a generate signal for bits i to j. For example, the equation

C2 = G1+( P1.G0)+( P1.P0. c0) can be rewritten as

c₈= G _7,4 + (P_7,4 .G_7,0) +( P_7,4 .P_3,0.c0)

This more general notation is useful in creating wider adders.

Short answer

After rewriting the equationfor a carry-lookahead logic for a 16-bit adder using the mentioned notation:

• C4=G3+(P3.G2)+( P3.P2.G1)+( P3.P2.P1.G0)+ ( P3.P2.P1.P0.c0) can be rewritten as c16= G _7,12+( P_15,12. G _7,8)+( P_15,12. P_11,8. G _7,4)+( P_15,12. P_11,8. P_7,4. G_7,0)+ ( P_15,12. P_11,8. P_7,4. P_3,0.c0)
• C3=G2+(P2.G1)+(P2.P1.G0)+(P2.P1.P0.c0)can be rewritten as

c12= G _7,8+( P_11,8. G _7,4)+( P_11,8. P_7,4. G_7,0)+( P_11,8. P_7,4. P_3,0.C0)

• C2 = G1+( P1.G0)+( P1.P0. c0) can be rewritten as

c8= G _7,4 + (P_7,4.G_7,0) +( P_7,4.P_3,0.c0)

• C1=G0+(P0.c0) can be rewritten as

c4= G_7,0+( P_3,0. c0)

Step-by-step solution

Define the concept.

The specified figure of 16 bit carry look-ahead adder:

For G0,

=G_7,0

=g7+(p7.g2)+(p7.p2.g1)+( p7.p2.p1.g0)

For G1,

G1

= G _7,4

=g7+(p7.g6)+(p7.p6.g5)+( p7.p6.p5.g4)

For G2,

G2

= G _7,8

=g11+(p11.g10)+(p11.p10.g9)+( p11.p10.p9.g8)

For G3,

G3

= G _7,12

=g15+(p15.g14)+(p15.p14.g13)+( p15.p14.p13.g12)

Determine the calculation.

The carry out c16 is simplified as,

C4=G3+(P3.G2)+( P3.P2.G1)+( P3.P2.P1.G0)+ ( P3.P2.P1.P0.c0)

The carry out c12 is simplified as,

C3=G2+(P2.G1)+(P2.P1.G0)+(P2.P1.P0.c0)

The carry out c8 is simplified as,

C2 = G1+( P1.G0)+( P1.P0. c0)

The carry out c4 is simplified as,

C1=G0+(P0.c0)

According to the mentioned-condition

G3 can be written as G _7,12

G2 can be written as G _7,8

G1 can be written as G _7,4

G0 can be written as G_7,0

P3 can be written as P_15,12

P2 can be written as P_11,8

P1 can be written as P_7,4

P0 can be written as P_3,0

Hence for the 16 bit carry look-ahead adder,

• C4=G3+(P3.G2)+( P3.P2.G1)+( P3.P2.P1.G0)+ ( P3.P2.P1.P0.c0) can be rewritten as c16= G _7,12+( P_15,12. G _7,8)+( P_15,12. P_11,8. G _7,4)+( P_15,12. P_11,8. P_7,4. G_7,0)+ ( P_15,12. P_11,8. P_7,4. P_3,0.c0)
• C3=G2+(P2.G1)+(P2.P1.G0)+(P2.P1.P0.c0)can be rewritten as

c12= G _7,8+( P_11,8. G _7,4)+( P_11,8. P_7,4. G_7,0)+( P_11,8. P_7,4. P_3,0.C0)

• C2 = G1+( P1.G0)+( P1.P0. c0) can be rewritten as

c8= G _7,4 + (P_7,4.G_7,0) +( P_7,4.P_3,0.c0)

• C1=G0+(P0.c0) can be rewritten as

c4= G_7,0+( P_3,0. c0)