Question

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1.1 Let's compare the performance of “ripple carry adder" and "carry lookahead adder". Assume that hardware corresponding to any Boolean equation containing only OR or AND terms, such as F = A. B or F = A + B, takes one time unit T. Equations that consist of the OR of several AND terms, e.g. F = A. B + CD, would take two time units, i.e., 27. The reason is that it would take T to produce the AND terms and then an additional T to produce the result of the OR. Calculate the performance ratio between “4-bit ripple carry adder" and "4-bit carry lookahead adder". If the terms in equations are further defined by other equations, then add the appropriate delays for those intermediate equations, and continue recursively until the actual input bits of the adder are used in an equation.

Answer 1

1.1 Ripple carry adder:

In ripple carry adders, for each adder block, the two bits that are to be added are available instantly. However, each adder block waits for the carry to arrive from its previous block. So, it is not possible to generate the sum and carry of any block until the input carry is known. The $i^{th}$ block waits for the $i-1^{th}$ block to produce its carry. So there will be a considerable time delay which is carry propagation delay.

Sum1 = A1⊕B1⊕Cin and Carry1 = A1B1⊕B1Cin⊕CinA1

x⊕y = x.!y + !x.y here we are ignoring time for not gate so XOR operation taking 2T unit of time.

sum would take 4T and carry would take 5T so overall time for 1 bit calculation is 5T.

Total time for 4 bit ripple carry adder = 20 T

Carry Look-ahead Adder :

A carry look-ahead adder reduces the propagation delay by introducing more complex hardware. In this design, the ripple carry design is suitably transformed such that the carry logic over fixed groups of bits of the adder is reduced to two-level logic.

Gi is calculated as Gi = Ai. Bi.

Carry propagated Pi is associated with the propagation of carry from Ci to Ci+1. It is calculated as Pi = Ai ⊕ Bi.

Using the Gi and Pi terms the Sum Si and Carry Ci+1 are given as below –

• Si = Pi ⊕ Gi. -- 2T unit of time
• Ci+1 = Ci.Pi +Gi. --2T unit of time

Therefore, the carry bits C1, C2, C3, and C4 can be calculated as

• C1 = C0.P0+G0. -- 2T unit of time
• C2 = C1.P1+G1 = ( C0.P0+G0).P1+G1. --4T unit of time
• C3 = C2.P2+G2 = (C1.P1+G1).P2+G2. --4T unit of time
• C4 = C3.P3+G3 = C0.P0.P1.P2.P3 + P3.P2.P1.G0 + P3.P2.G1 + G2.P3 + G3. --5T unit of time

Total Time = 7T

performance ratio = 4 bit Ripple carry adder/ 4 bit Carry Look-ahead Adder = 20T/ 7T = 2.857

1.2

Time for 16 bit Ripple carry adder= 16*5T = 80T

Time for four concatenated 4 bit carry lookahead = 4*5T = 20T

performance ratio = 16 bit Ripple carry adder/ four concatenated 4 bit carry lookahead = 80T/ 20T = 4