ECE 341. Lecture # 6

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1 ECE 34 Lecture # 6 Instructor: Zeshan Chishti zeshan@pdx.edu October 5, 24 Portland State University

2 Lecture Topics Design of Fast Adders Carry Looakahead Adders (CLA) Blocked Carry-Lookahead Adders Multiplication of Unsigned Numbers Array Multiplier Sequential Circuit Multiplier Reference: Chapter 9: Sections 9.2 and 9.3

3 CLA Delay Calculation Consider the expression: c i+ = G i + P i G i- + P i P i- G i P i P i-.p G + P i P i-.p c All the G i and P i functions can be obtained in parallel in one gate delay AND terms in each c i+ calculation require one additional gate delay ORing the AND terms in each c i+ calculation requires one additional gate delay Therefore, Total delay in calculating carry outputs = + + = 3 gate delays Sum outputs require one additional XOR delay after carries are computed Total delay in calculating sum outputs = 3 + = 4 gate delays n-bit CLA requires 4 gate delays independent of n

4 CLA Fan-in Limitation Performing n-bit CLA in 4 gate delays, independent of n, good only in theory In practice, CLA is limited by fan-in constraints c i+ = G i + P i G i- + P i P i- G i P i P i-.p G + P i P i-.p c OR gate & last AND gate in the expression for c i+ require i+2 inputs, each For a 4-bit CLA, the MSB carry-out (c 4 ) requires a fan-in of 5 5 is the practical fan-in limit for most gates In order to add operands larger than 4-bits, we can cascade multiple CLAs Cascade of CLAs is called Blocked Carry-Lookahead adder

5 Blocked Carry-Looakhead Adder x 3-28 c c. c bit 4-bit CLA y 3-28 x 7-4 CLA y 7-4 c 4 x 3- y 3-4-bit CLA c 32-bit Blocked CLA composed of eight 4-bit CLA blocks s 28 s 29 s 3 s 3 s 7 s 6 s 5 s 4 s 3 s 2 s s After input operands (X, Y and c ) are applied to the 32-bit adder: All the P i and G i terms in each CLA calculated in parallel in gate delay c 4 available after 3 gate delays c 8 available 2 gate delays after c 4 = 3 + (*2) = 5 gate delays c 2 available (2*2) gate delays after c 4 = 3 + (2*2) = 7 gate delays c 6 available after (3*2) gate delays after c 4 = 3 + (3*2) = 9 gate delays c 32 available after (7*2) gate delays after c 4 = 3 + (7*2) = 7 gate delays s 28, s 29, s 3, s 3 available after 7+ = 8 gate delays Carry-outs ripple from one CLA block to the next. Can we avoid this rippling?

6 Faster Blocked Carry-Lookahead adder Key Idea: Generate the carry outputs c 4, c 8, c 2, of CLA blocks in parallel, similar to how c, c 2, c 3, c 4 are generated in parallel within a CLA block Carry-out from a 4-bit block can be given as: c 4 = G 3 + P 3 G 2 + P 3 P 2 G + P 3 P 2 P G + P 3 P 2 P P c This can be re-written as: c 4 = G + P c where G = G 3 + P 3 G 2 + P 3 P 2 G + P 3 P 2 P G and P = P 3 P 2 P P We can similarly compute G, G 2, G 3.. G i & P i are first-level generate & propagate functions, where i denotes the CLA block G k = implies that the kth CLA block generates a carry P k = implies that the kth CLA block propagates a carry Carry-out of kth block = G k + P k G k- + P k P k- G k P k P k-.p G + P k P k- P c For example: c 6 = G 3 + P 3 G 2 + P 3 P 2 G + P 3 P 2 P G + P 3 P 2 P P c

7 Blocked CLA with First-level Propagates and Generates x 5-2 y 5-2 x -8 y -8 x 7-4 y 7-4 x 3- y 3- c 6 c 2 c 8 4-bit adder 4-bit adder 4-bit adder 4-bit adder c 4 c s 5-2 s -8 s 7-4 s 3- G 3 I P 3 I G 2 I P 2 I G I P I G I P I Carry-lookahead logic After input operands (X, Y and c ) are applied to the above 6-bit adder: P i and G i terms within each CLA calculated in parallel in gate delay First-level generates (G k ) available after + 2 = 3 gate delays Carry-outs of CLA blocks (c 4, c 8, c 2, c 6 ) available after = 5 gate delays Carries within CLA blocks (such as c 5 ) available after = 7 gate delays Sum outputs (such as s 5 ) available after 7 + = 8 gate delays Compare this with the blocked CLA formed by cascading, where c 5 and s 5 required 9 and gate delays respectively

8 Multiplication

9 Multiplication of Unsigned Numbers Product of two n-bit numbers is at most a 2n-bit number Unsigned multiplication can be viewed as addition of shifted versions of the multiplicand.

10 Multiplication of Unsigned Numbers (contd.) In hand multiplication, we add the shifted versions of multiplicand at the end (column-by-column) Alternative would be to accumulate partial products at each stage (row-by-row) Multiplication logic for two n-bit numbers can be implement as follows : Initialize the partial product PP to a value of Start from the LSB of multiplier and proceed towards MSB, one bit at a time. For each bit position of the multiplier, perform the following step: If the i th bit of the multiplier is, shift the multiplicand by i bit positions and add it to PPi in order to obtain PP(i+) After n steps, the partial product PPn represents the final product

11 Multiplication of Unsigned Numbers Bit of incoming partial product (PPi) j th multiplicand bit i th multiplier bit carry out FA carry in Bit of outgoing partial product (PP(i+)) Typical multiplication cell

12 Combanitorial Array Multiplier Multiplicand Each box represents a multiplication cell PP (PP) m m m 3 m 2 p q q PP2 p PP3 p 2 q 2 q 3 Product = P, 7,P 6,..P p 7 p 6 p 5 p 4 p 3 Multiplicand is shifted by displacing it through an array of adders

13 Combinatorial Array Multiplier (cont.) Array multipliers are highly inefficient: Need n n-bit adders => number of gate counts is proportional to n 2 Impractical for large numbers such as 32-bit or 64-bit numbers typically used in computers Perform only one function, namely, unsigned integer product Solution: Improve gate efficiency by using a mixture of combinatorial array techniques and sequential techniques Instead of n n-bit adders, use one n-bit adder Use a register to hold the accumulated partial product This is called a sequential multiplier

14 Sequential Multiplication Recall the rule for generating partial products: If the i th bit of the multiplier is, add the appropriately shifted multiplicand to the current partial product. Multiplicand is shifted left when being added to the partial product Key Observation: Adding a left-shifted multiplicand to an unshifted partial product is equivalent to adding an unshifted multiplicand to a right-shifted partial product

15 Sequential Circuit Multiplier Register A (initially ) Shift right C a n a q q - n - Multiplier Q Add/Noadd control n-bit Adder MUX Control sequencer m n - m Multiplicand M

16 Sequential Multiplication Algorithm Initialization: Load multiplicand in M register, multiplier in Q register Initialize C and A registers to all zeroes Repeat the following steps n times, where n is the number of bits in the multiplier If (LSB of Q register == ) A = A + M (carry-out goes to C register) Treat the C, A and Q registers as one contiguous register and shift that register s contents right by one bit position After the completion of n steps Register A contains high-order half of product Register Q contains low-order half of product

17 Sequential Multiplication Example M C A Q Initial configuration

18 Sequential Multiplication Example M Initial configuration C A Q A += M Shift First cycle

19 Sequential Multiplication Example M Initial configuration C A Q A += M Shift First cycle A += M Shift Second cycle

20 Sequential Multiplication Example M Initial configuration C A Q A += M Shift First cycle A += M Shift Second cycle No add Shift Third cycle

21 Sequential Multiplication Example M Initial configuration C A Q A += M Shift First cycle A += M Shift Second cycle No add Shift Third cycle A += M Shift Fourth cycle Product

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