Guy Even †. February 1, Abstract. We present a self-contained and detailed description of the parallel-prefix adder of Ladner and Fischer. Very little. Abstract. Ladner –Fischer adder is one of the parallel prefix adders. Parallel prefix adders are used to speed up the process of arithmetic operation. Download scientific diagram | Modified Ladner Fischer Adder from publication: Implementation of Efficient Parallel Prefix Adders for Residue Number System | In .

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Partial products are generated with Radix-4 modified Booth recoding. Fisdher present, the combination of CSD Canonic Signed-Digit coefficient encoding technique with the SW-based PPAs seems to provide the practical hardware implementation of fast constant-coefficient multipliers.

Figure 13 shows a bit carry-skip adder consisting of seven variable-size blocks. These hardware algorithms are also used to generate multipliers, constant-coefficient multipliers and multiply accumulators. The PPA stage then performs multi-operand addition for all the generated partial products and produces their sum in carry-save form.

To reduce the hardware complexity, we allow the lsdner of 2,2 counters in addition to 3,2 counters. Figure 15 shows an array for operand, producing 2 outputs, where CSA indicates a carry-save adder having three multi-bit inputs and two multi-bit outputs.

These expressions allow us to calculate all the carries in parallel from the operands. Given the matrix of partial product bits, the number of bits in each column is reduced to minimize the number of 3,2 and 2,2 counters. Redundant binary RB addition tree has a more regular structure than an ordinary CSA tree made of 3,2 counters because the RB partial products are added up in the binary tree form by RB addfr.

The structure a illustrates a typical situation, where the MAC is used to perform a multiply-add operation in an iterative fashion. A multiply accumulator is generated by a combination of hardware algorithms for multipliers and constant-coefficient multipliers. Once the incoming carry is known, we need only to select the correct set of outputs out of the two sets without waiting for padner carry to further propagate through the k positions.

Figure 5 is the parallel prefix graph of a Ladner-Fischer adder. The adder structure is divided into blocks of consecutive stages with a simple ripple-carry scheme.

The above idea is applied to each of groups separately. One set assumes that the incoming carry into the group is 0, the other assumes that it is 1. The main idea behind carry look-ahead addition is an attempt to generate all incoming carries in parallel and avoid waiting fischr the correct carry propagates from the stage FA of the adder where it has been generated.

One set assumes that the eventual incoming carry will be zero, while the other assumes that it will be one.

The Wallace tree guarantees the lowest overall delay but gischer the largest number of wiring tracks vertical feedthroughs between adjacent bit-slices.

A ripple-block carry look-ahead adder RCLA consists of N m-bit blocks arranged in such a way that carries within blocks are generated by carry look-ahead but carries between blocks are rippled. The basic idea in the conditional sum adder is to generate two sets of outputs for a given group of operand bits, say, k bits.

Dadda tree is based on 3,2 counters. The complexity of multiplier structures significantly varies with the coefficient value R. The underlying strategy of the carry-select adder is similar to that fkscher the conditional-sum adder. The n-operand array consists of n-2 carry-save adder. The fundamental carry operator is represented as Figure 4.

## Hardware algorithms for arithmetic modules

When the incoming carry into the group is assigned, its final value is avder out of the two sets. AMG provides multiply accumulators in the form: The PPG stage first generates partial products from the multiplicand and multiplier in parallel. Figure 17 shows an operand balanced delay tree, where CSA indicates a carry-save adder having three multi-bit inputs and two multi-bit outputs.

Figure 19 shows an adder 4;2 compressor tree, where 4;2 indicates a carry-save adder having four multi-bit inputs and two multi-bit outputs. Figure 18 shows an operand overturned-stairs tree, where CSA indicates a carry-save adder having three multi-bit inputs and two multi-bit outputs. Figure 8 is the parallel prefix graph of a Han-Carlson adder.

This reduces the ripple-carry delay through these blocks. A ladneg prefix adder can be represented as a parallel prefix graph consisting of carry operator nodes.

### Hardware algorithms for arithmetic modules

The RB addition tree is closely related to 4;2 compressor tree. AMG provides constant-coefficient multipliers in the form: Array is a straightforward way to accumulate partial products using a number of adders. This signal can be used to allow an incoming carry to skip all the stages within the block and generate a block-carry-out.

Overturned-stairs tree requires smaller number of wiring tracks compared with the Wallace tree and has lower overall delay compared with the balanced delay tree. This process can, in principle, be continued until a group of size 1 is reached.

The carry-save form is converted to the corresponding binary output by an FSA. Figure 16 shows an operand Wallace tree, where CSA indicates a carry-save adder having three multi-bit inputs and two multi-bit outputs.

Figure 14 compares the delay information of true paths and that of false paths in the case of Hitachi 0. In this generator, the group lengths follow the simple arithmetic progression 1, 1, 2, 3, Therefore, let Gi and Pi denote the generation and propagation at the ith stage, we have: Each set includes k sum bits and an outgoing carry.

This adder is the extreme case of maximum logic depth and minimum area. A constant-coefficient multiplier is given as a part of MACs as follow.