Booth's Algorithm

AI Summary

An AI-generated summary of this video lecture.

This lecture introduces Booth's algorithm, a method for optimizing binary multiplication by reducing the number of additions and subtractions required. Named after Andrew Donald Booth (1918–2009), the algorithm examines pairs of multiplier bits to determine whether to add, subtract, or shift. The instructor demonstrates the process using a concrete example of multiplying 7 by 3, utilizing a step-by-step table and flowchart to illustrate the four-cycle execution. Key registers involved include A (accumulator), Q (multiplier), Q-1 (auxiliary bit), and M (multiplicand). The algorithm relies on arithmetic shift right operations and decision logic based on the values of Q0 and Q-1 to control addition or subtraction of M from A.

Chapters

1. 0:00 – 2:00 00:00-02:00

   The video begins with an introduction to Booth's algorithm and its creator, Andrew Donald Booth. The slide displays biographical dates (1918–2009) and defines the algorithm's purpose: optimizing binary multiplication by reducing additions and subtractions. The instructor explains that the process involves examining multiplier bits to decide whether to add, subtract, or leave the multiplicand unchanged before shifting. A flowchart is introduced to visualize the logic, and a specific example of multiplying 7 by 3 is presented using a step-by-step table to illustrate the algorithm's execution over four cycles.

2. 2:00 – 5:00 02:00-05:00

   The instructor demonstrates the Booth algorithm execution for 7 x 3 using a detailed table and flowchart. Initial values are set in registers A, Q, Q-1, and M (0000 0011 0 0111). The first cycle involves the operation A <- A - M followed by an arithmetic shift right. The second cycle shows a shift-only operation because Q0 and Q-1 are both 1. The third cycle performs A <- A + M, followed by another shift. The instructor points to the flowchart decision diamond checking Q0 and Q-1 values to determine operations, tracing the algorithm steps through cycles while highlighting binary subtraction and addition logic.

3. 5:00 – 8:40 05:00-08:40

   The demonstration continues with the final cycles of Booth's algorithm for 7 x 3. The screen displays the execution of four cycles involving addition, subtraction, and arithmetic shift operations on registers A, Q, and M. The instructor highlights specific bit changes during addition/subtraction and shifting operations across the cycles. The final result of the multiplication is highlighted at the bottom of the table, confirming the correctness of the algorithm. The visual evidence shows the flow of data between registers A and Q, with the final answer circled in red to emphasize the outcome.

Booth's algorithm is a significant optimization in arithmetic logic units for binary multiplication. By analyzing adjacent bits of the multiplier (Q0 and Q-1), it reduces the number of arithmetic operations compared to standard multiplication methods. The algorithm uses four main registers: A (accumulator), Q (multiplier), Q-1 (auxiliary bit initialized to 0), and M (multiplicand). The core logic involves checking the pair Q0, Q-1: if 01, subtract M from A; if 10, add M to A; otherwise, do nothing. After each operation (or lack thereof), an arithmetic shift right is performed on A and Q combined, preserving the sign bit. The example of 7 x 3 demonstrates this process clearly over four cycles, showing how the algorithm handles both positive and negative operations to arrive at the correct product efficiently. This method is particularly useful in hardware implementations where minimizing addition/subtraction cycles improves performance.