Show the step-by-step multiplication process using Booth...

Show the step-by-step multiplication process using Booth algorithm (as in Table 10-3) when the following binary numbers are multiplied. Assume 5-bit registers that hold signed numbers. The multiplicand in both cases is +15 .
a. $(+15) \times(+13)$
b. $(+15) \times(-13)$

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Key Concepts

Booth Algorithm

Booth's algorithm is an efficient method for performing multiplication on signed binary numbers represented in two’s complement form. It minimizes the number of addition and subtraction steps by encoding sequences of ones, allowing for a more optimized multiplication process compared to standard shift-add approaches.

Two's Complement Representation

Two’s complement is a system for representing positive and negative integers in binary. It simplifies the design of arithmetic circuits and enables a single algorithm to handle both additions and subtractions, which is crucial for correctly implementing signed multiplication in Booth's algorithm.

Arithmetic Shifting

Arithmetic shifting is a technique used to shift the bits of a number while preserving its sign. In the context of Booth's algorithm, arithmetic right shifts are used to properly align partial products and maintain the integrity of signed numbers during the multiplication process.

Signed Binary Multiplication

Signed binary multiplication deals with multiplying numbers that can be positive or negative. Techniques such as Booth's algorithm are designed to efficiently combine bit-level operations, two's complement arithmetic, and shifting operations to accurately compute the product of signed integers.

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