Question

Calculate the time necessary to perform a multiply using the approach given in Figure 3.3 and 3.4 if an integer is 8 bits wide and each step of the operation takes 4-time units. Assume that in step 1a an addition is always performed -either the multiplicand will be added, or a zero will be. Also assume that the registers have already been initialized (you are just counting how long it takes to do the multiplication loop itself). If this is being done in hardware, the shifts of the multiplicand and multiplier can be done simultaneously. If this is being done in software, they will have to be done one after the other. Solve for each case.

Short answer

The time necessary to perform a multiply using the approach given in Figure 3.3 and 3.4 is as follows:

For case 1 -Hardware: 96 time units

For case 2-Software: 160 time units

Step-by-step solution

Determine the Multiplication Algorithm:

1. First, the multiplier will be checked for 0 and 1.

1a. If the multiplier is 1, then the multiplicand will be added to the product and then the left shift 1 bit in the multiplicand register

2. If the multiplier is 0, then the left shift for 1 bit in the multiplicand register.

3. Then Shift the multiplier register 1 bit to the right.

4. Then the repetition numbers will be checked, if it is less than 32 it again works from first. If it has completed 32 repetitions then the result will be produced.

Determine the time units for multiplication in hardware

Given that the each step of the operation will take 4 time units.

To perform addition it takes 1 cycle, to perform shift operation left or right it takes one cycle. To make the decision regarding the result it takes 1 cycle. So it takes around cycles for a loop operation with time units.

It is given that the integer is 8-bit wide which is “a”.

Therefore, the time units taken for hardware is as follows:

$$\begin{gathered} \left(3\times a\right)\times t=(3\times 8)\times 4 \\ =24\times 4 \\ =96timeunits \end{gathered}$$

Determine the time units for multiplication in Software

For Software:

Given that the each step of the operation will take 4 time units.

To perform a decision on what to be added takes 1 cycle, to add it take 1 cycle. To perform the left shift it takes 1 cycle and 1 cycle to perform the right shift. To make the decision regarding the result it takes 1 cycle. So it takes around cycles for a loop operation with time units.

It is given that the integer is 8-bit wide which is “t”.

Therefore, the time units taken for hardware is as follows:

$$\begin{gathered} \left(5\times a\right)\times t=\left(5\times 8\right)\times 4 \\ =40\times 4 \\ =160timeunits \end{gathered}$$