Unsigned Binary Multiplication


Unsigned Binary multiplication can be done using two approaches:

  • Paper Approach
  • Hardware Approach


    • Paper approach

    The paper approach is the easier of the two approaches but this should be learnt first to get a better understanding of the Hardware approach. The paper method consists of a multiplicand, multiplier and partial products. Each bit in the multiplier multiplies the multiplicand and the result of each bit is known as the partial product. If the multiplier bit is 1, the partial product is the same as the multiplicand, but if the multiplier is 0 then the partial product is also 0.

    To calculate each partial bit, start from the right most bit in the multiplier and multiply each bit in the multiplicand from right to left. Shift one space to the left and repeat the above process again until each bit in the multiplier multiplies the multiplicand. The partial products are then added together to find the result. This process may be confusing at first but use the example below to get a better understanding of this approach.

    Example 1

               1010         Multiplicand (10 decimal)
            x 0100        Multiplier ( 4 decimal)
              0000         Partial Products
            0000           shift one place left and multiply
          1010             shift two places left and multiply
        0000               shift three places left and multiply
        0101000         Product (40 decimal)

    Example 2

               1000         Multiplicand (8 decimal)
            x 1001        Multiplier ( 9 decimal)
              1000         Partial Products
            0000           shift one place left and multiply
          0000             shift two places left and multiply
        1000               shift three places left and multiply
        01001000        Product (72 decimal)

    Hardware Approach

    For the hardware approach the accumulator is represented by A and the carry is represented by C. The accumulator stores the sum of the previous accumulator value and the newly calculated partial product. It adds the partial products simultaneously as they are calculated. When the multiplier bit or least significant bit for the multiplier is zero (0), only the shift operation is executed. When the multiplier bit or least significant bit for the multiplier is one (1), the shift and addition operations are executed with the addition done first.

    Shift Operation

    The shift operation works by the following two steps:

  • Take the least significant bit from the left column which happens to be the carry column.
  • Move the bit from the left column to the right column.

    • The shift function remains the same throughout the calculation and is not done for the multiplicand column.

    Addition Operation

    The addition operation is the sum of the newly calculated partial product and the value is stored in the accumulator. The final result is the multiplier and accumulator combined. The length of the final result is the sum of the multiplier and multiplicand combined.

    Algorithm (M x Q):
  • A (n-bit) = 0;
  • C (1-bit) = 0;
  • Counter = n;
    • Repeat these steps until Counter = 0:
  • If Q0 (LSB of Q) = 1, then CA = A + M
  • Shift C, A, and Q one bit to the right, simultaneously.
  • Counter = Counter - 1
  • Stop. The result is in A, Q.
    •

    Example 1

    M = 1011 and Q = 1101

    C A Q
    0 0000 1101 Initial Values
    0
    0    		 1011
    0101    		 1101
    1110    		 Add
    Shift    		 First Cycle
    0 0010 1111 Shift Second Cycle
    0
    0    		 1101
    1111    		 0110
    1111    		 Add
    Shift    		 Third Cycle
    1
    0    		 0001
    1111    		 1000
    1111    		 Add
    Shift    		 Fourth Cycle (Product in A,Q)



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