2 bit Binary multiplier

Binary multiplication process: A Binary Multiplier is a digital circuit used in digital electronics to multiply two binary numbers and provide the result as output. The method used to multiply two binary numbers is similar to the method taught to school children for multiplying decimal numbers which is based on calculating partial product, shifting them and adding them together. Similar approach is used to multiply two binary numbers. Long multiplicand is multiplied by 0 or 1 which is much easier than decimal multiplication as product by 0 or 1 is 0 or same number respectively. Figure 1 below shows the block diagram of a 2-bit binary multiplier. The two numbers A1A0 and B1B0 are multiplied together to produce a 4-bit output P3P2P1P0. (The maximum product term can be 3 * 3 = 9, which is 1001, a 4-bit number). 

Figure 1: 2-bit Binary Multiplier Block Diagram

Let us take an example of multiplying two binary numbers as follows. The process is similar to multiplying two decimal numbers, with a difference that the resulting numbers are all binary.

       110 = 6

X     011 = 3
-----------------------------

                                                             1 1 0                 ; 110 X 1

                                                          1 1 0 x                 ; 110 X 1

                                                       0 0 0 x x                 ; 110 X 0

 ------------------------------

 1 0 0 1 0 =18

Now, we have seen that multiplying a number with binary ‘0’produces all zeroes, and with ‘1’ reproduces the number. So, multiplying two binary numbers is a straightforward job. It can be implemented without much difficulty using shifters, AND gates and adders.

2-bit binary multiplier circuit implementation: Let us implement a two bit binary multiplier. Let the two binary numbers be A₁A₀ and B₁B₀. The multiplication table will, then, look as:

                                                          A₁         A₀

                                           X           B₁          B₀

_{-------------------------------------------------------------------}

                                                      B₀A₁       B₀A₀

                                     B₁A₁         B₁A₀             x

_{-------------------------------------------------------------------}

                             P3       P2         P1            P0                                  

Thus, we get the partial products as:

P0 = A0*B0

P1 = A0*B1 xor  A1 * B0                  ; carry generated here goes to next stage

P2 = A1*B1   xor  (A0*B1) * (A1*B0)

P3 = A1*B1   and  (A0*B1) * (A1*B0)

 

Two-bit binary multiplier circuit diagram

Thus, we can see that a 2-bit binary multiplier can be implemented using two half-adders only.

Characteristics of a binary multiplication: As mentioned above, a binary multiplier is used to multiply binary numbers. In general, the characteristics of binary multiplication are as follows:

• To multiply two binary numbers, AND gates, shifters and adders are required.
• Product of N*M bit binary numbers in of (N+M) bits.
•  N*M AND gates are required to generate partial products of two M*N bit binary numbers.
• Number of adders required =  N+M-2
• Speed limiting factor here is to sum up  partial products.

Also read:

• Build an XOR/XNOR gate using 2:1 multiplexer
• Carry look ahead adder
• Digital counters
• Implement 3 and 4 variable functions using 8:1 multiplexers