Implement 3 and 4 variable function using 8:1 MUX

Three variable  function can be easily implemented using 8:1 multiplexer. connect 3 input lines to select lines of mux and connect 8 inputs of mux to logic 0 or 1 according to function output. For example, let us say Function is

                                      F(X,Y,Z) = Σ(0,1,3,6)

then X,Y,Z will be connected to select lines of Mux and I0 , I1, I3 and I6 will be connected to logic 1(VDD) and other will be connected to logic 0

For a 4 variable function, there are 16 possible combinations. To implement 4 variable function using 8:1 MUX, use 3 input as select lines of MUX and remaining 4th input and function will determine ith input of mux . Let us demonstrate it with an example :

                                  F(A,B,C,D) = Σ(1,5,7,9,10,11,12)

A	B	C	D	Decimal Equivalent	F
0	0	0	0	0	0
1	0	0	0	8	0
0	0	0	1	1	1
1	0	0	1	9	1
0	0	1	0	2	0
1	0	1	0	10	1
0	0	1	1	3	0
1	0	1	1	11	1
0	1	0	0	4	0
1	1	0	0	12	1
0	1	0	1	5	1
1	1	0	1	13	0
0	1	1	0	6	0
1	1	1	0	14	0
0	1	1	1	7	1
1	1	1	1	15	0

The 4 variable function represented using 8:1 mux

                                  ABar = ~A (inverted A)

As shown in figure, B,C,D are used as select lines and A will be used input  of Mux. from Truth table, if B,C,D  are 0 then output F is 0 irrespective of status of A so I0 = 0. For I5(BCD = 101) output depend upon A.

                                     for A = 0,  F  = 1
                                     for A = 1,  F =  0
                          Hence F = ~A (for BCD = 101)
                                    I4, (B = 1, C = 0, D = 0),  F = A
                                    I1, (B = 0, C =  0,D  = 1), F = 1 (irrespective of status of A)

similarly All other inputs can be inferred in the same way.  Thus we can conclude that to implement n variable function, we need 2^(n-1) to 1 MUX and an inverter. n-1 input lines shall be used as select lines and rest one will be used for input of MUX.

Also read:

• Binary to thermometer encoder
• Clock gating checks at a multiplexer
• Lockup latch - principle and application
• Carry look ahead adder
• Interesting problem - BCD multiply by 5 circuit

Race condition in digital circuits

In general, race condition refers to a situation in which the result depends upon the sequence in which events happen. In particular, a race condition occurs when a system/device is designed assuming a particular sequence of events without taking steps to ensure it. In logic gates, it happens when the inputs arrive at the gate in a sequence not assumed while deriving the function. This results, sometimes, in the anomalous behavior of the logic gate because of the unexpected dependence on the sequence of arrival of inputs. Since, the function of the logic gate is calculated assuming static inputs; the order of arrival may have impact on the output. We can consider race condition as a situation in which two or more signals are racing to have their effect on the output. This is how; race condition derives its name.

For a very simple example, consider the following logic function-

Z = A . A’

Figure 1: An example digital circuit having race condition

Theoretically, the output will always be ‘0’. However, this is true only if the non-inverted portion of the signal has greater delay in reaching the AND gate (we have considered ‘0’ -> ‘1’ transition of the input signal ‘A’). As shown in the figure above, the signal passing through inverter will have some delay of its own. If the signal through inverter reaches the AND gate later than the one without inverter, there will be a glitch as shown in the figure 2. For ‘1’ -> ‘0’ transition, the opposite will happen.

Figure 2: Glitch caused at the output of AND gate due to inverted signal being delayed than non-inverted signal

What can be the consequences: If the condition for race is violated, the design may enter an undefined state, the one which might not have been considered while designing. Hence, the whole system will malfunction in such a scenario leading to failure. It might be the case that some elements in the design enter metastable state, which can further cause problems. Hence, it is very important to give proper consideration to race conditions.

How can we prevent race condition: Normally, in digital designs, as long as setup and hold checks are met for each flip-flop, we do not need to worry about the race conditions in the design under consideration. But, there are some conditions like the output of the above AND gate might be used as a clock for some flop. In that case, we need to consider the race condition. We need to have proper sequencing of signals so as to have the desired output. Had the above case been in reality, it might be wise to delay the non-inverted signal. Race conditions in the design can be eliminated by using proper design techniques (e.g. K-map). Using these, designers and recognize and eliminate the race conditions before they cause problems.

Read also:

• Data checks: all about data setup and data hold check
• Synchronization schemes in VLSI design
• Ensuring glitch-free propagation of clock
• Design Quiz: multiply by 2 clock circuit
• Delay line based time to digital converter