about digital system

Question

Draw the circuit by following the steps for the given function.

f(a,b,c,d)= Σ1(1,4,5,7,9,12,13)

a) Fill the truth table.

b) Draw the circuit using 4:1 multiplexer and other required logic gates. Explain the steps.

Answer 1

We're given f(a,b,c,d) = ∑m(1,4,5,7,9,12,13)

What this means is that the function, f, outputs a 1 whenever its inputs, a, b, c, and d, create the one of the numbers that its summing up.

So if we make a truth table using this, it would look like this:

# | a b c d || f

0 | 0 0 0 0 || 0

1 | 0 0 0 1 || 1

2 | 0 0 1 0 || 0

3 | 0 0 1 1 || 0

4 | 0 1 0 0 || 1

5 | 0 1 0 1 || 1

6 | 0 1 1 0 || 0

7 | 0 1 1 1 || 1

8 | 1 0 0 0 || 0

9 | 1 0 0 1 || 1

10 | 1 0 1 0 || 0

11 | 1 0 1 1 || 0

12 | 1 1 0 0 || 1

13 | 1 1 0 1 || 1

14 | 1 1 1 0 || 0

15 | 1 1 1 1 || 0

Now, a 4:1 multiplexer has 2 inputs as a select, and I've divided the truth table into 4 different sections based on the first two inputs, a and b.

We're going to use a and b as our selects, and the inputs for the 4:1 mux will be some functions of the other two inputs, c and d.

The 4:1 mux has 4 different inputs that it chooses from based on the selects. This means that when a and b are 00, the input comes from the 00 pin on the mux, and would have this truth table:

c d || f

0 0 || 0

0 1 || 1

1 0 || 0

1 1 || 0 (similarly for a and b being 10)

As you can see, this is the same as the first section of our whole truth table, and the output is only 1 when c and d are 01. This is like taking the AND of c' and d. So our input to the 00 pin of the multiplexer would be c'd.

In the second section of the truth table, we see that the output is only zero when c and d are 10. This is like taking the OR of c' and d. Try writing that truth table off to the side to be sure ;). This means that the input to pin 01 (since a and b are 01) of the mux would be c' + d.

In our third section, we see the same pattern as in our first section. The output is 1 only when c and d are 01. This means that pin 10 is just like pin 00 and has the input c'd.

In our fourth section of the truth table, we see that d doesn't seem to affect the output. Whenever c is 0, the output is 1, regardless of what d is. So, for pin 11, the input would be c'.

This gives us everything we need for the 4:1 multiplexer:

Select 1: a

Select 2: b

Pin 00: c'd

Pin 01: c' + d

Pin 10: c'd

Pin 11: c'