1. Combinational Functions and Circuits
Soon Tee Teoh

2. Decoders From truth table, circuit for 2x4 decoder is:
Note: Each output is a 2-variable minterm (X'Y', X'Y, XY' or XY)
F0 = X'Y'
F1 = X'Y
F2 = XY'
F3 = XY X Y
From Figure 3-18, page 122

3. Decoders Design a 3x8 decoder. F1 = x'y'z x z y F0 = x'y'z' F2 = x'yz'

4. Decoders
Convert binary information from n input lines to 2n output lines.
Known as n-to-m-line decoder (m = 2n).
May be used to generate the 2n minterms of n input variables.

5. Decoders
In general, for an n-bit code, a decoder could select up to 2n lines: :
n-bit
code
n to 2n
decoder 2n
output lines

6. Decoder Expansion
Build a decoder with n+1 inputs using n-input decoder.
A2A1A0
000
010
001
011
100
110
101
111 A0 A1
From Figure 3-19, page 122 A2

7. Decoders: Implementing Functions
A Boolean function, in sum-of-minterms form, can be implemented using:
a decoder to generate the minterms, and
an OR gate to form the sum.
Any combinational circuit with n inputs and m outputs can be implemented with an n:2n decoder with m OR gates.
Good when circuit has many outputs, and each function is expressed with few minterms.

8. Decoders: Implementing Functions
Example: Full adder
S(x, y, z) =  m(1,2,4,7)
C(x, y, z) =  m(3,5,6,7)
3x8
Dec S2 S1 S0 x y z 1 2 3 4 5 6 7 S C
From Figure 3-22, page 127

9. Decoders: Implementing Functions
F1 = A’B’CD + A’BC’D + ABCD
Exercise: What are F2 and F3 ?

10. Multiplexer
Selects from one of many input lines and directs it to a single output line.
Selection is controlled by selection inputs.
For a 2n-to-1 multiplexer, there are 2n data input lines and n selection lines whose bit combination determines which input is selected.
2n Data
Inputs
Data
Output n
Selection Inputs

11. Making a Multiplexer using a Decoder
2-to-4
decoder I0 S0 I1 Y S1 I2 I3
What is Y when S0 is 1 and S1 is 0?

12. Selecting m-bit data
Until now, we have examined single-bit data selected by a MUX. What if we want to select m-bit data/words?
Combine MUX blocks in parallel with common select and enable signals

13. Implementing Combinational Function using Multiplexer
For function with n variables, we need n-1 selection inputs for mux
Example: F(X,Y,Z) = X’Y’Z + X’YZ’ + XYZ’ + XYZ
There are n=3 inputs, thus we need a 2-to-1 MUX
The first n-1 (=2) variables serve as the selection lines
The remaining variable goes into input lines
(See pp of your text)

14. Example Z Y X F
F=0 1
F=1
F= X´
F= X 1 X´ X F
Z Y

15. Example Y X Z F
F=Z 1
F=0
F= Z´
F= 1 Z Z´ 1 F
Y X