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UNIT - IIMinimization and Design of
Combinational Circuits
P.VIDYA SAGAR ( ASSOCIATE PROFESSOR)
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT II
Minimization and Design of Combinational Circuits: Introduction, The
Minimization of switching function using theorem, The Karnaugh Map
Method-Up to Five Variable Maps, Don’t Care Map Entries, Tabular
Method.
Design of Combinational Logic: Adders, Subtractors, comparators,
Multiplexers, Demultiplexers, Decoders, Encoders and Code converters,
Hazards and Hazard Free Relations.
UNIT - II
2019/2/172
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IICombinational Circuits
It consists of input variables, logic gates and output variables.
Output is function of input only
i.e. no feedback
When input changes, output changes (after a delay)
2019/2/17
•••
•••
n inputs m outputsCombinational
Circuits
3
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IICombinational Circuits
Analysis
Given a circuit, find out its function
Function may be expressed as:
Boolean function
Truth table
Design
Given a desired function, determine its circuit
Function may be expressed as:
Boolean function
Truth table
2019/2/17
?
?
?
C
BA
C
BA
BA
CA
CB
F1
F2
4
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT II
5
Analysis procedure
To obtain the output Boolean functions from a logic diagram, proceed as follows:
1.Label all gate outputs that are a function of input variables with arbitrary symbols.
Determine the Boolean functions for each gate output.
2.Label the gates that are a function of input variables and previously labeled gates with
other arbitrary symbols. Find the Boolean functions for these gates.
3.Repeat the process outlined in step 2 until the outputs of the circuit are obtained.
4.By repeated substitution of previously defined functions, obtain the output Boolean
functions in terms of input variables.
2019/2/17
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIAnalysis Procedure
Boolean Expression Approach
2019/2/17
F1=T2+T3= AB'C'+A'BC'+A'B'C+ABC
F2=AB+AC+BC
C
BA
C
BA
BA
CA
CB
F1
F2
T2=ABCT1=A+B+C
F2=AB+AC+BC
F’2=(A’+B’)(A’+C’)(B’+C’)
T3=AB'C'+A'BC'+A'B'C
6
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT II
C
BA
C
BA
BA
CA
CB
F1
F2
Analysis Procedure
We can obtain the truth table directly from the logic diagram
Truth Table Approach
2019/2/17
A B C F1 F2
0 0 0= 0
= 0
= 0
= 0
= 0
= 0
= 0
= 0
= 0
= 0
= 0
= 0
T2 = 0
T1 = 0
0
0
0
F2 = 0
F’2 = 1T3 = 0
0 0 0
7
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIFull Adder Circuit
2019/2/178
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT II
C
BA
C
BA
BA
CA
CB
F1
F2
Analysis Procedure
Truth Table Approach
2019/2/17
= 0
= 0
= 1
= 0
= 0
= 1
= 0
= 0
= 0
= 1
= 0
= 1
0
1
0
0
0
0
1
1
1
A B C F1 F2
0 0 0 0 0
0 0 1 1 0
9
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT II
C
BA
C
BA
BA
CA
CB
F1
F2
Analysis Procedure
Truth Table Approach
2019/2/17
= 0
= 1
= 0
= 0
= 1
= 0
= 0
= 1
= 0
= 0
= 1
= 0
0
1
0
0
0
0
1
1
1
A B C F1 F2
0 0 0 0 0
0 0 1 1 0
0 1 0 1 0
10
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT II
C
BA
C
BA
BA
CA
CB
F1
F2
Analysis Procedure
Truth Table Approach
2019/2/17
= 0
= 1
= 1
= 0
= 1
= 1
= 0
= 1
= 0
= 1
= 1
= 1
0
1
0
0
1
1
0
0
0
A B C F1 F2
0 0 0 0 0
0 0 1 1 0
0 1 0 1 0
0 1 1 0 1
11
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT II
C
BA
C
BA
BA
CA
CB
F1
F2
Analysis Procedure
Truth Table Approach
2019/2/17
= 1
= 1
= 1
= 1
= 1
= 1
= 1
= 1
= 1
= 1
= 1
= 1
1
1
1
1
1
1
0
0
1
A B C F1 F2
0 0 0 0 0
0 0 1 1 0
0 1 0 1 0
0 1 1 0 1
1 0 0 1 0
1 0 1 0 1
1 1 0 0 1
1 1 1 1 1
B
0 1 0 1
A 1 0 1 0
C
B
0 0 1 0
A 0 1 1 1
C
F1=AB'C'+A'BC'+A'B'C+ABC F2=AB+AC+BC
12
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIDesign Procedure
Given a problem statement:
Determine the number of inputs and outputs
Derive the truth table
Simplify the Boolean expression for each output
Produce the required circuit
Example:
Design a circuit to convert a “BCD” code to “Excess -3” code
2019/2/17
4-bits
0-9 values 4-bits
Value+3
?
13
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIDesign Procedure
BCD-to-Excess 3 Converter
2019/2/17
A B C D w x y z
0 0 0 0 0 0 1 1
0 0 0 1 0 1 0 0
0 0 1 0 0 1 0 1
0 0 1 1 0 1 1 0
0 1 0 0 0 1 1 1
0 1 0 1 1 0 0 0
0 1 1 0 1 0 0 1
0 1 1 1 1 0 1 0
1 0 0 0 1 0 1 1
1 0 0 1 1 1 0 0
x x x x x x x x
x x x x x x x x
x x x x x x x x
x x x x x x x x
x x x x x x x x
x x x x x x x x
C
1 1 1B
Ax x x x
1 1 x x
D
C
1 1 1
1B
Ax x x x
1 x x
D
C
1 1
1 1B
Ax x x x
1 x x
D
C
1 1
1 1B
Ax x x x
1 x x
D
w = A+BC+BD x = B’C+B’D+BC’D’
y = C’D’+CD z = D’
14
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT II
2019/2/1715
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIDesign Procedure
BCD-to-Excess 3 Converter
2019/2/17
w
x
D
C
z
y
B
A
w = A + B(C+D)
x = B’(C+D) + B(C+D)’
y = (C+D)’ + CD
z = D’
A B C D w x y z
0 0 0 0 0 0 1 1
0 0 0 1 0 1 0 0
0 0 1 0 0 1 0 1
0 0 1 1 0 1 1 0
0 1 0 0 0 1 1 1
0 1 0 1 1 0 0 0
0 1 1 0 1 0 0 1
0 1 1 1 1 0 1 0
1 0 0 0 1 0 1 1
1 0 0 1 1 1 0 0
x x x x x x x x
x x x x x x x x
x x x x x x x x
x x x x x x x x
x x x x x x x x
x x x x x x x x16
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IISeven-Segment Decoder (digital clock)
2019/2/17
a
b
c
g
e
d
f
?
w
x
y
z
abcdefg
w x y z a b c d e f g
0 0 0 0 1 1 1 1 1 1 0
0 0 0 1 0 1 1 0 0 0 0
0 0 1 0 1 1 0 1 1 0 1
0 0 1 1 1 1 1 1 0 0 1
0 1 0 0 0 1 1 0 0 1 1
0 1 0 1 1 0 1 1 0 1 1
0 1 1 0 1 0 1 1 1 1 1
0 1 1 1 1 1 1 0 0 0 0
1 0 0 0 1 1 1 1 1 1 1
1 0 0 1 1 1 1 1 0 1 1
1 0 1 0 x x x x x x x
1 0 1 1 x x x x x x x
1 1 0 0 x x x x x x x
1 1 0 1 x x x x x x x
1 1 1 0 x x x x x x x
1 1 1 1 x x x x x x x
y
1 1 1
1 1 1x
wx x x x
1 1 x x
z
BCD code
a = w + y + xz + x’z’ b = . . .c = . . .
d = . . .17
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIBinary Adder
Half Adder
Adds 1-bit plus 1-bit
Produces Sum and Carry
2019/2/17
HAx
y
S
C
x y C S
0 0 0 0
0 1 0 1
1 0 0 1
1 1 1 0
x
+ y
───
C S
x
y
S
C
18
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIBinary Adder
Full Adder
Adds 1-bit plus 1-bit plus 1-bit
Produces Sum and Carry
2019/2/17
x y z C S
0 0 0 0 0
0 0 1 0 1
0 1 0 0 1
0 1 1 1 0
1 0 0 0 1
1 0 1 1 0
1 1 0 1 0
1 1 1 1 1
x
+ y
+ z
───
C S
FAxyz
S
C
y
0 1 0 1
x 1 0 1 0
z
y
0 0 1 0
x 0 1 1 1
z
S = xy'z'+x'yz'+x'y'z+xyz = x y z
C = xy + xz + yz
19
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIBinary Adder
Full Adder
2019/2/17
x
y
z
S
C
xy
xz
yz
xyzxyzxyzxyz
xyz
x
y
z
xy
xz
yz
S
C
S = xy'z'+x'yz'+x'y'z+xyz = x y z
C = xy + xz + yz
Implementation of full adder in sum of products form
20
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIBinary Adder
Implementation of Full Adder with two half adder and an OR gate.
2019/2/17
x
y
z
S
C
HAxy
z
HAS
C
21
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIBinary Adder
2019/2/17
c3 c2 c1 .
+ x3 x2 x1 x0
+ y3 y2 y1 y0
────────
Cy S3 S2 S1 S0
FA
x3 x2 x1 x0
FAFAFA
y3 y2 y1 y0
S3 S2 S1 S0
C4 C3 C2 C1
0
Binary Adder
x3x2x1x0 y3y2y1y0
S3S2S1S0
C0C4
Carry
Propagate
Addition
22
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT II
Subscript i 3 2 1 0
Input carry 0 1 1 0 Ci
1 0 1 1 xi
0 0 1 1 yi
Sum 1 1 1 0 Si
Output carry 0 0 1 1 Ci+1
2019/2/1723
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT 2019/2/17
Carry propagation
When the correct outputs are available
The critical path counts :
(A0, B0, C0) → C1 → C2 → C3 → (C4, S3)
When 4-bits full-adder → 8 gate levels (n-bits: 2n gate
levels)
Figure 4.10 Full Adder with P and G Shown
24
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT 2019/2/17
Binary Adders (Parallel)
Reduce the carry propagation delay
Employ faster gates
Look-ahead carry (more complex mechanism, yet
faster)
Carry propagate: Pi = AiBi
Carry generate: Gi = AiBi
Sum: Si = PiCi
Carry: Ci+1 = Gi + PiCi
C0 = Input carry
C1 = G0+P0C0
C2 = G1+P1C1 = G1+P1(G0+P0C0) = G1+P1G0+P1P0C0
C3 = G2+P2C2 = G2+P2G1+P2P1G0+ P2P1P0C0
25
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT 2019/2/17
Carry Look-ahead Adder (1/2) Logic diagram
Fig. 4.11 Logic Diagram of Carry Look-ahead Generator
26
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT 2019/2/17
Carry Look-ahead Adder (2/2)
4-bit carry-look ahead adder
Propagation delay of C3, C2 and C1
are equal.
Fig. 4.12 4-Bit Adder with Carry Look-ahead
27
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIBinary Subtractor
Use 2’s complement with binary adder
x – y = x + (-y) = x + y’ + 1
2019/2/17
Binary Adder
A3 A2 A1 A0 B3 B2 B1 B0
S3 S2 S1 S0
CiCy 1
x3 x2 x1 x0 y3 y2 y1 y0
F3 F2 F1 F0
28
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIBinary Adder/Subtractor
M: Control Signal (Mode)
M = 0 F = x + y
M = 1 F = x – y
F = x + y’+ 1
2019/2/17
Binary Adder
A3 A2 A1 A0 B3 B2 B1 B0
S3 S2 S1 S0
CiCy
Mx3 x2 x1 x0 y3 y2 y1 y0
F3 F2 F1 F0
y 0 = yy 1 = y'
29
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIOverflow Discussion
Overflow
The storage is limited.
Add two positive numbers and obtain a negative number
number Add two negative numbers and obtaina positive
V = 0,no overflow;
V = 1, overflow
Example:
2019/2/1730
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIBCD Adder
2019/2/17
4-bits plus 4-bits
Operands : 0 to 9
This is called BCD adder. In order to add two decimal digits with a
possible carry-in of 1, then the maximum sum is 19. The following
table shows the sum when performed in binary and compared to the
sum when performed in BCD. In both cases five outputs are needed.
+ x3 x2 x1 x0
+ y3 y2 y1 y0
────────
Cy S3 S2 S1 S0
31
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT II
2019/2/1732
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT II
From the table, when the sum is greater than 9, a correction is needed. The correction is
adding 6 to the binary sum. The BCD adder will then consist of the 4-bit binary adder. A
second 4-bit binary adder is needed to add 6 to the sum when it is greater than 9.
01010 + 0110 = 10000
The required logic circuit needed to detect if correction is needed can be obtained as follows:
C = K + Z8Z4 +Z8Z2
2019/2/1733
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIBCD Adder
2019/2/1734
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIBinary Multiplier
To multiply two 2-bit binary numbers B1 B0 and A1 A0, we may use half adders & AND gates.
2019/2/1735
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT II
2019/2/1736
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT II
For J multiplier bits and K multiplicand bits, we need (J x K) AND gates and (J – 1) K-bit
adders to produce a product of J + K bits.
In the previous example, K = 4, J = 3, so we need 12 AND gates and 2 four-bit adders to
produce a product of seven bits.
2019/2/1737
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIMagnitude Comparator
Compare 4-bit number to 4-bit number
3 Outputs: < , = , >
Expandable to more number of bits
0010, 1000 (Using XNOR Gates for equality)
2019/2/17
Magnitude
Comparator
A3A2A1A0 B3B2B1B0
A<B A=B A>B
33333 BABAx
22222 BABAx
11111 BABAx
00000 BABAx
0123)( xxxxBA
00123112322333)( BAxxxBAxxBAxBABA
00123112322333)( BAxxxBAxxBAxBABA
38
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIMagnitude Comparator
2019/2/1739
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIDecoders
Is a combinational circuit that converts binary information from n input lines to a maximum of
2n unique output lines :
Extract “Information” from the code
n-to-m line decoder ( n inputs, m<= 2n
output)
Binary Decoder
Example: 2-bit Binary Number
2019/2/17
Binary
Decoder
x1
x0
Only one
lamp will
turn on
0
0
1
0
0
0
40
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIDecoders
2-to-4 Line Decoder
2019/2/17
I1 I0 Y0 Y1 Y2 Y3
0 0 1 0 0 0
0 1 0 1 0 0
1 0 0 0 1 0
1 1 0 0 0 1
Bin
ary
Dec
od
erI1
I0
Y0
Y1
Y2
Y3
I1
I0
Y3
Y2
Y1
Y0
013 IIY 012 IIY
011 IIY 010 IIY
41
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIDecoders
3-to-8 Line Decoder (Binary to Octal conversion)
2019/2/17
Bin
ary
Dec
od
erI2
I1
I0
Y0
Y1
Y2
Y3
Y4
Y5
Y6
Y7
I2
I0
Y7
Y6
Y5
Y4
Y3
Y2
Y1
Y0
I1
012 III
012 III
012 III
012 III
012 III
012 III
012 III
012 III
42
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT II
2019/2/1743
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIDecoders
“Enable” Control
2019/2/17
Bin
ary
Dec
od
erI1
I0
E
Y0
Y1
Y2
Y3
E I1 I0 Y0 Y1 Y2 Y3
0 x x 0 0 0 0
1 0 0 1 0 0 0
1 0 1 0 1 0 0
1 1 0 0 0 1 0
1 1 1 0 0 0 1
EI0
Y3
Y2
Y1
Y0
I1
44
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT II4 x16 decoder constructed with two 3 x 8 decoders
2019/2/1745
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIDecoders
Expansion
2019/2/17
E I1 I0 D7 D6 D5 D4 D3 D2 D1 D0
0 0 0 0 0 0 0 0 0 0 1
0 0 1 0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0 1 0 0
0 1 1 0 0 0 0 1 0 0 0
1 0 0 0 0 0 1 0 0 0 0
1 0 1 0 0 1 0 0 0 0 0
1 1 0 0 1 0 0 0 0 0 0
1 1 1 1 0 0 0 0 0 0 0
I2 I1 I0
Bin
arD
Dec
od
erI0
I1
E
D3
D2
1
D0
D7
D6
D5
D4
D3
D2
D1
D0Bin
ary
Dec
od
erI0
I1
E
D3
D2
D1
D0
3 x8 decoder constructed with two 2 x 4 decoders
46
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIDecoders
Active-High / Active-Low
2019/2/17
I1 I0 Y0 Y1 Y2 Y3
0 0 1 0 0 0
0 1 0 1 0 0
1 0 0 0 1 0
1 1 0 0 0 1
I1 I0 Y0 Y1 Y2 Y3
0 0 0 1 1 1
0 1 1 0 1 1
1 0 1 1 0 1
1 1 1 1 1 0
Bin
ary
Dec
od
erI1
I0
Y0
Y1
Y2
Y3
I1
I0
Y3
Y2
Y1
Y0
Bin
ary
Dec
od
erI1
I0
Y0
Y1
Y2
Y3
47
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIImplementation Using Decoders
Each output is a minterm
All minterms are produced
Sum the required minterms
Example: Full Adder
S(x, y, z) = ∑(1, 2, 4, 7)
C(x, y, z) = ∑(3, 5, 6, 7)
2019/2/17
x y z C S
0 0 0 0 0
0 0 1 0 1
0 1 0 0 1
0 1 1 1 0
1 0 0 0 1
1 0 1 1 0
1 1 0 1 0
1 1 1 1 1
48
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT II
A function with a long list of minterms requires an OR gate with a large number of inputs.
If the number of minterms in the function is greater than 2n/2, then F’ can be expressed with
fewer minterms.
So we use a NOR gate to sum the minterms of F’.
The output of NOR gate complements this sum and generates the normal output F.
2019/2/1749
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIImplementation Using Decoders with NAND gates
2019/2/17
I2
I1
I0
Y7
Y6
Y5
Y4
Y3
Y2
Y1
Y0
BinaryDecoder
x
y
z
S C
50
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIEncoders
Encoder is a digital circuit that performs the inverse operation of a decoder Generates a
unique binary code from several input lines.
Generally encoders produce2-bit, 3-bit or 4-bit code. n bit encoder has 2n input lines
Put “Information” into code (it generates the binary code corresponding to the input value).
Binary Encoder
Example: 4-to-2 Binary Encoder
2019/2/17
x0 x1 x2 x3 y1 y0
1 0 0 0 0 0
0 1 0 0 0 1
0 0 1 0 1 0
0 0 0 1 1 1
Binary
Encoder
y1
y0
x0
x1
x2
x3
Only one
switch
should be
activated
at a time
51
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIEncoders
Octal-to-Binary Encoder (8-to-3)
2019/2/17
I7 I6 I5 I4 I3 I2 I1 I0 Y2 Y1 Y0
0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0 0 0 1
0 0 0 0 0 1 0 0 0 1 0
0 0 0 0 1 0 0 0 0 1 1
0 0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 1 0 1
0 1 0 0 0 0 0 0 1 1 0
1 0 0 0 0 0 0 0 1 1 1
Bin
ary
En
cod
er Y2
Y1
Y0
I7
I6
I5
I4
I3
I2
I1
I0
13570
23671
45672
IIIIY
IIIIY
IIIIY
I7
I6
I5
I4
I3
I2
I1
I0
Y2
Y1
Y0
52
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIPriority encoder
If two inputs are active simultaneously, the output produces an undefined combination. We
can establish an input priority to ensure that only one input is encoded.
Another ambiguity in the octal-to-binary encoder is that an output with all 0’s is generated
when all the inputs are 0; the output is the same as when D0 is equal to 1.
2019/2/17
V=0no valid inputs
V=1valid inputs
X’s in output columns represent
don’t-care conditions X’s in the
input columns are useful for
representing a truth table in
condensed form. Instead of listing
all 16 minterms of four variables.
53
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIPriority Encoders
4-Input Priority Encoder ( V is a valid bit indicator)
2019/2/17
Pri
ori
ty
En
cod
er V
y
x
D3
D2
D1
D0
54
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT II
1
2019/2/1755
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT II
2019/2/1756
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIEncoder / Decoder Pairs
2019/2/17
Y2
Y1
Y0
I7
I6
I5
I4
I3
I2
I1
I0
I2
I1
I0
Y7
Y6
Y5
Y4
Y3
Y2
Y1
Y0
Binary
Encoder
Binary
Decoder
57
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIMultiplexers (Data Selector)
It Selects binary information from one of many
input lines and directs it to a single output line.
2019/2/17
MUX Y
I0
I1
I2
I3 S1 S0
S1 S0 Y
0 0 I0
0 1 I1
1 0 I2
1 1 I3
58
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT II2-to-1 MUX
2019/2/1759
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIMultiplexers
2-to-1 MUX or 2 x 1 MUX
4-to-1 MUX
2019/2/17
MUX Y
S
I1
I0
S
Y
MUX Y
I0
I1
I2
I3
S1 S0
I1
I0
S1
YI2
I3
S0
I0
I1
60
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIMultiplexers
Quad 2-to-1 MUX
2019/2/17
MUX Y0I0
I1 S
MUX Y1I0
I1 S
MUX Y2I0
I1 S
MUX Y3I0
I1 S
A0
A1
A2
A3
B0
B1
B2
B3
S
MUX
A0
A1
A2
A3
S E
Y3
Y2
Y1
Y0B0
B1
B2
B3
(two 4-bits input, one 4-bits output)
61
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT II
2019/2/1762
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIImplementation Using Multiplexers
Example
F(x, y, z) = ∑(1, 2, 6, 7)
2019/2/17
MUX Y
I0
I1
I2
I3 S1 S0
x y z F
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 0
1 0 0 0
1 0 1 0
1 1 0 1
1 1 1 1
x y
FF = zz
F = z
z
F = 0
0
F = 1
1
63
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT II
MUX Y
I0
I1
I2
I3
I4
I5
I6
I7S2 S1 S0
Implementation Using Multiplexers
Example
F(A, B, C, D) = ∑(1, 3, 4, 11, 12, 13, 14, 15)
2019/2/17
A B C D F
0 0 0 0 0
0 0 0 1 1
0 0 1 0 0
0 0 1 1 1
0 1 0 0 1
0 1 0 1 0
0 1 1 0 0
0 1 1 1 0
1 0 0 0 0
1 0 0 1 0
1 0 1 0 0
1 0 1 1 1
1 1 0 0 1
1 1 0 1 1
1 1 1 0 1
1 1 1 1 1 A B C
F
F = D
D
F = D D
F = D D
F = 00
F = 0
F = D
F = 1
F = 1
0
D
1
1
64
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT II
Y
I0
I1
I2
I3
I4
I5
I6
I7
S2 S1 S0
Multiplexer Expansion
8-to-1 MUX using Dual 4-to-1 MUX & one 2x1 Mux
2019/2/17
MUX Y
I0
I1
I2
I3 S1 S0
MUX Y
I0
I1
I2
I3 S1 S0
MUX YI0
I1S
0 01
65
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIDeMultiplexers
2019/2/17
DeMUXI
Y3
Y2
Y1
Y0
S1 S0
S1 S0 Y3 Y2 Y1 Y0
0 0 0 0 0 I
0 1 0 0 I 0
1 0 0 I 0 0
1 1 I 0 0 0
I
Y3
Y2
Y1
Y0
S0
S1
66
VIDYA SAGAR POTHARAJU , Department of Electronics and Communication Engineering, VBIT
STLD UNIT IIDeMultiplexers / Decoders
2019/2/17
Bin
ary
Dec
od
erI1
I0
E
Y0
Y1
Y2
Y3
E I1 I0 Y3 Y2 Y1 Y0
0 x x 0 0 0 0
1 0 0 0 0 0 1
1 0 1 0 0 1 0
1 1 0 0 1 0 0
1 1 1 1 0 0 0
DeMUXI
Y0
Y1
Y2
Y3
S1 S0
S1 S0 Y3 Y2 Y1 Y0
0 0 0 0 0 I
0 1 0 0 I 0
1 0 0 I 0 0
1 1 I 0 0 0
67