CHAPTER 4 Combinational Logic Design – Multiplexers (Sections 4.5)
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Transcript of CHAPTER 4 Combinational Logic Design – Multiplexers (Sections 4.5)
CHAPTER 4CHAPTER 4
Combinational Logic Design – Combinational Logic Design – Multiplexers Multiplexers
(Sections 4.5)(Sections 4.5)
MultiplexerMultiplexer ““Selects” binary information from one of many Selects” binary information from one of many
input lines and directs it to a single output line.input lines and directs it to a single output line. Also know as the “selector” circuit,Also know as the “selector” circuit, Selection is controlled by a particular set of inputs Selection is controlled by a particular set of inputs
lines whose output depends on the combination of lines whose output depends on the combination of the data input lines.the data input lines.
For a 2For a 2nn-to-1 multiplexer, there are 2-to-1 multiplexer, there are 2nn data input data input lines and lines and n n selection lines whose bit combination selection lines whose bit combination determines which input is selected.determines which input is selected.
Multiplexer (cont.)Multiplexer (cont.)
4-to-1 MUX4-to-1 MUX
A
B
Y=A’B’D0+A’BD1+AB’D2+ABD3
=∑miDii=0
2n-1
A B Y
00
0
0
D0
1
1
11
D1
D2
D3
8-to-1 MUX8-to-1 MUX74LS1574LS15
55
YYww
8-to-1 MUX8-to-1 MUX
Y= C’B’A’D0+ C’B’AD1+ C’BA’D2+C’BAD3+ CB’A’D4+ CB’AD5+ CBA’D6+ CBAD7
=∑miDii=0
i=2n-1
Until now, we have examined single-bit Until now, we have examined single-bit data selected by a MUX. What if we want data selected by a MUX. What if we want to select m-bit data/words?to select m-bit data/words? Combine MUX blocks in parallel with Combine MUX blocks in parallel with common select and enable signalscommon select and enable signals
Example: Construct a logic circuit that Example: Construct a logic circuit that selects between 2 sets of 4-bit inputs (see selects between 2 sets of 4-bit inputs (see next slide for solution).next slide for solution).
Multiplexer ExpansionsMultiplexer Expansions
Example: Quad 2-to-1 MUXExample: Quad 2-to-1 MUX Uses four 2-to-1 Uses four 2-to-1
MUXs with common MUXs with common select (S) and enable select (S) and enable (E).(E).
Select line chooses Select line chooses between Abetween Aii’s and B’s and Bii’s. ’s.
The selected four-The selected four-wire digital signal is wire digital signal is sent to the Ysent to the Yii’s’s
Enable line turns Enable line turns MUX on and off (E=1 MUX on and off (E=1 is on).is on).
Example: Quad 4-to-1 MUXExample: Quad 4-to-1 MUX
74LS153(P43)74LS153(P43)
Multiplexer ExpansionsMultiplexer Expansions
A32-to-A32-to-1multiplexer using 1multiplexer using two two 74xx150ICs(P144)74xx150ICs(P144)
Multiplexer ExpansionsMultiplexer Expansions
A 32-to-1 A 32-to-1 multiplexer using multiplexer using four 8-to-1 four 8-to-1 multiplexers and a multiplexers and a 2-to-4 2-to-4 decoder(P145)decoder(P145)
Implementing Boolean functions with Implementing Boolean functions with MultiplexersMultiplexers
E.g. Using an 8-to-1 multiplexer to realize the Boolean E.g. Using an 8-to-1 multiplexer to realize the Boolean function F=f(x,y,z)=∑(1,2,4,5,7)function F=f(x,y,z)=∑(1,2,4,5,7)
Y= C’B’A’D0+ C’B’AD1+ C’BA’D2+C’BAD3+ CB’A’D4+ CB’AD5+ CBA’D6+ CBAD7
=∑miDii=0
i=2n-1
F=f(x,y,z)=∑(1,2,4,5,7)
=x’y’z+x’yz’+xy’z’+xy’z+xyz
C=x,B=y,A=z
D0=D3=D6=0
D1= D2= D4= D5= D7=1
Implementing Boolean functions with Implementing Boolean functions with MultiplexersMultiplexers
C=x,B=y,A=z
D0=D3=D6=0
D1= D2= D4= D5= D7=1
Using an 4-to-1 multiplexer to realize the Using an 4-to-1 multiplexer to realize the Boolean function F=f(x,y,z)=∑(1,2,4,5,7)Boolean function F=f(x,y,z)=∑(1,2,4,5,7)
D3
D2
D1
D0
4-1
MUX
AB
F
1 1
1
1
XYZ 00 01 11 10
0
1
1
DD00=Z=Z
DD11=Z’=Z’
DD22=1=1
DD33=Z=Z
X Y
F=f(x,y,z)=∑(1,2,4,5,7)=x’y’z+x’yz’+xy’z’+xy’z+xyz
ZZ’1Z
ZXZ YDD00=X=X
DD11=X’=X’
DD22=1=1
DD33=X=X
DD00=Y=Y
DD11=Y’=Y’
DD22=Y’=Y’
DD33=1=1
YY’Y1
XX’1X
F=X’Y’DF=X’Y’D00+X’YD+X’YD11+XY’D+XY’D22+XYD+XYD33
Implementing Boolean functions Implementing Boolean functions with Multiplexerswith Multiplexers
Exe. implement function using a 4-to-1 MUX
F(X,Y,Z) = Σm(1,2,6,7) F(A,B,C) = m(1,3,5,6).
Implementing Boolean functions with Implementing Boolean functions with MultiplexersMultiplexers
•F(X,Y,Z) = X’Y’Z + X’YZ’ + XYZ’ + XYZ = Σm(1,2,6,7)•There are n=3 inputs, thus we need a 2222-to-1 MUX-to-1 MUX•The first n-1 (=2) inputs serve as the selection linesThe first n-1 (=2) inputs serve as the selection lines
Implementing Boolean functions with Implementing Boolean functions with MultiplexersMultiplexers
1
1
1 1
ABC 00 01 11 10
0
1
F(A,B,C) = F(A,B,C) = m(1,3,5,6).m(1,3,5,6).
D3
D2
D1
D0
4-1
MUX
AB
AB
F
When A=B=0, F=D0=C
When A=0, B=1, F=D1=C
When A=1, B=0, F=D2=CWhen A=B=1, F=D3=C’
Implementing Boolean functions with Implementing Boolean functions with MultiplexersMultiplexers
00111111
11001111
11110011
00000011
11111100
00001100
11110000
00000000
FFCCBBAA
When A=B=0, F=CWhen A=B=0, F=C
When A=0, B=1, When A=0, B=1, F=CF=CWhen A=1, B=0, When A=1, B=0, F=CF=CWhen A=B=1, When A=B=1, F=C’F=C’
MUX implementation of MUX implementation of F(A,B,C) = F(A,B,C) = m(1,3,5,6)m(1,3,5,6)
AA
BB
CC
CC
CC
C’C’
FF
E.g. Consider the following Boolean E.g. Consider the following Boolean expression given in sum-of-product form: expression given in sum-of-product form:
F(x1,x2,x3)=xF(x1,x2,x3)=x11’’xx22
’’+x+x11xx22’’+x+x11xx33
Derive a circuit for using only 2-to-1 Derive a circuit for using only 2-to-1 multiplexers.multiplexers.
Implementing Boolean functions with Implementing Boolean functions with MultiplexersMultiplexers
Implementing Boolean functions with Implementing Boolean functions with MultiplexersMultiplexers
F(x1,x2,x3)=xF(x1,x2,x3)=x11’’xx22
’’+x+x11xx22’’+x+x11xx33
= x= x11’’xx22
’ ’ xx3 3 ’’ + x+ x11
’’xx22’ ’ xx3 3 + x+ x11xx22
’ ’ xx3 3 ’’
+ x+ x11xx22’ ’ xx3 3 + x+ x11xx22xx33
=∑(0,1,4,5,7)=∑(0,1,4,5,7)X1X2
X300 01 11 10
0
1 1
1 1
11
DD00=X=X22’’
D1=XD1=X22’+X’+X33
How to derive it only from How to derive it only from function?function?
DD00=X=X22’’
D1=XD1=X22’+X’+X33
Implementing Boolean functions with Implementing Boolean functions with MultiplexersMultiplexers
D1
D0
2-1
MUX
A
X1
F
X2’
≥1X3
Implementing Boolean functions with Implementing Boolean functions with MultiplexersMultiplexers
Exe. F(A,B,C,D)=∑m(1,2,4,9,10,11,12,14,15)Exe. F(A,B,C,D)=∑m(1,2,4,9,10,11,12,14,15)
Derive a circuit for using 4-to-1 Derive a circuit for using 4-to-1 multiplexers.multiplexers.
MUX as a Universal GateMUX as a Universal Gate We can construct OR, AND, and NOT gates We can construct OR, AND, and NOT gates
using 2-to-1 MUXs. Thus, 2-to-1 MUX is a using 2-to-1 MUXs. Thus, 2-to-1 MUX is a universal gate.universal gate.
ORORNOTNOT ANDAND
z = xz = x11+ x+ x11’x’x0 0
= = xx11xx00’ + ’ + xx11xx00 + + xx11’x’x0 0 = = xx11 + x + x0 0
z = 0x + 1x’ = x’z = 0x + 1x’ = x’ z = xz = x11xx00 + 0x + 0x00’ = x’ = x11xx00
11
xx11
HomeworkHomework
P179: 25.(1), 26.(2)P179: 25.(1), 26.(2)