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Logic and Computer Design Fundamentals
Lecture 14 – Combinational
Functional Blocks
Charles Kime & Thomas Kaminski
© 2004 Pearson Education, Inc.
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Overview





Functions and functional blocks
Enabling
Decoding
Encoding
Selecting
Chapter 4
2
Functions and Functional Blocks
 Higher-level functions
• Common in real designs
• Simplify design process
• Add hierarchy and replication
 Implemented within a functional block.
• In the past, many functional blocks were
implemented as SSI, MSI, and LSI chips.
 Today, they are often simply parts within a
VLSI circuits.
Chapter 4
3
Enabling Function
 Enabling permits an input signal to pass
through to an output
 Disabling blocks an input signal from passing
through to an output, replacing it with a fixed
value
 The value on the output when it is disable can
be Hi-Z (as for three-state buffers and
transmission gates), 0 , or 1 ENX
F
 When disabled, 0 output
 When disabled, 1 output
 See Example 4-2 in text
(a)
X
F
EN
(b)
Chapter 4
4
Decoding
 Decoding - the
• Conversion of n-bit input to m-bit output
• Given n m  2n
 Circuits that perform decoding are called
decoders
• Called n-to-m line decoders, where m  2n, and
• Generate 2n (or fewer) minterms for the n input
variables
Chapter 4
5
Decoder Examples
 1-to-2-Line Decoder A
D0 D1
D0 5 A
0
1
 2-to-4-Line Decoder
A1 A0
A0
1
0
0
1
D1 5 A
A
(a)
(b)
D0 D1 D2 D3
A1
0
0
1
1
0
1
0
1
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
D0 5 A1 A0
D1 5 A1 A0
(a)
D2 5 A1 A0
 Note that the 2-4-line
made up of 2 1-to-2line decoders and 4 AND gates.
D3 5 A1 A0
(b)
Chapter 4
6
Decoder Expansion - Example
 Result
Chapter 4
7
Decoder with Enable
 In general, attach m-enabling circuits to the outputs
 See truth table below for function
• Note use of X’s to denote both 0 and 1
• Combination containing two X’s represent four binary combinations
 Alternatively, can be viewed as distributing value of signal
EN
EN to 1 of 4 outputs
A
 In this case, called a
A
demultiplexer
D
1
0
0
EN A 1 A 0
0
1
1
1
1
X
0
0
1
1
X
0
1
0
1
D0 D1 D2 D3
0
1
0
0
0
(a)
0
0
1
0
0
0
0
0
1
0
D1
0
0
0
0
1
D2
D3
(b)
Chapter 4
8
Encoding
 Encoding - the opposite of decoding
• Conversion of m-bit input to n-bit output
 Circuits that perform encoding are called encoders
• An encoder has 2n (or fewer) input lines and n output lines
which generate the binary code corresponding to the input
values
• Typically, an encoder converts a code containing exactly
one bit that is 1 to a binary code corres-ponding to the
position in which the 1 appears.
Chapter 4
9
Encoder Example
 A decimal-to-BCD encoder
• Inputs: 10 bits corresponding to decimal
digits 0 through 9, (D0, …, D9)
• Outputs: 4 bits with BCD codes
• Function: If input bit Di is a 1, then the
output (A3, A2, A1, A0) is the BCD code for i,
 The truth table could be formed, but
alternatively, the equations for each of the
four outputs can be obtained directly.
Chapter 4
10
Encoder Example (continued)
 Input Di is a term in equation Aj if bit Aj is 1
in the binary value for i.
0000
0001
 Equations:
A3 = D8 + D9
A2 = D4 + D5 + D6 + D7
A1 = D2 + D3 + D6 + D7
A0 = D1 + D3 + D5 + D7 + D9
0010
0011
0100
0101
0110
0111
1000
1001
Chapter 4
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Priority Encoder
 What if multiple inputs are true?
• Encoder just designed does not work.
 Instead, use a priority encoder.
 Encodes either
• Most significant input position
• Or least significant input position
Chapter 4
12
Priority Encoder Example
 Priority encoder with 5 inputs (D4, D3, D2, D1, D0) - highest priority to
most significant 1 present - Code outputs A2, A1, A0 and V where V
indicates at least one 1 present.
Outputs
No. of Minterms/Row
Inputs
D4
D3
D2
D1
D0
A2
A1
A0
V
1
0
0
0
0
0
X
X
X
0
1
0
0
0
0
1
0
0
0
1
2
0
0
0
1
X
0
0
1
1
4
0
0
1
X
X
0
1
0
1
8
0
1
X
X
X
0
1
1
1
16
1
X
X
X
X
1
0
0
1
 Xs in input part of table represent 0 or 1; thus table entries correspond to
product terms instead of minterms. The column on the left shows that all
32 minterms are present in the product terms in the table
Chapter 4
13
Priority Encoder Example (continued)
 Could use a K-map to get equations, but
can be read directly from table and
manually optimized if careful:
A2 = D4
A1 = D4 D3 + D4 D3D2 = D4F1, F1 = (D3 + D2)
A0 = D4 D3 + D4 D3 D2 D1 = D4 (D3 + D2D1)
V = D4 + F1 + D1 + D0
Chapter 4
14
Selecting
 Selecting of data or information is a critical
function in digital systems and computers
 Circuits that perform selecting have:
• A set of information inputs from which the selection
is made
• A single output
• A set of control lines for making the selection
 Logic circuits that perform selecting are called
multiplexers
 Selecting can also be done by three-state logic
or transmission gates
Chapter 4
15
Multiplexers
 A multiplexer selects one input line and
transfers it to output
• n control inputs (Sn - 1, … S0) called selection
inputs
• m  2n information inputs (I2n - 1, … I0)
• output Y
Chapter 4
16
2-to-1-Line Multiplexer
 Since 2 = 21, n = 1
 The single selection variable S has two values:
• S = 0 selects input I0
• S = 1 selects input I1
 The equation:
Y = SI0 + SI1
 The circuit:
Enabling
Circuits
Decoder
I0
Y
S
I1
Chapter 4
17
2-to-1-Line Multiplexer (continued)
 Note the regions of the multiplexer circuit shown:
• 1-to-2-line Decoder
• 2 Enabling circuits
• 2-input OR gate
 To obtain a basis for multiplexer expansion, we combine
the Enabling circuits and OR gate into a 2  2 AND-OR
Enabling
Decoder
circuit:
Circuits
• 1-to-2-line decoder
• 2  2 AND-OR
 In general, for an
2n-to-1-line
I0
Y
S
multiplexer:
I1
• n-to-2n-line decoder
• 2n  2 AND-OR
Chapter 4
18
Example: 4-to-1-line Multiplexer
 2-to-22-line decoder
 22  2 AND-OR
Decoder
S1
4 3 2 AND-OR
S0
S1
S0
Decoder
I0
I1
Y
Y
I2
I3
Chapter 4
19
Multiplexer Width Expansion
 Select “vectors of bits” instead of “bits”
 Use multiple copies of 2n  2 AND-OR in
parallel
 Example:
4-to-1-line
quad multiplexer
Chapter 4
20
Other Selection Implementations
 Three-state logic in place of AND-OR
S0
I0
I1
S1
Y
I2
I3
(b)
 Gate input cost = 14 compared to 22 (or
18) for gate implementation
Chapter 4
21
Other Selection Implementations
 Transmission Gate Multiplexer
 Gate input
cost = 8
compared
to 14 for
3-state logic
and 18 or 22
for gate logic
S0
S1
I0
I1
I2
TG
(S0 5 0)
TG
(S1 5 0)
TG
(S0 5 1)
Y
TG
(S0 5 0)
TG
(S1 5 1)
I3
TG
(S0 5 1)
Chapter 4
22
Summary





Functions and functional blocks
Enabling
Decoding
Encoding
Selecting with multiplexers
Chapter 4
23
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Chapter 4
24

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