Logic System Design KTU Chapter-4.ppt

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Chapter 4 Combinational Logic
 Logic circuits for digital systems may be
combinational or sequential.
 A combinational circuit consists of input variables,
logic gates, and output variables.

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4-2. 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.

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4-2. Analysis procedure
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.

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Example
F2 = AB + AC + BC; T1 = A + B + C; T2 = ABC; T3 = F2’T1;
F1 = T3 + T2
F1 = T3 + T2 = F2’T1 + ABC = A’BC’ + A’B’C + AB’C’ + ABC

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Derive truth table from logic
diagram
 We can derive the truth table in Table 4-1 by using
the circuit of Fig.4-2.

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4-3. Design procedure
1. Table4-2 is a Code-Conversion example, first, we
can list the relation of the BCD and Excess-3
codes in the truth table.

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Karnaugh map
2. For each symbol of the Excess-3 code, we use 1’s
to draw the map for simplifying Boolean function.

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Circuit implementation
z = D’; y = CD + C’D’ = CD + (C + D)’
x = B’C + B’D + BC’D’ = B’(C + D) + B(C + D)’
w = A + BC + BD = A + B(C + D)

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4-4. Binary Adder-Subtractor
 A combinational circuit that performs the addition of two
bits is called a half adder.
 The truth table for the half adder is listed below:
S = x’y + xy’
C = xy
S: Sum
C: Carry

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Implementation of Half-Adder

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Full-Adder
 One that performs the addition of three bits(two
significant bits and a previous carry) is a full adder.

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Simplified Expressions
S = x’y’z + x’yz’ + xy’z’ + xyz
C = xy + xz + yz
C

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Full adder implemented in
SOP

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Another implementation
 Full-adder can also implemented with two half
adders and one OR gate (Carry Look-Ahead adder).
S = z ⊕ (x ⊕ y)
= z’(xy’ + x’y) + z(xy’ + x’y)’
= xy’z’ + x’yz’ + xyz + x’y’z
C = z(xy’ + x’y) + xy = xy’z + x’yz + xy

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Binary adder
 This is also called
Ripple Carry
Adder ,because of
the construction
with full adders are
connected in
cascade.

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Carry Propagation
 Fig.4-9 causes a unstable factor on carry bit, and produces a
longest propagation delay.
 The signal from Ci to the output carry Ci+1, propagates
through an AND and OR gates, so, for an n-bit RCA, there
are 2n gate levels for the carry to propagate from input to
output.

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Carry Propagation
 Because the propagation delay will affect the output signals
on different time, so the signals are given enough time to
get the precise and stable outputs.
 The most widely used technique employs the principle of
carry look-ahead to improve the speed of the algorithm.

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Boolean functions
Pi = Ai ⊕ Bi steady state value
Gi = AiBi steady state value
Output sum and carry
Si = Pi ⊕ Ci
Ci+1 = Gi + PiCi
Gi : carry generate Pi : carry propagate
C0 = input carry
C1 = G0 + P0C0
C2 = G1 + P1C1 = G1 + P1G0 + P1P0C0
C3 = G2 + P2C2 = G2 + P2G1 + P2P1G0 + P2P1P0C0
 C3 does not have to wait for C2 and C1 to propagate.

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Logic diagram of
carry look-ahead generator
 C3 is propagated at the same time as C2 and C1.

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4-bit adder with carry lookahead
 Delay time of n-bit CLAA = XOR + (AND + OR) + XOR

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Binary subtractor
M = 1subtractor ; M = 0adder

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Overflow
 It is worth noting Fig.4-13 that binary numbers in the
signed-complement system are added and subtracted by the
same basic addition and subtraction rules as unsigned
numbers.
 Overflow is a problem in digital computers because the
number of bits that hold the number is finite and a result
that contains n+1 bits cannot be accommodated.

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Overflow on signed and unsigned
 When two unsigned numbers are added, an overflow is
detected from the end carry out of the MSB position.
 When two signed numbers are added, the sign bit is treated
as part of the number and the end carry does not indicate
an overflow.
 An overflow cann’t occur after an addition if one number is
positive and the other is negative.
 An overflow may occur if the two numbers added are both
positive or both negative.

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4-5 Decimal adder
BCD adder can’t exceed 9 on each input digit. K is the carry.

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Rules of BCD adder
 When the binary sum is greater than 1001, we obtain a
non-valid BCD representation.
 The addition of binary 6(0110) to the binary sum converts it
to the correct BCD representation and also produces an
output carry as required.
 To distinguish them from binary 1000 and 1001, which also
have a 1 in position Z8, we specify further that either Z4 or
Z2 must have a 1.
C = K + Z8Z4 + Z8Z2

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Implementation of BCD adder
 A decimal parallel
adder that adds n
decimal digits needs
n BCD adder stages.
 The output carry
from one stage
must be connected
to the input carry of
the next higher-
order stage.
If =1
0110

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4-6. Binary multiplier
 Usually there are more bits in the partial products and it is necessary to
use full adders to produce the sum of the partial products.
And

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4-bit by 3-bit binary multiplier
 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.
 K=4 and J=3, we need 12
AND gates and two 4-bit
adders.

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4-7. Magnitude comparator
 The equality relation of each
pair of bits can be expressed
logically with an exclusive-NOR
function as:
A = A3A2A1A0 ; B = B3B2B1B0
xi=AiBi+Ai’Bi’ for i = 0, 1, 2, 3
(A = B) = x3x2x1x0

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Magnitude comparator
 We inspect the relative
magnitudes of pairs of MSB. If
equal, we compare the next
lower significant pair of digits
until a pair of unequal digits is
reached.
 If the corresponding digit of A is
1 and that of B is 0, we conclude
that A>B.
(A>B)=
A3B’3+x3A2B’2+x3x2A1B’1+x3x2x1A0B’0
(A<B)=
A’3B3+x3A’2B2+x3x2A’1B1+x3x2x1A’0B0

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4-8. Decoders
 The decoder is called n-to-m-line decoder, where
m≤2n
.
 the decoder is also used in conjunction with other
code converters such as a BCD-to-seven_segment
decoder.
 3-to-8 line decoder: For each possible input
combination, there are seven outputs that are
equal to 0 and only one that is equal to 1.

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Implementation and truth table

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Decoder with enable input
 Some decoders are constructed with NAND gates, it
becomes more economical to generate the decoder
minterms in their complemented form.
 As indicated by the truth table , only one output can be
equal to 0 at any given time, all other outputs are equal to 1.

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Demultiplexer
 A decoder with an enable input is referred to as a
decoder/demultiplexer.
 The truth table of demultiplexer is the same with
decoder.
Demultiplexer
D0
D1
D2
D3
E
A B

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3-to-8 decoder with enable
implement the 4-to-16 decoder

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Implementation of a Full Adder
with a Decoder
 From table 4-4, we obtain the functions for the combinational circuit in
sum of minterms:
S(x, y, z) = ∑(1, 2, 4, 7)
C(x, y, z) = ∑(3, 5, 6, 7)

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4-9. Encoders
 An encoder is the inverse operation of a decoder.
 We can derive the Boolean functions by table 4-7
z = D1 + D3 + D5 + D7
y = D2 + D3 + D6 + D7
x = D4 + D5 + D6 + D7

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Priority 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.
 The discrepancy tables on Table 4-7 and Table 4-8 can
resolve aforesaid condition by providing one more output to
indicate that at least one input is equal to 1.

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Priority encoder
V=0no valid inputs
V=1valid 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.

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4-input priority encoder
 Implementation of
table 4-8
x = D2 + D3
y = D3 + D1D’2
V = D0 + D1 + D2 + D3
0
0
0
0

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4-10. Multiplexers
S = 0, Y = I0 Truth Table S Y Y = S’I0 + SI1
S = 1, Y = I1 0 I0
1 I1

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Quadruple 2-to-1 Line Multiplexer
 Multiplexer circuits can be combined with common selection inputs to
provide multiple-bit selection logic. Compare with Fig4-24.
I0
I1
Y

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Boolean function implementation
 A more efficient method for implementing a Boolean
function of n variables with a multiplexer that has n-1
selection inputs.
F(x, y, z) = (1,2,6,7)

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4-input function with a
multiplexer
F(A, B, C, D) = (1, 3, 4, 11, 12, 13, 14, 15)

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Three-State Gates
 A multiplexer can be constructed with three-state gates.

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4-11. HDL for combinational
circuits
 A module can be described in any one of the
following modeling techniques:
1. Gate-level modeling using instantiation of primitive gates
and user-defined modules.
2. Dataflow modeling using continuous assignment
statements with keyword assign.
3. Behavioral modeling using procedural assignment
statements with keyword always.

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Gate-level Modeling
 A circuit is specified by its logic gates and their interconnection.
 Verilog recognizes 12 basic gates as predefined primitives.
 The logic values of each gate may be 1, 0, x(unknown), z(high-
impedance).

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Gate-level description on Verilog
code
The wire declaration is for internal connections.

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Design methodologies
 There are two basic types of design methodologies: top-
down and bottom-up.
 Top-down: the top-level block is defined and then the sub-
blocks necessary to build the top-level block are
identified.(Fig.4-9 binary adder)
 Bottom-up: the building blocks are first identified and then
combined to build the top-level block.(Example 4-2 4-bit
adder)

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A bottom-up hierarchical description

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Three state gates
Gates statement: gate name(output, input, control)
>> bufif1(OUT, A, control);
A = OUT when control = 1, OUT = z when control = 0;
>> notif0(Y, B, enable);
Y = B’ when enable = 0, Y = z when enable = 1;

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2-to-1 multiplexer
 HDL uses the keyword tri to
indicate that the output has
multiple drivers.
module muxtri (A, B, select, OUT);
input A,B,select;
output OUT;
tri OUT;
bufif1 (OUT,A,select);
bufif0 (OUT,B,select);
endmodule

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Dataflow modeling
 It uses a number of operators that act on operands to
produce desired results. Verilog HDL provides about 30
operator types.
Table 4-10
Symbol Operation
+ binary addition
− binary subtraction
& bit-wise AND
| bit-wise OR
^ bit-wise XOR
~ bit-wise NOT
== equality
> greater than
< less than
{ } concatenation
?: conditional

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Dataflow modeling
 A continuous assignment is
a statement that assigns a
value to a net.
 The data type net is used
in Verilog HDL to represent
a physical connection
between circuit elements.
 A net defines a gate output
declared by an output or
wire.

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Dataflow description of 4-bit adder
HDL Example 4-4
//Dataflow description of 4-bit adder
module binary_adder (A,B,Cin,SUM,Cout);
input [3:0] A,B;
input Cin;
output [3:0] SUM;
output Cout;
assign {Cout,SUM} = A + B +Cin;
endmodule

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Data flow description of a 4-bit
comparator

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Dataflow description of 2-1 multiplexer
 Conditional operator( ? : )
 Condition? true-expression : false-expression;

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Behavioral modeling
 It is used mostly to describe sequential circuits, but can be
used also to describe combinational circuits.
 Behavioral descriptions use the keyword always followed by
a list of procedural assignment statements.
 The target output of procedural assignment statements
must be of the reg data type. Contrary to the wire data type,
where the target output of an assignment may be
continuously updated, a reg data type retains its value until
a new value is assigned.

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Behavioral description of 2-1 multiplexer

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Writing a simple test bench
 In addition to the always statement, test benches use the initial
statement to provide stimulus to the circuit under test.
 The always statement executes repeatedly in a loop. The initial
statement executes only once starting from simulation time=0 and may
continue with any operations that are delayed by a given number of
time units as specified by the symbol #.
For example:
initial
begin
A = 0; B = 0;
#10 A = 1;
#20 A = 0; B = 1;
end

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Stimulus and design modules
interaction
 The signals of test bench as inputs to the design module are
reg data type, the outputs of design module to test bench
are wire data type.

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Gate Level of Verilog Code of Fig.4-2

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Test Bench of the Figure 4-2

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