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CENG 241Digital Design 1
Lecture 6
Amirali Baniasadiamirali@ece.uvic.ca
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Decimal adder
When dealing with decimal numbers BCD code is used.
A decimal adders requires at least 9 inputs and 5 outputs.
BCD adder: each input does not exceed 9, the output can not exceed 19
How are decimal numbers presented in BCD?
Decimal Binary BCD 9 1001 1001 19 10011 (0001)(1001) 1
9
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Decimal Adder
Decimal numbers should be represented in binary code number. Example: BCD adder
Suppose we apply two BCD numbers to a binary adder then:
The result will be in binary and ranges from 0 through 19.
Binary sum: K(carry) Z8 Z4 Z2 Z1 BCD sum : C(carry) S8 S4 S2 S1
For numbers equal or less than 1001 binary and BCD are identical. For numbers more than 1001, we should add 6(0110) to binary to get
BCD. example: 10011(binary) = 11001(BCD) =19 ADD 6 to correct.
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BCD adder
Decides to add 6?
Adds 6
Numbers that need correction (add 6) are:01010 (10)01011 (11)01100 (12)01101 (13)01110 (14)01111 (15)10000 (16)10001 (17)10010 (18)10011 (19)
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BCD adder
Numbers that need correction (add 6) are:K Z8 Z4 Z2 Z10 1 0 1 0 (10)0 1 0 1 1 (11)0 1 1 0 0 (12)0 1 1 0 1 (13)0 1 1 1 0 (14)0 1 1 1 1 (15)1 0 0 0 0 (16)1 0 0 0 1 (17)1 0 0 1 0 (18)1 0 0 1 1 (19)
C = K + Z8Z4 +Z8Z2
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Magnitude Comparators
Compares two numbers, determines their relative magnitude.
We look at a 4-bit magnitude comparator;
A=A3A2A1A0, B=B3B2B1B0
Two numbers are equal if all bits are equal. A=B if A3=B3 AND A2=B2 AND A1=B1 AND A0=B0
Xi= AiBi + Ai’Bi’ ; Ai=Bi Xi=1 (remember exclusive NOR?)
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Magnitude Comparators
How do we know if A>B?
1.Compare bits starting from the most significant pair of digits
2.If the two are equal, compare the next lower significant bits
3.Continue until a pair of unequal digits are reached 4.Once the unequal digits are reached, A>B if Ai=1 and
Bi=0, A<B if Ai=0 and Bi = 1
A>B = A3B3’+X3A2B2’+X3X2A1B1’+X3X2X1A0B0’ A<B = A3’B3+X3A2’B2+X3X2A1’B1+X3X2X1A0’B0
Xi=1 if Ai=Bi
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Magnitude Comparators
A3=B3 ?
X3A2’B2
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Decoders
A decoder converts binary information from n input lines to a maximum of 2n output lines
Also known as n-to-m line decoders where m< 2n
Example 3-to-8 decoders.
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Decoders: Truth Table
X Y Z D0 D1 D2 D3 D4 D5 D6 D7
0 0 0 1 0 0 0 0 0 0 0
0 0 1 0 1 0 0 0 0 0 0
0 1 0 0 0 1 0 0 0 0 0
0 1 1 0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 1 0 0 0
1 0 1 0 0 0 0 0 1 0 0
1 1 0 0 0 0 0 0 0 1 0
1 1 1 0 0 0 0 0 0 0 1
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Decoders: AND implementation
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2-to-4 Decoder: NAND implementation
Decoder is enabled when E=0
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How to build bigger decoders?
We can combine two 3-to-8 decoders to build a 4-to-16 decoder.
Generates from 0000 to 0111
Generates from 1000 to 1111
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A decoder provides the 2n minterms of n input variables.
Any function is can be expressed in sum of minterms.
Use a decoder to make the minterms and an external OR gate to make the sum.
Example: consider a full adder. S(x,y,z) = Σ(1,2,4,7) C(x,y,z) = Σ (3,5,6,7)
Combinational Logic implementation
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Combinational Logic implementation
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Encoders
Encoders perform the inverse operation of a decoder:
Encoders have 2n input lines and n output line.
Output lines generate the binary code corresponding to the input value.
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Encoders: Truth Table
Outputs Inputs
X Y Z D0 D1 D2 D3 D4 D5 D6 D7 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 1
z=D1+D3+D5+D7 y=D2+D3+D6+D7 x=D4+D5+D6+D7
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Priority Encoders
Encoder limitations: If two inputs are active, the output is undefined. Solution: we need to take into account priority. What if all inputs are 0? Solution: we need a valid bit
Input Output D0 D1 D2 D3 x y v 0 0 0 0 X X 0 1 0 0 0 0 0 1 X 1 0 0 0 1 1 X X 1 0 1 0 1 X X X 1 1 1 1
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Priority Encoders: Map
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Priority Encoders: Circuit
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Multiplexers
Multiplexer: selects one binary input from many selections
example: 2-to-1 MUX
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4-to-1 MUX
Directs 1 of the 4 inputs to the output
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Multi-bit selection logic
Multiplexers can be combined with common selection inputs to support multi-bit selection logic
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Implementing Boolean functions w/ MUX
General rules for implementing any Boolean function with n variables:
Use a multiplexer with n-1 selection inputs and 2 n-1 data inputs
List the truth tabel Apply the first n-1 variables to the selection inputs
of multiplexer For each combination evaluate the output as a function
of the last variable. The function can be 0, 1 the variable or the complement
of the variable.
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Implementing Boolean functions w/ MUX
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Implementing Boolean functions w/ MUX
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Summary
Reading up to page 154
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