Transcript Lecture 03 - people.vcu.edu

Lecture 3 ENGRE 254
1/14/09
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Lecture 1 review
Digital signals assume two values represented by “0” and “1”.
Typically a “0” represents a voltage near 0 volts, and a “1”
represents a voltage near 5 volts (3.3 volts for newer devices).
Two types of digital circuits:
• Combinational
-Simpler
-Output a function of present input.
-Consists of logic gates without feedback.
• Sequential
-The output is a function of not only the present input,
and current state which is determined by past events.
- Consists of flip-flops or gates with feedback.
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3
Non-decimal arithmetic.
• Same rules as decimal.
• Different addition, subtraction,
multiplication tables.
• Binary Examples
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1
0
1
1 1 0 1
 0 1 0 1
1 0 0 1 0
0
1
10
0
10
1 0 0 1
 0 1 0
1 1 0
0
1
1
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Binary multiplication and division
example
1 0 1
 0 1 1
1 0 1
1 0 1
0 0 0
0 1 1 1 1
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101 1111
101
101
101
0
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Base 5 multiplication example
X
0
1
2
3
4
0
0
0
0
0
0
1
0
1
2
3
4
2
0
2
4
11
13
3
0
3
11
14
22
4
0
4
13
22
31
3 4
 1 4
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Representing negative numbers
• Sign-magnitude. +199.1210, -199.1210, -1012
– This is what people normally use.
– Complicates digital logic.
• Complement number systems.
– May use radix or radix-1 (actually rn or rn – 1 for n digit
words).
– In base 10 we use 10’s complement or 9’s complement.
– In base 2 we use 2’s complement or 1’s complement.
– Simplifies digital logic. Eliminates need for separate
hardware for addition and subtraction.
– The 2’s complement has become the standard.
– We will concentrate on 2’s complement.
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9’s and 10’s complement.
184910
10  10000  1849  8151
1849  9999  1849  8150
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10
8
9
One’s Complement
1710  000100012 ,
1210  00001100 2
0001000112  111111112  000100012
 111011102
00001100  111111112  000011002
1
2
 111100112
Rule: Complement each bit.
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Two’s complement
• To find 2’s complement.
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000100012  10000000  00010001  111011112
or better
00010001  00010001  1  111011112
• Easy way. Starting at rightmost bit, copy
through first 1 then complement each bit.
2
2
1
2
00010001  ?,
2
2
00001100  ?
2
2
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2’s complement
• With n bits we can represent binary values
form 0 to 2n-1.
– For example with 3 bits: 000, 001, …, 111.
– With 3 bits 111 + 001 = 000. (Overflow!)
• For an n bit number B, B2 = 2n – B
• -B = -2n + B2
• A – B = -2n + A + B2
Rule: To subtract A – B take two’s
complement of B and add. Ignore contents
of 2n bit position.
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Examples
17  000100012
12  000011002
17  12  29  000111012
17  12  5  000001012
 17  12  5  000001012  11111011
2
2
 17  12  29  11100011
2
2
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Two’s complement example (using 6 bits)
1710
 1210
510
0100012
 0011002
0 001012
0100012
2
 11010 02
1 0001012

Consider other examples such as:
12-17, -12-(-17), -17-(-12), etc.
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Binary codes for decimal numbers.
• Decimal numbers are represented in a digital
system using a string of bits.
– This can be done in various ways.
• A set of n-bit strings where different strings
represent different numbers or other things is
called a code.
• A particular combination of n-bit values is called a
code word.
• See Table 2-9 on page 49 for example codes for
decimal digits.
• A common and convenient code for decimal
numbers is BCD.
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Lab 2: Wire up the circuit below and verify that it converts 2421
code to BCD. For example an input of “1111” produces output of
“1001”.
1
1
1
0
1
1
0
1
1
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BCD
• BCD is a binary representation of decimal
numbers in a form that can easily be
interpreted by a person.
• Suppose we want to store 149210 in a 16 bit
computer word.
– In binary 149210 = 0000 0101 1101 01002
– In BCD 149210 is stored as
– 0001 0100 1001 0010BCD
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BCD addition
• Suppose 1492 and 1066 are stored in BCD.
1492  0001 0100 1001 0010
1066  0001 0000 0110 0110
If we add these in binary we get an incorrect result.
Rule to add BCD numbers start at least significant
BCD position (4 bits) add in binary if there is a
carry or the result is grater than 9 add 6. Repeat
for each BCD position.
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BCD addition example
0001 0100 1001 0010
0001 0000 0110 0110
____ 1000 = 8 < 9 no carry OK
1111
= 15 > 9 must add 6
0110
___1 0101
Now OK (9 + 6 = 15)
____ 0101
= 5 < 9 no carry OK
0010 = 2 < 9 no carry OK
Final BCD Answer = 0010 0101 0101 1000
149210 + 106610 = 255810
Conclusion BCD addition is easy we just have to know
when to correct by adding 6.
Some microprocessors have a decimal adjust addition
(DAA) instruction for automatically performing BCD
addition correction.
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Gray Code
The most common example of a unit distance code (Successive
values differ by only one bit). See Table 2-10 page 52.
Binary code
Gray code
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Decimal
Binary
Gray
0
000
000
1
001
001
2
010
011
3
011
010
4
100
110
5
101
111
6
110
101
7
111
100
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Gray Code conversion
Binary to Gray
• The binary value B = bn…b2 b1 b0 can be
converted to Gray code G = gn…g2 g1 g0.
• With gi = bi+1  bi or G = B  B/2
Examples:
If B =110 then G = 110  011 = 101
If B = 10110111then
G = 10110111
 1011011
= 11101100
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Gray to Binary
From gi = bi  bi+1 it follows that bi = gi  bi+1
•
•
•
•
•
•
•
•
•
•
Example: Let G = 01011111. Then using b8 = 0.
b7 = g7  b8 = 0  0 = 0
b6 = g6  b7 = 1  0 = 1
b5 = g5  b6 = 0  1 = 1
b4 = g4  b5 = 1  1 = 0
b3 = g3  b4 = 1  0 = 1
b2 = g2  b3 = 1  1 = 0
b1 = g1  b2 = 1  0 = 1
b0 = g0  b1 = 1  1 = 0
B = 01101010
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Character Codes
• ASCII most common code for representing
alphanumeric characters.
– 7 bit code. Typically stored in a byte. The 8th bit can be
used for parity or extended characters.
– See Table 2-11 on page 54.
• Microsoft has introduced Unicode, a 16-bit code,
and MBCC to provided windows support for
extended characters (Japanese etc.).
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n-cubes and distance
• The values of an n-bit code can be considered as
the coordinates of vertices of an n-dimensional
Boolean hypercube.
– See figure 2-8 on page 57.
• Useful for visualizing coding concepts.
– See figure 2-9 on page 57.
• Hamming distances – The number of bits in which
the coordinates of two points differ.
– See figure 2-10 page 59.
• m-subcube – a set of 2m vertices in which n-m bits
of the coordinates have the same value.
– See Figure 2-8 Page 57
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