Digital Electronics and Computer Interfacing

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Digital Electronics and Computer Interfacing

• Tim Mewes
• 3. Digital Electronics

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3.1 Digital Information

• Information is stored in two distinct physical
  states
• Charge state of a capacitor (DRAM)
• Magnetization direction (Hard disk, MRAM)
• The two states are referred to as
• TRUE/FALSE (Boolean)
• 1/0
• On/Off
• High/Low

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3.1 Digital Information

• Information can be transmitted using
• discrete Voltage levels
• TTL 1 2.0 V or greater 0 0.8 V or less
• CMOS 1 3.7 V or greater 0 1.3 V or less
• Light
• typical wavelengths 850, 1310, or 1550 nm
• Radio frequencies
• Bluetooth 2.4GHz

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3.2 Digital number representation

• 3.2.1 Unsigned integers
• use base 2 (binary) representation
• 167101?270?261?250?240?231?221?211?20
  1010 01112
• Each digit in the binary representation is called
  a bit
• Eight bits are called a byte
• The largest possible number that can be
  represented by n-bits is 2n-1 (255 in case of a
  byte)
• The leftmost bit is also called most significant
  bit
• The rightmost bit is also called least
  significant bit

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3.2.1 Unsigned integers

• How to convert from base 2 to base 10?
• Example 11001001, n8 bits
• Most significant bit 1?2n-11?27128 ? 128
• 1?2n-21?2664 ? 64
• 0?2n-30?250 ? 0
• 0?2n-40?240 ? 0
• 1?2n-51?238 ? 8
• 0?2n-60?220 ? 0
• 0?2n-70?210 ? 0
• 1?2n-81?201 ? 1

Sum 201
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3.2.1 Unsigned integers

• How to convert from base 10 to base 2?
• Example 100 with n8 bits
• Most significant bit 2n-127128 gt 100 ? 0
• 2n-22664 lt 100 ? 1
• 100-64 36 2n-32532 lt 36 ? 1
• 36-32 4 2n-42416 gt 4 ? 0
• 2n-5238 gt 4 ? 0
• 2n-6224 4 ? 1
• 4-4 0 2n-7212 gt 0 ? 0
• 2n-8201 gt 0 ? 0

10010011001002
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3.2.2 Signed integers

• Sign-and-magnitude
• Use the most significant bit to represent the
  sign
• 0 represents , 1 represents -
• For n-bits numbers from -2n-11 to 2n-1-1 can be
  represented
• Advantage similar to the way we usually indicate
  the sign of a number
• Disadvantage arithmetic calculations tricky
• Zero has two representations -0101000 00002
• 0100000 00002

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3.2.2 Signed integers

• Ones complement
• Negative numbers are represented by complementing
  all the bits (1 ? 0) of the binary representation
  of the magnitude of the number
• 4210 0010 10102-4210 1101 01012
• Zero still has two representations

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3.2.2 Signed integers

• Twos complement
• For negative numbers calculate the ones
  complement and add 1 to the result
• 4210 0010 10102Ones complement
  -4210 1101 01012
• Twos complement -4210 1101 01102
• Zero has only one representation
• Range for n bits -2n-1 to 2n-1-1 (-128 to 127
  for a byte)
• Advantage convenient for computer arithmetic

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3.2.3 Comparison (4-Bit)
Base 10 Unsigned Integer Sign-and-magnitude Ones complement Twos comlement
8 1000 - - -
7 0111 0111 0111 0111
6 0110 0110 0110 0110
5 0101 0101 0101 0101
4 0100 0100 0100 0100
3 0011 0011 0011 0011
2 0010 0010 0010 0010
1 0001 0001 0001 0001
0 0000 0000 0000 0000
-0 - 1000 1111 -
-1 - 1001 1110 1111
-2 - 1010 1101 1110
-3 - 1011 1100 1101
-4 - 1100 1011 1100
-5 - 1101 1010 1011
-6 - 1110 1001 1010
-7 - 1111 1000 1001
-8 - - - 1000
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3.3 Gates

• A logic gate is an arrangement of switches to
  calculate operations using Boolean logic in
  digital circuits
• The output of a gate only depends on its inputs
  and not its history

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3.3 Gates
A
Q
A
Q
B
A
Q
B
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3.3 Gates
A
Q
B
A
Q
B
A
Q
B
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3.4 Boolean algebra

• associativity
• commutativity
• absorption
• complements
• distributivity
• De Morgans theorem

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3.4 Boolean algebra

• How many gates do we really need?

Just one either NAND or NOR (universal gates)!
One can build all other gates using for example
only NAND
NOT
AND
OR
XOR