digital logic rcs
DESCRIPTION
TRANSCRIPT
Digital Logic
Digital Systems Digital systems operate on discrete elements
of information Numbers (eg pocket calculator)
Letters (eg word processor) Pictures (eg digital cameras)
For a digital systems to operate on a continuous data it needs to quantize (digitize) that data first
Covert data into digital representation Digital systems
Cell phone MP3 music player hellip etc
digital camera Ramzi Sh Alqrainy
NumbersEach number is represented by a string of digits in which
the position of each digit has an associated weight
Example
In general any decimal number D of the form
Has the value
Ramzi Sh Alqrainy
What is the range of values of an n-bit number in radix r Minimum value 0
Maximum value rn-1 Number of different values rn
What is the range of values of an 4-bit binary number Minimum value 0
Maximum value 24-1=15 Number of different values 24 = 16
-------------------------------------------------------------------------------
What is the range of values of an 2-bit decimal number Minimum value 0
Maximum value 102-1=99 Number of different values 102 = 100
Ramzi Sh Alqrainy
Number Base Conversions Binary to octal conversion
Starting at the binary point and working left separate the bits into groups of three and replace each group with the corresponding octal digit
(1010011100)2=001 010 011 100(=1234)8
Binary to hexadecimalseparate the bits into groups of four and replace each group with the corresponding hexadecimal digit
(1010011100)2 = 0010 0100 1100( = 29C)16
Ramzi Sh Alqrainy
Conversion of fractionsStarting at the binary point group the binary digits that lie to the right into groups of three or four
(010111)2 = 0101 110( = 056)8
(010111)2 = 01011 1000( = 0 B6)16
Conversion to binary numbersReplace each octal or hexadecimal digit with the corresponding 3 or 4 bit binary string
(4 5 5 6)8
(9 6 E)16
(2414)10=(100101101110)2
Ramzi Sh Alqrainy
COMPLEMENT Complement are used in digital computer for simplifying the
subtraction operation
There are two types of complement for each base-r system
1 -Diminished Radix Complement (r-1)rsquos
2 -Radix Complement rrsquos
the two types are referred to as the 2rsquos complement and 1rsquos complement for binary number
10rsquos complement and 9rsquos complement for decimal number
Ramzi Sh Alqrainy
Observation Subtraction from (rnndash1) will never require a borrow
Diminished radix complement can be computed digit-by-digit
For binary 1 ndash 0 = 1 and 1 ndash 1 = 0
Ex The 1rsquos complement of 1011000 is 0100111
The 1rsquos complement of 0101101 is 1010010
The 1rsquos complement of a binary number is formed by changing 1rsquos to 0rsquos and 0rsquos to 1rsquos
Ramzi Sh Alqrainy
Signed Numbers with Complements 3-bit number
DecimalSigned 2s
complementSigned 1s
complementSigned
Magnitude
+3011011011
+2010010010
+1001001001
0000000000
-0----111100
-1111110101
-2110101110
-3101100111
-4100--------
Ramzi Sh Alqrainy
Signed NumbersHow are signed numbers handled in base 10
~ Plus or minus sign placed in front of numberCan we do that for binary numbers
~ Sign needs to be represented in digital system
~ Only choice are lsquo0rsquo and lsquo1rsquolsquo 0 rsquoindicatesrsquo+lsquo lsquo 1 rsquoindicatesrsquondashlsquo
Examples on five-bit numbers011011110100000 10000
+13 ndash13 +0 ndash0Ramzi Sh Alqrainy
BCD AdditionAddition is done BCD digit by BCD digit
~ 4 bits at the time
Can we use normal binary addition
Problem digits adding up to more than 9
~ Binary addition will result in invalid BCD codes
~ 1010 hellip 1111 are not valid
Solution check if resulting value is greater than 9
~if so add 6
~6 will offset the invalid BCD codes and generate the carry
Ramzi Sh Alqrainy
BCD AdditionExample 184 + 576BCD carry 1 1
0001 1000 0100 184
+ 0101 0111 0110+ 576
Binary sum 0111 10000 1010
Add 6 0110 0110
BCD sum 0111 0110 0000 760
Ramzi Sh Alqrainy
Decimal ArithmeticEverything needs to be 4-bit aligned
1048697 lsquo+rsquo represented by 0 )=lsquo0000rsquo( 1048697 lsquondashrsquo represented by 9 )=lsquo1001rsquo(
1048707 Signed magnitude representation or complements 1048697 Signed magnitude hardly use
1048697 10rsquos complement most common1048707 Example 375 + (ndash240)1048697 Negative numbers represented by 10rsquos complement
1048697 10rsquos complement of 240 is 104 ndash 240 = 97601048697 Addition of all digits anddiscard of end carry 0 375
+ 9 760 0 130
1048697 Sign of result automatically correct
Ramzi Sh Alqrainy
Other Decimal CodesDecimals can be encoded in many ways in 4 bits1048707 Weighted codes
Each bit position is assigned a weighting factor Value of code is sum of weights where bits are lsquo1rsquo
Is BCD a weighted code raquo Yes Itrsquos a 8421 code
1048707 Other weighted code raquo 2421 code )yields non-unique coding(
raquo 84-2-1 code1048707 Self-complementing codes
9rsquos complement by exchanging 1rsquos with 0rsquos and 0rsquos with 1rsquos
Excess-3 and 2421 codes are self-complementing
Ramzi Sh Alqrainy
Other Decimal Codes
Ramzi Sh Alqrainy
Gray Code Imaging you code a 2-bit number with two light
switches connected to light bulbs ~ Can you count binary without causing ambiguity
Probably not ~ Switching from 01 to 10 cannot be done simultaneously on both bits
Either 01-gt00-gt10 or 01-gt11-gt10
This brief error or ambiguity can cause problems
Gray code changes only one bit between consecutivenumbers
Ramzi Sh Alqrainy
Gray Code
Ramzi Sh Alqrainy
ASCII Table
Ramzi Sh Alqrainy
Error-Detecting CodeCommunication between systems can be ldquonoisyrdquo
1048697 Environmental conditions can cause bit flips Error-detecting code
1048697 If one bit is flipped the code becomes invalid 1048697 Communication system can detect that and retransmit
bullldquoParity bitrdquo is an extra bit included with a message to make the total number of 1rsquos either even or odd
1048697 ldquoEven parityrdquo choose parity bit such that of 1rsquos is even 1048697 ldquoOdd parityrdquo choose parity bit such that of 1rsquos is odd
Example 1048697 Data=1000001 even parity=0 odd parity=1 1048697 Data=1010100 even parity=1 odd parity=0
Only one parity bit is used
Ramzi Sh Alqrainy
Binary Storage and Registers How is information stored on a digital system
1048697 Bits are stored in ldquobinary cells rdquo 1048697 Binary cell can have two stable states lsquo0rsquo and lsquo1rsquo
Binary cells are grouped into registers 1048697 n cells make up n-bit register
Size of registers is typically predefined 1048697 Simple microcontroller 8 bits = 1 byte
1048697 Pentium 32 bits = 4 bytes 1048697 Mac G5 64 bits = 8 bytes
Digital system can usually process entire registers
1048697 ldquoRegister transferrdquo operation specify processingRamzi Sh Alqrainy
Register Example
Ramzi Sh Alqrainy
Register Transfer Operations
Ramzi Sh Alqrainy
Digital Systems Digital systems operate on discrete elements
of information Numbers (eg pocket calculator)
Letters (eg word processor) Pictures (eg digital cameras)
For a digital systems to operate on a continuous data it needs to quantize (digitize) that data first
Covert data into digital representation Digital systems
Cell phone MP3 music player hellip etc
digital camera Ramzi Sh Alqrainy
NumbersEach number is represented by a string of digits in which
the position of each digit has an associated weight
Example
In general any decimal number D of the form
Has the value
Ramzi Sh Alqrainy
What is the range of values of an n-bit number in radix r Minimum value 0
Maximum value rn-1 Number of different values rn
What is the range of values of an 4-bit binary number Minimum value 0
Maximum value 24-1=15 Number of different values 24 = 16
-------------------------------------------------------------------------------
What is the range of values of an 2-bit decimal number Minimum value 0
Maximum value 102-1=99 Number of different values 102 = 100
Ramzi Sh Alqrainy
Number Base Conversions Binary to octal conversion
Starting at the binary point and working left separate the bits into groups of three and replace each group with the corresponding octal digit
(1010011100)2=001 010 011 100(=1234)8
Binary to hexadecimalseparate the bits into groups of four and replace each group with the corresponding hexadecimal digit
(1010011100)2 = 0010 0100 1100( = 29C)16
Ramzi Sh Alqrainy
Conversion of fractionsStarting at the binary point group the binary digits that lie to the right into groups of three or four
(010111)2 = 0101 110( = 056)8
(010111)2 = 01011 1000( = 0 B6)16
Conversion to binary numbersReplace each octal or hexadecimal digit with the corresponding 3 or 4 bit binary string
(4 5 5 6)8
(9 6 E)16
(2414)10=(100101101110)2
Ramzi Sh Alqrainy
COMPLEMENT Complement are used in digital computer for simplifying the
subtraction operation
There are two types of complement for each base-r system
1 -Diminished Radix Complement (r-1)rsquos
2 -Radix Complement rrsquos
the two types are referred to as the 2rsquos complement and 1rsquos complement for binary number
10rsquos complement and 9rsquos complement for decimal number
Ramzi Sh Alqrainy
Observation Subtraction from (rnndash1) will never require a borrow
Diminished radix complement can be computed digit-by-digit
For binary 1 ndash 0 = 1 and 1 ndash 1 = 0
Ex The 1rsquos complement of 1011000 is 0100111
The 1rsquos complement of 0101101 is 1010010
The 1rsquos complement of a binary number is formed by changing 1rsquos to 0rsquos and 0rsquos to 1rsquos
Ramzi Sh Alqrainy
Signed Numbers with Complements 3-bit number
DecimalSigned 2s
complementSigned 1s
complementSigned
Magnitude
+3011011011
+2010010010
+1001001001
0000000000
-0----111100
-1111110101
-2110101110
-3101100111
-4100--------
Ramzi Sh Alqrainy
Signed NumbersHow are signed numbers handled in base 10
~ Plus or minus sign placed in front of numberCan we do that for binary numbers
~ Sign needs to be represented in digital system
~ Only choice are lsquo0rsquo and lsquo1rsquolsquo 0 rsquoindicatesrsquo+lsquo lsquo 1 rsquoindicatesrsquondashlsquo
Examples on five-bit numbers011011110100000 10000
+13 ndash13 +0 ndash0Ramzi Sh Alqrainy
BCD AdditionAddition is done BCD digit by BCD digit
~ 4 bits at the time
Can we use normal binary addition
Problem digits adding up to more than 9
~ Binary addition will result in invalid BCD codes
~ 1010 hellip 1111 are not valid
Solution check if resulting value is greater than 9
~if so add 6
~6 will offset the invalid BCD codes and generate the carry
Ramzi Sh Alqrainy
BCD AdditionExample 184 + 576BCD carry 1 1
0001 1000 0100 184
+ 0101 0111 0110+ 576
Binary sum 0111 10000 1010
Add 6 0110 0110
BCD sum 0111 0110 0000 760
Ramzi Sh Alqrainy
Decimal ArithmeticEverything needs to be 4-bit aligned
1048697 lsquo+rsquo represented by 0 )=lsquo0000rsquo( 1048697 lsquondashrsquo represented by 9 )=lsquo1001rsquo(
1048707 Signed magnitude representation or complements 1048697 Signed magnitude hardly use
1048697 10rsquos complement most common1048707 Example 375 + (ndash240)1048697 Negative numbers represented by 10rsquos complement
1048697 10rsquos complement of 240 is 104 ndash 240 = 97601048697 Addition of all digits anddiscard of end carry 0 375
+ 9 760 0 130
1048697 Sign of result automatically correct
Ramzi Sh Alqrainy
Other Decimal CodesDecimals can be encoded in many ways in 4 bits1048707 Weighted codes
Each bit position is assigned a weighting factor Value of code is sum of weights where bits are lsquo1rsquo
Is BCD a weighted code raquo Yes Itrsquos a 8421 code
1048707 Other weighted code raquo 2421 code )yields non-unique coding(
raquo 84-2-1 code1048707 Self-complementing codes
9rsquos complement by exchanging 1rsquos with 0rsquos and 0rsquos with 1rsquos
Excess-3 and 2421 codes are self-complementing
Ramzi Sh Alqrainy
Other Decimal Codes
Ramzi Sh Alqrainy
Gray Code Imaging you code a 2-bit number with two light
switches connected to light bulbs ~ Can you count binary without causing ambiguity
Probably not ~ Switching from 01 to 10 cannot be done simultaneously on both bits
Either 01-gt00-gt10 or 01-gt11-gt10
This brief error or ambiguity can cause problems
Gray code changes only one bit between consecutivenumbers
Ramzi Sh Alqrainy
Gray Code
Ramzi Sh Alqrainy
ASCII Table
Ramzi Sh Alqrainy
Error-Detecting CodeCommunication between systems can be ldquonoisyrdquo
1048697 Environmental conditions can cause bit flips Error-detecting code
1048697 If one bit is flipped the code becomes invalid 1048697 Communication system can detect that and retransmit
bullldquoParity bitrdquo is an extra bit included with a message to make the total number of 1rsquos either even or odd
1048697 ldquoEven parityrdquo choose parity bit such that of 1rsquos is even 1048697 ldquoOdd parityrdquo choose parity bit such that of 1rsquos is odd
Example 1048697 Data=1000001 even parity=0 odd parity=1 1048697 Data=1010100 even parity=1 odd parity=0
Only one parity bit is used
Ramzi Sh Alqrainy
Binary Storage and Registers How is information stored on a digital system
1048697 Bits are stored in ldquobinary cells rdquo 1048697 Binary cell can have two stable states lsquo0rsquo and lsquo1rsquo
Binary cells are grouped into registers 1048697 n cells make up n-bit register
Size of registers is typically predefined 1048697 Simple microcontroller 8 bits = 1 byte
1048697 Pentium 32 bits = 4 bytes 1048697 Mac G5 64 bits = 8 bytes
Digital system can usually process entire registers
1048697 ldquoRegister transferrdquo operation specify processingRamzi Sh Alqrainy
Register Example
Ramzi Sh Alqrainy
Register Transfer Operations
Ramzi Sh Alqrainy
NumbersEach number is represented by a string of digits in which
the position of each digit has an associated weight
Example
In general any decimal number D of the form
Has the value
Ramzi Sh Alqrainy
What is the range of values of an n-bit number in radix r Minimum value 0
Maximum value rn-1 Number of different values rn
What is the range of values of an 4-bit binary number Minimum value 0
Maximum value 24-1=15 Number of different values 24 = 16
-------------------------------------------------------------------------------
What is the range of values of an 2-bit decimal number Minimum value 0
Maximum value 102-1=99 Number of different values 102 = 100
Ramzi Sh Alqrainy
Number Base Conversions Binary to octal conversion
Starting at the binary point and working left separate the bits into groups of three and replace each group with the corresponding octal digit
(1010011100)2=001 010 011 100(=1234)8
Binary to hexadecimalseparate the bits into groups of four and replace each group with the corresponding hexadecimal digit
(1010011100)2 = 0010 0100 1100( = 29C)16
Ramzi Sh Alqrainy
Conversion of fractionsStarting at the binary point group the binary digits that lie to the right into groups of three or four
(010111)2 = 0101 110( = 056)8
(010111)2 = 01011 1000( = 0 B6)16
Conversion to binary numbersReplace each octal or hexadecimal digit with the corresponding 3 or 4 bit binary string
(4 5 5 6)8
(9 6 E)16
(2414)10=(100101101110)2
Ramzi Sh Alqrainy
COMPLEMENT Complement are used in digital computer for simplifying the
subtraction operation
There are two types of complement for each base-r system
1 -Diminished Radix Complement (r-1)rsquos
2 -Radix Complement rrsquos
the two types are referred to as the 2rsquos complement and 1rsquos complement for binary number
10rsquos complement and 9rsquos complement for decimal number
Ramzi Sh Alqrainy
Observation Subtraction from (rnndash1) will never require a borrow
Diminished radix complement can be computed digit-by-digit
For binary 1 ndash 0 = 1 and 1 ndash 1 = 0
Ex The 1rsquos complement of 1011000 is 0100111
The 1rsquos complement of 0101101 is 1010010
The 1rsquos complement of a binary number is formed by changing 1rsquos to 0rsquos and 0rsquos to 1rsquos
Ramzi Sh Alqrainy
Signed Numbers with Complements 3-bit number
DecimalSigned 2s
complementSigned 1s
complementSigned
Magnitude
+3011011011
+2010010010
+1001001001
0000000000
-0----111100
-1111110101
-2110101110
-3101100111
-4100--------
Ramzi Sh Alqrainy
Signed NumbersHow are signed numbers handled in base 10
~ Plus or minus sign placed in front of numberCan we do that for binary numbers
~ Sign needs to be represented in digital system
~ Only choice are lsquo0rsquo and lsquo1rsquolsquo 0 rsquoindicatesrsquo+lsquo lsquo 1 rsquoindicatesrsquondashlsquo
Examples on five-bit numbers011011110100000 10000
+13 ndash13 +0 ndash0Ramzi Sh Alqrainy
BCD AdditionAddition is done BCD digit by BCD digit
~ 4 bits at the time
Can we use normal binary addition
Problem digits adding up to more than 9
~ Binary addition will result in invalid BCD codes
~ 1010 hellip 1111 are not valid
Solution check if resulting value is greater than 9
~if so add 6
~6 will offset the invalid BCD codes and generate the carry
Ramzi Sh Alqrainy
BCD AdditionExample 184 + 576BCD carry 1 1
0001 1000 0100 184
+ 0101 0111 0110+ 576
Binary sum 0111 10000 1010
Add 6 0110 0110
BCD sum 0111 0110 0000 760
Ramzi Sh Alqrainy
Decimal ArithmeticEverything needs to be 4-bit aligned
1048697 lsquo+rsquo represented by 0 )=lsquo0000rsquo( 1048697 lsquondashrsquo represented by 9 )=lsquo1001rsquo(
1048707 Signed magnitude representation or complements 1048697 Signed magnitude hardly use
1048697 10rsquos complement most common1048707 Example 375 + (ndash240)1048697 Negative numbers represented by 10rsquos complement
1048697 10rsquos complement of 240 is 104 ndash 240 = 97601048697 Addition of all digits anddiscard of end carry 0 375
+ 9 760 0 130
1048697 Sign of result automatically correct
Ramzi Sh Alqrainy
Other Decimal CodesDecimals can be encoded in many ways in 4 bits1048707 Weighted codes
Each bit position is assigned a weighting factor Value of code is sum of weights where bits are lsquo1rsquo
Is BCD a weighted code raquo Yes Itrsquos a 8421 code
1048707 Other weighted code raquo 2421 code )yields non-unique coding(
raquo 84-2-1 code1048707 Self-complementing codes
9rsquos complement by exchanging 1rsquos with 0rsquos and 0rsquos with 1rsquos
Excess-3 and 2421 codes are self-complementing
Ramzi Sh Alqrainy
Other Decimal Codes
Ramzi Sh Alqrainy
Gray Code Imaging you code a 2-bit number with two light
switches connected to light bulbs ~ Can you count binary without causing ambiguity
Probably not ~ Switching from 01 to 10 cannot be done simultaneously on both bits
Either 01-gt00-gt10 or 01-gt11-gt10
This brief error or ambiguity can cause problems
Gray code changes only one bit between consecutivenumbers
Ramzi Sh Alqrainy
Gray Code
Ramzi Sh Alqrainy
ASCII Table
Ramzi Sh Alqrainy
Error-Detecting CodeCommunication between systems can be ldquonoisyrdquo
1048697 Environmental conditions can cause bit flips Error-detecting code
1048697 If one bit is flipped the code becomes invalid 1048697 Communication system can detect that and retransmit
bullldquoParity bitrdquo is an extra bit included with a message to make the total number of 1rsquos either even or odd
1048697 ldquoEven parityrdquo choose parity bit such that of 1rsquos is even 1048697 ldquoOdd parityrdquo choose parity bit such that of 1rsquos is odd
Example 1048697 Data=1000001 even parity=0 odd parity=1 1048697 Data=1010100 even parity=1 odd parity=0
Only one parity bit is used
Ramzi Sh Alqrainy
Binary Storage and Registers How is information stored on a digital system
1048697 Bits are stored in ldquobinary cells rdquo 1048697 Binary cell can have two stable states lsquo0rsquo and lsquo1rsquo
Binary cells are grouped into registers 1048697 n cells make up n-bit register
Size of registers is typically predefined 1048697 Simple microcontroller 8 bits = 1 byte
1048697 Pentium 32 bits = 4 bytes 1048697 Mac G5 64 bits = 8 bytes
Digital system can usually process entire registers
1048697 ldquoRegister transferrdquo operation specify processingRamzi Sh Alqrainy
Register Example
Ramzi Sh Alqrainy
Register Transfer Operations
Ramzi Sh Alqrainy
What is the range of values of an n-bit number in radix r Minimum value 0
Maximum value rn-1 Number of different values rn
What is the range of values of an 4-bit binary number Minimum value 0
Maximum value 24-1=15 Number of different values 24 = 16
-------------------------------------------------------------------------------
What is the range of values of an 2-bit decimal number Minimum value 0
Maximum value 102-1=99 Number of different values 102 = 100
Ramzi Sh Alqrainy
Number Base Conversions Binary to octal conversion
Starting at the binary point and working left separate the bits into groups of three and replace each group with the corresponding octal digit
(1010011100)2=001 010 011 100(=1234)8
Binary to hexadecimalseparate the bits into groups of four and replace each group with the corresponding hexadecimal digit
(1010011100)2 = 0010 0100 1100( = 29C)16
Ramzi Sh Alqrainy
Conversion of fractionsStarting at the binary point group the binary digits that lie to the right into groups of three or four
(010111)2 = 0101 110( = 056)8
(010111)2 = 01011 1000( = 0 B6)16
Conversion to binary numbersReplace each octal or hexadecimal digit with the corresponding 3 or 4 bit binary string
(4 5 5 6)8
(9 6 E)16
(2414)10=(100101101110)2
Ramzi Sh Alqrainy
COMPLEMENT Complement are used in digital computer for simplifying the
subtraction operation
There are two types of complement for each base-r system
1 -Diminished Radix Complement (r-1)rsquos
2 -Radix Complement rrsquos
the two types are referred to as the 2rsquos complement and 1rsquos complement for binary number
10rsquos complement and 9rsquos complement for decimal number
Ramzi Sh Alqrainy
Observation Subtraction from (rnndash1) will never require a borrow
Diminished radix complement can be computed digit-by-digit
For binary 1 ndash 0 = 1 and 1 ndash 1 = 0
Ex The 1rsquos complement of 1011000 is 0100111
The 1rsquos complement of 0101101 is 1010010
The 1rsquos complement of a binary number is formed by changing 1rsquos to 0rsquos and 0rsquos to 1rsquos
Ramzi Sh Alqrainy
Signed Numbers with Complements 3-bit number
DecimalSigned 2s
complementSigned 1s
complementSigned
Magnitude
+3011011011
+2010010010
+1001001001
0000000000
-0----111100
-1111110101
-2110101110
-3101100111
-4100--------
Ramzi Sh Alqrainy
Signed NumbersHow are signed numbers handled in base 10
~ Plus or minus sign placed in front of numberCan we do that for binary numbers
~ Sign needs to be represented in digital system
~ Only choice are lsquo0rsquo and lsquo1rsquolsquo 0 rsquoindicatesrsquo+lsquo lsquo 1 rsquoindicatesrsquondashlsquo
Examples on five-bit numbers011011110100000 10000
+13 ndash13 +0 ndash0Ramzi Sh Alqrainy
BCD AdditionAddition is done BCD digit by BCD digit
~ 4 bits at the time
Can we use normal binary addition
Problem digits adding up to more than 9
~ Binary addition will result in invalid BCD codes
~ 1010 hellip 1111 are not valid
Solution check if resulting value is greater than 9
~if so add 6
~6 will offset the invalid BCD codes and generate the carry
Ramzi Sh Alqrainy
BCD AdditionExample 184 + 576BCD carry 1 1
0001 1000 0100 184
+ 0101 0111 0110+ 576
Binary sum 0111 10000 1010
Add 6 0110 0110
BCD sum 0111 0110 0000 760
Ramzi Sh Alqrainy
Decimal ArithmeticEverything needs to be 4-bit aligned
1048697 lsquo+rsquo represented by 0 )=lsquo0000rsquo( 1048697 lsquondashrsquo represented by 9 )=lsquo1001rsquo(
1048707 Signed magnitude representation or complements 1048697 Signed magnitude hardly use
1048697 10rsquos complement most common1048707 Example 375 + (ndash240)1048697 Negative numbers represented by 10rsquos complement
1048697 10rsquos complement of 240 is 104 ndash 240 = 97601048697 Addition of all digits anddiscard of end carry 0 375
+ 9 760 0 130
1048697 Sign of result automatically correct
Ramzi Sh Alqrainy
Other Decimal CodesDecimals can be encoded in many ways in 4 bits1048707 Weighted codes
Each bit position is assigned a weighting factor Value of code is sum of weights where bits are lsquo1rsquo
Is BCD a weighted code raquo Yes Itrsquos a 8421 code
1048707 Other weighted code raquo 2421 code )yields non-unique coding(
raquo 84-2-1 code1048707 Self-complementing codes
9rsquos complement by exchanging 1rsquos with 0rsquos and 0rsquos with 1rsquos
Excess-3 and 2421 codes are self-complementing
Ramzi Sh Alqrainy
Other Decimal Codes
Ramzi Sh Alqrainy
Gray Code Imaging you code a 2-bit number with two light
switches connected to light bulbs ~ Can you count binary without causing ambiguity
Probably not ~ Switching from 01 to 10 cannot be done simultaneously on both bits
Either 01-gt00-gt10 or 01-gt11-gt10
This brief error or ambiguity can cause problems
Gray code changes only one bit between consecutivenumbers
Ramzi Sh Alqrainy
Gray Code
Ramzi Sh Alqrainy
ASCII Table
Ramzi Sh Alqrainy
Error-Detecting CodeCommunication between systems can be ldquonoisyrdquo
1048697 Environmental conditions can cause bit flips Error-detecting code
1048697 If one bit is flipped the code becomes invalid 1048697 Communication system can detect that and retransmit
bullldquoParity bitrdquo is an extra bit included with a message to make the total number of 1rsquos either even or odd
1048697 ldquoEven parityrdquo choose parity bit such that of 1rsquos is even 1048697 ldquoOdd parityrdquo choose parity bit such that of 1rsquos is odd
Example 1048697 Data=1000001 even parity=0 odd parity=1 1048697 Data=1010100 even parity=1 odd parity=0
Only one parity bit is used
Ramzi Sh Alqrainy
Binary Storage and Registers How is information stored on a digital system
1048697 Bits are stored in ldquobinary cells rdquo 1048697 Binary cell can have two stable states lsquo0rsquo and lsquo1rsquo
Binary cells are grouped into registers 1048697 n cells make up n-bit register
Size of registers is typically predefined 1048697 Simple microcontroller 8 bits = 1 byte
1048697 Pentium 32 bits = 4 bytes 1048697 Mac G5 64 bits = 8 bytes
Digital system can usually process entire registers
1048697 ldquoRegister transferrdquo operation specify processingRamzi Sh Alqrainy
Register Example
Ramzi Sh Alqrainy
Register Transfer Operations
Ramzi Sh Alqrainy
Number Base Conversions Binary to octal conversion
Starting at the binary point and working left separate the bits into groups of three and replace each group with the corresponding octal digit
(1010011100)2=001 010 011 100(=1234)8
Binary to hexadecimalseparate the bits into groups of four and replace each group with the corresponding hexadecimal digit
(1010011100)2 = 0010 0100 1100( = 29C)16
Ramzi Sh Alqrainy
Conversion of fractionsStarting at the binary point group the binary digits that lie to the right into groups of three or four
(010111)2 = 0101 110( = 056)8
(010111)2 = 01011 1000( = 0 B6)16
Conversion to binary numbersReplace each octal or hexadecimal digit with the corresponding 3 or 4 bit binary string
(4 5 5 6)8
(9 6 E)16
(2414)10=(100101101110)2
Ramzi Sh Alqrainy
COMPLEMENT Complement are used in digital computer for simplifying the
subtraction operation
There are two types of complement for each base-r system
1 -Diminished Radix Complement (r-1)rsquos
2 -Radix Complement rrsquos
the two types are referred to as the 2rsquos complement and 1rsquos complement for binary number
10rsquos complement and 9rsquos complement for decimal number
Ramzi Sh Alqrainy
Observation Subtraction from (rnndash1) will never require a borrow
Diminished radix complement can be computed digit-by-digit
For binary 1 ndash 0 = 1 and 1 ndash 1 = 0
Ex The 1rsquos complement of 1011000 is 0100111
The 1rsquos complement of 0101101 is 1010010
The 1rsquos complement of a binary number is formed by changing 1rsquos to 0rsquos and 0rsquos to 1rsquos
Ramzi Sh Alqrainy
Signed Numbers with Complements 3-bit number
DecimalSigned 2s
complementSigned 1s
complementSigned
Magnitude
+3011011011
+2010010010
+1001001001
0000000000
-0----111100
-1111110101
-2110101110
-3101100111
-4100--------
Ramzi Sh Alqrainy
Signed NumbersHow are signed numbers handled in base 10
~ Plus or minus sign placed in front of numberCan we do that for binary numbers
~ Sign needs to be represented in digital system
~ Only choice are lsquo0rsquo and lsquo1rsquolsquo 0 rsquoindicatesrsquo+lsquo lsquo 1 rsquoindicatesrsquondashlsquo
Examples on five-bit numbers011011110100000 10000
+13 ndash13 +0 ndash0Ramzi Sh Alqrainy
BCD AdditionAddition is done BCD digit by BCD digit
~ 4 bits at the time
Can we use normal binary addition
Problem digits adding up to more than 9
~ Binary addition will result in invalid BCD codes
~ 1010 hellip 1111 are not valid
Solution check if resulting value is greater than 9
~if so add 6
~6 will offset the invalid BCD codes and generate the carry
Ramzi Sh Alqrainy
BCD AdditionExample 184 + 576BCD carry 1 1
0001 1000 0100 184
+ 0101 0111 0110+ 576
Binary sum 0111 10000 1010
Add 6 0110 0110
BCD sum 0111 0110 0000 760
Ramzi Sh Alqrainy
Decimal ArithmeticEverything needs to be 4-bit aligned
1048697 lsquo+rsquo represented by 0 )=lsquo0000rsquo( 1048697 lsquondashrsquo represented by 9 )=lsquo1001rsquo(
1048707 Signed magnitude representation or complements 1048697 Signed magnitude hardly use
1048697 10rsquos complement most common1048707 Example 375 + (ndash240)1048697 Negative numbers represented by 10rsquos complement
1048697 10rsquos complement of 240 is 104 ndash 240 = 97601048697 Addition of all digits anddiscard of end carry 0 375
+ 9 760 0 130
1048697 Sign of result automatically correct
Ramzi Sh Alqrainy
Other Decimal CodesDecimals can be encoded in many ways in 4 bits1048707 Weighted codes
Each bit position is assigned a weighting factor Value of code is sum of weights where bits are lsquo1rsquo
Is BCD a weighted code raquo Yes Itrsquos a 8421 code
1048707 Other weighted code raquo 2421 code )yields non-unique coding(
raquo 84-2-1 code1048707 Self-complementing codes
9rsquos complement by exchanging 1rsquos with 0rsquos and 0rsquos with 1rsquos
Excess-3 and 2421 codes are self-complementing
Ramzi Sh Alqrainy
Other Decimal Codes
Ramzi Sh Alqrainy
Gray Code Imaging you code a 2-bit number with two light
switches connected to light bulbs ~ Can you count binary without causing ambiguity
Probably not ~ Switching from 01 to 10 cannot be done simultaneously on both bits
Either 01-gt00-gt10 or 01-gt11-gt10
This brief error or ambiguity can cause problems
Gray code changes only one bit between consecutivenumbers
Ramzi Sh Alqrainy
Gray Code
Ramzi Sh Alqrainy
ASCII Table
Ramzi Sh Alqrainy
Error-Detecting CodeCommunication between systems can be ldquonoisyrdquo
1048697 Environmental conditions can cause bit flips Error-detecting code
1048697 If one bit is flipped the code becomes invalid 1048697 Communication system can detect that and retransmit
bullldquoParity bitrdquo is an extra bit included with a message to make the total number of 1rsquos either even or odd
1048697 ldquoEven parityrdquo choose parity bit such that of 1rsquos is even 1048697 ldquoOdd parityrdquo choose parity bit such that of 1rsquos is odd
Example 1048697 Data=1000001 even parity=0 odd parity=1 1048697 Data=1010100 even parity=1 odd parity=0
Only one parity bit is used
Ramzi Sh Alqrainy
Binary Storage and Registers How is information stored on a digital system
1048697 Bits are stored in ldquobinary cells rdquo 1048697 Binary cell can have two stable states lsquo0rsquo and lsquo1rsquo
Binary cells are grouped into registers 1048697 n cells make up n-bit register
Size of registers is typically predefined 1048697 Simple microcontroller 8 bits = 1 byte
1048697 Pentium 32 bits = 4 bytes 1048697 Mac G5 64 bits = 8 bytes
Digital system can usually process entire registers
1048697 ldquoRegister transferrdquo operation specify processingRamzi Sh Alqrainy
Register Example
Ramzi Sh Alqrainy
Register Transfer Operations
Ramzi Sh Alqrainy
Conversion of fractionsStarting at the binary point group the binary digits that lie to the right into groups of three or four
(010111)2 = 0101 110( = 056)8
(010111)2 = 01011 1000( = 0 B6)16
Conversion to binary numbersReplace each octal or hexadecimal digit with the corresponding 3 or 4 bit binary string
(4 5 5 6)8
(9 6 E)16
(2414)10=(100101101110)2
Ramzi Sh Alqrainy
COMPLEMENT Complement are used in digital computer for simplifying the
subtraction operation
There are two types of complement for each base-r system
1 -Diminished Radix Complement (r-1)rsquos
2 -Radix Complement rrsquos
the two types are referred to as the 2rsquos complement and 1rsquos complement for binary number
10rsquos complement and 9rsquos complement for decimal number
Ramzi Sh Alqrainy
Observation Subtraction from (rnndash1) will never require a borrow
Diminished radix complement can be computed digit-by-digit
For binary 1 ndash 0 = 1 and 1 ndash 1 = 0
Ex The 1rsquos complement of 1011000 is 0100111
The 1rsquos complement of 0101101 is 1010010
The 1rsquos complement of a binary number is formed by changing 1rsquos to 0rsquos and 0rsquos to 1rsquos
Ramzi Sh Alqrainy
Signed Numbers with Complements 3-bit number
DecimalSigned 2s
complementSigned 1s
complementSigned
Magnitude
+3011011011
+2010010010
+1001001001
0000000000
-0----111100
-1111110101
-2110101110
-3101100111
-4100--------
Ramzi Sh Alqrainy
Signed NumbersHow are signed numbers handled in base 10
~ Plus or minus sign placed in front of numberCan we do that for binary numbers
~ Sign needs to be represented in digital system
~ Only choice are lsquo0rsquo and lsquo1rsquolsquo 0 rsquoindicatesrsquo+lsquo lsquo 1 rsquoindicatesrsquondashlsquo
Examples on five-bit numbers011011110100000 10000
+13 ndash13 +0 ndash0Ramzi Sh Alqrainy
BCD AdditionAddition is done BCD digit by BCD digit
~ 4 bits at the time
Can we use normal binary addition
Problem digits adding up to more than 9
~ Binary addition will result in invalid BCD codes
~ 1010 hellip 1111 are not valid
Solution check if resulting value is greater than 9
~if so add 6
~6 will offset the invalid BCD codes and generate the carry
Ramzi Sh Alqrainy
BCD AdditionExample 184 + 576BCD carry 1 1
0001 1000 0100 184
+ 0101 0111 0110+ 576
Binary sum 0111 10000 1010
Add 6 0110 0110
BCD sum 0111 0110 0000 760
Ramzi Sh Alqrainy
Decimal ArithmeticEverything needs to be 4-bit aligned
1048697 lsquo+rsquo represented by 0 )=lsquo0000rsquo( 1048697 lsquondashrsquo represented by 9 )=lsquo1001rsquo(
1048707 Signed magnitude representation or complements 1048697 Signed magnitude hardly use
1048697 10rsquos complement most common1048707 Example 375 + (ndash240)1048697 Negative numbers represented by 10rsquos complement
1048697 10rsquos complement of 240 is 104 ndash 240 = 97601048697 Addition of all digits anddiscard of end carry 0 375
+ 9 760 0 130
1048697 Sign of result automatically correct
Ramzi Sh Alqrainy
Other Decimal CodesDecimals can be encoded in many ways in 4 bits1048707 Weighted codes
Each bit position is assigned a weighting factor Value of code is sum of weights where bits are lsquo1rsquo
Is BCD a weighted code raquo Yes Itrsquos a 8421 code
1048707 Other weighted code raquo 2421 code )yields non-unique coding(
raquo 84-2-1 code1048707 Self-complementing codes
9rsquos complement by exchanging 1rsquos with 0rsquos and 0rsquos with 1rsquos
Excess-3 and 2421 codes are self-complementing
Ramzi Sh Alqrainy
Other Decimal Codes
Ramzi Sh Alqrainy
Gray Code Imaging you code a 2-bit number with two light
switches connected to light bulbs ~ Can you count binary without causing ambiguity
Probably not ~ Switching from 01 to 10 cannot be done simultaneously on both bits
Either 01-gt00-gt10 or 01-gt11-gt10
This brief error or ambiguity can cause problems
Gray code changes only one bit between consecutivenumbers
Ramzi Sh Alqrainy
Gray Code
Ramzi Sh Alqrainy
ASCII Table
Ramzi Sh Alqrainy
Error-Detecting CodeCommunication between systems can be ldquonoisyrdquo
1048697 Environmental conditions can cause bit flips Error-detecting code
1048697 If one bit is flipped the code becomes invalid 1048697 Communication system can detect that and retransmit
bullldquoParity bitrdquo is an extra bit included with a message to make the total number of 1rsquos either even or odd
1048697 ldquoEven parityrdquo choose parity bit such that of 1rsquos is even 1048697 ldquoOdd parityrdquo choose parity bit such that of 1rsquos is odd
Example 1048697 Data=1000001 even parity=0 odd parity=1 1048697 Data=1010100 even parity=1 odd parity=0
Only one parity bit is used
Ramzi Sh Alqrainy
Binary Storage and Registers How is information stored on a digital system
1048697 Bits are stored in ldquobinary cells rdquo 1048697 Binary cell can have two stable states lsquo0rsquo and lsquo1rsquo
Binary cells are grouped into registers 1048697 n cells make up n-bit register
Size of registers is typically predefined 1048697 Simple microcontroller 8 bits = 1 byte
1048697 Pentium 32 bits = 4 bytes 1048697 Mac G5 64 bits = 8 bytes
Digital system can usually process entire registers
1048697 ldquoRegister transferrdquo operation specify processingRamzi Sh Alqrainy
Register Example
Ramzi Sh Alqrainy
Register Transfer Operations
Ramzi Sh Alqrainy
COMPLEMENT Complement are used in digital computer for simplifying the
subtraction operation
There are two types of complement for each base-r system
1 -Diminished Radix Complement (r-1)rsquos
2 -Radix Complement rrsquos
the two types are referred to as the 2rsquos complement and 1rsquos complement for binary number
10rsquos complement and 9rsquos complement for decimal number
Ramzi Sh Alqrainy
Observation Subtraction from (rnndash1) will never require a borrow
Diminished radix complement can be computed digit-by-digit
For binary 1 ndash 0 = 1 and 1 ndash 1 = 0
Ex The 1rsquos complement of 1011000 is 0100111
The 1rsquos complement of 0101101 is 1010010
The 1rsquos complement of a binary number is formed by changing 1rsquos to 0rsquos and 0rsquos to 1rsquos
Ramzi Sh Alqrainy
Signed Numbers with Complements 3-bit number
DecimalSigned 2s
complementSigned 1s
complementSigned
Magnitude
+3011011011
+2010010010
+1001001001
0000000000
-0----111100
-1111110101
-2110101110
-3101100111
-4100--------
Ramzi Sh Alqrainy
Signed NumbersHow are signed numbers handled in base 10
~ Plus or minus sign placed in front of numberCan we do that for binary numbers
~ Sign needs to be represented in digital system
~ Only choice are lsquo0rsquo and lsquo1rsquolsquo 0 rsquoindicatesrsquo+lsquo lsquo 1 rsquoindicatesrsquondashlsquo
Examples on five-bit numbers011011110100000 10000
+13 ndash13 +0 ndash0Ramzi Sh Alqrainy
BCD AdditionAddition is done BCD digit by BCD digit
~ 4 bits at the time
Can we use normal binary addition
Problem digits adding up to more than 9
~ Binary addition will result in invalid BCD codes
~ 1010 hellip 1111 are not valid
Solution check if resulting value is greater than 9
~if so add 6
~6 will offset the invalid BCD codes and generate the carry
Ramzi Sh Alqrainy
BCD AdditionExample 184 + 576BCD carry 1 1
0001 1000 0100 184
+ 0101 0111 0110+ 576
Binary sum 0111 10000 1010
Add 6 0110 0110
BCD sum 0111 0110 0000 760
Ramzi Sh Alqrainy
Decimal ArithmeticEverything needs to be 4-bit aligned
1048697 lsquo+rsquo represented by 0 )=lsquo0000rsquo( 1048697 lsquondashrsquo represented by 9 )=lsquo1001rsquo(
1048707 Signed magnitude representation or complements 1048697 Signed magnitude hardly use
1048697 10rsquos complement most common1048707 Example 375 + (ndash240)1048697 Negative numbers represented by 10rsquos complement
1048697 10rsquos complement of 240 is 104 ndash 240 = 97601048697 Addition of all digits anddiscard of end carry 0 375
+ 9 760 0 130
1048697 Sign of result automatically correct
Ramzi Sh Alqrainy
Other Decimal CodesDecimals can be encoded in many ways in 4 bits1048707 Weighted codes
Each bit position is assigned a weighting factor Value of code is sum of weights where bits are lsquo1rsquo
Is BCD a weighted code raquo Yes Itrsquos a 8421 code
1048707 Other weighted code raquo 2421 code )yields non-unique coding(
raquo 84-2-1 code1048707 Self-complementing codes
9rsquos complement by exchanging 1rsquos with 0rsquos and 0rsquos with 1rsquos
Excess-3 and 2421 codes are self-complementing
Ramzi Sh Alqrainy
Other Decimal Codes
Ramzi Sh Alqrainy
Gray Code Imaging you code a 2-bit number with two light
switches connected to light bulbs ~ Can you count binary without causing ambiguity
Probably not ~ Switching from 01 to 10 cannot be done simultaneously on both bits
Either 01-gt00-gt10 or 01-gt11-gt10
This brief error or ambiguity can cause problems
Gray code changes only one bit between consecutivenumbers
Ramzi Sh Alqrainy
Gray Code
Ramzi Sh Alqrainy
ASCII Table
Ramzi Sh Alqrainy
Error-Detecting CodeCommunication between systems can be ldquonoisyrdquo
1048697 Environmental conditions can cause bit flips Error-detecting code
1048697 If one bit is flipped the code becomes invalid 1048697 Communication system can detect that and retransmit
bullldquoParity bitrdquo is an extra bit included with a message to make the total number of 1rsquos either even or odd
1048697 ldquoEven parityrdquo choose parity bit such that of 1rsquos is even 1048697 ldquoOdd parityrdquo choose parity bit such that of 1rsquos is odd
Example 1048697 Data=1000001 even parity=0 odd parity=1 1048697 Data=1010100 even parity=1 odd parity=0
Only one parity bit is used
Ramzi Sh Alqrainy
Binary Storage and Registers How is information stored on a digital system
1048697 Bits are stored in ldquobinary cells rdquo 1048697 Binary cell can have two stable states lsquo0rsquo and lsquo1rsquo
Binary cells are grouped into registers 1048697 n cells make up n-bit register
Size of registers is typically predefined 1048697 Simple microcontroller 8 bits = 1 byte
1048697 Pentium 32 bits = 4 bytes 1048697 Mac G5 64 bits = 8 bytes
Digital system can usually process entire registers
1048697 ldquoRegister transferrdquo operation specify processingRamzi Sh Alqrainy
Register Example
Ramzi Sh Alqrainy
Register Transfer Operations
Ramzi Sh Alqrainy
Observation Subtraction from (rnndash1) will never require a borrow
Diminished radix complement can be computed digit-by-digit
For binary 1 ndash 0 = 1 and 1 ndash 1 = 0
Ex The 1rsquos complement of 1011000 is 0100111
The 1rsquos complement of 0101101 is 1010010
The 1rsquos complement of a binary number is formed by changing 1rsquos to 0rsquos and 0rsquos to 1rsquos
Ramzi Sh Alqrainy
Signed Numbers with Complements 3-bit number
DecimalSigned 2s
complementSigned 1s
complementSigned
Magnitude
+3011011011
+2010010010
+1001001001
0000000000
-0----111100
-1111110101
-2110101110
-3101100111
-4100--------
Ramzi Sh Alqrainy
Signed NumbersHow are signed numbers handled in base 10
~ Plus or minus sign placed in front of numberCan we do that for binary numbers
~ Sign needs to be represented in digital system
~ Only choice are lsquo0rsquo and lsquo1rsquolsquo 0 rsquoindicatesrsquo+lsquo lsquo 1 rsquoindicatesrsquondashlsquo
Examples on five-bit numbers011011110100000 10000
+13 ndash13 +0 ndash0Ramzi Sh Alqrainy
BCD AdditionAddition is done BCD digit by BCD digit
~ 4 bits at the time
Can we use normal binary addition
Problem digits adding up to more than 9
~ Binary addition will result in invalid BCD codes
~ 1010 hellip 1111 are not valid
Solution check if resulting value is greater than 9
~if so add 6
~6 will offset the invalid BCD codes and generate the carry
Ramzi Sh Alqrainy
BCD AdditionExample 184 + 576BCD carry 1 1
0001 1000 0100 184
+ 0101 0111 0110+ 576
Binary sum 0111 10000 1010
Add 6 0110 0110
BCD sum 0111 0110 0000 760
Ramzi Sh Alqrainy
Decimal ArithmeticEverything needs to be 4-bit aligned
1048697 lsquo+rsquo represented by 0 )=lsquo0000rsquo( 1048697 lsquondashrsquo represented by 9 )=lsquo1001rsquo(
1048707 Signed magnitude representation or complements 1048697 Signed magnitude hardly use
1048697 10rsquos complement most common1048707 Example 375 + (ndash240)1048697 Negative numbers represented by 10rsquos complement
1048697 10rsquos complement of 240 is 104 ndash 240 = 97601048697 Addition of all digits anddiscard of end carry 0 375
+ 9 760 0 130
1048697 Sign of result automatically correct
Ramzi Sh Alqrainy
Other Decimal CodesDecimals can be encoded in many ways in 4 bits1048707 Weighted codes
Each bit position is assigned a weighting factor Value of code is sum of weights where bits are lsquo1rsquo
Is BCD a weighted code raquo Yes Itrsquos a 8421 code
1048707 Other weighted code raquo 2421 code )yields non-unique coding(
raquo 84-2-1 code1048707 Self-complementing codes
9rsquos complement by exchanging 1rsquos with 0rsquos and 0rsquos with 1rsquos
Excess-3 and 2421 codes are self-complementing
Ramzi Sh Alqrainy
Other Decimal Codes
Ramzi Sh Alqrainy
Gray Code Imaging you code a 2-bit number with two light
switches connected to light bulbs ~ Can you count binary without causing ambiguity
Probably not ~ Switching from 01 to 10 cannot be done simultaneously on both bits
Either 01-gt00-gt10 or 01-gt11-gt10
This brief error or ambiguity can cause problems
Gray code changes only one bit between consecutivenumbers
Ramzi Sh Alqrainy
Gray Code
Ramzi Sh Alqrainy
ASCII Table
Ramzi Sh Alqrainy
Error-Detecting CodeCommunication between systems can be ldquonoisyrdquo
1048697 Environmental conditions can cause bit flips Error-detecting code
1048697 If one bit is flipped the code becomes invalid 1048697 Communication system can detect that and retransmit
bullldquoParity bitrdquo is an extra bit included with a message to make the total number of 1rsquos either even or odd
1048697 ldquoEven parityrdquo choose parity bit such that of 1rsquos is even 1048697 ldquoOdd parityrdquo choose parity bit such that of 1rsquos is odd
Example 1048697 Data=1000001 even parity=0 odd parity=1 1048697 Data=1010100 even parity=1 odd parity=0
Only one parity bit is used
Ramzi Sh Alqrainy
Binary Storage and Registers How is information stored on a digital system
1048697 Bits are stored in ldquobinary cells rdquo 1048697 Binary cell can have two stable states lsquo0rsquo and lsquo1rsquo
Binary cells are grouped into registers 1048697 n cells make up n-bit register
Size of registers is typically predefined 1048697 Simple microcontroller 8 bits = 1 byte
1048697 Pentium 32 bits = 4 bytes 1048697 Mac G5 64 bits = 8 bytes
Digital system can usually process entire registers
1048697 ldquoRegister transferrdquo operation specify processingRamzi Sh Alqrainy
Register Example
Ramzi Sh Alqrainy
Register Transfer Operations
Ramzi Sh Alqrainy
Signed Numbers with Complements 3-bit number
DecimalSigned 2s
complementSigned 1s
complementSigned
Magnitude
+3011011011
+2010010010
+1001001001
0000000000
-0----111100
-1111110101
-2110101110
-3101100111
-4100--------
Ramzi Sh Alqrainy
Signed NumbersHow are signed numbers handled in base 10
~ Plus or minus sign placed in front of numberCan we do that for binary numbers
~ Sign needs to be represented in digital system
~ Only choice are lsquo0rsquo and lsquo1rsquolsquo 0 rsquoindicatesrsquo+lsquo lsquo 1 rsquoindicatesrsquondashlsquo
Examples on five-bit numbers011011110100000 10000
+13 ndash13 +0 ndash0Ramzi Sh Alqrainy
BCD AdditionAddition is done BCD digit by BCD digit
~ 4 bits at the time
Can we use normal binary addition
Problem digits adding up to more than 9
~ Binary addition will result in invalid BCD codes
~ 1010 hellip 1111 are not valid
Solution check if resulting value is greater than 9
~if so add 6
~6 will offset the invalid BCD codes and generate the carry
Ramzi Sh Alqrainy
BCD AdditionExample 184 + 576BCD carry 1 1
0001 1000 0100 184
+ 0101 0111 0110+ 576
Binary sum 0111 10000 1010
Add 6 0110 0110
BCD sum 0111 0110 0000 760
Ramzi Sh Alqrainy
Decimal ArithmeticEverything needs to be 4-bit aligned
1048697 lsquo+rsquo represented by 0 )=lsquo0000rsquo( 1048697 lsquondashrsquo represented by 9 )=lsquo1001rsquo(
1048707 Signed magnitude representation or complements 1048697 Signed magnitude hardly use
1048697 10rsquos complement most common1048707 Example 375 + (ndash240)1048697 Negative numbers represented by 10rsquos complement
1048697 10rsquos complement of 240 is 104 ndash 240 = 97601048697 Addition of all digits anddiscard of end carry 0 375
+ 9 760 0 130
1048697 Sign of result automatically correct
Ramzi Sh Alqrainy
Other Decimal CodesDecimals can be encoded in many ways in 4 bits1048707 Weighted codes
Each bit position is assigned a weighting factor Value of code is sum of weights where bits are lsquo1rsquo
Is BCD a weighted code raquo Yes Itrsquos a 8421 code
1048707 Other weighted code raquo 2421 code )yields non-unique coding(
raquo 84-2-1 code1048707 Self-complementing codes
9rsquos complement by exchanging 1rsquos with 0rsquos and 0rsquos with 1rsquos
Excess-3 and 2421 codes are self-complementing
Ramzi Sh Alqrainy
Other Decimal Codes
Ramzi Sh Alqrainy
Gray Code Imaging you code a 2-bit number with two light
switches connected to light bulbs ~ Can you count binary without causing ambiguity
Probably not ~ Switching from 01 to 10 cannot be done simultaneously on both bits
Either 01-gt00-gt10 or 01-gt11-gt10
This brief error or ambiguity can cause problems
Gray code changes only one bit between consecutivenumbers
Ramzi Sh Alqrainy
Gray Code
Ramzi Sh Alqrainy
ASCII Table
Ramzi Sh Alqrainy
Error-Detecting CodeCommunication between systems can be ldquonoisyrdquo
1048697 Environmental conditions can cause bit flips Error-detecting code
1048697 If one bit is flipped the code becomes invalid 1048697 Communication system can detect that and retransmit
bullldquoParity bitrdquo is an extra bit included with a message to make the total number of 1rsquos either even or odd
1048697 ldquoEven parityrdquo choose parity bit such that of 1rsquos is even 1048697 ldquoOdd parityrdquo choose parity bit such that of 1rsquos is odd
Example 1048697 Data=1000001 even parity=0 odd parity=1 1048697 Data=1010100 even parity=1 odd parity=0
Only one parity bit is used
Ramzi Sh Alqrainy
Binary Storage and Registers How is information stored on a digital system
1048697 Bits are stored in ldquobinary cells rdquo 1048697 Binary cell can have two stable states lsquo0rsquo and lsquo1rsquo
Binary cells are grouped into registers 1048697 n cells make up n-bit register
Size of registers is typically predefined 1048697 Simple microcontroller 8 bits = 1 byte
1048697 Pentium 32 bits = 4 bytes 1048697 Mac G5 64 bits = 8 bytes
Digital system can usually process entire registers
1048697 ldquoRegister transferrdquo operation specify processingRamzi Sh Alqrainy
Register Example
Ramzi Sh Alqrainy
Register Transfer Operations
Ramzi Sh Alqrainy
Signed NumbersHow are signed numbers handled in base 10
~ Plus or minus sign placed in front of numberCan we do that for binary numbers
~ Sign needs to be represented in digital system
~ Only choice are lsquo0rsquo and lsquo1rsquolsquo 0 rsquoindicatesrsquo+lsquo lsquo 1 rsquoindicatesrsquondashlsquo
Examples on five-bit numbers011011110100000 10000
+13 ndash13 +0 ndash0Ramzi Sh Alqrainy
BCD AdditionAddition is done BCD digit by BCD digit
~ 4 bits at the time
Can we use normal binary addition
Problem digits adding up to more than 9
~ Binary addition will result in invalid BCD codes
~ 1010 hellip 1111 are not valid
Solution check if resulting value is greater than 9
~if so add 6
~6 will offset the invalid BCD codes and generate the carry
Ramzi Sh Alqrainy
BCD AdditionExample 184 + 576BCD carry 1 1
0001 1000 0100 184
+ 0101 0111 0110+ 576
Binary sum 0111 10000 1010
Add 6 0110 0110
BCD sum 0111 0110 0000 760
Ramzi Sh Alqrainy
Decimal ArithmeticEverything needs to be 4-bit aligned
1048697 lsquo+rsquo represented by 0 )=lsquo0000rsquo( 1048697 lsquondashrsquo represented by 9 )=lsquo1001rsquo(
1048707 Signed magnitude representation or complements 1048697 Signed magnitude hardly use
1048697 10rsquos complement most common1048707 Example 375 + (ndash240)1048697 Negative numbers represented by 10rsquos complement
1048697 10rsquos complement of 240 is 104 ndash 240 = 97601048697 Addition of all digits anddiscard of end carry 0 375
+ 9 760 0 130
1048697 Sign of result automatically correct
Ramzi Sh Alqrainy
Other Decimal CodesDecimals can be encoded in many ways in 4 bits1048707 Weighted codes
Each bit position is assigned a weighting factor Value of code is sum of weights where bits are lsquo1rsquo
Is BCD a weighted code raquo Yes Itrsquos a 8421 code
1048707 Other weighted code raquo 2421 code )yields non-unique coding(
raquo 84-2-1 code1048707 Self-complementing codes
9rsquos complement by exchanging 1rsquos with 0rsquos and 0rsquos with 1rsquos
Excess-3 and 2421 codes are self-complementing
Ramzi Sh Alqrainy
Other Decimal Codes
Ramzi Sh Alqrainy
Gray Code Imaging you code a 2-bit number with two light
switches connected to light bulbs ~ Can you count binary without causing ambiguity
Probably not ~ Switching from 01 to 10 cannot be done simultaneously on both bits
Either 01-gt00-gt10 or 01-gt11-gt10
This brief error or ambiguity can cause problems
Gray code changes only one bit between consecutivenumbers
Ramzi Sh Alqrainy
Gray Code
Ramzi Sh Alqrainy
ASCII Table
Ramzi Sh Alqrainy
Error-Detecting CodeCommunication between systems can be ldquonoisyrdquo
1048697 Environmental conditions can cause bit flips Error-detecting code
1048697 If one bit is flipped the code becomes invalid 1048697 Communication system can detect that and retransmit
bullldquoParity bitrdquo is an extra bit included with a message to make the total number of 1rsquos either even or odd
1048697 ldquoEven parityrdquo choose parity bit such that of 1rsquos is even 1048697 ldquoOdd parityrdquo choose parity bit such that of 1rsquos is odd
Example 1048697 Data=1000001 even parity=0 odd parity=1 1048697 Data=1010100 even parity=1 odd parity=0
Only one parity bit is used
Ramzi Sh Alqrainy
Binary Storage and Registers How is information stored on a digital system
1048697 Bits are stored in ldquobinary cells rdquo 1048697 Binary cell can have two stable states lsquo0rsquo and lsquo1rsquo
Binary cells are grouped into registers 1048697 n cells make up n-bit register
Size of registers is typically predefined 1048697 Simple microcontroller 8 bits = 1 byte
1048697 Pentium 32 bits = 4 bytes 1048697 Mac G5 64 bits = 8 bytes
Digital system can usually process entire registers
1048697 ldquoRegister transferrdquo operation specify processingRamzi Sh Alqrainy
Register Example
Ramzi Sh Alqrainy
Register Transfer Operations
Ramzi Sh Alqrainy
BCD AdditionAddition is done BCD digit by BCD digit
~ 4 bits at the time
Can we use normal binary addition
Problem digits adding up to more than 9
~ Binary addition will result in invalid BCD codes
~ 1010 hellip 1111 are not valid
Solution check if resulting value is greater than 9
~if so add 6
~6 will offset the invalid BCD codes and generate the carry
Ramzi Sh Alqrainy
BCD AdditionExample 184 + 576BCD carry 1 1
0001 1000 0100 184
+ 0101 0111 0110+ 576
Binary sum 0111 10000 1010
Add 6 0110 0110
BCD sum 0111 0110 0000 760
Ramzi Sh Alqrainy
Decimal ArithmeticEverything needs to be 4-bit aligned
1048697 lsquo+rsquo represented by 0 )=lsquo0000rsquo( 1048697 lsquondashrsquo represented by 9 )=lsquo1001rsquo(
1048707 Signed magnitude representation or complements 1048697 Signed magnitude hardly use
1048697 10rsquos complement most common1048707 Example 375 + (ndash240)1048697 Negative numbers represented by 10rsquos complement
1048697 10rsquos complement of 240 is 104 ndash 240 = 97601048697 Addition of all digits anddiscard of end carry 0 375
+ 9 760 0 130
1048697 Sign of result automatically correct
Ramzi Sh Alqrainy
Other Decimal CodesDecimals can be encoded in many ways in 4 bits1048707 Weighted codes
Each bit position is assigned a weighting factor Value of code is sum of weights where bits are lsquo1rsquo
Is BCD a weighted code raquo Yes Itrsquos a 8421 code
1048707 Other weighted code raquo 2421 code )yields non-unique coding(
raquo 84-2-1 code1048707 Self-complementing codes
9rsquos complement by exchanging 1rsquos with 0rsquos and 0rsquos with 1rsquos
Excess-3 and 2421 codes are self-complementing
Ramzi Sh Alqrainy
Other Decimal Codes
Ramzi Sh Alqrainy
Gray Code Imaging you code a 2-bit number with two light
switches connected to light bulbs ~ Can you count binary without causing ambiguity
Probably not ~ Switching from 01 to 10 cannot be done simultaneously on both bits
Either 01-gt00-gt10 or 01-gt11-gt10
This brief error or ambiguity can cause problems
Gray code changes only one bit between consecutivenumbers
Ramzi Sh Alqrainy
Gray Code
Ramzi Sh Alqrainy
ASCII Table
Ramzi Sh Alqrainy
Error-Detecting CodeCommunication between systems can be ldquonoisyrdquo
1048697 Environmental conditions can cause bit flips Error-detecting code
1048697 If one bit is flipped the code becomes invalid 1048697 Communication system can detect that and retransmit
bullldquoParity bitrdquo is an extra bit included with a message to make the total number of 1rsquos either even or odd
1048697 ldquoEven parityrdquo choose parity bit such that of 1rsquos is even 1048697 ldquoOdd parityrdquo choose parity bit such that of 1rsquos is odd
Example 1048697 Data=1000001 even parity=0 odd parity=1 1048697 Data=1010100 even parity=1 odd parity=0
Only one parity bit is used
Ramzi Sh Alqrainy
Binary Storage and Registers How is information stored on a digital system
1048697 Bits are stored in ldquobinary cells rdquo 1048697 Binary cell can have two stable states lsquo0rsquo and lsquo1rsquo
Binary cells are grouped into registers 1048697 n cells make up n-bit register
Size of registers is typically predefined 1048697 Simple microcontroller 8 bits = 1 byte
1048697 Pentium 32 bits = 4 bytes 1048697 Mac G5 64 bits = 8 bytes
Digital system can usually process entire registers
1048697 ldquoRegister transferrdquo operation specify processingRamzi Sh Alqrainy
Register Example
Ramzi Sh Alqrainy
Register Transfer Operations
Ramzi Sh Alqrainy
BCD AdditionExample 184 + 576BCD carry 1 1
0001 1000 0100 184
+ 0101 0111 0110+ 576
Binary sum 0111 10000 1010
Add 6 0110 0110
BCD sum 0111 0110 0000 760
Ramzi Sh Alqrainy
Decimal ArithmeticEverything needs to be 4-bit aligned
1048697 lsquo+rsquo represented by 0 )=lsquo0000rsquo( 1048697 lsquondashrsquo represented by 9 )=lsquo1001rsquo(
1048707 Signed magnitude representation or complements 1048697 Signed magnitude hardly use
1048697 10rsquos complement most common1048707 Example 375 + (ndash240)1048697 Negative numbers represented by 10rsquos complement
1048697 10rsquos complement of 240 is 104 ndash 240 = 97601048697 Addition of all digits anddiscard of end carry 0 375
+ 9 760 0 130
1048697 Sign of result automatically correct
Ramzi Sh Alqrainy
Other Decimal CodesDecimals can be encoded in many ways in 4 bits1048707 Weighted codes
Each bit position is assigned a weighting factor Value of code is sum of weights where bits are lsquo1rsquo
Is BCD a weighted code raquo Yes Itrsquos a 8421 code
1048707 Other weighted code raquo 2421 code )yields non-unique coding(
raquo 84-2-1 code1048707 Self-complementing codes
9rsquos complement by exchanging 1rsquos with 0rsquos and 0rsquos with 1rsquos
Excess-3 and 2421 codes are self-complementing
Ramzi Sh Alqrainy
Other Decimal Codes
Ramzi Sh Alqrainy
Gray Code Imaging you code a 2-bit number with two light
switches connected to light bulbs ~ Can you count binary without causing ambiguity
Probably not ~ Switching from 01 to 10 cannot be done simultaneously on both bits
Either 01-gt00-gt10 or 01-gt11-gt10
This brief error or ambiguity can cause problems
Gray code changes only one bit between consecutivenumbers
Ramzi Sh Alqrainy
Gray Code
Ramzi Sh Alqrainy
ASCII Table
Ramzi Sh Alqrainy
Error-Detecting CodeCommunication between systems can be ldquonoisyrdquo
1048697 Environmental conditions can cause bit flips Error-detecting code
1048697 If one bit is flipped the code becomes invalid 1048697 Communication system can detect that and retransmit
bullldquoParity bitrdquo is an extra bit included with a message to make the total number of 1rsquos either even or odd
1048697 ldquoEven parityrdquo choose parity bit such that of 1rsquos is even 1048697 ldquoOdd parityrdquo choose parity bit such that of 1rsquos is odd
Example 1048697 Data=1000001 even parity=0 odd parity=1 1048697 Data=1010100 even parity=1 odd parity=0
Only one parity bit is used
Ramzi Sh Alqrainy
Binary Storage and Registers How is information stored on a digital system
1048697 Bits are stored in ldquobinary cells rdquo 1048697 Binary cell can have two stable states lsquo0rsquo and lsquo1rsquo
Binary cells are grouped into registers 1048697 n cells make up n-bit register
Size of registers is typically predefined 1048697 Simple microcontroller 8 bits = 1 byte
1048697 Pentium 32 bits = 4 bytes 1048697 Mac G5 64 bits = 8 bytes
Digital system can usually process entire registers
1048697 ldquoRegister transferrdquo operation specify processingRamzi Sh Alqrainy
Register Example
Ramzi Sh Alqrainy
Register Transfer Operations
Ramzi Sh Alqrainy
Decimal ArithmeticEverything needs to be 4-bit aligned
1048697 lsquo+rsquo represented by 0 )=lsquo0000rsquo( 1048697 lsquondashrsquo represented by 9 )=lsquo1001rsquo(
1048707 Signed magnitude representation or complements 1048697 Signed magnitude hardly use
1048697 10rsquos complement most common1048707 Example 375 + (ndash240)1048697 Negative numbers represented by 10rsquos complement
1048697 10rsquos complement of 240 is 104 ndash 240 = 97601048697 Addition of all digits anddiscard of end carry 0 375
+ 9 760 0 130
1048697 Sign of result automatically correct
Ramzi Sh Alqrainy
Other Decimal CodesDecimals can be encoded in many ways in 4 bits1048707 Weighted codes
Each bit position is assigned a weighting factor Value of code is sum of weights where bits are lsquo1rsquo
Is BCD a weighted code raquo Yes Itrsquos a 8421 code
1048707 Other weighted code raquo 2421 code )yields non-unique coding(
raquo 84-2-1 code1048707 Self-complementing codes
9rsquos complement by exchanging 1rsquos with 0rsquos and 0rsquos with 1rsquos
Excess-3 and 2421 codes are self-complementing
Ramzi Sh Alqrainy
Other Decimal Codes
Ramzi Sh Alqrainy
Gray Code Imaging you code a 2-bit number with two light
switches connected to light bulbs ~ Can you count binary without causing ambiguity
Probably not ~ Switching from 01 to 10 cannot be done simultaneously on both bits
Either 01-gt00-gt10 or 01-gt11-gt10
This brief error or ambiguity can cause problems
Gray code changes only one bit between consecutivenumbers
Ramzi Sh Alqrainy
Gray Code
Ramzi Sh Alqrainy
ASCII Table
Ramzi Sh Alqrainy
Error-Detecting CodeCommunication between systems can be ldquonoisyrdquo
1048697 Environmental conditions can cause bit flips Error-detecting code
1048697 If one bit is flipped the code becomes invalid 1048697 Communication system can detect that and retransmit
bullldquoParity bitrdquo is an extra bit included with a message to make the total number of 1rsquos either even or odd
1048697 ldquoEven parityrdquo choose parity bit such that of 1rsquos is even 1048697 ldquoOdd parityrdquo choose parity bit such that of 1rsquos is odd
Example 1048697 Data=1000001 even parity=0 odd parity=1 1048697 Data=1010100 even parity=1 odd parity=0
Only one parity bit is used
Ramzi Sh Alqrainy
Binary Storage and Registers How is information stored on a digital system
1048697 Bits are stored in ldquobinary cells rdquo 1048697 Binary cell can have two stable states lsquo0rsquo and lsquo1rsquo
Binary cells are grouped into registers 1048697 n cells make up n-bit register
Size of registers is typically predefined 1048697 Simple microcontroller 8 bits = 1 byte
1048697 Pentium 32 bits = 4 bytes 1048697 Mac G5 64 bits = 8 bytes
Digital system can usually process entire registers
1048697 ldquoRegister transferrdquo operation specify processingRamzi Sh Alqrainy
Register Example
Ramzi Sh Alqrainy
Register Transfer Operations
Ramzi Sh Alqrainy
Other Decimal CodesDecimals can be encoded in many ways in 4 bits1048707 Weighted codes
Each bit position is assigned a weighting factor Value of code is sum of weights where bits are lsquo1rsquo
Is BCD a weighted code raquo Yes Itrsquos a 8421 code
1048707 Other weighted code raquo 2421 code )yields non-unique coding(
raquo 84-2-1 code1048707 Self-complementing codes
9rsquos complement by exchanging 1rsquos with 0rsquos and 0rsquos with 1rsquos
Excess-3 and 2421 codes are self-complementing
Ramzi Sh Alqrainy
Other Decimal Codes
Ramzi Sh Alqrainy
Gray Code Imaging you code a 2-bit number with two light
switches connected to light bulbs ~ Can you count binary without causing ambiguity
Probably not ~ Switching from 01 to 10 cannot be done simultaneously on both bits
Either 01-gt00-gt10 or 01-gt11-gt10
This brief error or ambiguity can cause problems
Gray code changes only one bit between consecutivenumbers
Ramzi Sh Alqrainy
Gray Code
Ramzi Sh Alqrainy
ASCII Table
Ramzi Sh Alqrainy
Error-Detecting CodeCommunication between systems can be ldquonoisyrdquo
1048697 Environmental conditions can cause bit flips Error-detecting code
1048697 If one bit is flipped the code becomes invalid 1048697 Communication system can detect that and retransmit
bullldquoParity bitrdquo is an extra bit included with a message to make the total number of 1rsquos either even or odd
1048697 ldquoEven parityrdquo choose parity bit such that of 1rsquos is even 1048697 ldquoOdd parityrdquo choose parity bit such that of 1rsquos is odd
Example 1048697 Data=1000001 even parity=0 odd parity=1 1048697 Data=1010100 even parity=1 odd parity=0
Only one parity bit is used
Ramzi Sh Alqrainy
Binary Storage and Registers How is information stored on a digital system
1048697 Bits are stored in ldquobinary cells rdquo 1048697 Binary cell can have two stable states lsquo0rsquo and lsquo1rsquo
Binary cells are grouped into registers 1048697 n cells make up n-bit register
Size of registers is typically predefined 1048697 Simple microcontroller 8 bits = 1 byte
1048697 Pentium 32 bits = 4 bytes 1048697 Mac G5 64 bits = 8 bytes
Digital system can usually process entire registers
1048697 ldquoRegister transferrdquo operation specify processingRamzi Sh Alqrainy
Register Example
Ramzi Sh Alqrainy
Register Transfer Operations
Ramzi Sh Alqrainy
Other Decimal Codes
Ramzi Sh Alqrainy
Gray Code Imaging you code a 2-bit number with two light
switches connected to light bulbs ~ Can you count binary without causing ambiguity
Probably not ~ Switching from 01 to 10 cannot be done simultaneously on both bits
Either 01-gt00-gt10 or 01-gt11-gt10
This brief error or ambiguity can cause problems
Gray code changes only one bit between consecutivenumbers
Ramzi Sh Alqrainy
Gray Code
Ramzi Sh Alqrainy
ASCII Table
Ramzi Sh Alqrainy
Error-Detecting CodeCommunication between systems can be ldquonoisyrdquo
1048697 Environmental conditions can cause bit flips Error-detecting code
1048697 If one bit is flipped the code becomes invalid 1048697 Communication system can detect that and retransmit
bullldquoParity bitrdquo is an extra bit included with a message to make the total number of 1rsquos either even or odd
1048697 ldquoEven parityrdquo choose parity bit such that of 1rsquos is even 1048697 ldquoOdd parityrdquo choose parity bit such that of 1rsquos is odd
Example 1048697 Data=1000001 even parity=0 odd parity=1 1048697 Data=1010100 even parity=1 odd parity=0
Only one parity bit is used
Ramzi Sh Alqrainy
Binary Storage and Registers How is information stored on a digital system
1048697 Bits are stored in ldquobinary cells rdquo 1048697 Binary cell can have two stable states lsquo0rsquo and lsquo1rsquo
Binary cells are grouped into registers 1048697 n cells make up n-bit register
Size of registers is typically predefined 1048697 Simple microcontroller 8 bits = 1 byte
1048697 Pentium 32 bits = 4 bytes 1048697 Mac G5 64 bits = 8 bytes
Digital system can usually process entire registers
1048697 ldquoRegister transferrdquo operation specify processingRamzi Sh Alqrainy
Register Example
Ramzi Sh Alqrainy
Register Transfer Operations
Ramzi Sh Alqrainy
Gray Code Imaging you code a 2-bit number with two light
switches connected to light bulbs ~ Can you count binary without causing ambiguity
Probably not ~ Switching from 01 to 10 cannot be done simultaneously on both bits
Either 01-gt00-gt10 or 01-gt11-gt10
This brief error or ambiguity can cause problems
Gray code changes only one bit between consecutivenumbers
Ramzi Sh Alqrainy
Gray Code
Ramzi Sh Alqrainy
ASCII Table
Ramzi Sh Alqrainy
Error-Detecting CodeCommunication between systems can be ldquonoisyrdquo
1048697 Environmental conditions can cause bit flips Error-detecting code
1048697 If one bit is flipped the code becomes invalid 1048697 Communication system can detect that and retransmit
bullldquoParity bitrdquo is an extra bit included with a message to make the total number of 1rsquos either even or odd
1048697 ldquoEven parityrdquo choose parity bit such that of 1rsquos is even 1048697 ldquoOdd parityrdquo choose parity bit such that of 1rsquos is odd
Example 1048697 Data=1000001 even parity=0 odd parity=1 1048697 Data=1010100 even parity=1 odd parity=0
Only one parity bit is used
Ramzi Sh Alqrainy
Binary Storage and Registers How is information stored on a digital system
1048697 Bits are stored in ldquobinary cells rdquo 1048697 Binary cell can have two stable states lsquo0rsquo and lsquo1rsquo
Binary cells are grouped into registers 1048697 n cells make up n-bit register
Size of registers is typically predefined 1048697 Simple microcontroller 8 bits = 1 byte
1048697 Pentium 32 bits = 4 bytes 1048697 Mac G5 64 bits = 8 bytes
Digital system can usually process entire registers
1048697 ldquoRegister transferrdquo operation specify processingRamzi Sh Alqrainy
Register Example
Ramzi Sh Alqrainy
Register Transfer Operations
Ramzi Sh Alqrainy
Gray Code
Ramzi Sh Alqrainy
ASCII Table
Ramzi Sh Alqrainy
Error-Detecting CodeCommunication between systems can be ldquonoisyrdquo
1048697 Environmental conditions can cause bit flips Error-detecting code
1048697 If one bit is flipped the code becomes invalid 1048697 Communication system can detect that and retransmit
bullldquoParity bitrdquo is an extra bit included with a message to make the total number of 1rsquos either even or odd
1048697 ldquoEven parityrdquo choose parity bit such that of 1rsquos is even 1048697 ldquoOdd parityrdquo choose parity bit such that of 1rsquos is odd
Example 1048697 Data=1000001 even parity=0 odd parity=1 1048697 Data=1010100 even parity=1 odd parity=0
Only one parity bit is used
Ramzi Sh Alqrainy
Binary Storage and Registers How is information stored on a digital system
1048697 Bits are stored in ldquobinary cells rdquo 1048697 Binary cell can have two stable states lsquo0rsquo and lsquo1rsquo
Binary cells are grouped into registers 1048697 n cells make up n-bit register
Size of registers is typically predefined 1048697 Simple microcontroller 8 bits = 1 byte
1048697 Pentium 32 bits = 4 bytes 1048697 Mac G5 64 bits = 8 bytes
Digital system can usually process entire registers
1048697 ldquoRegister transferrdquo operation specify processingRamzi Sh Alqrainy
Register Example
Ramzi Sh Alqrainy
Register Transfer Operations
Ramzi Sh Alqrainy
ASCII Table
Ramzi Sh Alqrainy
Error-Detecting CodeCommunication between systems can be ldquonoisyrdquo
1048697 Environmental conditions can cause bit flips Error-detecting code
1048697 If one bit is flipped the code becomes invalid 1048697 Communication system can detect that and retransmit
bullldquoParity bitrdquo is an extra bit included with a message to make the total number of 1rsquos either even or odd
1048697 ldquoEven parityrdquo choose parity bit such that of 1rsquos is even 1048697 ldquoOdd parityrdquo choose parity bit such that of 1rsquos is odd
Example 1048697 Data=1000001 even parity=0 odd parity=1 1048697 Data=1010100 even parity=1 odd parity=0
Only one parity bit is used
Ramzi Sh Alqrainy
Binary Storage and Registers How is information stored on a digital system
1048697 Bits are stored in ldquobinary cells rdquo 1048697 Binary cell can have two stable states lsquo0rsquo and lsquo1rsquo
Binary cells are grouped into registers 1048697 n cells make up n-bit register
Size of registers is typically predefined 1048697 Simple microcontroller 8 bits = 1 byte
1048697 Pentium 32 bits = 4 bytes 1048697 Mac G5 64 bits = 8 bytes
Digital system can usually process entire registers
1048697 ldquoRegister transferrdquo operation specify processingRamzi Sh Alqrainy
Register Example
Ramzi Sh Alqrainy
Register Transfer Operations
Ramzi Sh Alqrainy
Error-Detecting CodeCommunication between systems can be ldquonoisyrdquo
1048697 Environmental conditions can cause bit flips Error-detecting code
1048697 If one bit is flipped the code becomes invalid 1048697 Communication system can detect that and retransmit
bullldquoParity bitrdquo is an extra bit included with a message to make the total number of 1rsquos either even or odd
1048697 ldquoEven parityrdquo choose parity bit such that of 1rsquos is even 1048697 ldquoOdd parityrdquo choose parity bit such that of 1rsquos is odd
Example 1048697 Data=1000001 even parity=0 odd parity=1 1048697 Data=1010100 even parity=1 odd parity=0
Only one parity bit is used
Ramzi Sh Alqrainy
Binary Storage and Registers How is information stored on a digital system
1048697 Bits are stored in ldquobinary cells rdquo 1048697 Binary cell can have two stable states lsquo0rsquo and lsquo1rsquo
Binary cells are grouped into registers 1048697 n cells make up n-bit register
Size of registers is typically predefined 1048697 Simple microcontroller 8 bits = 1 byte
1048697 Pentium 32 bits = 4 bytes 1048697 Mac G5 64 bits = 8 bytes
Digital system can usually process entire registers
1048697 ldquoRegister transferrdquo operation specify processingRamzi Sh Alqrainy
Register Example
Ramzi Sh Alqrainy
Register Transfer Operations
Ramzi Sh Alqrainy
Binary Storage and Registers How is information stored on a digital system
1048697 Bits are stored in ldquobinary cells rdquo 1048697 Binary cell can have two stable states lsquo0rsquo and lsquo1rsquo
Binary cells are grouped into registers 1048697 n cells make up n-bit register
Size of registers is typically predefined 1048697 Simple microcontroller 8 bits = 1 byte
1048697 Pentium 32 bits = 4 bytes 1048697 Mac G5 64 bits = 8 bytes
Digital system can usually process entire registers
1048697 ldquoRegister transferrdquo operation specify processingRamzi Sh Alqrainy
Register Example
Ramzi Sh Alqrainy
Register Transfer Operations
Ramzi Sh Alqrainy
Register Example
Ramzi Sh Alqrainy
Register Transfer Operations
Ramzi Sh Alqrainy
Register Transfer Operations
Ramzi Sh Alqrainy