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Dr. Alaaeldin A. Amin Computer Engineering Department E-mail: amin@ccse.kfupm.edu.sa http://www.ccse.kfupm.edu.sa/~amin. Review - Outline. Machine Number Systems (Fixed radix Positional) Fixed Point Number Representation Base Conversion Representation of Signed Numbers

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Dr. Alaaeldin A. Amin

Computer Engineering Department

E-mail: amin@ccse.kfupm.edu.sa

http://www.ccse.kfupm.edu.sa/~amin

Review - Outline
• Machine Number Systems (Fixed radix Positional)
• Fixed Point Number Representation
• Base Conversion
• Representation of Signed Numbers
• Signed Magnitude (Sign and Magnitude)
• Complement Representation

• Precision Extension
• Arithmetic Shifts

COE 522 Dr. A. Amin

Machine Number Systems?

X = xk-1 xk-2 ... x1 x0 .x-1 x-2 ... x-m

=

• Since Numbers Are Stored in Registers of Fixed Length
• There is a Finite Number of Distinct Values that Can be Represented in a Register
• Let Xmax & Xmin Denote The Largest & Smallest Representable Numbers
• Xmax= rk-1For an Integral # of k-Digits
• Xmax= 1-r-mFor a Fractional # of m-Digits
• Xmax= rk-r-mFor a # of k- Integral Digits and m- FractionalDigits
• Xmin = 0
• Number of Possible Distinct Values = rk+m

Fractional Part

Integral Part

COE 522 Dr. A. Amin

• Let X be an Integer  XB in Radix B System
•  XA in Radix A System

Assumptions

• XB has n digits

XB = (bn-1………..b2 b1 b0)B ,

where bi is a digit in radix B system,

i.e. bi {0, 1, ….. , “B-1”}

• XA has m digits

XA =(am-1………..a2 a1 a0)A

where ai is a digit in radix A system,

i.e. ai {0, 1, ….., “A-1”}

Problem Statement

XB =(bn-1………..b2 b1 b0)B (am-1………..a2 a1 a0)A

Unknowns

Knowns

COE 522 Dr. A. Amin

Not Divisible by A

Not Divisible by A

Divisible by A

Divisible by A

XB = am-1*Am-1+……+ a2*A2 + a1*A1 + a0*A0

• Where ai {0-(A-1)}
• Accordingly, dividing XB by A, the remainder will be a0.

XB = Q0.A+a0

Where, Q0 = am-1*Am-2+…+ a2*A1 + a1*A0

Likewise  Q0 = Q1A+a1 Get a1

Q1 = Q2A+a2 Get a2

…………………………………

Qm-3 = Qm-2A+am-2 Get am-2

Qm-2 = Qm-1A+am-1 Get am-1

Where Qm-1= 0  Stopping Criteria

COE 522 Dr. A. Amin

Sign Digit

0 +ive

1 -ive

Representation of Signed Numbers

Three Main Systems

1. Signed Magnitude (Sign and Magnitude)

2. Complement Representation

Signed Magnitude

• Independent Representation of The Sign and The Magnitude
• Symmetric Range of Representation

{ -(2n-1 -1) :+(2n-1 -1) }

• Two Representations for 0  {+0 , -0}

 Nuisance for Implementation

• Multiplication & Division Less Problematic
• For Base R Sign Digit May Be
• 0  +ive Numbers
• R-1  -ive Numbers {Inefficient Space Utilization}

COE 522 Dr. A. Amin

Representation of Signed Numbers
• Alternatively
• 0, 1, …, (R/2 - 1)  +ive Numbers
• R/2, R/2+1, …., (R-1)  -ive Numbers  More Complex Sign Detection if r not power of 2

COE 522 Dr. A. Amin

Representation of Signed Numbers

Complement Representation

• Positive Numbers (+N) Are Represented in Exactly the Same Way as in Signed Magnitude System
• Negative Numbers (-N) Are Represented by the Complement of N (N`)

Define the Complement of a number N (N`) as:

N` = M -N

Where M= Some Constant

• This representation satisfies the Basic Requirement That:

-(-N ) = ( N` )` = M- (M-N) = N

Adding 2 Numbers; X (+ive) and Y (-ive) :

IF Y > X

• Result Z is -ive, i.e. Complement Represented:

Z = X + (M-Y) =

= M -(Y-X) = Correct Answer

in Complement Form

+ive

COE 522 Dr. A. Amin

M Chosen To Eliminate/Simplify Correction Step

+ive

+ive

Representation of Signed Numbers

IF Y < X

• Result Z is +ive:

Z = X + (M-Y)

= M +(X-Y)

Correct Answer = X - Y

Requirements of M:

1- Simple Complement Calculations

• Let xi be the Complement of a single Digit xi
• IF X = xk-1xk-2 ... x1 x0 .x-1 x-2 ... x-m
• Define X = xk-1xk-2 ... x1 x0 .x-1 x-2 ... x-m

IF Y < X

• Result Z is +ive:

Z = X + (M-Y)

= M +(X-Y)

Correct Answer = X - Y

Requirements of M:

1- Simple Complement Calculations

• Let xi be the Complement of a single Digit xi
• IF X = xk-1xk-2 ... x1 x0 .x-1 x-2 ... x-m
• Define X = xk-1xk-2 ... x1 x0 .x-1 x-2 ... x-m

M Chosen To Eliminate/Simplify Correction Step

COE 522 Dr. A. Amin

Representation of Signed Numbers

X xk-1xk-2 ... x1 x0 .x-1 x-2 ... x-m

+ X xk-1xk-2 ... x1 x0 .x-1 x-2 ... x-m

(r-1) (r-1) …….. . ………. (r-1)

+ ulp1

10 0 … 0 0 . 0 0 .….. 0 = r k

Thus, X + X + ulp = r k …………... (1)

where ulp = unit in least position

= r -m(for #`s with m Fractional digits)

= 1(for INTEGERS, i.e when m=0)

Eqn. (1) Can Be ReWritten As

r k - X = X + ulp …… (2)

COE 522 Dr. A. Amin

(Beyond Word Size)

Representation of Signed Numbers

N ` = r k - N

= N + ulp (Simple Computation)

• Computing N ` is independent of k
• Computing the Previous Addition Requires No Corrections

Z = M + (X-Y) = r k+ (X-Y)

Diminished Radix Complement: M= r k-ulp

N ` = r k - ulp - N

= N (Simplest Computation)

Z = M + (X-Y) = r k+ (X-Y) -ulp

COE 522 Dr. A. Amin

Sign Digit

0 +ive

1 -ive

Representation of Signed Numbers

2`s Complement

• Asymmetric Range of Representation

{ -(2k-1) :+(2k-1 -ulp) }

- Negation Can Lead to OverFlow

COE 522 Dr. A. Amin

Integral Part

Fractional Part

Sign Bit

2`s Complement Numbers System

X = xk-1 xk-2 ... x1 x0 .x-1 x-2 ... x-m

=

= Xif X is +ive

= -2k-1+(2k-1 - X )= Xif X is -ive

Precision/Range Extension of 2`s Comp #`s

• Extending # N from n-bits to k`-bits (k`> k)
• +ive N Pad with 0`s to the Left of the integral part And/Or to the right of Fractional Part.
• -ive N Pad with 0`s to the right of Fractional Part And/Or Extend Sign Bit to the Left of the integral part.
• In General
• Pad with 0`s to the right of Fractional Part And/Or Extend Sign Bit to the Left of the integral part.

...xk-1 xk-1xk-1 xk-2 ... x1 x0 .x-1 x-2 .... x-m000…..

COE 522 Dr. A. Amin

-ive N

N`k = 2k - N N`k` = 2k` - N

N`k` - N`k = 2k` - 2k = 2k (2k`-k - 1 )

Shift Left k-bits

(k`-k) 1`s

1111..111000..0

Sign Extension

(k`-k) 1`s

k 0`s

2`s Complement System

Features

• Leftmost Bit indicates Sign
• The Sign Bit is Part of a Truncated -String
• Asymetric Range(-2k-1:+(2k-1 -ulp))
• Overflow may result on Complementing
• Single Zero Representation 0000000 (+0)

COE 522 Dr. A. Amin

Sign Digit

0 +ive

1 -ive

Representation of Signed Numbers

1`s Complement

• Symmetric Range of Representation

{ -(2k-1 -ulp) :+(2k-1 -ulp) }

- 2 Representations of 0

(+0 = 0000000and-0 = 1111111)

COE 522 Dr. A. Amin

Integral Part

Fractional Part

Sign Bit

1`s Complement Numbers System

X = xk-1 xk-2 ... x1 x0 .x-1 x-2 ... x-m

=

= Xfor +ive X

= -(2k-1-ulp) + X ) = X` for -ive X

Precision/Range Extension of 2`s Comp #`s

• Extending # N from n-bits to k`-bits (k`> k)
• +ive N Pad with 0`s to the Left of the integral part And/Or to the right of Fractional Part.
• -ive N Pad with Sign Bit to the right of Fractional Part And/Or to the Left of the integral part.
• In General
• Pad with Sign Bit to the right of Fractional Part And/Or to the Left of the integral part.

..xk-1 xk-1xk-1 xk-2 .. x1 x0 .x-1 x-2 .. x-mxk-1 xk-1 …..

COE 522 Dr. A. Amin

1`s Complement Numbers System

Features

• Leftmost Bit indicates Sign
• The Sign Bit is Part of a Truncated -String
• Symetric Range(-(2k-1 -ulp) :+(2k-1 -ulp))
• Two Zero Representation (+0 , -0 )

Arithmetic Shifts

Effect

• Left Shift Multiply Number by radix r
• Right Shift Divide Number by radix r

(a) Shifting Unsigned or +ive Numbers

• Shift-In 0`s (for both Left & Right Shifts)

(b) Shifting -ive Numbers

• Depends on the Representation Method

Signed Magnitude:

• Sign Bit is maintained (Unchanged)
• Only The Magnitude is shifted and 0`s are Shifted-In for both Directions (Left & Right)

COE 522 Dr. A. Amin

Arithmetic Shifts

2`s Complement:

• Left Shifts:0`s are Shifted-In
• Right Shifts:Sign Bit Extended (Shifted Right)

1`s Complement:

• Left & Right Shifts:Sign Bit is Shifted-In

Right Shifts

Left Shifts

COE 522 Dr. A. Amin