Arithmetic

Question 1

Define Algebra

Answer 1

Any special system of notation adapted
to the study of a special system of relationship.

Question 2

Features of Boolean Algebra

Answer 2

• Simply: AND;NOT; 0; 1

• Distributive: a(b c) == ab ac
• Commutative: a b == b a
• Associative: (a b) c == a (b c)

Question 3

How is an N-bit Ripple Adder Built?

Answer 3

• An N-bit ripple adder can be
assembled by cascading full-
adders.
• All combinatorial
• Individual bit delays are dif-
ferent, but total delay propor-
tional to N.
• All stages are identical (apart
from LSB).

Question 4

How are unsigned integers represented in Boolean

Answer 4

• Stores values in [0; 2n 􀀀 1].
• e.g. The pattern 1101 represents the natural
number 20 + 22 + 23 = 1 + 4 + 8 = 13.

Question 5

How are signed Integers represented?

Answer 5

Signed integer numbers

• Sign magnitude
• Two’s complement
• One’s complement
• Biased oset

Question 6

Describe Sign Magnitude

Answer 6

• Stores values in [-2^n-1 - 1; 2^n-1 - 1].
• The most significant bit (MSB, at n-1) represents
the sign of the integer.
• e.g. The 8-bit pattern 10001101 represents the
integer number -13.

Question 7

Describe Two’s Complement

Answer 7

• Stores values in [-2^n-1; 2^n-1 - 1] (no -0).
• Value defined such that X + (-X) = 2ⁿ. This is
convenient:
- Addition: An unsigned adder is suitable.
- Negation: Complement each bit and add 1.

Question 8

Describe One’s Complement

Answer 8

• Stores values in [-2^n-1 - 1; 2^n-1 - 1] (we lose one
due to -0).
• Value defined such that inverting the bit pattern
returns the negative of the original value.
• Rarely used (Some ADCs use this).

Question 9

Describe Biased Offset

Answer 9

• Think unsigned but subtract a constant from all
values.
• Range depends on the bias (but we don’t lose one
due to 0).

Question 10

Describe general floating-point numbers implementation?

Answer 10

• Standard form as you know it: 2:56X 10⁴.
• Value = (-1)^S X 1:F X 2^E
• The exponential field E is biased such that all
legitimate numbers satisfy E >= 0.

Question 11

What are the four binary formats?

Answer 11

Question 12

Binary Formats with Width and Bias

Answer 12

Question 13

What are the special values of floating-point numbers?

Answer 13

Question 14

How do special values affect computation?

Answer 14

• Computation must continue under certain special
values - they propagate with certain identities (e.g.
f(NaN)=NaN, 1 + (+infinity) = +infinity)

• Denormalised numbers are susceptible to creeping
underflow: the numbers have decreasing precision
as they get smaller. These numbers are treated
specially: Value = (-1)^S X 0:F.

Question 15

What are the attributes of black box leaf-level operations?

Answer 15

• Static attributes: size, speed
• Dynamic attributes: power, electromagnetic
  compatibility (emc)

Question 16

Describe NAND and AND Gates

Answer 16

• Size is approximately proportional to fan-in.
• AND is bigger than NAND due to extra inversion.

Question 17

What affects gate delay?

Answer 17

• Transport delay (di erent for 1 → 0 and 0 → 1):
• Depends on the values of other inputs:
• Depends on gate type.
• Depends on signal value.
• Depends on load:

However, we normally combine all this as a single ddelay constant

Question 18

What is inertia and how does it relate to inertial delay?

Answer 18

if two events occur on an input of the component with an interval time less than the defined delay, the output will not reflect either input event,

Inertia is Indisposition to motion, exertion, or
change

Question 19

Feature of Cascading Full Adders

Answer 19

• Size proportional to n (good)
• Delay proportional to n (awful)
• NB: Assuming constant delay d

Question 20

Features of Carry Lookahead Logic

Answer 20

• Size proportional to n2 (awful)
• Delay constant (good)

Question 21

What are the steps for floating-point addition subtraction?

Answer 21

1. Handle NaN and +- infinity propagation.
2. Check for zero operands (saves time).
3. Subtract exponents such that |E_A - E_B| = d.
4. Align the fractions: right-shift the fraction of the
   smaller exponent by d (you should now have two
   numbers with the same exponent, but one may not
   be in standard form).
5. Add/Subtract the fraction components.
6. Normalise the number (bring it into standard form,
   while adjusting the exponent parts).
7. Rounding, as appropriate.

Question 22

Unsigned Multiplication

Answer 22

Combinatorial
• Area proportional to n² (massive)
• Single Cycle

Question 23

Features of Booth Multiplication

Answer 23

Computes P = A X B, with P starting at zero. Method (repeat n times):
• Add P to (B or 0) depending on the LSB of A.
• Replace P with the sum.
• Right-shift P&A.
Smaller area, but n cycles.

Question 24

Features of Signed Multiplication

Answer 24

• The right-shift of P&A is arithmetic (i.e. the
left-hand bit loops around), instead of logical.
• Register A is extended one bit to the right.
• +- ALU: Extra circuit is needed.

Question 25

Steps for Floating Point Multiplication

Answer 25

1. Handle NaN and +- infinity propagation.
2. Check for zero operands (saves time).
3. Evaluate product sign ().
4. Fixed-point unsigned multiply/divide the fractions.
5. Sum the exponents
6. Normalise the number (bring it into standard form,
   while adjusting the exponent parts).
7. Rounding, as appropriate.

Question 26

What is an ALU?

Answer 26

• A selectable function unit
• Usually two data inputs of equal size, with a
function-select bus. The compiler needs to
know how this bus is coded.
• ALUs typically support logical and arithmetic
operations.