ELEC1303 - Lecture 15

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ELEC1303 - Lecture 14
Arithmetic Circuits - Slides cont.

Signed Overflow

• When working with $n$-bit signed numbers, the range of values is:
  $-2^{n-1}\text{to}{2}^{2-1}-1$
  e.g. for a 4-bit system, the range is from -8 to +7
• Overflow occurs when result of a sum/subtraction is outside the range
  • Only occurs when terms are both positive or both negative
  • Gives incorrect result
• Overflow only occurs in two situations:
  • If you add two positive numbers and get a negative result
  • If you add two negative numbers and get a positive result

Sign Extension

• Sign extension is where we assume numbers have an infinite number of 0s in from of them
  • Helpful for lining up decimal numbers
• However, you must be careful in extending signed binary numbers because the leftmost bit is the sign and not part of the magnitude!
  • The proper way to extend a signed binary number is to replicate the sign bit, so the sign is preserved

Two’s Complement Adder

We can use a series of full adders to do subtraction

• To find A - B:
  • Complement each bit of B
  • Set the adder’s carry in to 1

• Only differences between adder and subtractor circuits are:
  • The subtractor has to negate B3 B2 B1 B0
  • The subtractor sets the initial carry in to 1, instead of 0

Adder-Subtractor Circuit

• Building upon this idea, we can use XOR gates instead of inverters
  • Lets us switch between add and subtract mode
    i.e. A + B → A - B

• When Sub = 0, circuit is in adder mode
• When Sub = 1, circuit is in subtractor mode

Comparators

• Comparators checks if bits are equal (A = B)
• Composed of XNOR gates to check any number of bits
  • Compare each digit of two terms with the XNOR gate, then AND all the outputs

Binary Multiplication

• Binary multiplication: 1x1=1, otherwise = 0

• Multiplication consists of two steps:
  • Evaluation of partial products
  • Accumulation of the shifted partial products
• Since we always multiply by either 0 or 1, the partial products are always either 0000 or the multiplicand (1101 in this example)

Multipliers are very complex circuits (see slides)

Shifting Bits

• Shifting bits to the left (e.g. 101 → 1010) results in multiplication
  • Shift by $n$-bits is multiplication by $2^n$
• Shifting bits to the right (e.g. 100110 → 10011) results in division
  • Shift by $n$-bits is division by $2^n$

4-Bit Combinational Shifter:

• Settings for $s_1{s}_0$:
  • 00 → pass through
  • 01 → shift left and fill with 0
  • 10 → shift right and fill with 0
  • 11 → rotate right
    i.e. in${}_3$in${}_2$in${}_1$in${}_0$ → in${}_0$in${}_3$in${}_2$in${}_1$

Shift Registers

• A shift register is a cascade of flip-flops sharing the same clock
  • Allows data to be shifted from each flip-flop to its neighbour
• All bits are shifted simultaneously at the active edge of the clock

Parallel-In Serial-Out Shift Register

• Two control functions:
  • s = 0 → shift (works like above)
  • s = 1 → load data (update D${}_n$ for each flip-flop)

Universal Shift Register

• Four control functions:
  • s = 00 → no change in value
  • s = 01 → shift right (right-shift serial input)
  • s = 10 → shift left (left-shift serial input)
  • s = 11 → parallel load 𝑛𝑛 input bits

Multi-Function Arithmetic Logic Unit (ALU)

• An ALU can do arithmetic and logical operations

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ELEC1303 - Lecture 16