Fixed and Floating Point Representations [7M]

1. Fixed Point Representation:

In fixed point, binary point is fixed at one position (usually after sign bit / between integer and fractional part). Used for integers and fractional numbers where range is limited.

• Format: S | Integer part | Fractional part . e.g. Q15 format: 1 sign + 15 fractional bits. Value = $I.F$ where $F$ has weight $2^{-k}$.
• Signed representation: Sign-magnitude, 1's complement, but practically 2's complement is used. e.g. 8-bit 2's complement : $+5 = 00000101$, $-5 = 11111011$. Range for $n$-bits: $-2^{n-1}$ to $+2^{n-1}-1$.
• Fractional fixed point: e.g. S = 0.101 ($\approx 0.625$ in $Q_{3}$). Value = $-b_{0}.b_{-1}b_{-2}...$ in 2's complement : $V = -b_{0} + \sum_{i=1}^{k} b_{-i}2^{-i}$.
• Advantages: Simple hardware, fast ALU, exact representation within range.
• Disadvantages: Very limited dynamic range. Overflow easily. Small numbers lose precision. Trade-off between Range and Precision: $range \approx 2^{n}$, $resolution = 2^{-f}$ where $f$=fractional bits.

Used in DSP, embedded systems (no FPU), address arithmetic.

2. Floating Point Representation:

Based on scientific notation: $N = \pm M \times B^{E}$ where $M$ = mantissa/significand, $B$ = base (2), $E$ = exponent.

Standard: IEEE 754

• Single precision (32-bit): 1-bit Sign (S) | 8-bit Exponent (E) | 23-bit Mantissa (M)
  Value = $(-1)^S \times 1.M \times 2^{E-127}$
  Bias = 127. Normalized form: leading 1 is hidden (1.M).

• Double precision (64-bit): 1 | 11 | 52 , Bias = 1023. Value = $(-1)^S \times 1.M \times 2^{E-1023}$

• Example: $-12.5_{10} = -1100.1_2 = -1.1001 \times 2^3$
  $S=1$, $E = 3+127 = 130 = 10000010_2$, $M = 1001000...0$ (23 bits)
  Encoding: 1 10000010 10010000000000000000000 = 0xC1480000

• Special cases:

  • $E=0, M=0$ => $0$ ($\pm 0$)
  • $E=0, M
    eq0$=> Denormalized:$(-1)^S \times 0.M \times 2^{-126}$ (gradual underflow)
  • $E=255, M=0$ => $\pm\infty$
  • $E=255, M
    eq0$ => NaN

• Advantages: Huge dynamic range ($\approx 10^{\pm38}$ single, $10^{\pm308}$ double), automatic scaling.

• Disadvantages: Complex hardware (align exponents for add), rounding error, not exact (e.g. 0.1), slower.

Comparison:

Feature	Fixed Point	Floating Point
Range	Small, fixed	Very large
Precision	Uniform	Precision decreases with magnitude
HW Cost	Low	High (FPU)
Speed	High	Lower
Use	Counters, DSP, address	Scientific computing, graphics

Modern CPUs provide both; GPU/FPU handles IEEE 754 while ALU handles fixed point integers.