DSP Number Formats and Their Considerations

Key Questions in DSP Number Formats

• Dynamic Range: Range of distinct numbers that can be represented in an n-bit number format.
• Accuracy: Degree of error resulting from calculations.

Dynamic Range

• Fixed Point Representation:

  • Dynamic range is 2^n.
  • Headroom is defined as the ability to handle signal excess without clipping.
  • Example with 16-bit fixed point: Ranges between 2^15 - 1 to -2^15 for integers or rounded 1 to -1 for fractions.
  • Overflow occurs when the signal exceeds the range, causing undefined behavior.
  • Clipping is used to manage overflow by holding signal values at extreme range limits.

• Challenges:

  • Intermediate signal processing stages may require headroom to prevent clipping.
  • To accommodate, fixed point DSP processors use larger registers (e.g., Motorola 56k family has 56-bit accumulators for 24-bit audio).

Accuracy in Fixed Point Numbers

• Addition/Subtraction:

  • Perfectly accurate with no accumulated error.
  • Overflow still a concern but assumed handled.

• Division:

  • Error-prone due to discarded fractional components.
  • Relative error is higher with smaller numbers, significant in low-level audio signals.
  • Error quantification: up to 0.5 LSB for rounding, 1 LSB for truncation.
  • Example: Dividing 3 by 2 results in 1.5, truncated to 1.

• Error Accumulation:

  • Single calculation errors are insignificant, but recursive processes can accumulate errors.
  • Multiply accumulate operations use larger registers, but recursive filter paths can create unacceptable errors in pro-quality audio.

Use of Fixed Point vs. Floating Point

• Fixed Point:

  • Preferred for storing digital audio and playback in low-powered devices.
  • Efficient for non-complex signal processing.

• Floating Point:

  • Preferred in modern computer audio for signal processing.
  • Used in single or double precision form.
  • Advantages and complexities will be explored in the next discussion.