Fixedpoint

Fixed-point is a numeric representation method in which a real number is stored as an integer with a fixed scaling factor between the integer and the fractional part. The value is typically the stored integer divided by a power of two, enabling arithmetic to be performed with integer operations. This makes fixed-point attractive for hardware without floating-point units and for real-time software where deterministic behavior matters.

Format and notation: Fixed-point formats are commonly described using Q notation, such as Q1.15 (a 16-bit signed word with 1 sign bit and 15 fractional bits) or Q1.31 in 32-bit words. The number of fractional bits determines the precision and the range of the integer part. All arithmetic uses the same scaling.

Arithmetic: Addition and subtraction operate directly on the scaled integers. Multiplication yields a product with twice the fractional bits, so a right shift by the fractional width restores the scale, with rounding or truncation as needed. Division can be implemented by scaling before integer division. Overflow and underflow are common concerns; many implementations use saturation or guard bits.

Conversion: To encode a real value, multiply by 2^n and round to the nearest integer; to decode, divide by 2^n.

Applications: Fixed-point is widespread in embedded systems, digital signal processing, audio and video processing, control systems, and financial calculations where fixed decimal places are essential.

Pros and cons: It offers deterministic, low-latency performance and low hardware cost, but has a limited dynamic range and quantization errors, and requires careful scaling. See also: floating-point, Q notation, saturation arithmetic.