Q15

Q15 is a fixed-point numeric format commonly used in digital signal processing and embedded systems to represent fractional numbers with a 16-bit signed word. In this format, the most significant bit serves as the sign, and the remaining 15 bits hold the fractional part.

A Q15 value is typically stored as an integer n in the range [-32768, 32767], with the value computed as x = n / 32768.0. This yields a range from -1.0 to just under +1.0, and the minimum step size is 1/32768 (approximately 3.05e-5). For example, 0x8000 encodes -1.0, 0x0000 encodes 0.0, and 0x4000 encodes +0.5.

Arithmetic in Q15 uses fixed-point operations with saturation to avoid wraparound. Addition and subtraction are performed on the 16-bit integers with saturation to the range [-32768, 32767]. Multiplication requires a wider intermediate result (typically 32 bits, yielding a Q30 value) and is then shifted right by 15 bits to return to Q15, followed by saturation. Fixed-point libraries often provide saturated Q15 operations for robust results.

Common usage includes implementing DSP tasks on microcontrollers without floating-point units, particularly in audio processing, filtering, and fast Fourier transform pipelines. The format enables efficient memory usage and predictable performance on resource-constrained hardware. Converting between Q15 and floating point involves scaling by 32768 (and clamping during conversion) and the reverse for going back to floating-point representations.

Q15 is part of the broader family of Q-format fixed-point representations, which shares the principle of representing fractional numbers with a fixed number of fractional bits.