Fixed-Point Representation: Mastering Precision in FPGA Design

In digital design, especially when leveraging the power of FPGAs, efficiently representing real numbers with fractional components is a cornerstone challenge. While various methods exist, fixed-point representation stands out as a highly effective and frequently preferred approach due to its direct mapping to hardware and its inherent efficiency.

This post will dive deep into fixed-point numbers, exploring their structure, how they are represented, the critical distinction between signed and unsigned formats, and why mastering fixed-point is essential for optimal FPGA design.

What Exactly is Fixed-Point Representation?

At its heart, fixed-point representation is a method of representing real numbers where the position of the binary point (analogous to a decimal point in base-10) is fixed and predetermined. Unlike floating-point where the point "floats" based on an exponent, in fixed-point, you explicitly define how many bits are for the integer part and how many are for the fractional part.

Consider a binary number like 1011.0110. In a fixed-point system, we would decide beforehand where that binary point sits. If we allocate 4 bits for the integer part and 4 bits for the fractional part, this exact number fits perfectly.

The elegance of fixed-point lies in its simplicity for hardware. Arithmetic operations (addition, subtraction, multiplication) on fixed-point numbers can largely be performed using standard integer arithmetic units, followed by appropriate scaling or shifting to account for the implicit binary point.

The Qm.n Notation: Defining Your Precision

A common and intuitive notation for fixed-point numbers is Qm.n.

• 'Q' simply indicates a fixed-point number.
• 'm' represents the number of bits allocated for the integer part (including the sign bit for signed numbers).
• 'n' represents the number of bits allocated for the fractional part.

The total number of bits for the fixed-point number would be m+n.