Fixed-Point Converter

Convert decimal values into fixed-point binary, hexadecimal, and stored integers. Configure total word width, fractional bits, signed or unsigned interpretation, rounding, and overflow behavior. This calculator is useful for embedded programming, digital signal processing (DSP), microcontrollers, FPGA logic, and Q-format arithmetic.

Fixed-Point / Q-Format Calculator

Common format

Conversion direction

Total word width (bits)

Fractional bits

Interpretation

Decimal value

Enter a decimal number, such as 0.75, -1.25, or 12.5.

Rounding method

Overflow handling when encoding

Overflow policy affects encoding. Decoding always interprets the supplied word at the selected width.

Conversion result

Stored binary word

Signed stored integer

Unsigned raw code

Hexadecimal word

Exact represented value

Quantization error (represented − requested)

Resolution (one LSB)

Representable range

Scale factor

Format

Status

Calculation steps

Quick examples:

What Is Fixed-Point Representation?

Fixed-point representation stores a number as an integer with an implied binary point. The point is not physically stored in the word. Instead, the number of fractional bits determines the scale used to interpret the stored integer.

Core formula:
Represented value = stored integer ÷ 2^F
where F is the number of fractional bits.

For example, if F = 8, each integer step represents 1/256, or 0.00390625. A stored integer of 384 represents 384/256 = 1.5.

Signed and unsigned fixed-point values

In signed mode, the stored integer uses two's complement. For a word of W bits, its integer range is from −2^W−1 to 2^W−1 − 1. In unsigned mode, the range is from 0 to 2^W − 1. Dividing those endpoints by 2^F gives the representable real-value range.

Common fixed-point formats

Format	Total bits	Fraction bits	Example use
Q7	8	7	Compact coefficients and small values
Q15 / Q1.15	16	15	Audio and DSP coefficients
Q31 / Q1.31	32	31	Higher precision DSP calculations
Q8.8	16	8	General embedded calculations
Q16.16	32	16	Control systems and graphics

Q-format labels are not universal. Some conventions count the sign bit as part of the integer-bit count, while others do not. This calculator avoids that ambiguity by specifying the total word width, fractional bits, and signedness independently.

Rounding and quantization error

A decimal value is encoded by multiplying it by 2^F and converting the scaled value to an integer. If the result is not an integer, a rounding rule determines which stored integer is chosen. The difference between the represented value and the requested value is the quantization error.

What happens when a value is out of range?

Reject: show an error rather than encoding an invalid value.
Saturate: use the closest representable endpoint.
Wrap: reduce the integer code modulo 2^W, matching fixed-width word wrapping.

Frequently Asked Questions

How do I convert a decimal number to fixed-point binary?

Choose the word width and fractional-bit count, multiply the decimal value by 2^F, apply the selected rounding and overflow rules, then encode the resulting integer into the selected-width binary word.

What does Q15 mean?

In this calculator, Q15 means a signed 16-bit two's complement value with 15 fractional bits. Its scale is 32768, and its representable range is −1 through 1 − 1/32768. The positive value +1 cannot be represented in this format.

Why is the represented value different from my input?

Fixed-point formats have a finite resolution. If the input does not fall exactly on a representable step, rounding selects a nearby value. The calculator reports the quantization error.

When should I use fixed-point instead of floating-point?

Fixed-point can be useful on microcontrollers, DSPs, FPGAs, and embedded systems where predictable integer arithmetic, defined scaling, or hardware without floating-point support is important. The available range and precision must be chosen carefully.