Q-format is a notation used to describe the layout of bits in a fixed-point number system, specifying how many bits are used for the integer part and how many for the fractional part.

How do you convert float to fixed point?

To convert a float to fixed-point (Qn), multiply the float by 2^n and round the result to the nearest integer.

Advanced Q-format Calculator | Fixed-Point Converter Tool

Q-format Calculator

Fixed-Point Arithmetic Converter & Analyzer

Input Value (Float)

Word Length (Total Bits)

Fractional Bits (n) Q12

Bit Allocation

I (3)

F (12)

Format: Q3.12 (Signed)

Fixed-Point Value (Decimal)

Stored Integer

Hex Representation

0x----

Binary Representation

----

Actual Value (Reconverted)

Error / Precision Loss

Format Specs

Resolution

Range

Mastering Fixed-Point Arithmetic with Q-format

In the world of Embedded Systems and Digital Signal Processing (DSP), floating-point hardware (FPU) is a luxury. Microcontrollers often lack dedicated floating-point units, making math operations slow and energy-expensive. The solution? Fixed-Point Arithmetic.

This Q-format Calculator helps engineers visualize and convert real-world numbers into integer representations used by processors, ensuring precision requirements are met while maximizing performance.

💡 Why use Q-format?

Q-format allows a processor to perform math using standard integer instructions (ADD, SUB, MUL) while representing fractional numbers. For example, to represent 0.5 in Q15 format, we store the integer 16384. The CPU sees an integer, but the programmer knows it represents a fraction.

Understanding Qm.n Notation

Q-format is typically denoted as Q m.n, where:

S (Sign bit): Usually implied as 1 bit for Two's Complement arithmetic.
m (Integer bits): Determines the maximum integer value (Range) before saturation/overflow occurs.
n (Fractional bits): Determines the precision (Resolution) of the number.

The total number of bits is $W = 1 + m + n$. For a standard 16-bit signed integer, if we choose Q12 (12 fractional bits), we are left with $16 - 1 - 12 = 3$ integer bits.

Key Calculations

Here are the formulas used by this tool:

Resolution (Step Size): $R = 2^{-n}$. This is the smallest non-zero positive number you can represent.
Maximum Value: $Max = (2^{W-1} - 1) \times R$.
Minimum Value: $Min = -2^{W-1} \times R$.
Float to Fixed Conversion: $Fixed = \text{round}(Float \times 2^n)$.
Fixed to Float Conversion: $Float = Fixed \times 2^{-n}$.

Common Q-Formats

Depending on the application, different formats are standard:

Q15 (1.15): Used heavily in DSP for values between -1 and +0.999. Max precision for 16-bit audio processing.
Q31 (1.31): High-precision standard for 32-bit processors like ARM Cortex-M4/M7.
Q8.8: Balanced format for 16-bit systems requiring a range of ±128 and moderate precision (0.0039).

Frequently Asked Questions (FAQ)

What is quantization error? +

Quantization error is the difference between the actual floating-point number and its nearest fixed-point representation. Since there are finite bits, some numbers cannot be represented exactly (e.g., 0.1 cannot be perfectly represented in binary). This calculator shows this as "Precision Loss".

How do I handle overflow? +

In fixed-point math, you must carefully choose your Q-format. If you need to represent the number 500, you need at least 10 integer bits ($2^9 = 512$). If you try to store 500 in a Q15 format (which has range ±1), it will overflow and wrap around, causing calculation errors.

What is saturation arithmetic? +

Saturation arithmetic clamps values to the maximum or minimum limit instead of letting them wrap around upon overflow. This is critical in audio processing to avoid loud "pops" or glitches.

Read Also: