Defining Fixed-Point Notation
Fixed-point notation is a method for representing real numbers in a computer system where the position of the radix point (decimal point in base 10, binary point in base 2) is fixed. This means that a specific number of bits or digits are allocated to the integer part and a specific number to the fractional part, providing a consistent scale and precision for calculations without dynamic tracking of the radix point position.
How Fixed-Point Representation Works
In fixed-point notation, a number is typically stored as an integer, and it is implicitly understood that the radix point resides at a certain position within that integer. For example, if we have an 8-bit binary number and decide the last 3 bits represent the fractional part, then the first 5 bits represent the integer part. To interpret the number, you simply divide the stored integer value by 2 raised to the power of the number of fractional bits (e.g., 2^3 = 8 for 3 fractional bits).
Practical Example: Binary Fixed-Point
Consider an 8-bit fixed-point number where the lowest 4 bits are fractional (Q4 format). If the stored binary value is `01011100`, its integer equivalent is 92. With 4 fractional bits, the actual value is 92 / 2^4 = 92 / 16 = 5.75. The binary point is understood to be after the fourth bit from the right, making '0101' the integer part (5) and '1100' the fractional part (0.75 or 12/16).
Importance and Applications
Fixed-point notation is crucial in systems where computational resources are limited, such as embedded systems, digital signal processors (DSPs), and microcontrollers, because it avoids the complex hardware required for floating-point arithmetic. It offers predictable precision and performance, making it suitable for applications like audio processing, image filtering, and control systems where speed and consistent behavior are paramount, despite a more limited dynamic range compared to floating-point numbers.