Introduction to Fixed Point Representation
Fixed point representation is a method of storing numbers in binary format. It is widely used in DSP products for telecommunications. One reason to use to use fixed point format (rather than floating point) is for cost savings in the digital signal processing chips used for implementing a system. Another reason is to have greater precision than floating point for a given number of bits per number represented.
So what is fixed point? Fixed point refers to a method of representing numbers with a fractional part on an ALU that only handles integer operations. Companies will often market their DSP processors as either “fixed point” or “floating point” models. Fractional digits of a number are handled with integers by using an assumed scaling factor, such as . This is commonly called Q15 notation.
To understand how Q15 numbers are handled, it helps to understand the twos complement representation of integers. I won’t explain it here, but there are plenty of explanations on the web (on Wikipedia for example). Now assume we have a 16 bit twos complement integer a as follows, where s is a sign bit and is the digit at bit n:
For a positive integer a, the value of the number a is:
For a Q15 number, the weightings of each bit change. Let . Then the value of a positive Q15 number b is:
The range of numbers that can be handled is from -1 to 0.999969482421875, corresponding to the integers -32768 and 32767, which are the smallest and largest integers that can be represented in 16 bit twos complement.
Negative numbers are found by taking the twos complement of a given positive value, in the same manner as for integers. One way to do this is to flip all the bits (change all the ones to zeros, change all the zeros to ones) and then add one. For example, assume we have the Q15 number 0.5, which is 0x4000 in hexadecimal. Flipping the bits gives 0xBFFF, and adding one produces 0xC000. So a Q15 value of -0.5 is stored as 0xC000. One case that does not work is if one tries to negate a Q15 value of -1, since +1 is not possible in Q15. If you try flipping the bits and adding one, you will end up with a result of -1 instead of +1. This is an overflow case, and some DSP chips will produce the largest positive Q15 number (32767/32768) if -1 is negated.
Q15 isn’t the only possible representation of course. It is possible to use different scaling factors and different bit widths. For example, Q31 is common for 32 bit number widths. When using unsigned operations, it is possible to store a Q16 number in a 16 bit word. In this case we have:
It is also possible to mix integer and fractional parts in one word. For example, we could have a number with 3 integer bits and 12 fractional bits stored in a 16 bit word. Lets call this format Q3.12. The range of numbers in Q3.12 using 16 bit twos complement is to , or -8 to 7.999755859375.
And finally, there is no rule that says a Q15 number can’t be stored in a 32 bit register. In this case, there would be 17 bits used for the sign part of the number (for twos complement).
So in conclusion, there are a large variety of number formats that can be handled when using a processor or hardware with integer-based arithmetic. Fixed point representation relies on an assumed scaling factor and a given register width. These two variables determine where the decimal point is. Fixed point arithmetic was taught to me by my first boss during the first few days at my first full time job. As my boss used to say “real men know where to find the decimal point.” The difficulty in finding the decimal point will become more clear when I cover fixed point multiplication.