Fixed-point number representation

Fixed-point arithmetic is generally used when hardware cost, speed, or complexity is important. Finite-precision quantization issuesusually arise in fixed-point systems, so we concentrate on fixed-point quantization and error analysis in the remainder of this course.For basic signal processing computations such as digital filters and FFTs, the magnitude of the data, the internalstates, and the output can usually be scaled to obtain good performance with a fixed-point implementation.

Two's-complement integer representation

As far as the hardware is concerned, fixed-point number systems represent data as $B$ -bit integers. The two's-complement number system is usually used: $$k=\begin{cases}\text{binary integer representation} & \text{if $0\le k\le 2^{(B-1)}-1$}\\ \mathrm{bit-by-bit\; inverse}(-k)+1 & \text{if $-2^{(B-1)}\le k\le 0$}\end{cases}$$

Fractional fixed-point number representation

For the purposes of signal processing, we often regard the fixed-point numbers as binary fractions between $\left[-1 , 1\right)$ , by implicitly placing a decimal point after the sign bit.

Truncation error

Consider the multiplication of two binary fractions

Overflow error

Consider the addition of two binary fractions;

There are thus two types of fixed-point error: roundoff error, associated with data quantization and multiplication, andoverflow error, associated with data quantization and additions. In fixed-point systems, one must strike a balancebetween these two error sources; by scaling down the data, the occurence of overflow errors is reduced, but the relative sizeof the roundoff error is increased.