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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.
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: $$k=\begin{cases}\text{binary integer representation} & \text{if $0\le k\le 2^{(B-1)}-1$}\\ \mathrm{bit-by-bit\; inverse}(-k)+1 & \text{if $-2^{(B-1)}\le k\le 0$}\end{cases}$$
The most significant bit is known at the sign bit ; it is 0 when the number is non-negative; 1 when the number is negative.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.
or $$x=-{b}_{0}+\sum_{i=1}^{B-1} {b}_{i}2^{-i}$$ This interpretation makes it clearer how to implement digital filters in fixed-point, at least when the coefficients have amagnitude less than 1.Consider the multiplication of two binary fractions
Note that full-precision multiplication almost doubles thenumber of bits; if we wish to return the product to a $B$ -bit representation, we must truncate the $B-1$ least significant bits. However, this introduces truncation error (also known as quantization error , or roundoff error if the number is rounded to the nearest $B$ -bit fractional value rather than truncated). Note that this occurs after multiplication .Consider the addition of two binary fractions;
Note the occurence of wraparound overflow ; this only happens with addition . Obviously, it can be a bad problem.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.
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