General Formula for Two's-Complement, Integer Fixed-Point Numbers

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. General Formula for Two's-Complement, Integer Fixed-Point Numbers

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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General Formula for Two's-Complement, Integer Fixed-Point Numbers

Let $N$ denote the (even) number of bits. Then the value of a two's complement integer fixed-point number can be expressed in terms of its bits $\ as


We visualize the binary word containing these bits as

Each bit $b_i$ is of course either 0 or 1. Check that the $N=3$ table above is computed correctly using this formula. As an example, the numer 3 is expressed as

while the number -3 is expressed as

and so on.

The most-significant bit in the word, $b_0$, can be interpreted as the ''sign bit''. If $b_0$ is ''on'', the number is negative. If it is ''off'', the number is either zero or positive.

The least-significant bit is $b_{N-1}$. ''Turning on'' that bit adds 1 to the number, and there are no fractions allowed.

The largest positive number is when all bits are on except $b_0$, in which case $x=2^{N-1}-1$. The largest (in magnitude) negative number is$10\, i.e., $b_0=1$ and $b_i=0$ for all $i>0$. Table 4.5 shows some of the most common cases.

Table 4.5:Numerical range limits in $N$-bit two's-complement.
$N$ $x_{\ $x_{\
8 -128 127
16 -32768 32767
24 -8,388,608 8,388,607
32 -2,147,483,648 2,147,483,647

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