**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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 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 thebinary wordcontaining these bits as

Each bit is of course either 0 or 1. Check that the 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, , can be interpreted as the “sign bit”. If is “on”, the number is negative. If it is “off”, the number is either zero or positive.

The least-significant bit is . “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 , in which case . The largest (in magnitude) negative number is, i.e., and for all . Table 4.5 shows some of the most common cases.

Table 4.5:Numerical range limits in -bit two’s-complement.

8 -128 127 16 -32768 32767 24 -8,388,608 8,388,607 32 -2,147,483,648 2,147,483,647