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Two's-Complement, Integer Fixed-Point Numbers

Let $ N$ denote the number of bits. Then the value of a two's complement integer fixed-point number can be expressed in terms of its bits $ \{b_i\}_{i=0}^{N-1}$ as

$\displaystyle x = - b_0 \cdot 2^{N-1} + \sum_{i=1}^{N - 1} b_i \cdot 2^{N - 1 - i},\quad b_i\in\{0,1\} \protect$ (G.1)

We visualize the binary word containing these bits as

$\displaystyle x = [b_0\, b_1\, \cdots\, b_{N-1}]
$

Each bit $ b_i$ is of course either 0 or 1. Check that the $ N=3$ case in Table G.3 is computed correctly using this formula. As an example, the number 3 is expressed as

$\displaystyle 3 =[ 0 1 1 ] = - 0\cdot 4 + 1\cdot 2 + 1 \cdot 1
$

while the number -3 is expressed as

$\displaystyle -3 =[ 1 0 1 ] = - 1\cdot 4 + 0\cdot 2 + 1 \cdot 1
$

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\cdots0$ , i.e., $ b_0=1$ and $ b_i=0$ for all $ i>0$ . Table G.4 shows some of the most common cases.


Table G.4: Numerical range limits in $ N$ -bit two's-complement.
$ N$ $ x_{\mbox{\small min}}$ $ x_{\mbox{\small max}}$
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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``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 © 2014-04-21 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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