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Mathematics of the DFT
    Number Systems for Digital Audio
       Linear Number Systems
          Binary Integer Fixed-Point Numbers
             Two's-Complement, Integer Fixed-Point Numbers

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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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Next: Fractional Binary Fixed-Point Numbers
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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