One's Complement Fixed-Point Format

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). One's Complement Fixed-Point Format

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One’s Complement Fixed-Point Format

One’s Complement is a particular assignment of bit patterns to numbers. For example, in the case of 3-bit binary numbers, we have the assignments shown in Table 4.3.


Table 4.3:Three-bit one’s-complement binary fixed-point numbers.
BinaryDecimal
0000
0011
0102
0113
100-3
101-2
110-1
111-0


In general, $N$-bit numbers are assigned to binary counter values in the “obvious way” as integers from 0 to $2^{N-1}-1$, and then the negative numbers are assigned in reverse order, as shown in the example.

The term “one’s complement” refers to the fact that negating a number in this format is accomplished by simply complementing the bit pattern (inverting each bit).

Note that there are two representations for zero (all 0s and all 1s). This is inconvenient when testing if a number is equal to zero. For this reason, one’s complement is generally not used.

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