Little Endian Formula

GUIDE: Mathematics of the Discrete Fourier Transform (DFT).

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“Little Endian” Formula for Two’s-Complement, Integer Fixed-Point Numbers

The formula of the preceding section can be considered “big endian” because the bits are labeled “left to right” in the word as it is normally visualized. That is, $b_0$ is the most significant bit instead of the least significant bit. The “little endian” convention numbers bits in order of their significance instead.

For the “little-endian” formula, we simply reverse the bits $\ in the previous formula:

\

The binary word is now written

\

The actual bits in the word are visualized the same in both big and little endian labelings. For example, in the three-bit case, a decimal 3 is still a binary $011$ and a decimal -4 is still a binary $100$. Only the bit-symbol subscripts are different: $b_0 b_1 b_2$ vs. $b_2 b_1 b_0$.

Note that this representation extends to any “base” (currently 2). For example, positive decimal integers can be expanded as

\

and an $N$-digit integer is written as $d_N d_{N-1}\. This type of “positional notation” was a major “breakthrough invention” in ancient times that greatly facilitated the handling of large numbers. (Have you ever tried working extensively with Roman numerals?)

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