Binary encoding of quantization levels

Consider an n-bit binary number representing a full-scale range R. In other words, the range R is being quantized into 2ⁿ quantization levels. If R is unipolar, the quantized value x_Q lies in the range [0,R]. If it is bipolar, x_Q lies in the range [−R/2,R/2].

We shall denote the n-bit pattern as a vector b = [b_n-1, b_n-2, …, b₁, b₀] where b_n-1 is called the most significant bit (MSB) and b₀ is the least significant bit (LSB). There are many ways in which this n-bit pattern can be used to encode x_Q. Three most common ways are:

Here b_n−1 denotes the complement of b_n−1.

Example A.1

For R = 2 V and 3-bit (8-level) quantization, the correspondence between the binary representations and the quantized value are given in the following table.

The unipolar natural binary representation encodes levels in the range 0 to 2 V. Offset binary and 2′s complement encodes −1 V to 1 V.