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Binary to Gray code conversion for PSK and PAM

Posted By Krishna Sankar On May 11, 2008 @ 1:12 pm In Modulation | 14 Comments

In this post, let us try to understand Gray codes and their usage in digital communication. Quoting from Wiki entry on Gray code [Gray-Wiki ^[1]],

The reflected binary code, also known as Gray code after Frank Gray ^[2], is a binary numeral system ^[3] where two successive values differ in only one digit.

In a digital communication system, if the constellation symbols are Gray encoded, then the bit pattern representing the adjacent constellation symbols differ by only one bit. We will show in another post that having this encoding structure gives a lesser probability of error than the ‘natural binary ordering’. However, in this post, let us try to figure out the conversion of natural binary representation to Gray code.

Conversion from natural Binary to Gray code

Consider a $n$ bit binary number $b[n-1:0]$ with $j$ representing the index of the binary number. Let $g[n-1:0]$ be the equivalent Gray code.

1. For $j=n-1$ ,

$g[n-1] = b[n-1]$ i.e, the most significant bit (MSB) of the Gray code is same as the MSB of original binary number.

2. For $j=n-2\mbox{ to }0$ ,

$g[j]=b[j+1]\oplus b[j]$ i.e, $j^{th}$ bit of the Gray code is the exclusive-OR (XOR) of ${j}^{th}$ of the bit of the binary number and ${j+1}^{th}$ of the bit of the binary number.

Simulation

Simple Matlab/Octave code for doing the binary to Gray code conversion

clear; ip = [0:15]; % decimal equivalent of a four bit binary word op = bitxor(ip,floor(ip/2)); % decimal equivalent of the equivalent four bit gray word

Table : Natural Binary to Gray code

Input, decimal	Input, binary	Gray, decimal	Gray, Binary
0	0000	0	0000
1	0001	1	0001
2	0010	3	0011
3	0011	2	0010
4	0100	6	0110
5	0101	7	0111
6	0110	5	0101
7	0111	4	0100
8	1000	12	1100
9	1001	13	1101
10	1010	15	1111
11	1011	14	1110
12	1100	10	1010
13	1101	11	1011
14	1110	9	1001
15	1111	8	1000

Note

1. As can be seen from the Table above, each row differs from the row above and below by only one bit. Further, just to highlight that this behavior is indeed true for 16th row [1000] and the 1st row [0000] .

2. The conversion shown in the Table above can be used for general modulation schemes like M-PSK (Phase Shift Keying), M-PAM (Pulse Amplitude Modulation) etc.

3. However, for a general M-QAM modulation the binary to Gray code conversion is bit more complicated (and I need to figure that out). We will discuss the QAM case in a future post.

Thanks,

Krishna