Binary to Gray code conversion for PSK and PAM

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],

The reflected binary code, also known as Gray code after Frank Gray, is a binary numeral system 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 bit binary number with representing the index of the binary number. Let be the equivalent Gray code.

1. For ,

i.e, the most significant bit (MSB) of the Gray code is same as the MSB of original binary number.

2. For ,

i.e, bit of the Gray code is the exclusive-OR (XOR) of of the bit of the binary number and of the bit of the binary number.


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

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

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


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.


14 thoughts on “Binary to Gray code conversion for PSK and PAM

  1. i have a little problem in the conversion illustrated above???
    What to do if the binary number is not a simple binary number but a binary including the decimal???
    For example what would be the gray code of 110101.101101???
    Today i got this question in my engineering exam…???
    i am in a great doubt…so please help me???

  2. Yes, i agree (strange) !
    Disregarding the way of implementation of bitshift/floor under Matlab/Octave, it is well known that (Bit Shifting) is faster than any multiplication/division method, specially on microprocessors.

  3. Great job 😉
    i just wanted to mention that this snippet :

    op = bitxor(ip,floor(ip/2));

    can be more optimized by just doing a right-shift bit by 1.
    so the equivalent code would be :


    -1 means a 1 bit shift to the right (the negative sign means right).

    1. @O.S.O: Thanks for the tip. However, I did a quick experiment with octave using tic-toc and did not observe that using bitshift() instead of floor is faster.
      octave:51> clear all; tic; ip = [0:2^20-1];op = bitxor(ip,floor(ip/2));toc
      Elapsed time is 0.4 seconds.
      octave:52> clear all; tic;ip1 = [0:2^20-1];op1 = bitxor(ip1,bitshift(ip1,-1));toc
      Elapsed time is 0.46 seconds.


  4. @elizabeth: Well, you can use the steps 1, 2 in “Conversion from natural Binary to Gray code” to write your own code. No?

  5. i am doing engg…. thanks for the quick reference…. please also include how to do it in order… egs why does the 1001 follow should we write it on our own….. in a sequence such that we can make a tabulation table to draw the logic diagram of a code converter from binary to grey etc!

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