Gray code to Binary conversion for PSK and PAM

Posted by Krishna Pillai

May 12, 2008

Given that we have discussed Binary to Gray code conversion, let us discuss the Gray to BInary conversion.

Conversion from Gray code to natural Binary

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

1. For $j=n-1$ ,

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

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

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

Simulation model

Look Up Table based Matlab/Octave mode for Gray to Binary conversion
clear % binary to gray code conversion ipBin = [0:15] ; % decimal equivalent for a 4-bit binary opGray = bitxor(ipBin,floor(ipBin/2)) % Gray to Binary conversion [tt ind] = sort(opGray); % sorting Gray code elements to form the lookup table opBin = ind(opGray+1)-1; % picking elements from the array

The Trick from DSP-Guru to do Gray to Binary conversion, thanks to Jerry Avins.
clear % binary to Gray code ipBin = [0:15] opGray = bitxor(ipBin,floor(ipBin/2)) % Gray code to binary opBin = bitxor(opGray,floor(opGray/2^8)) ; opBin = bitxor(opBin,floor(opBin/2^4)) ; opBin = bitxor(opBin,floor(opBin/2^2)) ; opBin = bitxor(opBin,floor(opBin/2^1)) ;

Concluding thoughts

1. As on now, I do not understand how the DSP Guru trick for Gray code to Binary works. Of course, if I figure out, I will update this post.

2. Having discussed Binary to Gray code conversion and Gray to Binary conversion, we are now armed to discuss the bit-error rate probabilities for various modulation schemes (Recall: We have been discussing symbol error rate in Additive White Gaussian noise till date).

Thanks,

Krishna

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