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Gray code to Binary conversion for PSK and PAM

by Krishna Sankar on 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 be the equivalent Gray code for an bit binary number with representing the index of the bit.

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 Binary number is the exclusive-OR (XOR) of of the bit of the Gray code and 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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{ 18 comments… read them below or add one }

M.Ramya April 12, 2012 at 7:42 pm

how can we convert decimal to 2421 code

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Avatar of Krishna Sankar Krishna Sankar April 16, 2012 at 5:23 am

@M.Ramya: I have not tried it, but can the following link be of help to you?
http://mathforum.org/library/drmath/view/67281.html

Reply

shanu May 19, 2011 at 11:17 am

i want a c program for converting a digital signal into binary format??????? pls help me

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Avatar of Krishna Sankar Krishna Sankar May 23, 2011 at 2:34 am

@shanu: you might want to check dec2bin() function in matlab

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sneha September 1, 2010 at 10:44 am

i want conversion of gray code to binary code using kmaps???????if this is thereplz tel me???not there den tel me d reason

Reply

Avatar of Krishna Sankar Krishna Sankar September 5, 2010 at 6:26 am

@sneha: What is kmaps?

Reply

Bob July 30, 2010 at 2:12 am

Since you ask how it works, here is an explanation:

Notation: A single letter represents an 8 bit value
A single letter followed by a digit represents one bit in that 8 bit value.
Bits are numbered from right to left, so bit n has the value 2^n.

Start with the binary value b.
b=b7;b6;b5;b4;b3;b2;b1;b0

Compute its gray code, g = b ^ ( b >> 1 )
g=g7;g6;g5;g4;g3;g2;g1;g0
[0] g7=b7; g6=b7^b6; g5=b6^b5; g4=b5^b4; g3=b4^b3; g2=b3^b2; g1=b2^b1; g0=b1^b0

Now recover the original binary value from g in three steps
[1] x = g ^ (g >> 4)
x=x7;x6;x5;x4;x3;x2;x1;x0
x7=g7; x6=g6; x5=g5; x4=g4 x3=g3^g7 x2=g2^g6 x1=g1^g5; x0=g0^g4

[2] y = x ^ (x >> 2)
y=y7;y6;y5;y4;y3;y2;y1;y0
y7=x7; y6=x6; y5=x5^x7; y4=x4^x6; y3=x3^x5; y2=x2^x4; y1=x1^x3; y0=x0^x2

substitute eqn [1] into eqn [2]
[2a] y7=g7; y6=g6; y5=g5^g7; y4=g4^g6; y3=g3^g7^g5; y2=g2^g6^g4; y1=g1^g5^g3^g7; y0=g0^g4^g2^g6

[3] z = y ^ (y >> 1)
z7=y7; z6=y6^y7; z5=y5^y6; z4=y4^y5; z3=y3^y4; z2=y2^y3; z1=y1^y2; z0=y0^y1

Substitute eqn [2a] into eqn [3]
[3a] z7=g7; z6=g6^g7; z5=g5^g7^g6; z4=g4^g6^g5^g7; z3=g3^g7^g5^g4^g6; z2=g2^g6^g4^g3^g7^g5; z1=g1^g5^g3^g7^g2^g6^g4; z0=g0^g4^g2^g6^g1^g5^g3^g7

Substitute eqn [0] into eqn [3a]
[3b] z7=b7;
z6=b7^b6^b7;
z5=b6^b5^b7^b7^b6;
z4=b5^b4^b7^b6^b6^b5^b7;
z3=b4^b3^b7^b6^b5^b5^b4^b7^b6;
z2=b3^b2^b7^b6^b5^b4^b4^b3^b7^b6^b5;
z1=b2^b1^b6^b5^b4^b3^b7^b3^b2^b7^b6^b5^b4;
z0=b1^b0^b5^b4^b3^b2^b7^b6^b2^b1^b6^b5^b4^b3^b7

Rearrange the bits on the right
[3c] z7=b7;
z6=b6^b7^b7;
z5=b5^b6^b6^b7^b7;
z4=b4^b5^b5^b6^b6^b7^b7;
z3=b3^b4^b4^b5^b5^b6^b6^b7^b7;
z2=b2^b3^b3^b4^b4^b5^b5^b6^b6^b7^b7;
z1=b1^b3^b3^b2^b2^b4^b4^b5^b5^b6^b6^b7^b7;
z0=b0^b1^b1^b2^b2^b3^b3^b4^b4^b5^b5^b6^b6^b7^b7

Now, because x^x=0 and y^0=y, we can just remove each identical pair
[3c] z7=b7;
z6=b6;
z5=b5;
z4=b4;
z3=b3;
z2=b2;
z1=b1;
z0=b0;

And that’s how it works.

Reply

Avatar of Krishna Sankar Krishna Sankar July 30, 2010 at 6:36 am

@Bob: Thanks much. :)
I will try to digest and get back, for any clarifications…

Reply

rayan2 January 29, 2010 at 11:55 pm

use this formula
G=(N XOR 2*N)/2

Reply

Avatar of Krishna Sankar Krishna Sankar April 4, 2010 at 4:43 am

@rayan2: It does the same thing? How does that work?

Reply

jerraz September 7, 2009 at 2:22 pm

Hello!
I need convert 10-bits number from gray to binary. Could anybody help me?

Reply

Avatar of Krishna Sankar Krishna Sankar September 9, 2009 at 5:48 am

@jerraz: Hope the Lookup Table based Matlab code provided in the post helps.

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khalid_hassan September 3, 2009 at 1:53 pm

iam a student of heritage of 2nd yera I.T

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R.Ramya March 29, 2009 at 9:41 am

I cant understand this line particualr line:

[tt ind] = sort(opGray); % sorting Gray code elements to form the lookup table

here ind refers to ???

Reply

Avatar of Krishna Sankar Krishna Sankar April 4, 2009 at 8:49 am

@R.Ramya:
The variable ind refers to the index of the elements prior to sorting.
as a quick example,
a = [3 1 4];
[tt ind] = sort(a);
tt =
1 3 4
ind =
2 1 3

Hope this helps.

Reply

khushi August 24, 2012 at 3:51 pm

thanks a lot sir:)

Reply

R.Ramya March 29, 2009 at 7:51 am

hi krishna,
there is a typo in the article
it shuld be written as
1. For j=n-1;
b[n-1]=g[n-1];

instead of 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.

cheers.

Reply

Avatar of Krishna Sankar Krishna Sankar April 4, 2009 at 8:43 am

@R.Ramya: Thanks for noticing the typo. I corrected. :)

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