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Gray code to Binary conversion for PSK and PAM

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 $j=n-1$,

$b[n-1] = g[n-1]$ 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. ðŸ™‚

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

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

1. M.Ramya says:

how can we convert decimal to 2421 code

2. shanu says:

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

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

4. Bob says:

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.

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.

1. @Bob: Thanks much. ðŸ™‚
I will try to digest and get back, for any clarifications…

5. rayan2 says:

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

6. jerraz says:

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

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

7. R.Ramya says:

I cant understand this line particualr line:

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

here ind refers to ???

1. @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.

1. khushi says:

thanks a lot sir:)

8. R.Ramya says:

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];