Gray
Code in Matlab – from/to binary and decimal

The Gray
code (also known as reflected
binary code), is a
binary numerical system where two consecutive values differ in only one bit.
The Gray code code
was originally designed to prevent undesired
transient states or outputs from electro mechanical switches. Today,
this code
is used to facilitate error correction in digital communications and
digitaltoanalog converters.

In this article, we’re going to
develop a simple Matlab
algorithm to make conversions between (from/to) binary and gray
codes.
Convert a binary number to a Gray number
Let’s understand the algorithm to go from binary to Gray. See
the conversion from ‘11101’ binary to its equivalent in Gray code.
The most significant bit
(MSB) in Gray is taken directly
from the MSB in binary. The rest of the Gray bits comes from a xor
operation
between the precedent binary bit(b(i1)) and the current binary bit
(b(i)). In
the case shown in the figure above:
g(1) = b(1)
g(2) = b(1) xor b(2)
g(3) = b(2) xor b(3)
g(4) = b(3) xor b(4)
g(5) = b(4) xor b(5)
The xor
operation produces a 1 if the bits are different,
and produces a 0 if the bits are equal. So, a binary ‘11101’ becomes a
‘10011’
in Gray.
Let’s propose a code in
Matlab to do it.
function g =
bin2gray(b)
g(1) =
b(1);
for i = 2 :
length(b);
x
= xor(str2num(b(i1)), str2num(b(i)));
g(i)
= num2str(x);
end
The input parameter to
this function is a binary number
(expressed in a string), the output is the equivalent Gray number (also
expressed
as a string).
g =
bin2gray('11101')
g =
bin2gray('10010')
g =
bin2gray('11110')
g =
bin2gray('10000')
We get these
stringresults from Matlab:
g = 10011
g = 11011
g = 10001
g = 11000
We also can test a full sequence. We go from decimaltobinary (using
builtin function dec2bin)
and then to Gray (using our developed
function), for example:
for d = 0 :
15
b
= dec2bin(d);
g
= bin2gray(b);
disp({d
b g})
end
The results are (decimal,
binary, gray):
[0] '0'
'0'
[1] '1'
'1'
[2] '10'
'11'
[3] '11'
'10'
[4] '100'
'110'
[5] '101'
'111'
[6] '110'
'101'
[7] '111'
'100'
[8] '1000'
'1100'
[9] '1001'
'1101'
[10] '1010'
'1111'
[11] '1011'
'1110'
[12] '1100'
'1010'
[13] '1101'
'1011'
[14] '1110'
'1001'
[15] '1111'
'1000'
Convert a Gray number to a binary number
Now let’s understand the algorithm to go from Gray to
binary. See the conversion from ‘10011’ Gray to its binary equivalent.
The
most significant bit (MSB) in binary is taken directly
from the MSB in Gray. The rest of the binary bits comes from a xor
operation
between the precedent binary bit (b(i1)) and the current Gray bit
(g(i)). In
the case shown in the figure above:
b(1) = g(1)
b(2) = b(1) xor g(2)
b(3) = b(2) xor g(3)
b(4) = b(3) xor g(4)
b(5) = b(4) xor g(5)
Our proposed Matlab function to achieve the conversion is:
function b =
gray2bin(g)
b(1) =
g(1);
for i = 2 :
length(g);
x
= xor(str2num(b(i1)), str2num(g(i)));
b(i)
= num2str(x);
end
Let’s test our function:
b =
gray2bin('10011')
b =
gray2bin('11011')
b =
gray2bin('10001')
b =
gray2bin('11000')
We get
these stringresults:
b = 11101
b = 10010
b = 11110
b = 10000
And now we test a full
sequence, going decimaltobinary,
binarytogray, and graytobinary:
for d = 0 :
15
g
= bin2gray(dec2bin(d));
b
= gray2bin(g);
disp({d
g b})
end
We get (decimal, gray, binary):
[0]
'0'
'0'
[1] '1' '1'
[2] '11'
'10'
[3] '10'
'11'
[4] '110'
'100'
[5] '111'
'101'
[6] '101'
'110'
[7] '100'
'111'
[8] '1100'
'1000'
[9] '1101'
'1001'
[10] '1111'
'1010'
[11] '1110'
'1011'
[12] '1010'
'1100'
[13] '1011'
'1101'
[14] '1001'
'1110'
[15] '1000'
'1111'
From
'Gray Code' to home
From
'Gray Code' to 'Matlab Cookbook II'
