Documentation

# bin2gray

Convert positive integers into corresponding Gray-encoded integers

## Syntax

```y = bin2gray(x,modulation,M) [y,map] = bin2gray(x,modulation,M) ```

## Description

`y = bin2gray(x,modulation,M)` generates a Gray-encoded vector or matrix output `y` with the same dimensions as its input parameter `x`. `x` can be a scalar, vector, matrix, or 3-D array. `modulation` is the modulation type and must be `'qam'`, `'pam'`, `'fsk'`, `'dpsk'`, or `'psk'`. `M` is the modulation order that can be an integer power of 2.

`[y,map] = bin2gray(x,modulation,M)` generates a Gray-encoded output `y` with its respective Gray-encoded constellation map, `map`.

You can use map output to label a Gray-encoded constellation. The map output gives the Gray encoded labels for the corresponding modulation. See the example below.

### Note

If you are converting binary coded data to Gray-coded data and modulating the result immediately afterwards, you should use the appropriate modulation object or function with the `'Gray'` option, instead of `bin2gray`.

## Examples

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This example shows how to use the `bin2gray` and `gray2bin` functions to map integer inputs from a natural binary order symbol mapping to a Gray coded signal constellation and vice versa, assuming 16-QAM modulation. In addition, a visual representation of the difference between Gray and binary coded symbol mappings is shown.

Create a complete vector of 16-QAM integers.

```M= 16; x = (0:M-1)';```

Convert the input vector from a natural binary order to a Gray encoded vector using `bin2gray`.

`[y,mapy] = bin2gray(x,'qam',M);`

Convert the Gray encoded symbols, `y`, back to a binary ordering using `gray2bin`.

`z = gray2bin(y,'qam',M);`

Verify that the original data, `x`, and the final output vector, `z` are identical.

`isequal(x,z)`
```ans = logical 1 ```

To create a constellation plot showing the different symbol mappings, use the `qammod` function to find the complex symbol values.

`sym = qammod(x,M);`

Plot the constellation symbols and label them using the Gray, `y`, and binary, `z`, output vectors. The binary representation of the Gray coded symbols is shown in black while the binary representation of the naturally ordered symbols is shown in red. Set the axes scaling so that all points are displayed.

```scatterplot(sym,1,0,'b*'); for k = 1:16 text(real(sym(k))-0.3,imag(sym(k))+0.3,... dec2base(mapy(k),2,4)); text(real(sym(k))-0.3,imag(sym(k))-0.3,... dec2base(z(k),2,4),'Color',[1 0 0]); end axis([-4 4 -4 4])``` Observe that only a single bit differs between adjacent constellation points when using Gray coding.