# pskmod

Modulate signal using M-PSK method

## Syntax

``Y = pskmod(X,M)``
``Y = pskmod(X,M,phaseoffset)``
``Y = pskmod(X,M,phaseoffset,symorder)``
``Y = pskmod(X,M,Name=Value)``

## Description

````Y = pskmod(X,M)` modulates the input signal `X` using the M-Ary phase shift keying (M-PSK) method. `M` specifies the modulation order.```
````Y = pskmod(X,M,phaseoffset)` specifies the phase offset of the M-PSK constellation.```

example

````Y = pskmod(X,M,phaseoffset,symorder)` specifies the symbol order of the M-PSK constellation.```
````Y = pskmod(X,M,Name=Value)` specifies optional name-value arguments using any of the previous syntaxes. For example, `pskmod(Y,M,PlotConstellation=true)` modulates using modulation order `M` and plots the constellation. Specify name-value arguments after all other input arguments.```

## Examples

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Modulate and plot the constellations of QPSK and 16-PSK signals.

QPSK

Set the modulation order to 4.

`M = 4;`

Generate random data symbols.

`data = randi([0 M-1],1000,1);`

Modulate the data symbols.

`txSig = pskmod(data,M,pi/M);`

Pass the signal through white noise and plot its constellation.

```rxSig = awgn(txSig,20); scatterplot(rxSig)```

16-PSK

Change the modulation order from 4 to 16.

`M = 16;`

Generate random data symbols.

`data = randi([0 M-1],1000,1);`

Modulate the data symbols.

`txSig = pskmod(data,M,pi/M);`

Pass the signal through white noise and plot its constellation.

```rxSig = awgn(txSig,20); scatterplot(rxSig)```

Generate random symbols.

`dataIn = randi([0 3],1000,1);`

QPSK modulate the data.

`txSig = pskmod(dataIn,4,pi/4);`

Pass the signal through an AWGN channel.

`rxSig = awgn(txSig,10);`

Demodulate the received signal and compute the number of symbol errors.

```dataOut = pskdemod(rxSig,4,pi/4); numErrs = symerr(dataIn,dataOut)```
```numErrs = 3 ```

Set the modulation order, then create a data sequence containing a complete set of constellation points.

```M = 8; data = (0:M-1); phaseoffset = 0;```

Visualize the plot constellations of 8-PSK symbol mapping for modulated and demodulated gray-coded and binary-coded data.

```symgray = pskmod(data,M,phaseoffset,'gray',PlotConstellation=true, ... InputType='integer');```

```mapgray = pskdemod(symgray,M,phaseoffset,'gray',OutputType='integer'); symbin = pskmod(data,M,phaseoffset,'bin'); mapbin = pskdemod(symbin,M,phaseoffset,'bin',PlotConstellation=true, ... OutputType='bit');```

## Input Arguments

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Input signal, specified as a scalar, vector, or matrix of positive integers. The elements of `X` must have binary or integer values in the range [0, `M` – 1], where `M` is the modulation order.

Note

To process an input signal as binary elements, set the `InputType` name-value argument to `'bit'`. For binary inputs, the number of rows must be an integer multiple of log2(`M`). The function maps groups of log2(`M`) bits onto a symbol, with the first bit representing the MSB and the last bit representing the LSB.

#### Dependencies

For input signals of datatype `logical`, you must set `InputType` to `'bit'`.

Data Types: `double` | `single` | `int8` | `int16` | `int32` | `uint8` | `uint16` | `uint32` | `logical`

Modulation order, specified as an integer value greater than 1.

Data Types: `double`

Phase offset of the PSK constellation in radians, specified as a scalar.

Data Types: `double`

Symbol order, specified as `'gray'`, `'bin'`, or a vector. This argument specifies how the function assigns binary vectors to corresponding integers.

• `'gray'` — Use a Gray-coded ordering.

• `'bin'` — Use a binary-coded ordering.

• vector –– Use custom symbol ordering. The vector is of length `M` containing unique values in the range [0, `M` – 1]. The first element correlates to the constellation point corresponding to angle `phaseoffset`, with subsequent elements running counter-clockwise.

### Name-Value Arguments

Specify optional pairs of arguments as `Name1=Value1,...,NameN=ValueN`, where `Name` is the argument name and `Value` is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Example: ```Y = pskmod(X,M,phaseoffset,symorder,InputType='bit')```

Input type, specified as either `'integer'` or `'bit'`.

• `'integer'` –– Input signal consists of integers in the range [0, `M` – 1].

• `'bit'` –– Input signal consists of binary values and the number of rows must be an integer multiple of log2(`M`).

Data type of output, specified as either `'double'` or `'single'`.

Option to plot constellation, specified as logical `0` (`false`) or `1` (`true`). To plot the PSK constellation, set `'PlotConstellation'` to `true`.

Data Types: `logical`

## Output Arguments

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M-PSK modulated baseband signal, returned as a scalar, vector or matrix of complex values. The columns of `Y` represent independent channels. For integer inputs, the output `Y` has the same dimensions as the input signal `X`. For bit inputs, the number of rows in `Y` is the number of rows in `X` divided by log2(`M`).

## Algorithms

For higher-order PSK constellations, the complex baseband form for an M-ary PSK signal using binary-ordered symbol mapping is

`${s}_{m}\left(t\right)=\mathrm{exp}\left(j\left(\frac{2\pi m}{M}+\varphi \right)\right);\text{ }m\in \left\{0,1,\dots ,M-1\right\}.$`

When the input is configured for bits, groups of log2(M) bits represent the complex symbols for the configured symbol mapping. The mapping can be binary encoded, Gray encoded, or custom encoded.

Gray coding has the advantage that only one bit changes between adjacent constellation points, which results in better bit error rate performance.

This 8-PSK constellation uses Gray-coded symbol mapping.

For modulation orders beyond 4, the bit error rate performance of PSK in AWGN worsens. In this bit error rate plot for Gray-coded mapping, the QPSK and BPSK curves overlap one another.

## References

[1] Proakis, John G. Digital Communications. 4th ed. New York: McGraw Hill, 2001.

## Version History

Introduced before R2006a

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