## Phase Modulation

Phase modulation (PM) is a linear baseband modulation technique in which the message modulates the phase of a constant amplitude signal. Communications Toolbox™ software includes these functions, System objects, and blocks to modulate digital baseband signals with these modulation methods:

Binary, quadrature, and general phase shift keying (PSK)

Binary, quadrature, and general differential phase shift keying (DPSK)

Offset quadrature phase shift keying (OQPSK)

### BPSK

In binary phase shift keying (BPSK), the phase of a constant amplitude signal switches between two values corresponding to binary 1 and binary 0. The passband waveform of a BPSK signal is

$${s}_{n}(t)=\sqrt{\frac{2{E}_{b}}{{T}_{b}}}\mathrm{cos}\left(2\pi {f}_{c}t+{\varphi}_{n}\right),$$

where:

*E*is the energy per bit._{b}*T*is the bit duration._{b}*f*is the carrier frequency._{c}

In MATLAB^{®}, the baseband representation
of a BPSK signal is

$${s}_{n}(t)={e}^{-i{\varphi}_{n}}=\mathrm{cos}\left(\pi n\right).$$

The BPSK signal has two phases: 0 and *π*.

The probability of a bit error in an AWGN channel is

$${P}_{b}=Q\left(\sqrt{\frac{2{E}_{b}}{{N}_{0}}}\right),$$

where *N _{0}* is the noise power spectral
density.

### QPSK

In quadrature phase shift keying, the message bits are grouped into 2-bit symbols, which are transmitted as one of four phases of a constant amplitude baseband signal. This grouping provides a bandwidth efficiency that is twice as great as the efficiency of BPSK. The general QPSK signal is expressed as

$${s}_{n}(t)=\sqrt{\frac{2{E}_{s}}{{T}_{s}}}\mathrm{cos}\left(2\pi {f}_{c}t+(2n+1)\frac{\pi}{4}\right);\text{\hspace{1em}}n\in \{0,1,2,3\},$$

where *E _{s}* is the energy per symbol and

*T*is the symbol duration. The complex baseband representation of a QPSK signal is

_{s}$${s}_{n}(t)=\mathrm{exp}\left(j\pi \left(\frac{2n+1}{4}\right)\right);\text{\hspace{1em}}n\in \{0,1,2,3\}.$$

In this QPSK constellation diagram, each 2-bit sequence is mapped to one of four
possible states. The states correspond to phases of *π*/4, 3*π*/4, 5*π*/4, and 7*π*/4.

To improve bit error rate performance, the incoming bits can be mapped to a Gray-coded ordering.

**Binary-to-Gray Mapping**

Binary Sequence | Gray-Coded Sequence |
---|---|

00 | 00 |

01 | 01 |

10 | 11 |

11 | 10 |

The primary advantage of the Gray code is that only one of the two bits changes when moving between adjacent constellation points. Gray codes can be applied to higher-order modulations, as shown in this Gray-coded QPSK constellation.

The bit error probability for QPSK in AWGN with Gray coding is

$${P}_{b}=Q\left(\sqrt{\frac{2{E}_{b}}{{N}_{0}}}\right),$$

which is the same as the expression for BPSK. As a result, QPSK provides the same performance with twice the bandwidth efficiency.

### Higher-Order PSK

You can modulate and demodulate higher-order PSK constellations. The complex baseband form for an M-ary PSK signal using binary-ordered symbol mapping is

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

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 the following figure, the QPSK and BPSK curves overlap one another.

### DPSK

DPSK is a noncoherent form of phase shift keying that does not require a coherent reference signal at the receiver. With DPSK, the difference between successive input symbols is mapped to a specific phase. As an example, for binary DPSK (DBPSK), the modulation scheme operates such that the difference between successive bits is mapped to a binary 0 or 1. When the input bit is 1, the differentially encoded symbol remains the same as the previous symbol, while an incoming 0 toggles the output symbol.

The disadvantage of DPSK is that it is approximately 3 dB less energy efficient
than coherent PSK. The bit error probability for DBPSK in AWGN is *P _{b}* = 1/2
exp(

*E*/

_{b}*N*).

_{0}### OQPSK

Offset QPSK is similar to QPSK except that the time alignment of the in-phase and quadrature bit streams differs. In QPSK, the in-phase and quadrature bit streams transition at the same time. In OQPSK, the transitions have an offset of a half-symbol period as shown.

The in-phase and quadrature signals transition only on boundaries between symbols. These transitions occur at 1-second intervals because the sample rate is 1 Hz. The following figure shows the in-phase and quadrature signals for an OQPSK signal.

For OQPSK, the quadrature signal has a 1/2 symbol period offset (0.5 s).

The BER for an OQPSK signal in AWGN is identical to that of a QPSK signal. The BER is

$${P}_{b}=Q\left(\sqrt{\frac{2{E}_{b}}{{N}_{0}}}\right),$$

where *E _{b}* is the energy per bit and

*N*is the noise power spectral density.

_{0}## References

[1] Rappaport, Theodore S. *Wireless Communications: Principles and Practice.* Upper Saddle River, NJ: Prentice Hall, 1996, pp. 238–248.

[2] Viterbi, A.J. “An Intuitive Justification and a Simplified Implementation of the MAP Decoder for Convolutional Codes.” *IEEE Journal on Selected Areas in Communications* 16, no. 2 (February 1998): 260–64. https://doi.org/10.1109/49.661114.

## See Also

### Functions

### Objects

### Blocks

- Raised Cosine Transmit Filter | Raised Cosine Receive Filter | Bipolar to Unipolar Converter | Unipolar to Bipolar Converter | Data Mapper

## Related Examples

- Phase Modulation Examples
- Estimate BER of QPSK in AWGN with Reed-Solomon Coding
- Log-Likelihood Ratio (LLR) Demodulation