## Phase Modulation

Phase modulation is a linear baseband modulation technique in which the message modulates the phase of a constant amplitude signal. Communications Toolbox™ provides modulators and demodulators for these phase modulation techniques:

Phase shift keying (PSK) — Binary, quadrature, and general PSK

Differential phase shift keying (DPSK) — Binary, quadrature, and general DPSK

Offset QPSK (OQPSK)

To modulate input data with these techniques, you can use MATLAB^{®} functions, System objects, or Simulink^{®} blocks.

Modulation Scheme | MATLAB functions | System objects | Simulink blocks |
---|---|---|---|

Binary PSK (BPSK) | |||

Quadrature PSK (QPSK) | |||

General PSK | |||

Differential BPSK (DBPSK) | |||

Differential QPSK (DQPSK) | |||

General DPSK | |||

OQPSK |

### Baseband and Passband Simulation

Communications Toolbox supports *baseband* and
*passband* simulation methods; however, the phase shift
keying techniques support baseband simulation only.

A general passband waveform can be represented as

$${Y}_{1}(t)\sqrt{2}\mathrm{cos}(2\pi {f}_{c}t+\theta )-{Y}_{2}(t)\sqrt{2}\mathrm{sin}(2\pi {f}_{c}t+\theta )\text{\hspace{0.17em}},$$

where *f _{c}* is the carrier frequency and

*θ*is the initial phase of the carrier signal. This equation is equal to the real part of

$$[({Y}_{1}(t)+j{Y}_{2}(t)){e}^{j\theta}]\mathrm{exp}(j2\pi {f}_{c}t)\text{\hspace{0.17em}}.$$

In a baseband simulation, only the expression within the square brackets is
modeled. The vector *y* is a sampling of the complex signal

$$({Y}_{1}(t)+j{Y}_{2}(t)){e}^{j\theta}\text{\hspace{0.17em}}.$$

### 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

In MATLAB, you can modulate and demodulate higher-order PSK constellations. The complex baseband form for an M-ary PSK signal using natural 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}### Soft-Decision Demodulation

All Communications Toolbox demodulator functions, System objects and blocks can demodulate binary data using either hard decisions or soft decisions. Two soft-decision algorithms are available: exact log-likelihood ratio (LLR) and approximate LLR. Exact LLR provides the greatest accuracy but is slower, while approximate LLR is less accurate but more efficient.

#### Exact LLR Algorithm

The log-likelihood ratio (LLR) is the logarithm of the ratio of probabilities
of a 0 bit being transmitted versus a 1 bit being transmitted for a received
signal. The LLR for a bit, *b*, is defined as:

$$L(b)=\mathrm{log}\left(\frac{\mathrm{Pr}(b=0|r=(x,y))}{\mathrm{Pr}(b=1|r=(x,y))}\right)$$

Assuming equal probability for all symbols, the LLR for an AWGN channel can be expressed as:

$$L(b)=\mathrm{log}\left(\frac{{\displaystyle \sum _{s\in {S}_{0}}{e}^{-\frac{1}{{\sigma}^{2}}\left({(x-{s}_{x})}^{2}+{(y-{s}_{y})}^{2}\right)}}}{{\displaystyle \sum _{s\in {S}_{1}}{e}^{-\frac{1}{{\sigma}^{2}}\left({(x-{s}_{x})}^{2}+{(y-{s}_{y})}^{2}\right)}}}\right)$$

Variable | Description |
---|---|

$$r$$
| Received signal with coordinates (x,
y) |

$$b$$
| Transmitted bit (one of the K bits in an M-ary symbol, assuming all M symbols are equally probable) |

$${S}_{0}$$
| Ideal symbols or constellation points with bit 0, at the given bit position |

$${S}_{1}$$
| Ideal symbols or constellation points with bit 1, at the given bit position |

$${s}_{x}$$
| In-phase coordinate of ideal symbol or constellation point |

$${s}_{y}$$
| Quadrature coordinate of ideal symbol or constellation point |

$${\sigma}^{2}$$
| Noise variance of baseband signal |

$${\sigma}_{x}^{2}$$
| Noise variance along in-phase axis |

$${\sigma}_{y}^{2}$$
| Noise variance along quadrature axis |

**Note**

Noise components along the in-phase and quadrature axes are assumed to be independent and of equal power, that is, $${\sigma}_{x}^{2}={\sigma}_{y}^{2}={\sigma}^{2}/2$$.

#### Approximate LLR Algorithm

Approximate LLR is computed by using only the nearest constellation point to the received signal with a 0 (or 1) at that bit position, rather than all the constellation points as done in exact LLR. It is defined in [2] as:

$$L(b)=-\frac{1}{{\sigma}^{2}}\left(\underset{s\in {S}_{0}}{\mathrm{min}}\left({(x-{s}_{x})}^{2}+\text{}\text{}{(y-{s}_{y})}^{2}\right)-\underset{s\in {S}_{1}}{\mathrm{min}}\left({(x-{s}_{x})}^{2}+\text{}\text{}{(y-{s}_{y})}^{2}\right)\right)$$

## 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*. Vol.
16, No. 2, Feb. 1998, pp. 260–264