## Continuous-Phase Modulation

Continuous-phase modulation (CPM) is a linear baseband modulation technique in which the message modulates the frequency of a continuous-phase signal. The signal has memory because the phase of the carrier is constrained to be continuous. Communications Toolbox™ software includes these modulation and demodulation functions, System objects, and blocks to model continuous-phase frequency shift keying (CPFSK), continuous-phase modulation (CPM), Gaussian minimum shift keying (GMSK), and minimum shift keying (MSK).

### CPM

In CPM, the baseband representation of the modulated signal is

$$\begin{array}{l}s(t)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\mathrm{exp}\left[j\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}{\displaystyle \sum _{i\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0}^{n}{\alpha}_{i}{h}_{i}q(t-iT)}\right],\text{and}\\ nT\text{\hspace{0.17em}}t(n+1)T.\end{array}$$

where:

{

*α*_{i}} is a sequence of*M*-ary data symbols selected from the alphabet ±1, ±3, ±(*M*–1).*M*must have the form 2^{k}for some positive integer*k*, where*M*is the modulation order and specifies the size of the symbol alphabet.{

*h*_{i}} is a sequence of modulation indices.*h*_{i}moves cyclically through a set of indices {*h*_{0},*h*_{1},*h*_{2}, ...,*h*_{H-1}}. When*H*=1, only one modulation index exists,*h*_{0}, which is denoted as*h*.

#### CPM Pulse Shape Filtering

The CPM method uses pulse shaping to smooth the phase transitions of the modulated signal. The
function *q*(*t*) is the phase response obtained from the
frequency pulse, *g*(*t*), through this relation: $$q(t)={\displaystyle {\int}_{-\text{\hspace{0.17em}}\infty}^{t}g(t)dt}$$.

The specified frequency pulse shape corresponds to these pulse shape expressions for
*g*(*t*).

Pulse Shape | Expression |
---|---|

Rectangular | $$g(t)=\{\begin{array}{cc}\frac{1}{2LT},& 0\le t\le LT\\ 0& \text{otherwise}\end{array}$$ |

Raised cosine | $$g(t)=\{\begin{array}{cc}\frac{1}{2LT}\left[1-\mathrm{cos}\left(\frac{2\pi t}{LT}\right)\right],& 0\le t\le LT\\ 0& \text{otherwise}\end{array}$$ |

Spectral raised cosine | $$g(t)=\frac{1}{{L}_{\text{main}}T}\frac{\mathrm{sin}\left(\frac{2\pi t}{{L}_{\text{main}}T}\right)}{\frac{2\pi t}{{L}_{\text{main}}T}}\frac{\mathrm{cos}\left(\beta \frac{2\pi t}{{L}_{\text{main}}T}\right)}{1-{\left(\frac{4\beta}{{L}_{\text{main}}T}t\right)}^{2}},\text{\hspace{1em}}0\le \beta \le 1$$ |

Gaussian | $$\begin{array}{c}g(t)=\frac{1}{2T}\left\{Q\left[2\pi {B}_{b}\frac{t-{\scriptscriptstyle \frac{T}{2}}}{\sqrt{\mathrm{ln}2}}\right]-Q\left[2\pi {B}_{b}\frac{t+{\scriptscriptstyle \frac{T}{2}}}{\sqrt{\mathrm{ln}2}}\right]\right\},\text{\hspace{0.17em}}\text{where}\\ Q(t)={\displaystyle {\int}_{t}^{\infty}\frac{1}{\sqrt{2\pi}}{e}^{-{\tau}^{2}/2}d\tau}\end{array}$$ |

Tamed FM (tamed frequency modulation) | $$\begin{array}{l}g(t)={\scriptscriptstyle \frac{1}{8}}\left[{g}_{0}(t-T)+2{g}_{0}(t)+{g}_{0}(t+T)\right],\text{\hspace{0.17em}}\text{where}\\ {\text{g}}_{0}(t)\approx \frac{1}{T}\left[\frac{\mathrm{sin}({\scriptscriptstyle \frac{\pi t}{T}})}{{\scriptscriptstyle \frac{\pi t}{T}}}-\frac{{\pi}^{2}}{24}\frac{2\mathrm{sin}\left({\scriptscriptstyle \frac{\pi t}{T}}\right)-{\scriptscriptstyle \frac{2\pi t}{T}}\mathrm{cos}\left({\scriptscriptstyle \frac{\pi t}{T}}\right)-{\left({\scriptscriptstyle \frac{\pi t}{T}}\right)}^{2}\mathrm{sin}\left({\scriptscriptstyle \frac{\pi t}{T}}\right)}{{\left({\scriptscriptstyle \frac{\pi t}{T}}\right)}^{3}}\right]\end{array}$$ |

*L*_{main}is the main lobe pulse duration in symbol intervals.*β*is the roll-off factor of the spectral raised cosine.*B*_{b}is the product of the bandwidth and the Gaussian pulse.The duration of the pulse,

*LT*, is the pulse length in symbol intervals. As defined by the expressions, the spectral raised cosine, Gaussian, and tamed FM pulse shapes have infinite length. For all practical purposes,*LT*specifies the truncated finite length.*T*is the symbol durations.*Q(t)*is the complementary cumulative distribution function.

### CPFSK

MSK is a specific form of CPM (and CPFSK) in which the modulation index
*h* = 1/2 and *g*(*t*) is a
rectangular pulse of duration *T*. As described in Proakis ([2]), the signal waveforms may be expressed as

$$s(t)={A}_{c}\mathrm{cos}\left(2\pi {f}_{c}t+{D}_{f}{\displaystyle {\int}_{-\infty}^{t}m(\alpha )d\alpha}\right)$$

*A*_{c}represents the amplitude of the CPFSK signal.*f*_{c}is the base carrier frequency.*D*_{f}is a parameter that controls the frequency deviation of the modulated signal.The integral portion of the cosine argument results in the continuous phase characteristic of the CPFSK signal.

### GMSK

GMSK is a continuous phase scheme with no phase discontinuities because the frequency changes occur at the carrier zero-crossing points. For GMSK (and MSK), the frequency difference between the logical one and logical zero states is always equal to half the data rate. This difference can be expressed in terms of the modulation index. Specifically, an input symbol of 1 causes a phase shift of π/2 radians, which corresponds to a modulation index of 0.5.

GMSK applies a Gaussian pulse shaping filter, as described by the Gaussian filter equation in CPM Pulse Shape Filtering.

### MSK

MSK is a specific form of CPM (and CPFSK) in which the modulation index
*h* = 1/2 and *g*(*t*) is a
rectangular pulse of duration *T*. As described in Proakis ([2]), the signal waveforms may be expressed as

$$s(t)=A\mathrm{cos}\left[2\pi \left({f}_{c}+\frac{1}{4T}{I}_{n}\right)t-\frac{1}{2}n\pi {I}_{n}+{\theta}_{n}\right],\begin{array}{ccc}& & \end{array}nT\le t\le (n+1)T$$

## References

[1] Proakis, John G. *Digital Communications*.
5th ed. New York: McGraw Hill, 2007.

[2] Pasupathy, S., “Minimum Shift Keying: A Spectrally Efficient
Modulation.”*IEEE Communications Magazine*, July, 1979, pp.
14–22.

[3] Anderson, John B., Tor Aulin, and Carl-Erik Sundberg. *Digital Phase Modulation*. New York: Plenum Press, 1986.