Mary amplitude phase shift keying (APSK) modulation
Communications Toolbox / Modulation / Digital Baseband Modulation / APM
The MAPSK Modulator Baseband block modulates the input signal using Mary amplitude phase shift keying (APSK) modulation. The output is a baseband representation of the modulated signal. M, the Modulation Order for MAPSK, equals the sum of the elements in Constellation points per circle. For a description of MAPSK modulation, see Algorithms.
Note
MAPSK Modulator Baseband specifically applies to multiple ring PSK constellations. For a single ring PSK constellation, use MPSK Modulator Baseband.
Data Types 

Multidimensional Signals 

VariableSize Signals 

The block implements a pure APSK constellation.
A pure MAPSK constellation is composed of N_{C} concentric rings or contours, each with uniformly spaced PSK points. The MAPSK constellation set is
$$\chi =\{\begin{array}{cc}{R}_{1}\mathrm{exp}\left(j\left(\frac{2\pi}{{M}_{1}}i+{\theta}_{1}\right)\right),& i=0,\dots ,{M}_{1}1,\\ {R}_{2}\mathrm{exp}\left(j\left(\frac{2\pi}{{M}_{2}}i+{\theta}_{2}\right)\right),& i=0,\dots ,{M}_{2}1,\\ \vdots & \vdots \\ {R}_{{N}_{\text{C}}}\mathrm{exp}\left(j\left(\frac{2\pi}{{M}_{{N}_{\text{C}}}}i+{\theta}_{\text{Nc}}\right)\right),& i=0,\dots ,{M}_{{N}_{\text{C}}}1,\end{array}$$
where
The modulation order is equal to the sum of all M_{l} for l = 1, 2, ... , N_{C}.
N_{C} is the number of concentric rings. N_{C} ≥ 2.
M_{l} is the number of constellation points in the lth ring.
R_{l} is the radius of the lth ring.
θ_{l} is the phase offset of the lth ring.
$$j=\sqrt{1}$$
[1] Corazza, Giovanni E. Digital Satellite Communications. New York: Springer Science Business Media, LLC, 2007.
[2] Liu, Z., Q. Xie, K. Peng, and Z. Yang. "APSK Constellation with Gray Mapping." IEEE Communications Letters. Vol. 15, Number 12, December 2011, pp. 1271–1273.