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Capacity analysis is a fundamental and important issue for continuous phase modulation (CPM) signals. In the letter, we investigate the capacity formula of CPM MIMO systems. Using Finite State Machine (FSM), the CPM symbols can be modeled as Markov source by combining channel and CPM modulation. Thus the capacity of CPM signals can be derived in form of the erroneous probability and normalized CPM bandwidth. In addition, the capacity of CPM MIMO systems is derived over
*Gaussian* channels and
*Rayleigh* channels. Finally, numerical simulations are implemented according to various parameters such as modulation scheme, modulation index h, memory length
*L*, and antenna configuration.

CPM is a non-linear modulation with significant advantages for low power, low cost and spectrum efficiencies. Notwithstanding these favorable features, however, it is difficult to compute the capacity of a channel using CPM due to several factors such as the continuous phase and the nonlinearity of modulation [

Unfortunately, there is not closed-form formula so far even for point-to-point link. Let alone the analysis of CPM in MIMO environment. In this letter, we derive the upper bounds of capacity for CPM systems over Gaussian and Rayleigh band-limited channels. First of all, the symmetry information rate (SIR) is given in form of mutual information between input and output. Then SIR is calculated in terms of Markov source for M-ary CPM signals [

Capacity portrays the achievable rate at which information can be reliably transmitted over a communications channel. Unlike linear modulations such as PSK and QAM, the spectral properties of CPM signals generally depend on the complete statistical description.

The baseband CPM signal transmitted by source has complex form as [

and

where E_{s} is the power of CPM signal, T is the symbol period, h is the modulation index, _{n}} is the sequence of independent information symbols drawn from the alphabet {±1, ±3, ・・・}, θ_{0} is initial phase.

The function q(t) in (2) is defined as the phase smoothing response of the CPM signals.

The shape of g(t) in (3) defines a family of CPM schemes, where two widely used types such as rectangular pulse with pulse length L (LREC) and raised cosine pulse with pulse length L (LRC) are given in

The block diagram of a CPM modulation channel is illustrated in _{i}(k) Î {−(M − 1), ・・・, (M − 1)}. Herein the channel combined with CPM modulation model would be better identified as a discrete memory-less channel (DMC). As the inputs of CPM modulator are independent and uniformly distributed random variables, a generalized CPM scheme may be modeled using Finite State Machine

Modulation Scheme | Pulse-Shaping Function g(t) |
---|---|

LRC | |

LREC |

(FSM).

Thus, the SIR (bit/channel use) may be calculated in terms of Markov source for M-ary CPM signals (as shown in

where the noise w has zero-mean and variance N_{0}. The mutual information between the input signal X and the output signal Y is therefore obtained as the length N goes to infinity, i.e.,

Invoking information theory, the mutual information can be written as

To calculate each item in (6), we have

To compute (7) and (8), the probability of an erroneous decision over Gaussian channels is written as [

where γ_{Gauss} = E_{b}/N_{0}, and _{n} and

In Rayleigh fading channels, the CPM-MIMO system is comprised of N_{T} transmit antenna and N_{R} receive antenna. The received signal vectors are given by

where X are CPM signal vectors that are transmitted from N_{T} antennas, H is the channel matrix, in which each entry is independent and identical distributed (i.i.d) Rayleigh fading.

For the orthogonal design with N_{T} × N_{R} antennas, the received SNR can be given by expression in [

As the magnitude of h_{ntnr}, i.e. |h_{ntnr}|, is Rayleigh distributed, ρ = |h_{ntnr}|^{2} is exponentially distributed. Hence, the probability of an erroneous decision over Rayleigh channels can be calculated to

Since Q(・) function can be simplified as Q(r) ≤ (1/2)exp(−r^{2}/2), (13) can be approximately upper-bounded by

On the other hand, the normalized CPM bandwidth is confined according to the parameters of CPM signals [

As for LREC scheme, the Carson’s Rule bandwidth of the CPM signal is given by

As for LRC scheme, the Carson’s Rule bandwidth of the CPM signal is given by

Finally, the CPM capacity can be given by

In this section, we provide simulated results to analyze the channel capacity of CPM MIMO system.

As a comparison, the plots of channel capacity are given regardless of bandwidth. It is observed in

The capacity of CPM with M = 2, h = 0.5 for REC and RC (L = 1 and 2) over Gaussian channels are shown in

Then, our next concern is about CPM MIMO systems with N_{T} × N_{R} antennas over Rayleigh channels. Under the condition that M is same (as plotted in _{t}, N_{r}) = (2, 2) should be the largest due to its diversity gain. On the other hand, when the configuration (N_{t}_{,} N_{r}) of CPM MIMO systems is same, the capacity for M = 4 is entirely larger than that for M = 2.

In this paper, we investigate the capacity of CPM MIMO systems over band-li- mited channels. We give a formulation of CPM capacity in form of SIR and normalized CPM bandwidth. For this purpose, the erroneous probability of CPM MIMO systems over Gaussian channels and Rayleigh channels is given and derived respectively. Finally, the capacity of CPM MIMO systems is simulated and evaluated according to various parameters such as modulation scheme, modulation index h, memory length L, and antenna configuration.

This work is supported by Foundation project of department of education of Fujian Province (JAT160260), Li Shangda Discipline Construction Fund of Jimei University, Pre-research project of National Natural Science Foundation of China (XYK201406).

Lei, G.W. and Xiao, X.F. (2017) On the Capacity of CPM MIMO Systems over Band-Limited Channels. Int. J. Communications, Network and System Sciences, 10, 69-75. https://doi.org/10.4236/ijcns.2017.108B008