- Research
- Open Access
A phase trajectories optimization method for CPM signal based on Pan-function model
- Wei Xue^{1},
- Wenjing Shang^{1},
- Sergey B. Makarov^{2} and
- Yidong Xu^{1}
https://doi.org/10.1186/s13634-016-0351-1
© Xue et al. 2016
- Received: 20 July 2015
- Accepted: 18 April 2016
- Published: 5 May 2016
Abstract
The continuous phase modulation (CPM) signal has the characteristics of continuous phase, excellent spectral properties, less out-band radiation, and constant envelope. Thus, CPM technology is widely used in communication systems. The shape of frequency pulse can influence the bandwidth occupancy of CPM, and smoother phase trajectories are obtained by using smoother frequency pulses. In this paper, a mathematical Pan-function model of the optimized frequency pulse is established and solved by introducing Fourier series, which can provide smooth phase trajectories of CPM signal. The simulations of the CPM signal quadrature modulation and coherent demodulation are performed using the MATLAB software. Moreover, the spectral characteristics of the obtained optimized CPM signal were analyzed and compared with the minimum shift keying (MSK) signal and other CPM signals with smooth frequency pulses. The simulation results indicate that the proposed method provides an excellent bandwidth efficiency compared to other existing methods discussed in this paper. The whole system has been successfully downloaded to field programmable gate array (FPGA) devices. The operating results are consistent with expected results, verifying the correctness of this method.
Keywords
- CPM
- Optimization
- Fourier series
- Pan function
- Frequency pulse shaping
1 Introduction
The spectral characteristics of the digitally modulated signals are extremely important in the design of modern digital communication system. Continuous phase modulation (CPM) is one of the most important modulation techniques due to its efficient use of bandwidth. It has been widely applied in many communications, where multipath fading and nonlinear distortions make constant signal envelope and efficient bandwidth necessary.
If g(t)=0 for t>T, the CPM signal is called full response CPM. If g(t)≠0 for t>T, the modulated signal is called partial response CPM. An infinite variety of CPM signals can be generated by choosing different pulse shapes g(t) and by varying the modulation index h and alphabet size M. The rectangular pulse, raised cosine pulse, and Gaussian minimum-shift keying (GMSK) are commonly used CPM pulse shapes [2–4]. An appropriate pulse g(t) can be selected that can provide a continuous phase trajectory of modulated signal. This study aimed to achieve an optimized frequency pulse based on Pan-function model, for h=1/2 binary full response CPM, with characteristics of a higher spectral roll-off rate, less out-of-band radiation, and a constant envelope.
2 Foundation of the optimized mathematical model
2.1 Minimum energy radiation criterion outside the frequency bands
where \({a^{\left ({2n} \right)}(t)}\) is the 2n-order derivative of the symbol signal.
According to (15), the criterion for minimum out-band radiation can be transformed into the problem of unraveling the symbol function a(t) when minimizing the Pan-functional J [7].
2.2 Additional constraints of Pan-function
The mathematical Pan-function is added to the constraints of energy of a single-symbol signal, the boundary conditions, and peak-to-average power ratio (PAPR) of signal.
2.2.1 Restriction on the symbol boundary condition
2.2.2 Restriction on the single-symbol signal energy condition
2.2.3 Restriction on the PAPR of signal
where λ and μ are the Lagrange constants.
2.3 Optimized frequency pulse solving process using Fourier series
The solution of simultaneous equations is used to determine the unknown coefficients.
Coefficient of Fourier series and signal PAPR
m | n | a _{0} | a _{1} | a _{2} | a _{3} | a _{4} | PAPR |
---|---|---|---|---|---|---|---|
4 | 1 | 1.7550 | 0.6627 | −0.1291 | 0.0551 | −0.0306 | 1.44 |
2 | 1.6606 | 0.7875 | −0.0341 | 0.0066 | −0.0021 | 1.59 | |
3 | 1.5076 | 0.91996 | 0.12822 | −0.0285 | 0.0094 | 1.79 | |
4 | 1.4720 | 0.9396 | 0.1830 | −0.0174 | 0.0032 | 1.84 | |
5 | 1.3519 | 0.98337 | 0.3442 | 0.0281 | −0.0088 | 2.02 | |
6 | 1.3475 | 0.9897 | 0.3624 | 0.0404 | −0.0060 | 2.06 |
The optimized signals a(t) have excellent spectral characteristics. Therefore, smooth phase trajectories of CPM signal can be achieved using pulse a(t) as frequency pulse. The obtained CPM signal using proposed method has the characteristics of a higher out-of-band spectral roll-off rate and spectral efficiency.
The power spectra of the CPM signals selecting different frequency pulses under different values of n and PAPR are calculated, according to Table 1.
3 The modulation and demodulation scheme
3.1 The modulation scheme of system
3.2 The demodulation scheme of system
3.3 Simulation of modulation and demodulation
In Fig. 10, A is the baseband symbol signal transmitted in the in-phase channel, B is the baseband symbol signal transmitted in the orthogonal channel, and C is the quadrature-modulated output signal.
Figure 11 shows the simulation results of coherent demodulation when the E _{ b }/N _{0} is equal to 10 dB. In Fig. 11, D is the transmitted signal corrupted with additive white Gaussian noise (AWGN), which is the input to the receiver, E is the baseband signal demodulated in the in-phase channel, and F is the baseband signal demodulated in the orthogonal channel. The envelope of the output signal after the modulation is found to be constant and consistent with the expected results.
3.4 Hardware implementation
The modulation and demodulation of the CPM system is ultimately implemented on the field programmable gate array (FPGA) devices. FPGA have features of high integration, flexible programming, more pins, low power, and fast design speed. The hardware architectures have been implemented in Quartus II software which provides several tools for synthesizing the design, configuration techiques, performance analyses, including resource, speed, and power consumption. The whole system is divided into several small modules based on top-down design method and using verilog HDL hardware description language to design each module. The whole system also has been simulated in Modelsim SE10.0 simulation environment and successfully downloaded to the chip of the Altera Cyclone III EP3C55U484C6N.
For high carrier frequencies, direct synthesis of the CPM signal by computing phase, using a digital approach is impractical since maintaining an adequate sampling rate requires an extremely high operating frequency. Instead, one can resort to a quadrature implementation, where baseband I and Q signals containing the phase information are generated that vary much slower than the phase of the modulated carrier, thus making it feasible to implement them digitally. An efficient I-Q implementation of a CPM modulator skips some steps and instead generates the I and Q baseband signals directly from the binary data, thereby eliminating errors in filtering and phase and sine /cosine computation inherent in the conventional architecture.
In band-limited communication system, the modulated signal always run through filter to limit the bandwidth of the signal; however, the hardware resource consumption of filter includes the multipliers, adders, and registers. In this paper, the binary data output is passed through optimized waveform read-only memories (ROMs) whose outputs are applied to I and Q carriers. Also, the modulated signal passing through filter could introduce ISI and increase the complexity of the channel equalization.
4 Conclusions
In general, the constant continuous phase modulation systems use the pre-existing waveforms as frequency pulse, such as rectangular, cosine, raised cosine pulse, and other forms or through a filter (GMSK signal). There are few uses of the method of establishing a mathematical model base on some standards to obtain the optimized frequency pulse. In this paper, a Pan-function model is established to obtain optimized frequency pulse in accordance with the minimum out-of-band energy radiation criterion. The mathematical derivation is performed using Fourier series. It is shown that the phase trajectories of the CPM signal are smoothed using the proposed frequency pulse. The simulation of the signal modulation and demodulation scheme was performed using the MATLAB software. It is demonstrated that the CPM signal with proposed optimized frequency pulse has a higher spectral roll-off rate, less out-of-band radiation, and a constant envelope. The whole system has been successfully downloaded to FPGA devices. The operating results are consistent with expected results and it is verified the correctness of this method.
Declarations
Acknowledgements
This research has been supported by International Science and Technology Cooperation Program of China (2014 DFR10240), Harbin Science and Technology research projects (2013AE1BE003), Hei Long Jiang Postdoctoral Foundation (LBH-Z14066), and the Fundamental Research Funds for the Central Universities (China) [HEUCF1508].
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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