On the probability density function of the real and imaginary parts in WFRFT signals

Recently a Hybrid Carrier (HC) scheme based on Weighted-type Fractional Fourier Transform (WFRFT) was proposed and developed, which contains Single Carrier (SC) and Multi-Carrier (MC) synergetic transmission. The wide interest is primarily due to its appealing characteristics, such as the robust performances in different types of selective fading channels and a great deal of potential for secure communications. According to the literatures, the HC signal and SC or MC signal probability distributions are different. In particular, some benefits of this HC scheme are brought by the quasi-Gaussian distribution of WFRFT signals. However, until now researchers have only presented statistic properties through computer simulations, and the accurate expressions of signals are not derived yet. In this paper, we derive the accurate and rigorously established closed-form expressions of Probability Density Function (PDF) of WFRFT signal real and imaginary parts with a large number of QPSK subcarriers, and this PDF can describe the behavior of data modulated by WFRFT, avoiding the complex computation for extensive computer simulations. Furthermore, the components of PDF expression are described and analyzed, and it is revealed that the tendency of signal quasi-Gaussian changes with the increasing of the parameter α (α in (0, 1]). To validate the analytical results, extensive simulations have been conducted, showing a very good match between the analytical results and the real situations. The contribution of this paper may be useful to deduce the closed form expressions of Bit Error Ratio (BER), the Complementary Cumulative Distribution Function (CCDF) of Peak to Average Power Ratio (PAPR), and other analytical studies which adopt the PDF.