EURASIP Journal on Advances in Signal Processing
Volume 2010 (2010), Article ID 794246, 14 pages
doi:10.1155/2010/794246
Research Article

Spectral Correlation of Multicarrier Modulated Signals and Its Application for Signal Detection

Haijian Zhang, Didier Le Ruyet (EURASIP Member), and Michel Terré

Electronics and Communications Laboratory, CNAM, 75141 Paris, France

Received 31 January 2009; Revised 15 September 2009; Accepted 21 October 2009

Academic Editor: Ying-Chang Liang

Copyright © 2010 Haijian Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Spectral correlation theory for cyclostationary time-series signals has been studied for decades. Explicit formulas of spectral correlation function for various types of analog-modulated and digital-modulated signals are already derived. In this paper, we investigate and exploit the cyclostationarity characteristics for two kinds of multicarrier modulated (MCM) signals: conventional OFDM and filter bank based multicarrier (FBMC) signals. The spectral correlation characterization of MCM signal can be described by a special linear periodic time-variant (LPTV) system. Using this LPTV description, we have derived the explicit theoretical formulas of nonconjugate and conjugate cyclic autocorrelation function (CAF) and spectral correlation function (SCF) for OFDM and FBMC signals. According to theoretical spectral analysis, Cyclostationary Signatures (CS) are artificially embedded into MCM signal and a low-complexity signature detector is, therefore, presented for detecting MCM signal. Theoretical analysis and simulation results demonstrate the efficiency and robustness of this CS detector compared to traditionary energy detector.

1. Introduction

A cyclostationary process is an appropriate probabilistic model for the signals that undergo periodic transformation, such as sampling, modulating, multiplexing, and coding operations, provided that the signal is appropriately modeled as a stationary process before undergoing the periodic transformation [1]. Increasing demands on communication system performance indicate the importance of recognizing the cyclostationary character of communicated signals. The growing role of the cyclostationarity is illustrated by abundant works in the detection area and other signal processing areas. Spectral correlation is an important characteristic property of wide sense cyclostationarity, and a spectral correlation function is a generalization of the power spectral density (PSD) function. Recently, the spectral correlation function has been largely exploited for signal detection, estimation, extraction and classification mainly because different types of modulated signals have highly distinct spectral correlation functions and the fact stationary noise and interference exhibit no spectral correlation property. Furthermore, the spectral correlation function contains phase and frequency information related to timing parameters in modulated signals.

In [1, 2], explicit formulas of the CAF and SCF for various types of single carrier modulated signals are derived. The cyclostationary properties of OFDM have been analyzed in [3, 4], and the formulas of CAF and SCF of OFDM signal are derived by a mathematic deduce process in [3], whereas the authors in [4] provide a straightforward derivation of CAF and SCF for OFDM signal by a matrix-based stochastic method without involving complicated theory. For FBMC signals, the second-order syclostationarity properties of FBMC signal are exploited in [5, 6] for blind joint carrier-frequency offset (CFO) and symbol timing estimation.

The main objective of this article is to obtain the general formulas for calculating the CAF and SCF of MCM signals using a common derivation model. A particularly convenient method for calculating the CAF and SCF for many types of modulated signals is to model the signal as a purely stationary waveform transformed by a Linear Periodically Time-Variant (LPTV) transformation [7, 8]. Multicarrier modulated signal can be regarded as a special model with the multi-input transformed by LPTV transformation and one scalar output. By modeling MCM signal into a LPTV system it is convenient to analyze MCM signal using the known LPTV theory. With the help of the mature LPTV theory, herein we derive the explicit formulas for nonconjugate and conjugate cyclic autocorrelation function and spectral correlation function of OFDM and FBMC signals, which are very useful for blind MCM signals detection and classification.

Cognitive Radio (CR) has recently been proposed as a possible solution to improve spectrum utilization via dynamic spectrum access, and spectrum sensing has also been identified as a key enabling functionality to ensure that cognitive radios would not interference with primary users. We are interested in various efficient (low Signal-to-Noise Ratio, (SNR), detection requirement of licensed signal) and low-complex methods for the detection of free bands at the worst situation that we know only few information about the received signal. Cyclostationary based detector is efficient and more robust than energy detector [9], which is highly susceptible to noise uncertainty. In most of practical situations, it is not very likely that the cognitive radio has access to the nature of licensed signal, hence rendering noise estimation impossible. The worse thing is that energy detector cannot differentiate between modulated signals, noise and interference. Feature detector such as cyclostationarity is, therefore, proposed for signal detection in CR context. An inherent cyclostationary detection method, by detecting the presence of nonconjugate cyclostationarity in some non-zero cyclic frequency, is proposed in [3]. Although this detector exhibits good detection performance, it cannot achieve the low SNR requirement of CR system specified by FCC. In addition, the computation of the proposed cyclostationarity detection algorithm is complex.

Therefore, in order to alleviate the computation complexity and achieve better detection performance for low SNR level, we apply a conjugate cyclostationarity detector by inserting Cyclostationary Signature [10] (CS), which is realized by redundantly transmitting message symbols at some predetermined cyclic frequency based on the theoretical spectral analysis and the fact that most of the MCM signals and noise do not exhibit conjugate cyclostationarity. Previous works introducing artificially cyclostationarity for OFDM signal at the transmitter can be found in [10–12]. In this paper, the signal detection between FBMC signal and noise is investigated. We implement the spectral detection of FBMC signal embedded by CS using a low-complexity conjugate cyclostationarity detector considering both AWGN and Rayleigh fading environments in the CR domain. Experimental results are provided to show the efficiency and the robustness compared to the traditionary energy detector.

The remainder of this paper is organized as follows: Section 2 presents the basic definition of spectral correlation. The fundamental concepts of LPTV system are mentioned in Section 3. Through the aforementioned theoretical knowledge, Section 4 analyzes and derives the theoretical formulas of nonconjugate and conjugate cyclic autocorrelation and spectral correlation functions of OFDM and FBMC signals. In Section 5, corresponding spectral analysis for FBMC signals with CS is investigated. A low-complexity CS detector is presented in Section 6. Simulation results are given in Section 7. Finally, conclusions are drawn in Section 8.

2. Definition of Cyclic Spectral Correlation

A complete understanding of the concept of spectral correlation is given in the tutorial paper [8]. This section is a very brief review of the fundamental definitions for spectral correlation.

The probabilistic nonconjugate autocorrelation of a stochastic process 𝑥 ( 𝑡 ) is

𝑅 𝑥 𝑥 𝜏 ( 𝑡 , 𝜏 ) = 𝐸 𝑡 + 2 𝑥 ∗ 𝜏 𝑡 − 2 , ( 1 ) where the superscript asterisk denotes complex conjugation. 𝑥 ( 𝑡 ) is defined to be second-order cyclostationary (in the wide sense) if 𝑅 𝑥 ( 𝑡 , 𝜏 ) is the periodic function about 𝑡 with period 𝑇 0 and can be represented as a Fourier series:

𝑅 𝑥 ( 𝑡 , 𝜏 ) = 𝛼 𝑅 𝛼 𝑥 ( 𝜏 ) 𝑒 𝑗 2 𝜋 𝛼 𝑡 , ( 2 ) which is called periodic autocorrelation function, where the sum is taken over integer multiples of the fundamental frequency “ 1 / 𝑇 0 ”. The Fourier coefficients can be calculated as

𝑅 𝛼 𝑥 ( 𝜏 ) = l i m 𝑇 → ∞ 1 𝑇 𝑇 / 2 − 𝑇 / 2 𝑅 𝑥 ( 𝑡 , 𝜏 ) 𝑒 − 𝑗 2 𝜋 𝛼 𝑡 𝑑 𝑡 , ( 3 ) where 𝛼 = i n t e g e r / 𝑇 0 , and 𝑅 𝛼 𝑥 ( 𝜏 ) is called the cyclic autocorrelation function. The idealized cyclic spectrum function can be characterized as the Fourier transform

𝑆 𝛼 𝑥 ( 𝑓 ) = ∞ − ∞ 𝑅 𝛼 𝑥 ( 𝜏 ) 𝑒 − 𝑗 2 𝜋 𝑓 𝜏 𝑑 𝜏 . ( 4 )

In the nonprobabilistic approach, for a time-series 𝑥 ( 𝑡 ) that contains second-order periodicity, synchronized averaging applied to the lag product time-series “ 𝑦 ( 𝑡 ) ≜ 𝑥 ( 𝑡 + 𝜏 / 2 ) 𝑥 ∗ ( 𝑡 − 𝜏 / 2 ) ” yields

𝑅 𝑥 ( 𝑡 , 𝜏 ) ≜ l i m 𝑁 → ∞ 1 2 𝑁 + 1 𝑁 𝑛 = − 𝑁 𝑥 𝑡 + 𝑛 𝑇 0 + 𝜏 2 ⋅ 𝑥 ∗ 𝑡 + 𝑛 𝑇 0 − 𝜏 2 , ( 5 ) which is referred to as the limit periodic autocorrelation function. The nonprobabilistic counterpart of (3) is given by

𝑅 𝛼 𝑥 ( 𝜏 ) ≜ l i m 𝑇 → ∞ 1 𝑇 𝑇 / 2 − 𝑇 / 2 𝑥 𝜏 𝑡 + 2 𝑥 ∗ 𝜏 𝑡 − 2 𝑒 − 𝑗 2 𝜋 𝛼 𝑡 𝑑 𝑡 , ( 6 ) which is recognized as the limit cyclic autocorrelation function. The limit cyclic spectrum function can be characterized as the Fourier transform like  (4):

𝑆 𝛼 𝑥 ( 𝑓 ) = ∞ − ∞ 𝑅 𝛼 𝑥 ( 𝜏 ) 𝑒 − 𝑗 2 𝜋 𝑓 𝜏 𝑑 𝜏 . ( 7 )

The limit cyclic spectrum function is also called spectral correlation function. Fourier transform relation in (7) is called the cyclic Wiener relation.

In summary, the limit cyclic autocorrelation can be interpreted as a Fourier coefficient in the Fourier series expansion of the limit periodic autocorrelation like (2). If 𝑅 𝛼 𝑥 ( 𝜏 ) ≡ 0 for all 𝛼 ≠ 0 and 𝑅 𝑥 ( 𝜏 ) ≠ 0 , then 𝑥 ( 𝑡 ) is purely stationary. If 𝑅 𝛼 𝑥 ( 𝜏 ) ≠ 0 only for 𝛼 = i n t e g e r / 𝑇 0 for some period 𝑇 0 , then 𝑥 ( 𝑡 ) is purely cyclostationary with period 𝑇 0 . If 𝑅 𝛼 𝑥 ( 𝜏 ) ≠ 0 for values of 𝛼 that are not all integer multiples of some fundamental frequency 1 / 𝑇 0 , then 𝑥 ( 𝑡 ) is said to exhibit cyclostationary [1]. For modulated signals, the periods of cyclostationarity correspond to carrier frequencies, pulse rates, spreading code repetition rates, time-division multiplexing rates, and so on.

In paper [8], an useful modification of the CAF called conjugate cyclic autocorrelation function is given as

𝑅 𝛼 𝑥 ∗ ( 𝜏 ) = l i m 𝑇 → ∞ 1 𝑇 𝑇 / 2 − 𝑇 / 2 𝑅 ∗ 𝑥 ( 𝑡 , 𝜏 ) 𝑒 − 𝑗 2 𝜋 𝛼 𝑡 𝑑 𝑡 , ( 8 ) with 𝑅 ∗ 𝑥 ( 𝑡 , 𝜏 ) = 𝐸 [ 𝑥 ( 𝑡 + 𝜏 / 2 ) 𝑥 ( 𝑡 − 𝜏 / 2 ) ] , and the corresponding SCF called conjugate spectral correlation function is

𝑆 𝛼 𝑥 ∗ ( 𝑓 ) = ∞ − ∞ 𝑅 𝛼 𝑥 ∗ ( 𝜏 ) 𝑒 − 𝑗 2 𝜋 𝑓 𝜏 𝑑 𝜏 . ( 9 )

For a noncyclostationary signal, 𝑅 𝛼 𝑥 ( 𝜏 ) = 𝑅 𝛼 𝑥 ∗ ( 𝜏 ) = 𝑆 𝛼 𝑥 ( 𝑓 ) = 𝑆 𝛼 𝑥 ∗ ( 𝑓 ) = 0 for all 𝛼 ≠ 0 , and for a cyclostationary signal, any nonzero value of the frequency parameter 𝛼 , for which the nonconjugate and conjugate CAFs and SCFs differ from zero is called a cycle frequency. Both nonconjugate and conjugate CAFs and SCFs are discrete functions of the cycle frequency 𝛼 and are continuous in the lag parameter 𝜏 and frequency parameter 𝑓 , respectively.

3. LPTV System

LPTV is a special case of linear almost-periodically time-variant (LAPTV), which is introduced in [7]. A linear time-variant system with input 𝑥 ( 𝑡 ) , output 𝑦 ( 𝑡 ) , impulse response function ℎ ( 𝑡 , 𝑢 ) , and input-output relation

𝑦 ( 𝑡 ) = 𝑅 ℎ ( 𝑡 , 𝑢 ) 𝑥 ( 𝑢 ) 𝑑 𝑢 , ( 1 0 ) is said to be LAPTV if the impulse response function admits the Fourier series expansion:

ℎ ( 𝑡 , 𝑢 ) = 𝜎 ∈ 𝐺 ℎ 𝜎 ( 𝑡 − 𝑢 ) 𝑒 𝑗 2 𝜋 𝜎 𝑢 , ( 1 1 ) where 𝐺 is a countable set.

By substituting (11) into (10) the output 𝑦 ( 𝑡 ) can be expressed in the two equivalent forms

𝑦 ( 𝑡 ) = 𝜎 ∈ 𝐺 ℎ 𝜎 ( 𝑡 ) ⊗ 𝑥 ( 𝑡 ) 𝑒 𝑗 2 𝜋 𝜎 𝑡 , ( 1 2 ) 𝑦 ( 𝑡 ) = 𝜎 ∈ 𝐺 𝑔 𝜎 ( 𝑒 𝑡 ) ⊗ 𝑥 ( 𝑡 ) 𝑗 2 𝜋 𝜎 𝑡 , ( 1 3 ) where “ ⊗ ” denotes convolution operation, and

𝑔 𝜎 ( 𝑡 ) = ℎ 𝜎 ( 𝑡 ) 𝑒 − 𝑗 2 𝜋 𝜎 𝑡 . ( 1 4 )

From (12) it follows that a LAPTV system performs a linear time-invariant filtering of frequency-shifted version of the input signal. For this reason LAPTV is also referred to as frequency-shift filtering. Equivalently, form (13) it follows that a LAPTV system performs a frequency shift of linear time-invariant filtered versions of the input.

In the special case for which 𝐺 ≡ { 𝑘 / 𝑇 0 } 𝑘 ∈ 𝑍 for some period 𝑇 0 , the system becomes the linear periodically time-variant (LPTV).

LPTV transformation is defined as follows [8]:

𝑦 ( 𝑡 ) = ∞ − ∞ ̂ ̂ 𝐡 ( 𝑡 , 𝑢 ) 𝐱 ( 𝑢 ) 𝑑 𝑢 , ( 1 5 ) where ̂ 𝐱 is a 𝐿 -element column vector input ( 𝐿 is any non-zero positive integer) and 𝑦 ( 𝑡 ) is a scalar response. ̂ ̂ 𝐡 ( 𝑡 , 𝑢 ) = 𝐡 ( 𝑡 + 𝑇 0 , 𝑢 + 𝑇 0 ) is the periodically time-variant ( 𝐿 -element row vector) of impulse response functions that specify the transformation. The function ̂ 𝐡 ( 𝑡 + 𝜏 , 𝑡 ) is periodic in 𝑡 with a period 𝑇 0 for each 𝜏 represented by the Fourier series

̂ 𝐡 ( 𝑡 + 𝜏 , 𝑡 ) = ∞ 𝑛 = − ∞ ̂ 𝐠 𝑛 ( 𝜏 ) 𝑒 𝑗 2 𝜋 𝑛 𝑡 / 𝑇 0 , ( 1 6 ) where

̂ 𝐠 𝑛 1 ( 𝜏 ) = 𝑇 0 𝑇 0 / 2 − 𝑇 0 / 2 ̂ 𝐡 ( 𝑡 + 𝜏 , 𝑡 ) 𝑒 − 𝑗 2 𝜋 𝑛 𝑡 / 𝑇 0 𝑑 𝑡 . ( 1 7 )

The Fourier transform of function ̂ 𝐡 ( 𝑡 + 𝜏 , 𝑡 ) is defined as a system function:

𝐆 ( 𝑡 , 𝑓 ) = ∞ − ∞ ̂ 𝐡 ( 𝑡 , 𝑡 − 𝜏 ) 𝑒 − 𝑗 2 𝜋 𝑓 𝜏 𝑑 𝜏 , ( 1 8 ) which can be also represented by a Fourier series:

𝐆 ( 𝑡 , 𝑓 ) = ∞ 𝑛 = − ∞ 𝐆 𝑛 𝑛 𝑓 + 𝑇 0 𝑒 𝑗 2 𝜋 𝑛 𝑡 / 𝑇 0 , ( 1 9 ) where

𝐆 𝑛 ( 𝑓 ) = ∞ 𝑛 = − ∞ ̂ 𝐠 𝑛 ( 𝜏 ) 𝑒 − 𝑗 2 𝜋 𝑓 𝜏 𝑑 𝜏 . ( 2 0 )

By substitution of (15) and (16) into the definition of (3) and (4), it can be shown that the nonconjugate cyclic autocorrelation and cyclic spectrum of the input ̂ 𝐱 ( 𝑡 ) and output 𝑦 ( 𝑡 ) of the LPTV system are related by the formulas

𝑅 𝛼 𝑦 ( 𝜏 ) = ∞ 𝑛 , 𝑚 = − ∞ 𝐑 t r a c e 𝛼 − ( 𝑛 − 𝑚 ) / 𝑇 0 𝑥 ( 𝜏 ) ⋅ 𝑒 − 𝑗 𝜋 ( 𝑛 + 𝑚 ) 𝜏 / 𝑇 0 ⊗ ̂ 𝐫 𝛼 𝑛 𝑚 , 𝑆 ( − 𝜏 ) ( 2 1 ) 𝛼 𝑦 ( 𝑓 ) = ∞ 𝑛 , 𝑚 = − ∞ 𝐆 𝑛 𝛼 𝑓 + 2 ̂ 𝐒 𝛼 − ( 𝑛 − 𝑚 ) / 𝑇 0 𝑥 [ ] 𝑓 − 𝑛 + 𝑚 2 𝑇 0 ⋅ 𝐆 𝑇 𝑚 𝛼 𝑓 − 2 ∗ , ( 2 2 ) where “ ⊗ ” denotes convolution operation, the superscript symbol “ 𝑇 ” denotes matrix transposition, and “ ∗ ” denotes conjugation. 𝐑 𝛽 𝑥 is the matrix of cyclic cross-correlation of the elements of the vector ̂ 𝐱 ( 𝑡 )

𝐑 𝛽 𝑥 ( 𝜏 ) = L i m 𝑇 → ∞ 1 𝑇 𝑇 / 2 − 𝑇 / 2 ̂ 𝐱 ∗ 𝜏 𝑡 + 2 ̂ 𝐱 𝑇 𝜏 𝑡 − 2 𝑒 − 𝑗 2 𝜋 𝛽 𝑡 𝑑 𝑡 , ( 2 3 ) and ̂ 𝐫 𝛼 𝑛 𝑚 is the matrix of finite cyclic cross correlation

̂ 𝐫 𝛼 𝑛 𝑚 ( 𝜏 ) = ∞ − ∞ ̂ 𝐠 𝑇 𝑛 𝜏 𝑡 + 2 ̂ 𝐠 ∗ 𝑚 𝜏 𝑡 − 2 𝑒 − 𝑗 2 𝜋 𝛼 𝑡 𝑑 𝑡 . ( 2 4 )

Formulas (21) and (22) reveal that the cyclic autocorrelation and spectra of a modulated signal are each self-determinant characteristics under an LPTV transformation.

The conjugate cyclic autocorrelation and cyclic spectrum of the input ̂ 𝐱 ( 𝑡 ) and output 𝑦 ( 𝑡 ) of the LPTV system are obtained similarly:

𝑅 𝛼 𝑦 ∗ ( 𝜏 ) = ∞ 𝑛 , 𝑚 = − ∞ 𝐑 t r a c e 𝛼 − ( 𝑛 + 𝑚 ) / 𝑇 0 𝑥 ∗ ( 𝜏 ) ⋅ 𝑒 − 𝑗 𝜋 ( 𝑛 − 𝑚 ) 𝜏 / 𝑇 0 ⊗ ̂ 𝐫 𝛼 𝑛 𝑚 ∗ , 𝑆 ( − 𝜏 ) 𝑚 𝑜 𝑑 𝑢 𝑙 𝑎 𝑡 𝑒 𝑑 𝑠 𝑖 𝑔 𝑛 𝑎 𝑙 ( 2 5 ) 𝛼 𝑦 ∗ ( 𝑓 ) = ∞ 𝑛 , 𝑚 = − ∞ 𝐆 𝑛 𝛼 𝑓 + 2 ̂ 𝐒 𝛼 − ( 𝑛 + 𝑚 ) / 𝑇 0 𝑥 ∗ [ ] 𝑓 − 𝑛 − 𝑚 2 𝑇 0 ⋅ 𝐆 𝑇 𝑚 𝛼 𝑓 − 2 , 𝐑 ( 2 6 ) 𝛽 𝑥 ∗ ( 𝜏 ) = L i m 𝑇 → ∞ 1 𝑇 𝑇 / 2 − 𝑇 / 2 ̂ 𝐱 𝜏 𝑡 + 2 ̂ 𝐱 𝑇 𝜏 𝑡 − 2 𝑒 − 𝑗 2 𝜋 𝛽 𝑡 ̂ 𝐫 𝑑 𝑡 , ( 2 7 ) 𝛼 𝑛 𝑚 ∗ ( 𝜏 ) = ∞ − ∞ ̂ 𝐠 𝑇 𝑛 𝜏 𝑡 + 2 ̂ 𝐠 𝑚 𝜏 𝑡 − 2 𝑒 − 𝑗 2 𝜋 𝛼 𝑡 𝑑 𝑡 . ( 2 8 )

4. Spectral Correlation of MCM Signals

Generally, the carrier modulated passband MCM signal 𝑐 ( 𝑡 ) can be expressed as

𝑐 ( 𝑡 ) = R e 𝑦 ( 𝑡 ) 𝑒 𝑗 2 𝜋 𝑓 𝑐 𝑡 , ( 2 9 ) where R e denotes the real part of { ⋅ } , 𝑦 ( 𝑡 ) is the baseband envelope of the actual transmitted MCM signal, and 𝑓 𝑐 is the carrier frequency.

If the baseband envelope signal 𝑦 ( 𝑡 ) is cyclostationary, the spectral correlation function of its corresponding carrier modulated signal 𝑐 ( 𝑡 ) can be expressed as [13]

𝑆 𝛼 𝑐 1 ( 𝑓 ) = 4 𝑆 𝛼 𝑦 𝑓 − 𝑓 𝑐 + 𝑆 𝛼 𝑦 𝑓 + 𝑓 𝑐 + 𝑆 𝛼 − 2 𝑓 𝑐 𝑦 ∗ 𝑓 − 𝑓 𝑐 + 𝑆 𝛼 + 2 𝑓 𝑐 𝑦 ∗ 𝑓 + 𝑓 𝑐 , ( 3 0 ) where 𝑆 𝛼 𝑦 ( 𝑠 ) and 𝑆 𝛼 𝑦 ∗ ( 𝑠 ) are the nonconjugate and conjugate spectral correlation function of the complex envelope 𝑦 ( 𝑡 ) , respectively. We can observe that the spectral correlation of the carrier modulated signal 𝑐 ( 𝑡 ) is determined by the nonconjugate and conjugate spectral correlation of the complex envelope signal 𝑦 ( 𝑡 ) and is related to the double carrier frequency, so the problem of spectral analysis correlation analysis of passband carrier modulated signal can be reduced to the spectral correlation analysis of the complex baseband signal.

The spectral correlation analysis of MCM signals is the theoretical basis for further signal processing. In this section, we investigate two typical MCM signals: OFDM and FBMC signals. Other MCM signals share similar spectral correlation properties with these two signals.

4.1. Spectral Correlation of OFDM Signal Using LPTV

Figure 1 shows a filter bank based schematic baseband equivalent of transmultiplexer system, based on the LPTV theory 𝑀 parallel complex data streams are passed to 𝑀 subcarrier transmission filters. OFDM system is special filter bank based multicarrier system with the rectangular pulse filters. The baseband OFDM signal can be expressed as a sum of 𝑀 single carrier signals like  (15)

794246.fig.001
Figure 1: Baseband OFDM transmitter.

𝑦 ( 𝑡 ) = 𝑀 − 1 𝑘 = 0 0 𝑥 0 2 0 0 𝑑 ∞ 𝑙 = − ∞ 𝑎 𝑙 𝑘 𝑝 𝑡 − 𝑙 𝑇 𝑠 ⋅ 𝑒 𝑗 𝑘 ( 2 𝜋 𝑡 / 𝑇 0 ) ⋅ 𝑒 − 𝑗 𝑀 ( 𝜋 / 𝑇 0 ) 𝑡 = 𝑀 − 1 𝑘 = 0 𝑥 𝑘 ( 𝑡 ) ℎ 𝑘 ( 𝑡 ) , ( 3 1 ) where 𝑥 𝑘 ( 𝑡 ) is the element of the input vector of LPTV system and ℎ 𝑘 ( 𝑡 ) is the element of impulse response of LPTV:

𝑥 𝑘 ( 𝑡 ) = ∞ 𝑙 = − ∞ 𝑎 𝑘 𝑙 𝑇 𝑠 𝑝 𝑡 − 𝑙 𝑇 𝑠 ℎ , 𝑘 = 0 , 1 , … , 𝑀 − 1 , 𝑘 ( 𝑡 ) = 𝑒 𝑗 ( 𝑘 − 𝑀 / 2 ) 2 𝜋 𝑡 / 𝑇 0 , 𝑘 = 0 , 1 , … , 𝑀 − 1 , ( 3 2 ) for which 𝑎 𝑘 is the purely stationary data, 𝑇 𝑠 = 𝑇 0 + 𝑇 𝑔 is one OFDM symbol duration, where 𝑇 0 is the useful symbol duration and 𝑇 𝑔 is the length of the guard interval where the OFDM signal is extended cyclically. 𝑝 ( 𝑡 ) is the rectangular pulse function, and ℎ 𝑘 ( 𝑡 ) can be regarded as the periodic function in 𝑡 with the period 𝑇 0 for 𝑘 = 0 , 1 , … , 𝑀 − 1 .

Element of input vector 𝑥 𝑘 ( 𝑡 ) can also be regarded as an inherent LPTV transformation of data 𝑎 𝑘 with the time-invariant filters 𝑝 ( 𝑡 ) :

𝑥 𝑘 ( 𝑡 ) = 𝑎 0 ( 𝑡 ) ⊗ 𝑝 ( 𝑡 ) , ( 3 3 ) where

𝑎 0 ( 𝑡 ) = ∞ 𝑙 = − ∞ 𝑎 𝑘 𝑙 𝑇 𝑠 𝛿 𝑡 − 𝑙 𝑇 𝑠 . ( 3 4 )

Assuming 𝐸 [ 𝑎 𝑙 , 𝑘 𝑎 ∗ 𝑙 , 𝑘 ] = 𝜎 2 , each entity of matrix 𝐑 𝛼 𝑥 ( 𝜏 ) and ̂ 𝐒 𝛼 𝑥 ( 𝑓 ) in (21) and (22) reduce to

𝑅 𝛼 𝑥 𝑘 ( 𝜏 ) = 𝑅 𝛼 𝑎 0 ( 𝜏 ) ⊗ 𝑟 𝛼 𝑝 ( 𝑆 𝜏 ) , 𝛼 𝑥 𝑘 ( 𝑓 ) = 𝑆 𝛼 𝑎 0 ( 𝑓 ) ⋅ 𝑆 𝛼 𝑝 ( 𝑓 ) , ( 3 5 ) where 𝑆 𝛼 𝑝 ( 𝑓 ) is the Fourier transform of 𝑟 𝛼 𝑝 ( 𝜏 ) and

𝑟 𝛼 𝑝 ( 𝜏 ) = ∞ − ∞ 𝑝 𝜏 𝑡 + 2 𝑝 𝜏 𝑡 − 2 𝑒 − 𝑗 2 𝜋 𝛼 𝑡 𝑅 𝑑 𝑡 , 𝛼 𝑎 0 𝜎 ( 𝜏 ) = 2 𝑇 𝑠 ⋅ 𝛿 ( 𝜏 ) , 𝛼 = i n t e g e r 𝑇 𝑠 , 𝑆 𝛼 𝑎 0 𝜎 ( 𝑓 ) = 2 𝑇 𝑠 , 𝛼 = i n t e g e r 𝑇 𝑠 . ( 3 6 )

Other terms corresponding to the LPTV system can be similarly calculated:

̂ 𝑒 𝐡 ( 𝑡 , 𝑢 ) = 𝑗 ( − 𝑀 / 2 ) ( 2 𝜋 𝑡 / 𝑇 0 ) 𝑒 𝛿 ( 𝑡 − 𝑢 ) , 𝑗 ( 1 − 𝑀 / 2 ) ( 2 𝜋 𝑡 / 𝑇 0 ) 𝑒 𝛿 ( 𝑡 − 𝑢 ) , … , 𝑗 ( 𝑀 / 2 − 1 ) ( 2 𝜋 𝑡 / 𝑇 0 ) , 𝑒 𝛿 ( 𝑡 − 𝑢 ) 𝐆 ( 𝑡 , 𝑓 ) = 𝑗 ( − 𝑀 / 2 ) ( 2 𝜋 𝑡 / 𝑇 0 ) , 𝑒 𝑗 ( 1 − 𝑀 / 2 ) ( 2 𝜋 𝑡 / 𝑇 0 ) 𝑒 , … , 𝑗 ( 𝑀 / 2 − 1 ) ( 2 𝜋 𝑡 / 𝑇 0 ) , ̂ 𝐠 𝑛 ⎡ ⎢ ⎢ ⎣ 𝑔 ( 𝜏 ) = 0 𝑛 ⋮ 𝑔 𝑛 𝑀 − 1 ⎤ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎦ , 𝐆 𝛿 ( 𝜏 ) ⋯ 0 ⋮ 𝛿 ( 𝜏 ) ⋮ 0 … 𝛿 ( 𝜏 ) 𝑛 ⎡ ⎢ ⎢ ⎣ 𝐺 ( 𝑓 ) = 0 𝑛 ⋮ 𝐺 𝑛 𝑀 − 1 ⎤ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎦ , 𝑀 1 ⋯ 0 ⋮ 1 ⋮ 0 … 1 𝑛 = − 2 𝑀 , … , 2 − 1 f o r 𝑀 = 8 , 1 6 , … . ( 3 7 )

Substituting (31) ∼ (37) into (21) and (22), the nonconjugate cyclic autocorrelated and cyclic spectra of OFDM signal are transformed into

𝑅 𝛼 o f d m ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 𝜎 ( 𝜏 ) = 2 𝑇 𝑠 ⋅ 𝑇 s i n 𝜋 𝛼 𝑠 − | 𝜏 | ⋅ 𝜋 𝛼 s i n 𝜋 𝑀 𝜏 / 𝑇 0 s i n 𝜋 𝜏 / 𝑇 0 , 𝛼 = i n t e g e r 𝑇 𝑠 , | 𝜏 | < 𝑇 𝑠 , 0 , 𝛼 ≠ i n t e g e r 𝑇 𝑠 , 𝑆 ( 3 8 ) 𝛼 o f d m ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 𝜎 ( 𝑓 ) = 2 𝑇 𝑠 𝑀 / 2 − 1 𝑛 = − 𝑀 / 2 𝑃 𝛼 𝑓 + 2 − 𝑛 𝑇 0 ⋅ 𝑃 ∗ 𝛼 𝑓 − 2 − 𝑛 𝑇 0 , 𝛼 = i n t e g e r 𝑇 𝑠 , 0 , 𝛼 ≠ i n t e g e r 𝑇 𝑠 , ( 3 9 ) where 𝑇 𝑠 is the time length of one OFDM symbol, 𝑃 ( 𝑓 ) is the Fourier transform of 𝑝 ( 𝑡 ) . The magnitudes of nonconjugate CAF and SCF of OFDM signal are drawn in graphical terms as the heights of surfaces above a bifrequency plane in Figures 2 and 3.

794246.fig.002
Figure 2: 8-channel nonconjugate cyclic autocorrelation of OFDM signal.
794246.fig.003
Figure 3: 8-channel nonconjugate spectral correlation function of OFDM signal.

For the conjugate case, according to (25) and (26), the conjugate cyclic autocorrelation and cyclic spectra of OFDM signal are transformed into

𝑅 𝛼 o f d m ∗ ( ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1 𝜏 ) = 𝑇 𝑠 𝑀 / 2 − 1 𝑛 = − 𝑀 / 2 𝑟 𝛼 − ( 2 𝑛 / 𝑇 0 ) 𝑝 𝑎 ( 𝜏 ) ⋅ 𝐸 𝑙 , 𝑘 𝑎 𝑙 , 𝑘 , 𝛼 = i n t e g e r 𝑇 𝑠 , | 𝜏 | < 𝑇 𝑠 , 0 , 𝛼 ≠ i n t e g e r 𝑇 𝑠 , 𝑆 ( 4 0 ) 𝛼 o f d m ∗ = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 1 ( 𝑓 ) 𝑇 𝑠 𝑀 / 2 − 1 𝑛 = − 𝑀 / 2 𝑃 𝛼 𝑓 + 2 − 𝑛 𝑇 0 × 𝑃 ∗ 𝛼 𝑓 − 2 + 𝑛 𝑇 0 𝑎 ⋅ 𝐸 𝑙 , 𝑘 𝑎 𝑙 , 𝑘 , 𝛼 = i n t e g e r 𝑇 𝑠 , 0 , 𝛼 ≠ i n t e g e r 𝑇 𝑠 . ( 4 1 )

Consequently, the explicit spectral correlation function of the carrier modulated OFDM signal can be derived by substituting (39) and (41) into  (30):

𝑆 𝛼 𝑐 o f d m ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ( 𝑓 ) = 𝑀 / 2 − 1 𝑛 = − 𝑀 / 2 𝜎 2 𝑇 𝑠 𝑃 𝑓 − 𝑓 𝑐 + 𝛼 2 − 𝑛 𝑇 0 × 𝑃 ∗ 𝑓 − 𝑓 𝑐 − 𝛼 2 − 𝑛 𝑇 0 + 𝜎 2 𝑇 𝑠 𝑃 𝑓 + 𝑓 𝑐 + 𝛼 2 − 𝑛 𝑇 0 × 𝑃 ∗ 𝑓 + 𝑓 𝑐 − 𝛼 2 − 𝑛 𝑇 0 + 𝐴 𝑇 𝑠 𝑃 𝑓 − 𝑓 𝑐 + 𝛼 − 2 𝑓 𝑐 2 − 𝑛 𝑇 0 × 𝑃 ∗ 𝑓 − 𝑓 𝑐 − 𝛼 − 2 𝑓 𝑐 2 + 𝑛 𝑇 0 + 𝐴 𝑇 𝑠 𝑃 𝑓 + 𝑓 𝑐 + 𝛼 + 2 𝑓 𝑐 2 − 𝑛 𝑇 0 × 𝑃 ∗ 𝑓 + 𝑓 𝑐 − 𝛼 + 2 𝑓 𝑐 2 + 𝑛 𝑇 0 , 𝛼 = i n t e g e r 𝑇 𝑠 , 0 , 𝛼 ≠ i n t e g e r 𝑇 𝑠 , ( 4 2 ) where 𝐴 = 𝐸 [ 𝑎 𝑙 , 𝑘 𝑎 𝑙 , 𝑘 ] . Since 𝐸 [ 𝑎 𝑙 , 𝑘 𝑎 𝑙 , 𝑘 ] = 0 for MPSK ( 𝑀 ≠ 2 ) or QAM modulation types, given that 𝑎 𝑙 , 𝑘 is centered and i.i.d.. According to (41), it can be seen that the OFDM signal does not exhibit conjugate cyclostationarity, that is 𝑅 𝛼 o f d m ∗ ( 𝜏 ) = 𝑆 𝛼 o f d m ∗ ( 𝑓 ) = 0 , f o r a l l 𝛼 , 𝜏 , 𝑓 . The spectral correlation function of the carrier-modulated signal for MPSK ( 𝑀 ≠ 2 ) or QAM modulation can be simplified as

𝑆 𝛼 𝑐 o f d m ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝜎 ( 𝑓 ) = 2 𝑇 𝑠 𝑀 / 2 − 1 𝑛 = − 𝑀 / 2 𝑃 𝑓 − 𝑓 𝑐 + 𝛼 2 − 𝑛 𝑇 0 × 𝑃 ∗ 𝑓 − 𝑓 𝑐 − 𝛼 2 − 𝑛 𝑇 0 + 𝑃 𝑓 + 𝑓 𝑐 + 𝛼 2 − 𝑛 𝑇 0 × 𝑃 ∗ 𝑓 + 𝑓 𝑐 − 𝛼 2 − 𝑛 𝑇 0 , 𝛼 = i n t e g e r 𝑇 𝑠 , 0 , 𝛼 ≠ i n t e g e r 𝑇 𝑠 . ( 4 3 )

4.2. Spectral Correlation of FBMC Signal Using LPTV

An efficient FBMC scheme based on offset quadrature amplitude modulation (OQAM) system has been developed [14–19]. OQAM in [15] can achieve smaller intersymbol interference (ISI) and interchannel interference (ICI) without using the cyclic prefix by utilizing well designed pulse shapes. Saltzberg in [16] showed that by designing a transmit pulse-shape in a multichannel QAM system, and by introducing a half symbol space delay between the in-phase and quadrature components of QAM symbols, it is possible to achieve a baud-rate spacing between adjacent subcarrier channels and still recover the information symbol free of ISI and ICI. Further development was made by Hirosaki [17], who showed that the transmitter and receiver part of this modulation method could be implemented efficiently in a polyphase Discrete Fourier Transform (DFT) structure. Some new progress about OQAM system can be found in [18, 19].

The principle of OQAM multicarrier modulation system is to divide the transmission into 𝑀 independent transmissions using 𝑀 subcarriers. Instead of a Fast Fourier Transform (rectangular shape filters), a more normal filter bank 𝑝 ( 𝑡 ) is used. Subcarrier bands are spaced by the symbol rate 1 / 𝑇 0 ( 𝑇 0 is one OQAM symbol period). An introduced orthogonality condition between subcarriers guarantees that the transmitted symbols arrive at the receiver free of ISI and ICI, which is achieved through time staggering the in-phase and quadrature components of the subcarrier symbols by half a symbol period 𝑇 0 / 2 . The typical baseband OQAM transmitter system is shown in Figure 4.

794246.fig.004
Figure 4: Baseband OQAM transmitter.

Supposing the complex input symbols of OQAM system are

𝑥 𝑙 𝑘 = 𝑎 𝑙 𝑘 + 𝑗 𝑏 𝑙 𝑘 , ( 4 4 ) where 𝑎 𝑙 𝑘 and 𝑏 𝑙 𝑘 are, respectively, the real and imaginary parts of the kth subcarrier of the lth symbol. The complex-values baseband OQAM signal is defined as

𝑦 ( 𝑡 ) = 𝑀 − 1 ∞ 𝑘 = 0 𝑙 = − ∞ 𝑎 𝑙 𝑘 𝑝 𝑡 − 𝑙 𝑇 0 + 𝑗 𝑏 𝑙 𝑘 𝑝 𝑡 − 𝑙 𝑇 0 − 𝑇 0 2 ⋅ 𝑒 𝑗 ( 𝑘 − 𝑀 / 2 ) ( 2 𝜋 𝑡 / 𝑇 0 + 𝜋 / 2 ) . ( 4 5 )

From (45) and Figure 4 we can see that OQAM signal is a special model with 𝑀 -input ̂ 𝐱 ( 𝑡 ) transformed by LPTV transformation ̂ 𝐡 ( 𝑡 ) and one scalar output 𝑦 ( 𝑡 ) . The baseband OQAM signal (45) can also be expressed as a sum of 𝑀 single carrier signals like  (15)

𝑦 ( 𝑡 ) = 𝑀 − 1 𝑘 = 0 𝑥 𝑘 ( 𝑡 ) ℎ 𝑘 ( 𝑡 ) , ( 4 6 ) where 𝑥 𝑘 ( 𝑡 ) is the element of the input vector of LPTV system and ℎ 𝑘 ( 𝑡 ) is the element of impulse response of LPTV

𝑥 𝑘 ( 𝑡 ) = ∞ 𝑙 = − ∞ 𝑎 𝑘 𝑙 𝑇 0 𝑝 𝑡 − 𝑙 𝑇 0 + 𝑗 𝑏 𝑘 𝑙 𝑇 0 𝑝 𝑡 − 𝑙 𝑇 0 − 𝑇 0 2 , ℎ 𝑘 = 0 , 1 , … , 𝑀 − 1 , 𝑘 ( 𝑡 ) = 𝑒 𝑗 ( 𝑘 − 𝑀 / 2 ) ( 2 𝜋 𝑡 / 𝑇 0 + 𝜋 / 2 ) , 𝑘 = 0 , 1 , … , 𝑀 − 1 , ( 4 7 ) for which 𝑎 𝑘 and 𝑏 𝑘 are the purely stationary data, 𝑇 0 is one OQAM symbol duration, 𝑝 ( 𝑡 ) is the prototype filter bank pulse function, and ℎ 𝑘 ( 𝑡 ) can be regarded as the periodic function in 𝑡 with the period 𝑇 0 for 𝑘 = 0 , 1 , … , 𝑀 − 1 .

𝑥 𝑘 ( 𝑡 ) also can be regarded as a two-element vector LPTV transformation of input data 𝑎 𝑘 and 𝑏 𝑘 with the time-invariant filters 𝑝 ( 𝑡 ) and 𝑝 ( 𝑡 − 𝑇 0 / 2 ) :

𝑥 𝑘 𝑇 ( 𝑡 ) = 𝑎 0 ( 𝑡 ) ⊗ 𝑝 ( 𝑡 ) + 𝑏 0 ( 𝑡 ) ⊗ 𝑝 𝑡 − 0 2 , ( 4 8 ) where

𝑎 0 ( 𝑡 ) = ∞ 𝑙 = − ∞ 𝑎 𝑘 𝑙 𝑇 0 𝛿 𝑡 − 𝑙 𝑇 0 , 𝑏 0 ( 𝑡 ) = ∞ 𝑙 = − ∞ 𝑗 𝑏 𝑘 𝑙 𝑇 0 𝛿 𝑡 − 𝑙 𝑇 0 . ( 4 9 )

Assuming 𝐸 [ 𝑎 𝑙 , 𝑘 𝑎 ∗ 𝑙 , 𝑘 ] = 𝐸 [ 𝑏 𝑙 , 𝑘 𝑏 ∗ 𝑙 , 𝑘 ] = 𝜎 2 , each entity of matrices 𝐑 𝛼 𝑥 ( 𝜏 ) and ̂ 𝐒 𝛼 𝑥 ( 𝑓 ) in (21) and (22) reduces to

𝑅 𝛼 𝑥 𝑘 𝜎 ( 𝜏 ) = 2 𝑇 0 ⋅ 𝛿 ( 𝜏 ) ⊗ 𝑟 𝛼 𝑝 1 ( 𝜏 ) + 𝛿 ( 𝜏 ) ⊗ 𝑟 𝛼 𝑝 2 = 𝜎 ( 𝜏 ) 2 𝑇 0 ⋅ 𝑟 𝛼 𝑝 1 ( 𝜏 ) 1 + 𝑒 − 𝑗 𝜋 𝛼 𝑇 0 , 𝛼 = i n t e g e r 𝑇 0 , 𝑆 𝛼 𝑥 𝑘 𝜎 ( 𝑓 ) = 2 𝑇 0 ⋅ 𝑆 𝛼 𝑝 1 ( 𝑓 ) + 𝑆 𝛼 𝑝 2 = 𝜎 ( 𝑓 ) 2 𝑇 0 ⋅ 𝑆 𝛼 𝑝 1 ( 𝑓 ) 1 + 𝑒 − 𝑗 𝜋 𝛼 𝑇 0 , 𝛼 = i n t e g e r 𝑇 0 , ( 5 0 ) where 𝑆 𝛼 𝑝 1 ( 𝑓 ) is the Fourier Transform of 𝑟 𝛼 𝑝 1 ( 𝜏 ) and

𝑟 𝛼 𝑝 1 ( 𝜏 ) = ∞ − ∞ 𝑝 𝜏 𝑡 + 2 𝑝 𝜏 𝑡 − 2 𝑒 − 𝑗 2 𝜋 𝛼 𝑡 𝑟 𝑑 𝑡 , 𝛼 𝑝 2 ( 𝜏 ) = ∞ − ∞ 𝑝 𝜏 𝑡 + 2 − 𝑇 0 2 𝑝 𝜏 𝑡 − 2 − 𝑇 0 2 𝑒 − 𝑗 2 𝜋 𝛼 𝑡 𝑑 𝑡 = 𝑟 𝛼 𝑝 1 ( 𝜏 ) 𝑒 − 𝑗 𝜋 𝛼 𝑇 0 . ( 5 1 )

Other terms corresponding to the LPTV system can be similarly calculated:

̂ 𝑒 𝐡 ( 𝑡 , 𝑢 ) = 𝑗 ( − ( 𝑀 / 2 ) ) 2 𝜋 𝑡 / 𝑇 0 + ( 𝜋 / 2 ) 𝑒 𝛿 ( 𝑡 − 𝑢 ) , 𝑗 ( 1 − ( 𝑀 / 2 ) ) ( ( 2 𝜋 𝑡 / 𝑇 0 ) + ( 𝜋 / 2 ) ) 𝑒 ⋅ 𝛿 ( 𝑡 − 𝑢 ) , … , 𝑗 ( ( 𝑀 / 2 ) − 1 ) 2 𝜋 𝑡 / 𝑇 0 + ( 𝜋 / 2 ) , 𝑒 𝛿 ( 𝑡 − 𝑢 ) 𝐆 ( 𝑡 , 𝑓 ) = 𝑗 ( − ( 𝑀 / 2 ) ) ( ( 2 𝜋 𝑡 / 𝑇 0 ) + ( 𝜋 / 2 ) ) , 𝑒 𝑗 ( 1 − ( 𝑀 / 2 ) ) ( ( 2 𝜋 𝑡 / 𝑇 0 ) + ( 𝜋 / 2 ) ) 𝑒 , … , 𝑗 ( ( 𝑀 / 2 ) − 1 ) ( ( 2 𝜋 𝑡 / 𝑇 0 ) + ( 𝜋 / 2 ) ) , ̂ 𝐠 𝑛 ( ⎡ ⎢ ⎢ ⎣ 𝑔 𝜏 ) = 0 𝑛 ⋮ 𝑔 𝑛 𝑀 − 1 ⎤ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎣ 𝛿 0 ( 𝜏 ) ⋯ ⋮ 𝑒 𝑗 ( 𝜋 / 2 ) 𝑘 𝛿 ⋮ ( 𝜏 ) 0 … 𝑒 𝑗 ( 𝜋 / 2 ) ( 𝑀 − 1 ) 𝛿 ⎤ ⎥ ⎥ ⎦ , 𝐆 ( 𝜏 ) 𝑛 ⎡ ⎢ ⎢ ⎣ 𝐺 ( 𝑓 ) = 0 𝑛 ⋮ 𝐺 𝑛 𝑀 − 1 ⎤ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎣ 1 ⋯ 0 ⋮ 𝑒 𝑗 ( 𝜋 / 2 ) 𝑘 ⋮ 0 … 𝑒 𝑗 ( 𝜋 / 2 ) ( 𝑀 − 1 ) ⎤ ⎥ ⎥ ⎦ , 𝑀 𝑛 = − 2 𝑀 , … , 2 − 1 f o r 𝑀 = 8 , 1 6 , … . ( 5 2 )

By the substitution of (46) ∼ (52) into (21) and (22), the nonconjugate cyclic autocorrelation and cyclic spectra of OQAM signal are transformed into

𝑅 𝛼 o q a m ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ( 𝜏 ) = 2 𝜎 2 𝑇 0 ⋅ 𝑟 𝛼 𝑝 1 ( 𝜏 ) ⋅ s i n 𝜋 𝑀 𝜏 / 𝑇 0 s i n 𝜋 𝜏 / 𝑇 0 , 𝛼 = 2 ⋅ i n t e g e r 𝑇 0 , | 𝜏 | < 𝐾 𝑇 0 , 0 , 𝛼 ≠ 2 ⋅ i n t e g e r 𝑇 0 , 𝑆 ( 5 3 ) 𝛼 o q a m ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ( 𝑓 ) = 2 𝜎 2 𝑇 0 𝑀 / 2 − 1 𝑛 = − 𝑀 / 2 𝑃 𝛼 𝑓 + 2 − 𝑛 𝑇 0 𝑃 ∗ 𝛼 𝑓 −