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Constant envelope OFDM RadCom fusion system
EURASIP Journal on Wireless Communications and Networking volume 2018, Article number: 104 (2018)
Abstract
A joint radar and wireless communication (RadCom) fusion system is an important trend for future intelligent transportation, such as the Internet of Vehicles (IOV). However, traditional joint orthogonal frequency division multiplexing (OFDM)based RadCom fusion system suffers from the problem of high peaktoaverage power ratio (PAPR) of transmission signals, leading to nonlinear distortion in high power amplifiers (HPAs). To solve this problem, a new constant envelope OFDM RadCom approach is proposed in this paper. Taking phase modulation into consideration, the constant envelope OFDM RadCom system obtains 0 dB PAPR value, which results in linear high power amplifier being feasible. In the receiver side, both communication transmission and target detection are guaranteed by a phase demodulator and frequencydomain signal processing algorithm, respectively. Simulation results demonstrate that the constant envelope OFDM RadCom system is a feasible solution for future IOV with the capability of target detection and wireless communication simultaneously.
Introduction
For future Internet of Vehicles (IOV) systems, all vehicles adopt both wireless communication and radar equipment in order to improve the safety of its drivers [1]. In a modern vehicle, the communication equipment is used to receive radio signals and the radar equipment is used for the shortdistance warning. To avoid interference between the communication system and radar system, the spectrum has to be different. Therefore, it results in wasting frequency spectrum resources. Consequently, it is necessary to complete joint communication and radar function in a fusion system within the same frequency spectrum [2]. Such fusion systems, providing radar and communication functions on a single hardware platform with a single waveform, will be denoted in the following with the acronym “RadCom” [1, 2].
Recalling from traditional wireless communication and radar systems, typical radar signals are immovable, whereas traditional wireless communication signal is random due to datadriving transmitting signals. Taking spectrum efficiency and hardware cost into consideration, a unique fusion system consists of wireless communication and radar application, which accomplishes the essential tasks of environmental sensing and information transmission. Till now, RadCom fusion systems become recently an important trend for communication and radar technique development [3]. With such a suitable RadCom system, all vehicles on the road could interact in a cooperative communication and radar fusion network. In Fig. 1, the vehicle with RadCom system could finish information transmission and target detection simultaneously to promote the transportation safety. In addition, RadCom system can be applied in other scenarios like aeronautics and military.
Singlecarrier RadCom signals, in combination with spreadspectrum techniques, can be used for multipleuser communication [4,5,6], whereas its dynamic range remains limited for radar measurements [1]. For multicarrier RadCom waveforms, an additional multipleaccess technique is required for multipleuser communication operation [7], and dedicated frequencydomain processing techniques can be exploited to improve the performance of the radar application. In terms of broadband continuous waveform, orthogonal frequency division multiplexing (OFDM) is the key technology of LongTerm Evolution (LTE) and its performance can be promoted by encoding, e.g., the turbo code [8, 9].
Channel estimation is the key process of multicarrier systems [10]. For the application of multicarrier waveforms and frequencydomain modulation, frequencydomain equalization (FDE) can be used in multicarrier systems, and the complexity of channel estimation is reduced effectively. However, the peaktoaverage power ratio (PAPR) of traditional OFDM signals can be bigger than 7 dB, which is a rough index for the high power amplifier (HPA). In radio frequency front end, the efficiency of HPA is degenerated with increasing PAPR of input signals [11,12,13]. The high PAPR value of OFDM signals will lead to the serious nonlinear distortion of HPAs [14,15,16]. As a consequence, it is significant to decrease PAPR of OFDM transmitting signals for engineering practical applications.
In this paper, a new constant envelope OFDM (CEOFDM) RadCom system is proposed. This system can provide radar and wireless communication functions simultaneously. The PAPR of CEOFDM is 0 dB. In this article, both the phase modulator and the phase unwrap demodulator guarantee constant envelope communication signals [17, 18]. A unique frequencydomain processing algorithm ensures target information is obtained [19,20,21]. The constant amplitude of CEOFDM RadCom signals means that the common HPAs can satisfy the requirement for transmitting signals, and the OFDM high PAPR problem can be perfectly solved.
Ambiguity function is an evaluation standard for RadCom system, which is a cross field of communication and radar. The characteristic of signal energy concentration in timefrequency domain can be obtained by its ambiguity function. The characteristic of signal energy concentration has an effect on intersymbol interference (ISI) and intercarrier interference (ICI) in communication systems. The ambiguity function is also useful for investigating different radar waveforms and determining the range and Doppler resolution of the special waveform [22]. Observing the ambiguity function of signals, the signal characteristic, which not only includes the orthogonality and energy distribution of communication signals but also contains delay ambiguity and Doppler ambiguity of radar signals, can be obtained. In this paper, ambiguity function analysis of CEOFDM RadCom signals is provided.
This paper is organized as follows: Section 2 presents the CEOFDM RadCom fusion approach. Section 3 and Section 4 describe the CEOFDM RadCom communication analysis and the frequencydomain detection algorithm, respectively. The simulation method is introduced in Section 5. In Section 6 and Section 7, comprehensive simulation and analysis are presented, including bit error rate (BER) performance, ambiguity analysis, multiple target detection, detection threshold, and detection probability. Conclusions are presented in Section 8.
CEOFDM multicarrier RadCom systems
With the capability of wireless communication and target detection, some applications can benefit from the RadCom fusion systems, such as intelligent transportation applications. With a suitable RadCom system, all vehicles on the road could interact in an integrated radar and communication sensor network, which promotes transportation security. Therefore, the key of RadCom development is to design waveforms that are suitable for both wireless communications and radar functions [1, 2].
Compared with traditional OFDM system, the key of CEOFDM system is to solve OFDM high PAPR problem effectively. This section principally introduces a new CEOFDM multicarrier RadCom fusion approach. The constant envelope RadCom fusion approach is based on the CEOFDM system, in which the realvalued OFDM signals are modulated by a phase modulator to obtain constant amplitude signals. For promoting spectral effectiveness, the frequencydomain detection algorithm is combined in this fusion approach to ensure the realtime environment sensing detection. The CEOFDM RadCom fusion approach is able to accomplish environment detection and wireless communication simultaneously with the same CEOFDM transmitted signals. Straightforward, this fusion approach makes the continuous CEOFDM system append detection function within the same timefrequency resources. What is more, CEOFDM RadCom fusion approach can be implemented by common HPA, and this is helpful for saving hardware cost.
Figure 2 represents CEOFDM RadCom system architecture. During each Tsecond elementary symbol interval, after a quadrature amplitude modulation (QAM) modulator, the bit information is modulated into M_{ QAM }ary QAM symbols. And then, to make the output of inverse discrete Fourier transform (IDFT) realvalued, QAM symbols are rearranged. For obtaining realvalued OFDM signals {x[n]}, the input of N_{ DFT }point IDFT is a special data vector [23], which is conjugate symmetric and zeropadded
where \( {\left\{X\left[k\right]\right\}}_{k=1}^{N_{QAM}} \) are M_{ QAM }ary QAM symbols and \( {\mathbf{0}}_{1\times {N}_{ZP}} \) is a row vector composed of N_{ ZP } zeros. N_{ DFT } = 2N_{ QAM } + N_{ ZP } + 2 is the IDFT size. J = N_{ DFT }/(N_{ DFT } − N_{ ZP }) is the oversampling multiple. The two zeros at index k = 0 and k = N_{ QAM } + N_{ ZP } + 1 are used to ensure implementation for conjugate symmetry, and the remaining zeros guarantee the result of oversampling the timedomain sequence. Therefore, the output of the IDFT can be represented as
n = 0, 1, …, N_{ DFT } − 1, where \( j=\sqrt{1} \). For analysis, the continuous time form of x[n] can be expressed as
−T_{ CP } ≤ t < T. Where T_{ CP } is the cyclic prefix duration. It is clear that the OFDM signal is comprised of N = 2N_{ QAM } subcarriers. And the symbols on subcarriers are \( M=\sqrt{M_{QAM}} \)pulse amplitude modulation (PAM) data symbols. Therefore,
I[k] ∈ {±1, ±3, …, ±(M − 1)}, and the subcarrier can be written as
So, the subcarrier orthogonality condition holds:
Next, a phase modulator transforms the high PAPR realvalued OFDM sequence {x[n]} into the constant envelope sequence {s[n] = exp(jCx[n])}, where C is a scaling constant. Then, to avoid ISI, an N_{ CP } points cyclic prefix (CP) is needed to obtain \( {\left\{s\left[n\right]\right\}}_{n={N}_{CP}}^{N_{DFT}1} \), where s[n] = s[N_{ DFT } + n], n = − N_{ CP }, …, − 2, − 1. The discretetime constant envelope samples are then passed through a digitaltoanalog converter to obtain the continuous signal. Last, the continuous signal is amplified by HPA and transmitted into the channel.
The modulation index is h. The phase of CEOFDM signal is defined as φ[n] = 2πhC_{ norm }x[n], where C_{ norm } is a constant. To make the variance of phases of CEOFDM signals \( {\sigma}_{\varphi}^2={\left(2\pi h\right)}^2 \), C_{ norm } should be set as \( \sqrt{2/\left(N{\sigma}_{PAM}^2\right)} \), where \( {\sigma}_{PAM}^2=\left({M}^21\right)/3 \) is the variance of the independent and identically distributed MPAM symbols. Therefore, the lowpass equivalent representation of the discrete transmitted CEOFDM signal can be written as
−N_{ CP } ≤ n < N_{ DFT }, where A is the signal amplitude and θ is an arbitrary phase offset (PO), and it could be used to achieve continuous phase modulation (CPM) [24].
Figure 3 represents the constellation maps of CEOFDM and OFDM random timedomain signals. The constellation points of OFDM with 8.5 dB PAPR (the plus sign) distribute randomly in its constellation map. Meanwhile, the distance from the constellation points of CEOFDM (the cross) to zero is constant one, and the CEOFDM constellation points seem to form a unit circle.
Figure 4 represents amplitude comparison between CEOFDM timedomain signals and OFDM timedomain signals. The OFDM amplitude distribution is chaotic with 8.5 dB PAPR, and it approximately follows independent random distribution. However, the nonlinear distortion of HPAs, which is caused by high PAPR, could reduce the HPA efficiency [14, 25]. Figure 2 presents that the CEOFDM amplitude is constant, which means that the requirement of transmitting information can be satisfied by using common HPAs, and the high PAPR problem of traditional OFDM can also be solved [26, 27]. Furthermore, the constant envelope enhances transmission range. Therefore, taking the PAPR problem into consideration, CEOFDM waveform is one applicable RadCom system candidate.
The envelope of CEOFDM RadCom transmitted signals is constant, and the nonlinear distortion of HPAs will be eliminated. Consequently, the problem of high PAPR problem of OFDM is saved. The information demodulation and environment sensing are accomplished in the CEOFDM RadCom receiver. Later on, the information demodulation and performance analysis for wireless communication is represented in Section 3 and the frequencydomain detection process is introduced in Section 4.
The CEOFDM RadCom analysis in communication
Since the constant envelope transmitted signals are the result of phase modulation, the bandwidth of s(t), which is the continuous signal of s[n], is a function of the modulation index. Since the baseband subcarrier signals are centered at the frequencies ±nT Hz, n = 1, …, N/2, the doublesided bandwidth is defined as W ≡ N/T Hz. The root mean square (RMS) bandwidth of CEOFDM signals can be defined as [28]
and the bit rate is \( R={N}_{QAM}{\log}_2\sqrt{M}/T \) bps.
The received communication signal is \( r(t)={\int}_0^{\tau_{\mathrm{max}}}h\left(\tau \right) \) s(t − τ)dτ + ω(t), where h(τ) is the channel impulse response, τ_{max} is maximum propagation delay, and ω(t) is complexvalued additive white Gaussian noise (AWGN) with a power density spectrum Φ_{ ω }(f) = N_{0}. Provided that the channel with L paths is static in T seconds, the received samples are represented as
where h[l],s[n], and ω[n] represent the discrete time samples of channel impulse response, CEOFDM signals, and AWGN, respectively. The CP duration is fixed according to N_{ CP } ≥ L. Therefore, r[n] could be rewritten as
where (∗) represents the linear convolution, (⊙) denotes the circular convolution, and IDFT(⋅) means IDFT operation. S[k] and H[k] k = 0, …, N_{ DFT } − 1 are the discrete Fourier transform (DFT) of timedomain transmitted signals and channel impulse response, respectively. Therefore, provided that N_{ CP } ≥ L, the linear and circular convolutions are equivalent for the receive samples at the indexes n = 0, …, N_{ DFT } − 1.
Furthermore, the circular convolution in timedomain means the multiplication in frequencydomain. It makes the frequencydomain equalizer (FDE) work in CEOFDM fusion system. After CP samples are removed, FDE can be used to eliminate the channel distortion. Then, the following operations, which are inverse to those performed at the transmitter, are performed: a phase unwrap demodulator, followed by a DFT operation, then followed by a QAM demodulator.
For the simple AWGN case, the received signal is \( r\left[n\right]=s\left[n\right]{e}^{j{\varphi}_0}+\omega \left[n\right] \), where φ_{0} is a phase offset of channel. The output of the phase demodulator is \( \widehat{\varphi}\left[n\right]=\varphi \left[n\right]+\theta + \) φ_{0} + ξ[n], where \( \xi \left[n\right]=\arctan \left[\frac{A_{\omega}\left[n\right]\sin \left[{\varphi}_{\omega}\left[n\right]\varphi \left[n\right]\theta {\varphi}_0\right]}{A+{A}_{\omega}\left[n\right]\sin \left[{\varphi}_{\omega}\left[n\right]\varphi \left[n\right]\theta {\varphi}_0\right]}\right] \) is the nonlinear noise component and the envelope and phase of AWGN is A_{ ω }[n] = ω[n] and φ_{ ω }[n] = arctan[ω[n]].
The output of DFT following the phase demodulator is
k = 0, …, N_{ DFT } − 1. The signal component is
where C_{ norm } is a constant, which is defined previously. The phase offset component is
k = 1, …, N_{ DFT } − 1. The noise component is
Assuming a high carriertonoise ratio (CNR), A ≫ A_{ ω }[n], and ξ[n] is well approximated as zeromean Gaussian noise [17]. For the zeromean Gaussian noise ξ[n], N[k] can also be approximated as zeromean Gaussian random variable [17].
The output of DFT in receiver includes the signal component and AWGN in high CNR. The symbol error rate (SER) for the CEOFDM system can be simplified to the problem of determining the SER for conventional PAM [29]. Therefore, the SER and BER are straightforward to express that [18]
and
where \( Q(x)={\int}_x^{\infty }{e}^{{y}^2/2} dy/\sqrt{2\pi } \) is the Gaussian Q function and ε_{ b } = A^{2}T/(2Nlog_{2}M) is the energy per bit of the CEOFDM signal. The basic information demodulation and performance analysis of CEOFDM communication is presented. For promoting RadCom performance, an additional multipleaccess technique is required for multipleuser communication operation [30,31,32].
CEOFDM radar processing in frequency domain
The key issue of CEOFDM RadCom system is that the implementation of wireless communication and environment sensing depends on same constant envelope transmitted waveform. Meanwhile, the frequencydomain radar process is also an important part of CEOFDM RadCom fusion approach. The radar process is based on the CEOFDM signals which contain transmitted information of users. Therefore, the CEOFDM fusion approach is able to complete communication and detection within the same spectrum simultaneously. This section will reveal an algorithm that allows for completing the radar detection only with the condition of the frequencydomain signal, instead of the baseband signals.
To make radar processing easier, the timedomain CEOFDM multicarrier transmitted signal can be expressed as [7,8,9].
where S_{ Tx }(μ, n) is the DFT of s[n] without oversampling. n represents the individual subcarrier index and μ denotes the individual CEOFDM symbol index. N_{ frame } denotes the number of symbols. f_{ n } is the individual subcarrier frequency, T_{ OFDM } = T + T_{ CP } is the total OFDM symbol duration, and rect(⋅) describes a rectangular window of duration T_{ OFDM }.
CEOFDM RadCom signals are reflected by targets in the environment. Firstly, the CP of received signals is removed at the receiver. The timedomain signals without CP will be simultaneously sent to the communication processor and radar processor. The CEOFDM received signal can also be expressed as the similar form
where S_{ Rx }(μ, n) is the DFT of the received signals without CP.
With the orthogonality between individual subcarriers, the following relation must hold [8]:
where Δf expresses the orthogonality between subcarriers, which is the reciprocal of the elementary OFDM symbol duration.
For the ground transportation application, it is expected that the received signal is shifted in frequency due to a relative velocity between the radar platform and reflecting objects. In the case of the radar application between radar and target moving with relative velocity v_{ rel }, the Doppler frequency shift f_{ D } at the receiver is [1]
where λ is the wavelength, f_{ c } is the carrier frequency, and c is speed of light.
If a CEOFDM signal is reflected at an object in the range R with a Doppler shift f_{ D } due to a relative movement between the transmitter and the reflecting object, the received signal can be expressed as
with A(μ, n) being the signal amplitude. In order to make the different influence between Doppler and range more obvious, the received signal can be rearranged into
The receiver observes the received signal only in the elementary OFDM symbol duration to recover one CEOFDM symbol. When the appropriate CP duration has been chosen, the receiver will still cut the observed samples from the same CEOFDM symbol. Therefore, the time shift of the rectfunction in the above two formulas can be omitted. When the CEOFDM signal bandwidth is much smaller than the carrier frequency, Doppler frequency results in a uniform phase shift on every subcarrier [1]. Besides, for a fixed subcarrier index n, the Doppler effect introduces a linear phase shift, 2πf_{ D }T_{ OFDM }, between the consecutive frequency symbols. Furthermore, the phase change of the range is the same on one subcarrier in one frame time. When A(μ, n) is constant and ignored, the received signals can also be rewritten as:
And it is clear that the received frequencydomain symbols can be quantified as
It is clear that, the influence of range and the Doppler frequency from relative velocity is completely orthogonal in frequency domain. While the Doppler introduces a linear phase shift only along the time axle, the range also introduces a linear phase shift along the frequency axle. If the observation duration is short enough, the reflecting object remains within one range resolution cell. As a result, this orthogonality can be satisfied. Therefore, it must search for a suitable processing algorithm to recover range and Doppler independently.
In order to make the target detection process arrive at a more descriptive representation, a matrix is applied to denote the frequencydomain symbol frame in (25). In the matrix, every row represents a vector composed of timedomain samples in one subcarrier, whereas each column represents one CEOFDM symbol consisting of different subcarrier samples. That is, vertical axle is the frequency axle and abscissa axle is the time axle, which makes the frequencydomain symbols arranged in timefrequency space. This rearrangement can be used to represent both the transmitted and the received frequencydomain symbols.
It makes the row vector
and another row vector
where k_{ R }(n) = exp(−j2πnΔf2R/c), n = 0, …, N − 1, and k_{ D }(μ) = exp(j2πμT_{ OFDM }2v_{ rel }f_{ c }/c), μ = 0, …, N_{ frame } − 1. The vector \( {\overrightarrow{k}}_R \) and the vector \( {\overrightarrow{k}}_D \) describe the influence of range and the Doppler introduced by the reflecting object on the received frequencydomain symbols.
To obtain the range and relative velocity of the reflecting object, it first achieves an elementwise multiplication between the transmitted and received frequencydomain symbol matrices. With the same matrix representation, the quotient matrix of the frequencydomain symbols is provided
with (⋅)^{T} being the matrix transpose operation and (⋅) being the matrix multiplication operation. It shows clearly that influence of range and the Doppler of the reflecting object is orthogonal. To observe the relation of range and Doppler in element level, the above equation can be rewritten in terms of the range factor k_{ R }(n) and the Doppler factor k_{ D }(μ) as
In one CEOFDM symbol, the range R is translated into a linear phase shift between subcarriers on one frequencydomain symbol. Computing the IDFT of k_{ R }(n) is the most convenient way to obtain the range to the target
with p = 0, …, N − 1. It can be seen that, the two exponential terms in the above equation cancel each other out and result in unity, under the condition
where ⌊⋅⌋ represents rounddown operation. This means that a peak will occur at this index of p in the IDFT result within a target in the environment.
In a similar way, the relative velocity v_{ rel } is translated into a linear phase shift between the frequencydomain symbols on the time axle. The relative velocity v_{ rel } can be easily solved by applying a DFT
with l = 0, …, N_{ frame } − 1. When the condition has been satisfied,
It can be seen that the two exponential terms in the above equation cancel each other out, and a peak will occur at this index of l in the DFT result within a target in environment. Hence, the array process for range is to compute the IDFT of D_{ div } along the time axle. And then, along the spectrum axle, the DFT of the gotten matrix is a result of the array process.
The processing methods of range and relative velocity are completely independent from the transmitted information. The typical Fourier side lobes ensure a high dynamic range performance. All processing operations are linear, and it guarantees that this processing approach will operate for an unlimited number of targets with different ranges and Doppler frequency.
In summary, the received frequencydomain symbol matrix D_{ Rx } is indispensable for the detection processing algorithm and the key of the algorithm is the previous elementwise complex division. At last, the matrix result is a twodimensional radar image in range and relative velocity.
On account of the unique radar processing, peak value in the radar image cannot represent the processing gain of CEOFDM RadCom system. Specifically, the RadCom signals obtain a total power gain of N^{2} after a Fourier transform, which is a coherent deterministic process. However, after a Fourier transform, only a power gain of N is added for the noise, which is a random quantity. Therefore, for CEOFDM RadCom fusion system, the radar processing gain is expressed as
The radar function of CEOFDM RadCom fusion system is based on radar processing in frequency domain. And the main frequency domain processing lies on the computation of DFT, which is a standard procedure that can be efficiently implemented at present.
Simulation and analysis methods
The simulation of this article is based on the Matlab software and the Windows 7 desktop system. The parameters in Table 1 are satisfied. Following the structure of CEOFDM RadCom fusion system, the BER simulation results with turbo code (code length = 1008 and code rate = 1/3) have been obtained in AWGN channel and multipath channel. The ambiguity function simulation and multiple target detection of the CEOFDM RadCom system are also completed.
The NeymanPearson criterion states that P_{ d } is the biggest detection probability when a specific false alarm probability is satisfied with a given SNR. According to this criterion, the output maximum of the array processor is measured to obtain the detection threshold of 0.001 false alarm probability, when only 0 dB SNR AWGN is received. Since the special radar process algorithm requires the known transmitted signals, the detection threshold with different symbol modulation order and modulation index is simulated in Fig. 11. With the fixed false alarm probability, the detection threshold with any SNR is linearly correlated with the detection threshold with 0 dB SNR. The detection probability, with different detection threshold data in Fig. 11, is simulated within different SNR in Fig. 12 to verify the effectiveness of obtained detection threshold.
Simulation and analysis results for communication
The RadCom fusion system is able to exchange realtime mutual information and remove the external interference between wireless communication and radar systems. Specifically for the road safety, the CEOFDM RadCom system contributes the cars in not only information sharing, but also completing the range and relative velocity information detecting by other cars. Normally, the industrial, scientific, and medical (ISM) bands can be used for any device without requiring a license. The radar application requires at least 100 MHz bandwidth for sufficient detection resolution. The 24GHz ISM band is the lowest worldwide available ISM band that satisfies this requirement [1]. However, the achievable range for communications decreases with increasing carrier frequency due to the influence of free space attenuation and diffraction. Therefore, in this paper, the 24GHz ISM band is considered as a suitable frequency band for integrated RadCom fusion systems. In Table 1, a complete set of OFDM system parameters is provided, which satisfies general conditions for vehicles. The maximum absolute value of the Doppler shift f_{ D } is limited by the OFDM symbol duration. The maximum absolute Doppler shift can be measured as 1/(2T_{ OFDM }). Notice that, in this paper, performances of wireless communication and radar detection are separately discussed in two different sections. In this section, performances of wireless communications are presented.
Figure 5 presents the BER performance of CEOFDM with the rise of the energy per bit to noise power spectral density ratio (E_{ b }/N_{0}) in the presence of AWGN channels. According to [18], phase offset (PO) is a key factor for phase modulated signals. When M_{ QAM } = 4, Fig. 5 presents BER performance of CEOFDM RadCom with and without PO in AWGN channel. The turbo code (code length =1008 and code rate =1/3) is applied for improving BER performance.
It is clear that BER performance becomes worse when choosing a lower modulation index, i.e., 2πh = 0.5. With 2πh = 1, the best BER performance with turbo encoding is achieved, which is about 0.00001 with 4 dB E_{ b }/N_{0}. The BER performance curves with and without PO almost coincide. It proves that the phase unwrap demodulator can remove the phase offset in channel.
Since the inherent property of CEOFDM, 2πh = 1 means real value OFDM timedomain signals mapping into the phase space. When 2πh is bigger than 1, CEOFDM system performance also degenerates, because some phase value is over (−π, π), which cannot be recovered. As a result, these simulation results prove that the phase unwrap receiver could remove the phase offset and recover phase information.
In this paper, frequencydomain equalization is simplified with the conventional transmission used in OFDM systems. The circular convolution in timedomain leads to the vector multiplication in frequencydomain. Therefore, the channel distortion can be disported in a simple way. The frequencydomain minimum meansquared error (MMSE) equalizer definition is [33].
where (⋅)^{∗} represents the conjugate operation. The output of the equalizer can be simplified as \( \widehat{s}\left[n\right]=\mathrm{IDFT}\left\{R\left[k\right]C\left[k\right]\right\} \), n = 0, …, N_{ DFT } − 1.
With the turbo encoding (code length =1008 and code rate =1/3) and M_{ QAM } = 4, Fig. 6 shows the CEOFDM BER performance with MMSE equalizer in the presence of multipath fading channels with different numbers of multipath, such as 5 paths, 10 paths, and 64 paths. The CEOFDM BER performance with 5 paths about 1 dB is worse than that with 10 paths OFDM at 12 dB E_{ b }/N_{0}. Compared with 64 paths, channel response with 5 or 10 paths has greater influence on CEOFDM BER performance. When the number of channel multipath is 64, the CEOFDM BER performance is almost the same as 5 or 10 paths CEOFDM BER at E_{ b }/N_{0} = 5 dB and then it declines rapidly. When E_{ b }/N_{0} is 13 dB, the CEOFDM BER with 64 paths is around 0.0001. It proves that the BER performance of CEOFDM RadCom systems is improved with the increasing number of path, which means it is able to achieve multipath diversity. As the input of the phase modulator is realvalued OFDM signals, every CEOFDM symbol data is extended into the multipath frequency point. Hence, CEOFDM RadCom system exploits multipath gain.
The fact that CEOFDM exploits multipath diversity is interesting since traditional OFDM does not [34]. The diversity gain in CEOFDM is explained by viewing the Taylor series expansion of the CEOFDM transmitted signal
The higherorder terms of φ(t) lead to an nfold frequencydomain convolution and the frequencydomain spreading of the data symbols. Therefore, the promotion of performance in multipath channels results from the frequencydomain spreading.
Simulation and analysis results for detection
The ambiguity function graph
The ambiguity function is defined as [22]
The threedimension graph, which is obtained by χ(τ, f_{ d })^{2}, is called the ambiguity graph. Meanwhile, the ambiguity graph displays the delay ambiguity and Doppler ambiguity caused by the neighbor target with a single peak lying in the center of the ambiguity graph. In engineering application, the sectional view of given height, like − 3 dB, is often used to denote the ambiguity graph.
The ambiguity function of radar signals should be focused on a single peak, and other energy is distributed on delayDoppler plane averagely. The radar ambiguity function is always used as a tool to investigate different waveforms and to determine the range resolution and the Doppler resolution of the special waveform. Ambiguity function can estimate parameters about accuracy and resolution of range and relative velocity of single target. Multiple targets can be distinguished according to these parameters.
For wireless communication systems, ambiguity function is defined the same as the radar ambiguity function. For avoiding ISI and ICI, it requires signals to satisfy orthogonality, and the shaping pulse function needs the characteristic of timefrequency localization. Therefore, wireless communication ambiguity function are often used to observe the orthogonality and the timefrequency localization property of signals.
Assuming that various wireless signals are utilized in detection procedure, the detection performance value can be derived from its ambiguity graph. With the parameters in Table 1 being fulfilled, Fig. 7 represents the simulation result from a server and the MATLAB software, in which the 1152×256=294,912 length CEOFDM RadCom timedomain transmitted signals are calculated by ambiguity function. Figure 7 demonstrates that there is only one peak in the center of the ambiguity graph of CEOFDM RadCom transmitted signals. The range resolution and the Doppler resolution are small enough. This proves the CEOFDM RadCom signal fits common requirements of radar detection.
Detection in multiple target scenario
The process methods of range and relative velocity for CEOFDM RadCom system are completely independent from the transmitted information. Furthermore, all processing operations are linear, and it can operate for an unlimited number of targets with different ranges and Doppler frequency. In this section, three targets with different ranges and Doppler frequency are assumed in environment.
Table 2 defines three targets in environment with different range and relative velocity information: target 1 and target 2 with the same range and target 2 and target 3 with the same relative velocity. Figure 8 presents the radar display graph of detection with the three targets.
In 24GHz ISM band, with the parameters in Table 1 being fulfilled, the radar detection is simulated in Fig. 8 when signaltonoise ratio (SNR) is 1 dB. Three peaks that represent three targets can be explicitly found in the graph.
With three rhombus points, the − 6dB sectional view, in which the central coordinate point means the information of relative velocity and range, is demonstrated in Fig. 9. Three points can be acquired from the view. The estimated values in the figure are almost the same as the target parameters. This proves that the targets with the same range or relative velocity can be separated by the frequencydomain process. Hence, the frequencydomain radar process is able to operate for multiple target scenarios within appropriate resolution.
In dynamic range, the reflecting targets are denoted as peaks in radar display graph, which means fine detection performance of the CEOFDM RadCom system. The simulation environment is multipath fading channel with 1 dB SNR, and the sidelobe peak value is 0.1, which is not able to affect target detection due to the unique process property of CEOFDM RadCom system. Furthermore, the radar processing approach can achieve the target information without amplifying noise.
Detection threshold
Radar detection is a very complex process. When there is a target in environment, the probability that the target is successfully detected by the radar system is defined as the detection probability [35], P_{ d }. When there is no target in environment, the probability of false alarm for radar is the false alarm probability, P_{ fa }. The detection probability is an important parameter for radar systems. However, the detection probability is not fixed. In a real detection system, the detection probability depends on a detection threshold, which lies on the requirements for false alarm probability and SNR.
The NeymanPearson criterion is widely applied in radar signal detection. This criterion states that P_{ d } is the biggest detection probability when a specific false alarm probability is satisfied with a given SNR [35]. Therefore, the confirmation of detection probability for a radar system is complicated. The detection threshold is the basis of achieving detection probability. Following the NeymanPearson criterion, the detection threshold depends on one specific false alarm probability with a given SNR. The signals of common radar system are definite while the signals of CEOFDM RadCom system are random for the signal phase.
Figure 10 shows structure of the frequencydomain detection processor in the CEOFDM RadCom fusion system. In this detection algorithm, the reference signal is the CEOFDM transmitted signal s(t), which contains user random data in its phase. Reference signals and received signals are mapped into frequency symbols by DFT. After the elementwise division, the matrix will be passed an array processor, including range and velocity extraction. And the gotten matrix result could form the threedimension radar image. Then, the target detector will make a decision according to the detection threshold. If one peak is bigger than the detection threshold, the target detector outputs the range and velocity information about the peak, which means one target has been detected.
Since the user random data is transmitted, the reference signal varies with different data and modulation in the CEOFDM RadCom fusion system. For the unique radar process in frequency domain, the user data has no influence on detection. However, detection threshold is obtained when only AWGN is received. And the array processing is a linear procedure, so it has no effect on the detection threshold confirmation. In another word, the detection threshold only depends on reference signals with a specific P_{ fa } and a given SNR. The given SNR is denoted as α = E_{ s }/N_{0}, where E_{ s } is symbol energy and N_{0} is power spectral density (PSD) of AWGN. When input of detection receiver is AWGN, the output matrix of array processor can be expressed as
where D_{ div }, Ω_{ α }, and S_{ Tx } are the result matrix of elementary division, DFT of AWGN ,and DFT of reference signals, respectively. ω_{ α } is the AWGN matrix. IDFT(⋅) and DFT(⋅) are IDFT operation and DFT operation. The output value of array processor is r ∈ r_{ α }; the cumulative distribution function (CDF) of r can be expressed as
where f_{ r }(t) is probability density function (PDF) of r. The CDF of r is the probability that r is lower than the fixed value x. When the detection threshold is defined as D_{T, α}, the false alarm probability can be represented as
The detection threshold of a given false alarm probability can be obtained in the CDF results of output value of array processor. When received signals are AWGN, the output of elementary division can be expressed as
For the CEOFDM fusion system, the denominator of the equation defines the DFT of CEOFDM reference signal, which varies with modulation index h, normalization constant C_{ norm }, and realvalued OFDM signal x[i]. Since user random data is necessary, C_{ norm } and x[i] depend on the symbol modulation order M_{ QAM }. As a consequence, D_{ div } relates with modulation index h, symbol modulation order M_{ QAM }.
In the frequencydomain detection algorithm, the output of array processor is influenced by user random data. The detection threshold not only depends on the false alarm probability P_{ fa } and SNR, but also depends on the symbol modulation order M_{ QAM } and phase modulation index h. Although the confirmation of detection threshold is complicated, detection threshold is a necessity for detection probability, which is an important radar parameter. With a given false alarm probability and SNR, the detection threshold D_{T, α} can be obtained by simulating the CDFs of output value of array processor for CEOFDM fusion approach. Especially the modulation of 4QAM is the same as quadrature phase shift keying (QPSK). Therefore, QPSK represents the modulation with M_{ QAM } = 4.
Provided that the false alarm probability is 0.001 and SNR is 0 dB, the input of detection receiver is only AWGN and the parameters in Table 1 have been fulfilled. Figure 11 displays the simulation result about the detection threshold with a different symbol modulation order M_{ QAM } and modulation index h for CEOFDM RadCom fusion systems. Since all envelopes of reference signals are constant and the detection process is complex, the detection processing result is not normalized. Therefore, when the detection threshold is smaller, the detection probability will be higher with a given SNR and a given false alarm probability.
These five curves represent the threshold results of different reference signals in Fig. 11, which are generated from the transmitter with different input symbols. The modulation index h is a parameter, which relates with the stochastic distribution of phase of constant envelope signals in (−π, π]. When the phase modulation index is big enough, 2πh ≥ 1.2, the phase distribution of reference signals is almost random and the curves trend to be flat and the detection threshold trends to be a constant, about 13.8 dB. When modulation order is big, M_{ QAM } = 16 or 64, and phase modulation index is small, 2πh < 0.3, the phase value of reference signals is very small, which results in the high detection threshold.
It is unexpected that the threshold curves of QPSK reference signals and 2QAM reference signals at first declines with 2πh and then rises to about 14 dB. The smallest threshold value of QPSK reference signals is about 10 dB with 2πh = 0.5, and the smallest threshold value of 2QAM is around 7 dB with 2πh = 0.7. The low modulation order, M_{ QAM } = 2 or 4, and appropriate phase modulation index, 0.3 < 2πh < 0.9, result that the phase distribution of reference signals is more nonrandom. And the nonrandom phase distribution affects the variance of 1/S_{ Tx }, which is a part of D_{ div }. As a result, symbol modulation order M_{ QAM } and modulation index h are important parameters for the detection threshold of the CEOFDM RadCom fusion system.
Detection probability
When input of detection receiver is AWGN with a given α, provided that M_{ QAM } and 2πh are fixed, the output matrix of array processor can be expressed as
where c is proportion coefficient of input noise amplitude and ω_{ α } = cω_{α = 1}. Since IDFT operation and DFT operation are both linear operations, the change of α leads to the linear change of r_{ α }. With 0 dB SNR, α = 1 and N_{0, α = 1} is the PSD of AWGN. Since α = E_{ s }/(c^{2}N_{0, α = 1}) = 1/c^{2}, it is visible that \( {D}_{T,\alpha }={cD}_{T,1}={D}_{T,1}/\sqrt{\alpha } \), where D_{T, 1} can be obtained according to Fig. 11. It is clearly visible that the detection threshold with 0 dB SNR has a linear correlation with the threshold with other SNR. When the input of detection processor is the reflected CEOFDM signal and AWGN, detection probability for the CEOFDM RadCom system is defined as the probability that the maximum output value of detection processor is higher than D_{T, α}. Therefore, detection probability for the CEOFDM RadCom system could be expressed according to D_{T, α} and the PDF of r ∈ r_{ α }:
With P_{ fa } = 0.001 and the detection threshold D_{T, 1} in Fig. 11, the Fig. 12 represents the detection probability of the CEOFDM RadCom system for a different symbol modulation order M_{ QAM } and different modulation index h. The detection probability curves rise with the increasing SNR for CEOFDM RadCom system. When 99% detection probability is obtained, the smaller required SNR is better. When 2πh = 0.3, required SNR of 99% detection probability rises with M_{ QAM }. When 2πh = 0.7 or 1.0, required SNR of 99% detection probability is smallest with 2QAM and the required SNR with 16/64QAM are almost the same and biggest. When 2πh = 1.4, required SNR of 99% detection probability rises with QPSK/16QAM/64QAM are nearly the same, but the required SNR with 2QAM is biggest.
With a different modulation index and different symbol modulation order, the results of detection probability for CEOFDM RadCom system confirm the validity of corresponding detection threshold of P_{ fa } = 0.001 and 0 dB SNR in Fig. 11. The smallest required SNR of 99% detection probability, − 17 dB, is obtained under the condition 2QAM and 2πh = 0.7.
The CEOFDM RadCom system exploits CEOFDM signals with user information to complete environment sensing. The achievement of detection probability in this system is more complex than traditional radar system. The variation of reference signals must be taken into consideration, and it is significant to investigate RadCom system with a single waveform.
Conclusions
A joint radar and wireless communication fusion system is the technology development tendency of future intelligence wireless system. Focusing on high PAPR problem of traditional broadband multicarrier waveform, a new CEOFDM RadCom fusion approach is introduced in this paper. This RadCom fusion system is composed of CEOFDM wireless communication system and the frequencydomain radar processing. The PAPR of CEOFDM RadCom signals is constant 0 dB. The high PAPR problem of traditional broadband multicarrier waveform is solved effectively. The constant envelope process and information demodulation are described and analyzed. And the frequencydomain radar processing is introduced. Then, the CEOFDM RadCom fusion system is simulated with a series of parameters in a 24GHz ISM band. Simulation results demonstrate that CEOFDM RadCom system is able to communicate with high data rate and exploit multipath gain in multipath channel. Meanwhile, the ambiguity image of transmitted CEOFDM RadCom signals is achieved with the high resolution for detection. Finally, the detection threshold and the detection probability of CEOFDM RadCom fusion system is simulated and analyzed according to the NeymanPearson criterion. The further work will focus on enhancing the array detection algorithm and improving the RadCom technique for satellite traffic recognition and telemetry.
Abbreviations
 AWGN:

Additive white Gaussian noise
 BER:

Bit error rate
 CDF:

Cumulative distribution function
 CEOFDM:

Constant envelope orthogonal frequency division multiplexing
 CNR:

Carriertonoise ratio
 CP:

Cyclic prefix
 CPM:

Continuous phase modulation
 DFT:

Discrete Fourier transformation
 FDE:

Frequencydomain equalization
 HPA:

High power amplifier
 ICI:

Intercarrier interference
 IDFT:

Inverse discrete Fourier transform
 IOV:

Internet of Vehicles
 ISI:

Intersymbol interference
 ISM:

Industrial scientific and medical
 LTE:

LongTerm Evolution
 MMSE:

Minimum meansquared error
 OFDM:

Orthogonal frequency division multiplexing
 PAM:

Pulse amplitude modulation
 PAPR:

Peaktoaverage power ratio
 PDF:

Probability density function
 PO:

Phase offset
 PSD:

Power spectral density
 QAM:

Quadrature amplitude modulation
 QPSK:

Quadrature phase shift keying
 RadCom:

Radar and wireless communication
 RMS:

Root mean square
 SER:

Symbol error rate
 SNR:

Signaltonoise ratio
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Funding
This work is jointly supported by the MOST Program of International S&T Cooperation (Grant No.2016YFE0123200), National Natural Science Foundation of China (Grant No.61471100/61701503/61750110527), Science and Technology on Electronic Information Control Laboratory (Grant No.6142105040103), and Fundamental Research Funds for the Central Universities (Grant No.ZYGX2015J012).
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YH, SH, and SM conceived the proposed scheme. QL and DH conducted the detailed derivation to evaluate the performance of the proposed scheme and wrote the manuscript. YG and RS reviewed the manuscript. All authors have read and approved the final manuscript.
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Huang, Y., Hu, S., Ma, S. et al. Constant envelope OFDM RadCom fusion system. J Wireless Com Network 2018, 104 (2018). https://doi.org/10.1186/s1363801811056
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Keywords
 Constant envelope
 Radar
 Wireless communication
 OFDM