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Biorthogonal Fourier transform for multichirprate signal detection over dispersive wireless channel
EURASIP Journal on Wireless Communications and Networking volume 2018, Article number: 23 (2018)
Abstract
Biorthogonal Fourier transform (BFT), consistent with the matched signal transform (MST), has been introduced to demodulate the Mray chirprate signal which possesses good orthogonality in the BFT domain. Here, we analyze the characteristics of BFT detection in a further step, including the resolution capability of the multichirprate signal, the property of pulse compression, the closedform biterror rate in the additive white Gaussian noise (AWGN) channel, and the interference in the timefrequency dispersive channel. Even in the high Doppler environment, the shift in BFT detection is proven to be slight. In addition, we deduce that the orthogonality among received chirp rates in the BFT domain would be affected in the multipath dispersive environment. This causes the mutual interference among different chirp rates in a symbol and over symbols concurrently. The theoretical result shows that the chirp modulation parameter can be adjusted to obtain the tradeoff between time and frequency dispersion. By the multipath model of chirprate signal, an auxiliary parallel interference cancellation (PIC) method is further introduced in multipath environment. Simulations verify our analyzed performance of BFT detection in the AWGN, Doppler, and multipath channels. The proposed interference cancellation algorithms are also proven to be effective.
Introduction
Early chirp signals, i.e., linear frequency modulation (LFM) signals, are implemented in radar and communication. In radar, the impulsecompression characteristic of chirp is utilized to extract targets, while the chirp signal is invoked as spreadspectrum waveform to suppress interferences in communications [1]. In recent years, wireless sensor networks, as well as military ad hoc network, have the requirement of integration with self localization, sensing, and communication functions. In this context, chirp scheme for localization and communication is unanimously adopted by the IEEE 802.15.4a Working Group in 2005 as a supplemental physical layer standard of wireless sensor network [2]. Meanwhile, many efforts in signal processing devote to estimate range or localization with a chirp signal [3, 4].
In other fields, broadband wireless communications in highspeed vehicles, such as aircraft and highspeed trains, are much in demand. Conventional phasebased transceivers are unable to meet the needs of the large Doppler shift caused by moving speeds over 200 km/h unless they adopt complicated frequencyshift estimation and compensation. The chirp signal is a good candidate since it is a constantmodulus timefrequency signal and does not require phase detection. However, chirp suffers lowmodulation efficiency owing to its simplex spreadspectrum waveform.
Some efforts on improving the modulation efficiency of a chirp signal have been reported in recent years. Wysocki proposes a Walshcoded chirp modulation in [5] for multiple access. Segments with different phases in a symbol are designed to maintain orthogonality among multiuser signals. A multidimensional chirp modulation scheme with code division spread spectrum is presented in [6]. By overlapping subbands and modulating with pseudorandom sequences, the scheme improves modulation efficiency in frequency and time domain. Analog and digital correlators are both adopted in [7]. The correlation by the former reduces the timebandwidth product of a chirp signal and thereby reducing the complexity of poststage digital processing. To improve orthogonality, [8] introduces a twosegment structured symbol modulated by different chirp rates and Walsh codes. Submitting to the IEEE802.15.4a physical layer proposal, Nanotron Inc. develops a Mary modulation by foursegments linear chirps [9] and employs the multichoice precoding (MCP) technology to solve the multipath fading problem in noncoherent detection. Its modulation efficiency still needs to be improved, and MCP has poor performance in a fast fading channel due to the requirement of the channel information feedback from receiver to transmitter.
The progress in timefrequency (TF) signal processing is also applied to chirp detection. Recently, the TF dispersive channel and orthogonal frequencydivision multiplexing (OFDM) signal have been intensively studied [10]. The series of the filter bank multicarrier (FBMC) and TF filters for OQAM have been proposed to offer better performance to Doppler shift [11]. Here, we concern the various TFbased detectors developed for detecting chirp signals. They include the dechirping technique, the RadonWigner transform [12], the Radonambiguity transform [13], the chirplet transform [14], and the shorttime Fourier transform [15]. A multicarrier chirp communication system based on the fractional Fourier transform (FRFT) is developed in [16] and has been proven to be more reliable compared to fast Fourier transform (FFT) based OFDM in timefrequencyselective channels. In recent years, FRFTOFDM has received attention, and its performance has been analyzed in the dispersive environment, especially in the frequency offset or Doppler frequency shift analysis [17–19]. All of them reveal that FRFT detection performance is significantly better. Furthermore, the affine fourier transform as a generalized FRFT is introduced to multicarrier system in [20, 21]. In [22], the outofband power reduction methods are proposed to the weightedtype FRFTbased multicarrier system. The detectors mentioned above show better antiICI performance, that is proved to be feasible in groundtoair channels. However, the time dispersion (ISI) evaluation and performance analysis for the chirp multicarrier method is neglected.
Other analyses in [23] and [24] concentrate on the resolution of multicomponents in the shortterm FRFT and FRFT domain. The condition is deduced to obtain the maximal resolution performance. Tao [31] and Zhao [26] introduce the FRFT to estimate the multiple components of a chirp signal. However, simultaneous FRFT detectors with different orders should be applied to corresponding chirprate signals. This, however, results in high computing load and complexity in practice.
Matched signal transforms (MSTs) to exponential instantaneous frequency structures are proposed in [27], and further, a linear MST method is introduced to suppress LFM interference [28]. Wang [29, 30] presents a transform being consistent with linear MSTs, naming it the biorthogonal Fourier transform (BFT), to detect a chirp signal since it matches the chirp rate. Like the Fourier transform to a singlefrequency waveform, the impulsecompression effect of the chirp rate is achieved in the BFT domain. Unlike FRFT, the demodulation of the multichirprate signal requires only one BFT process to complete. So far, the demodulation performance of BFT or linear MST to chirprate signal, and the applicability of BFT in wireless doubly dispersive channels have not yet been analyzed in the literatures.
This paper deduces the output signaltonoiseratio (SNR) of BFT detection of chirprate signal, and the closedform biterrorrate (BER) result is obtained. Some characteristics of the discrete BFT are analyzed to design the multichirprate signal and the demodulation algorithm. In a practical environment, we demonstrate that the orthogonality among the chirprate signals in the BFT domain will be unfortunately affected by multipath propagation. However, by adjusting the chirp rate of the signaling scheme, we obtain a tradeoff between the tolerance of the time and frequency dispersion, i.e., ICI and ISI. In addition, we also developed an auxiliary parallel interference cancellation (PIC) method based on the dispersive model in BFT domain to mitigate the multipath interference. At the end of paper, the effect of the Doppler shift is analyzed and verified by simulations. With squarelaw BFT (BFT^{2}), outstanding detection performance exceeds that of FRFT and FSK in the channels with large Doppler shift.
The rest of this paper is organized as follows: firstly, multichirprate signal models in the time domain is given in Section 2. Secondly, Section 3 gives the definition of BFT and analyzes the BFT characteristics, including the closedform solution of BFT detection performance. Thirdly, BFT detection in frequencyoffset and multipath environments are discussed in Section 4 and Section 5, respectively. Fourthly, based on the multipath model in BFT domain derived in Section 5, a MMSE detection aided by the decisiondirected PIC is proposed in Section 6. Fifthly, the simulation results and some discussions are given in Section 7. Finally, the conclusions of our work are summarized in last section.
Multichirprate signal model
A multichirprate scheme is introduced to improve the modulation efficiency of a chirp spreadspectrum (CSS) signal [29, 31]. It maintains the frequency characteristics of the original LFM and avoids high demand of the phasemodulated signal for strict synchronization and equalization of the receiver.
At the nth symbol, the Mary chirprate signal model can be presented as
where M represents the number of chirp rates, i.e., the Mary modulated symbol. K_{ m } is the chirp rate, and f_{0} is the center frequency. T_{ s } is the modulated symbol period. b[ k]∈{−1,1} is the binary information bit. This multiple chirprate modulation can be viewed as a parallel Mwaymodulated signal combination in the chirprate domain. Different chirp rate represents different information bit in a symbol period, as shown in Fig. 1.
BFT on chirprate signal
The biorthogonal Fourier transform (BFT) and its inverse conversion algorithm is defined in [29, 30] (or refer to [28]):
Obviously, to the signal of linear frequency modulation f(t)=exp(jπK_{ m }t^{2}), it has BFT[f(t)]=2πδ(β−πK_{ m }), where the impulse position at πK_{ m } in the BFT domain reflects the chirp rate. Referring to the shortterm discrete Fourier transform (STDFT), the discrete BFT (DBFT) performs a circular convolution operation. Here, the oversampling frequency is denoted as f_{ c } and its interval is T_{ c }. The time and BFT domain are discretized as t=lT_{ c } and β=kΔβ, where l and k are integers. Furthermore, the Npoint symbol duration is defined as T_{ s }=NT_{ c } with N≫1, and Npoint chirprate duration is defined as Ω=NΔβ. Thus, it has the following relationship:
The DBFT and its inverse transform (DIBFT) can be derived as
The BFT is shown to be equivalent to the Fourier transform of a nonlinearly sampled or warped version of the signal. Therefore, the FFT can be used to improve its computing efficiency. It can be accomplished by firstly oversampling f(t) to reduce approximation errors in obtaining warped signal samples.
By analyzing the BFT and DBFT, we derive some characteristics and performances of the transform in multichirprate signal detection.
Theorem 1
The BFT impulse position is determined by the timebandwidth product of chirprate signal. When the timebandwidth product is an integer multiple of 2, the BFT impulse peak is at the discrete k.
Proof
Suppose chirprate signal is f(t)=A exp(jπK_{ m }t^{2}). Its discrete form is \(f({lT}_{c})=A\exp \left (j\pi K_{m} l^{2}T_{c}^{2}\right) \). Inserting it into the above DBFT expression, we obtain the peak position κ of F(k), which is given by
According to K_{ m }=±B_{ m }/T_{ s } where B_{ m } is the bandwidth of chirprate signal, we have κ=±B_{ m }T_{ s }/2. Therefore, the BFT peak is at the integer k when the chirp B_{ m }T_{ s } is a multiple of 2. □
By calculating the BFT at κ=±B_{ m }T_{ s }/2 (m=1,2,⋯,M), the BFT peak of the mth chirprate signal is obtained. This operation undoubtedly simplifies the BFT of chirprate signal and reduces computational load.
Lemma 1
When the chirprate signals have different integer B_{ m }T_{ s }/2, they are orthogonal to each other at the discrete k in BFT domain.
Proof
In a symbol duration T_{ s }, the BFT of f(t)=A exp(jπK_{ m }t^{2}), (0≤t<T_{ s }) is given by
which is a Sa(·) impulse. Thus, there is
With \(\Delta \beta \,=\,2\pi /T_{s}^{2}\), substituting \(\beta =\pi K_{m}\pm 2\pi k/T_{s}^{2}\, (k\neq 0, k=\pm 1,\pm 2,\cdots)\) into the above BFT result, it has
In addition, it has the peak \(\text {BFT}[f(t)]={AT}_{s}^{2} \) at β=πK_{ m }, that is k=β/Δβ=B_{ m }T_{ s }/2 in the discrete BFT. Therefore, the chirprate signals have the orthogonality in discrete BFT domain if they have different integer B_{ m }T_{ s }/2. □
According to the lemma, the multichirprate symbol can be demodulated, and the BFT impulses corresponding to different chirp rates can be resolved.
Theorem 2
In the AWGN channel, SNR of the BFT detection to chirprate signal depends on the timebandwidth product of the received signal where the bandwidth is the sampling bandwidth. The BER of BFT demodulation is
where E_{ b } is the received chirprate energy in a symbol period, and N_{0} is the singlesided power spectral density of AWGN.
Proof
In the continuous time domain, the peak amplitude of BFT output is \({AT}_{s}^{2}\) where A is the amplitude of the received signal. Owing to the IQ complex modulation in Eq. (1), the peak power by BFT demodulation is given by
In the complex zeromean AWGN environment, the noise can be decomposed into n(t)=a_{ n }(t) exp(jϕ(t)), where a_{ n }(t) is the amplitude with Rayleigh distribution, and ϕ(t)∈(0,2π] is the phase with uniform distribution, and they are statistically independent of each other [32]. Thus, the noise by BFT can be expressed as
Since ϕ(t) follows uniform distribution, (ϕ(t)−βt^{2}) also follows uniform distribution. Thus, n^{′}(t)=a_{ n }(t) exp(jϕ(t)−jβt^{2}) is still a Gaussian noise due to the independence of a_{ n }(t) and (ϕ(t)−βt^{2}), which obey Rayleigh distribution and uniform distribution, respectively. Then, the noise variance of the BFT can be represented as
By the front bandpass filter and sampler in the receiver, n^{′}(t) is the Gaussian noise with bandwidth B=f_{ c }. Therefore, its autocorrelation function is \(R_{n'}=\sigma _{n}^{2} S_{a}(\pi \tau /T_{c})\). We note that \(R_{n'}(\tau)\approx 0\phantom {\dot {i}\!}\) when τ>T_{ c }. In addition, when τ≤T_{ c }, i.e., τ≪T_{ s }, the item \(\frac {1}{6}\tau ^{3}\) is negligible with the oversampling times N≫1 and \((T_{s}\tau)^{2}\approx T_{s}^{2}\), \((T_{s}\tau)^{3}\approx T_{s}^{3}\). Thus, we have
The item \(R_{n'}(\tau)\cdot T_{s}^{2}\tau \) is an odd function of τ, so that its integral over [−T_{ s },T_{ s }] is zero.
We note that the BFT in Eq. (10) is a linear transform, so that N_{out}(β) is still a Gaussian noise. Therefore, the BER of BFT demodulation for the bipolar chirprate signal could be derived as
Without regarding to the processing gain g=T_{ s }/T_{ c }=N, the BER is expressed by \(E_{b}/N_{0}=A^{2}T_{s}/\sigma _{n}^{2}T_{c}\) as
□
Here, we further deduce the output SNR of the DBFT. The discretized chirprate signal \(f({lT}_{c})=A\exp \left (j\pi K_{m} l^{2} T_{c}^{2}\right) \) is transformed as
At k^{′}=B_{ m }T_{ s }/2, the amplitude of matching impulse by DBFT is derived to be S(k^{′})=(N+1)AT_{ s }.
By expressing the discrete zeromean AWGN as n(l)=A_{ l } exp(jX_{ l }), where A_{ l } meets Rayleigh distribution and X_{ l } meets uniform distribution in (0,2π]. Similar to the deduction in Eq. (11), substituting n(l) into the DBFT expression, we have
Shown in the expression, the output noise from the DBFT is a linear accumulation of Gaussian processes. Obviously, it is still a zeromean Gaussian noise, so that its variance is derived as
Here, we have the output SNR of the DBFT on chirprate signal when N≫1
This result is consistent with the SNR of the BFT to continuous signal, and the additional N in the numerator is the processing gain of the DBFT demodulator.
Frequency offset in BFT detection
At the receiver, there is a frequency offset between the received signal and local carrier. This offset is caused either by the difference oscillators in the transmitter and receiver, or by the Doppler shift. The detection performance may be severely degraded as the frequency offset is large and volatile, especially in an OFDM system. Not surprisingly, BFT detection of the chirprate signal is affected by frequency offset. The compressed impulse in the BFT domain is shifted and attenuated.
Supposing there is a frequency offset f_{ d }, the chirprate signal is given by
where ϕ_{ i } is the initial random phase of the ith symbol. By BFT, we have
At β=πK_{ m }, the discrete sample for the chirp rate in the BFT domain is deduced as
This is the result of DBFT detection with a frequencyoffset signal, thereby developing the attenuation expression \(F_{fo}(\beta =\pi K_{m})/{AT}_{s}^{2}\).
By Taylor series expansion, \(\sqrt {V}\) is extracted as
According to the range of V in \(\left [0,T_{s}^{2}\right ]\) and the integral in Eq. (20), the approaching point is near \(V_{0}=T_{s}^{2} \). The first two items in the series are left to approximate \(\sqrt {V}\); then, we have \(\sqrt {V}\approx T_{s}/2+ V/(2T_{s})\). Substituting it into Eq. (20), we obtain
As a result, the compressed impulse in the BFT domain is shifted by about πf_{ d }/T_{ s } from its original πK_{ m } value. Compared to the interval \(\Delta \beta =2\pi /T_{s}^{2}\) in the DBFT domain, this shift is small with f_{ d }≪1/T_{ s } in general.
BFT on multipath signal
Signal analysis in the BFT domain
In practice, the detector has to be confronted with wireless propagation environment. Here, we analyze the transform on the multipath chirprate signal. Without loss of generality, the model of the multipath channel is given by
where L is the number of paths and τ_{ l } is the delay of the lth path. A multipath chirprate signal can then be expressed as
We extract a path component of this signal and analyze it in the BFT domain. By the transform in t∈ [ 0,T_{ s }], we obtain
Similar to the deduction in Eq. (23), the integral above is simplified by Taylor series expansion. \(\sqrt {U}\approx T_{s}/2+U/(2T_{s}) \) is substituted in, and further derivation is given by
where
From the result, it is notable that the impulse in the BFT domain is shifted from πK_{ m } to πK_{ m }(1−τ_{ l }/T_{ s }). At the same time, the width of the impulse is also spread from \(2\pi /T_{s}^{2}\) to \(2\pi /\left (T_{s}^{2}\tau _{l}^{2}\right)\). The longer the delay, the looser the impulse. These results also present the offset and extension of the βdomain impulse when there exists symbol synchronization error.
On the other hand, the BFT of the multipath signal with cross delay τ_{ l } from the previous symbol is derived in the same way as
where
The BFT impulse from the cross signal is shifted to π(2K_{ m }−τ_{ l }/T_{ s }), which is far from πK_{ m } with τ_{ l }≪T_{ s }. Its amplitude is low and its width is spread due to small τ_{ l }. Therefore, the intersymbol interference can be negligible with small cross delay τ_{ l } in general.
Shown in the upper part of Fig. 2 is the timefrequency illustration of different delayed chirprate signals in a symbol duration T_{ s }. The lower part of the figure is their projections in the BFT domain. The blue impulse comes from the synchronized path component, and the red pulses correspond with the other multipath components of the same symbol. The black pulses are the BFT of the multipath components of the pervious chirprate symbols. In the illustration, the red and black pulses may be the interference in BFT domain to other chirprate components in the same symbol duration, including intrasymbol mutual interference and intersymbol interference, respectively. Therefore, the orthogonality in the BFT domain among the chirprate signals is affected in a practical multipath environment.
Despite having dispersion, the multipath chirp in BFT domain are found to concentrate when T_{ s }≫τ_{ l } in Eq. (27). Noting that the conclusion in the above section that T_{ s }≪1/f_{ d } is required, T_{ s } must be selected as a compromise for low BFT dispersion when Doppler shift and multipath delays coexist. Here, we measure the BFT dispersion performance by d, which is defined as
where E_{ i } is the energy of the desired component in the BFT domain, and E_{ k } is the energy of dispersed component. Figure 3 illustrates the performances of time and Doppler dispersions. In this example, the timebandwidth product is set to a constant 10, the symbol duration T_{ s } varies from 2 to 100μs, and the corresponding bandwidth varies from 5 to 0.1MHz. Four Doppler shifts {1kHz,5kHz,10kHz,30kHz} and three pathdelays {0.1μs,0.3μs,0.5μs} are measured to obtain the BFT dispersion performance. The crossing points present the compromises of the modulation parameters under specified double dispersive channel. On the other hand, increasing T_{ s }, i.e., decreasing B_{ m }, in concentrating time dispersion in the BFT domain will reduce the symbol rate. Multicarrier modulation could reconcile the original data rate and dispersion requirement of BFT detection.
Multipath and multichirprate signal model in the BFT domain
Through the above analysis, the multipath signal with a single chirp rate in the nth symbol period is obtained in the BFT domain as
where n_{ l }=⌊τ_{ l }/T_{ s }⌋,(τ_{ l }>T_{ s }), Δτl,m′=πK_{ m }τ_{ l }/T_{ s } and L^{′} is the number of multipaths whose delay does not exceed a symbol duration. The function δ(β) is an abbreviation of \(\text {Sa}\left [\left (T_{s}^{2}\tau _{l}^{2}\right)\beta /2\right ]e^{\Phi (\beta)} \).
Thus, the BFT of the received multichirprate signal in the multipath channel is \(R_{n}(\beta)=\sum _{m=0}^{M1} R_{n,m}(\beta)+ v(\beta)\), where v(β) is the Gaussian noise due to BFT being linear transform and chirp rate K_{ m } is assigned according to Theorem 1. Define Mary source vector b(n)=[ b_{0}(n),⋯,b_{ m }(n),⋯,b_{M−1}(n)]^{T}. By the normalized DBFT impulse of the chirprate signal, the DBFT output on received multichirprate signal is expressed as Eq. (31). In Eq. (31), αm,i′ denotes the mutual interference to the mth chirp rate from other ith chirprate component in a same symbol period. From the above analysis, the chirprate K_{ m } is just jammed by the chirprate K_{i≥m+1} components. αm,i,j′ is the intersymbol interference from the ith chirprate component before j symbols, and \(\mathbf {h}_{m}=[\alpha _{0},\alpha _{m,m,1},\alpha _{m,m,2},\cdots,\alpha _{m,m,N_{l}}]\phantom {\dot {i}\!}\) is the jamming vector from the same chirprate components due to multipath propagation.
Define the output vector by BFT of the received multipath signal as r (n)= [ r_{ n }(πK_{0}), r_{ n }(πK_{1}), ⋯,r_{ n }(πK_{M−1})]^{T}. According to Eq. (31), we obtain the vector model by BFT as Eq. (32), where \(\tilde {\mathbf {b}}(n) = [\mathbf {b}(n),\mathbf {b}(n1),\cdots,\mathbf {b}(nN_{L})]^{T}\) is the Mary source series.
Detection of multichirprate signal in multipath environment
Although there are a large number of interference terms in H, many of them are zero or minute, that is, the matrix H is sparse. From analysis of Eq. (27) and Fig. 2, the multipath impulses by BFT are on the left of πK_{ m } in the β domain. The further from πK_{ m }, the weaker the impulses are. Therefore, the mutual collisions among impulses are not severe in practice.
The signal model by BFT in the multipath channel is a typical model of mutual and intersymbol interference. Estimation of the matrix H is difficult. It is feasible to apply a training sequence and adaptive algorithms to suppress interference.
A minimum mean square error (MMSE) detector is commonly applied to mitigate interference. Because of the linear signal model in (32), MMSE detection is also available. Here, the M×(N_{ l }+1) weight matrix W acts as the filter to Mchannel detection. The optimal criterion in MMSE detection for the BFT of the Mchirprate signal is given by argW minE{b(n)−W^{H}r(n)^{2}}. MMSE detection can be implemented by applying a training sequence and an adaptive LMS/RLS algorithm. The optimal solution is W_{opt}=R^{−1}r, where R=E[r(n)r(n)^{H}].
Disturbed not only by the previous symbol but also by other chirprate signals in the same symbol, linear MMSE detection may be inadequate for the job in a deep fading channel. Here, a decisiondirected method and parallel interference cancellation (PIC) are introduced to construct a robust algorithm for the multichirprate signal in the multipath environment, which is called MMSEDDPIC for short. The algorithm block diagram is shown in Fig. 4.
The interference cancellation for a chirprate signal is given by
where \(\hat {b}_{j}\) is the decisiondirected result of channel j and u_{m,j} are the weights of the channel for cancellation. Define U_{ m }=[u_{m,1},⋯,u_{m,j},⋯,u_{m,M}]^{T} where (j≠m). By a decisiondirected method, this U_{ m } can be simultaneously adjusted with the above MMSE, which is given by
where \(\hat {\mathbf {d}}_{m}(n)=[\hat {b}_{1}(n),\cdots,\hat {b}_{m1}(n),\hat {b}_{m+1}(n),\cdots,\hat {b}_{M}(n)]\) are the decisions of other channels and 1_{ m } in constraint is a zeros vector except for its mth element being one. In this MMSE criterion, W_{ m } is the feedforward filter, and U_{ m } is the interference cancellation filter. The soft estimation \(y_{m}(n)=\mathbf {W}_{m}^{H}\mathbf {r}(n)\mathbf {U}_{m}^{H}\hat {\mathbf {d}}(n)\). The constraint \(\mathbf {W}_{m,\text {opt}}^{H}\cdot \mathbf {1}_{m} = 1\) in the criterion regards the impulse position as the signature of the chirp rate in the BFT domain.
Simulations
In this section, we give simulation results of the discrete BFT demodulation and compare them with the theoretical results. In simulations, a multichirpratemodulated signal is implemented. The parameters of modulation include a bandwidth of 10 MHz and a modulated symbol rate of 1 Msps. According to Theorem 1, applying orthogonal IQ modulations, the number of chirp rates can be set to M = 10, and the chirp rates are \(K_{m}~=~B_{m}/T_{s}~=~2k/T_{s}^{2}\), where k = ±1,± 2,⋯±5.
Under ideal conditions including synchronization and no noise, Fig. 5 illustrates a DBFT result for a 10chirpratemodulated signal. Consistent with Lemma 1, the DBFT of each chirprate signal is an impulse at the discrete β domain without mutual interference. Meanwhile, the result verifies the condition for orthogonality in a multichirprate signal.
Figure 6 shows the comparison of the BER performance of DBFT detection among single chirprate and 10chirprate signals; Fig. 6b, c shows the plots for the first and fourth rate signals, respectively. It must be pointed out that the E_{ b } in Fig. 6b, c is just composed of the energy of one chirprate component in a symbol period. From the results, we can see that the DBFT BER performances on singlechirprate and multichirprate signals are consistent. It also proves the orthogonality of the components in the multichirprate signal in the DBFT domain. At the same time, the identity of simulated and theoretical BERs verifies the correctness of the closedform solution of DBFT BER in Theorem 2.
Conventional chirprate demodulations include matched filter (DeChirp) and FRFT demodulations. In addition, a frequencyshift keying (FSK) signal is also a timefrequencymodulated signal. The BER performances of the three demodulation methods on timefrequency signal are given below.

Matching filter on a chirprate signal (DeChirp): \(P_{e}=Q (\sqrt {{2E_{b}}/{N_{0}}})\).

FRFT on a chirp binaryorthogonal keying (BOK) signal: \(P_{e}=Q(\sqrt {{E_{b}}/{2N_{0}}})\).

Matching filter on FSK signal: \(P_{e}=Q(\sqrt {{E_{b}}/{N_{0}}})\).
In Fig. 7, we compare their demodulation performance with BFT detection. Simulation results show that the BER performance of BFT is better than that of FSK coherent and FRFT noncoherent detection.
Because BFT detection is a linear transformation, frequency error, or offset will cause rotation of the signal phase, we apply squarelaw BFT processing (BFT^{2}) and unipolar chirprate onoff keying (OOK) signal to avoid deterioration in performance. However, the performance degrades due to this squarelaw detection. By Eq. (21), the decline of BFT sampling amplitude caused by offset is obtained to estimate the BER of BFT ^{2}
where γ is the input SNR of squarelaw detector.
From [25], the FRFT peak of the chirprate OOK signal under frequency offset is deduced as
where α is the timefrequency rotation angle in FRFT, A is the amplitude of signal, and f_{ d } is the Doppler shift. When  sinα≈T_{ s }/B, where B is the bandwidth of the chirp signal and then the BER can be approximated by
Under different conditions, including no frequency offset, f_{ d } = 100 kHz, and f_{ d } = 200 kHz, BFT^{2} detection performance is simulated and compared. The BER results of BFT^{2} in Fig. 7 are very close, which indicate that detection performance declines slightly even with large Doppler shift or frequency error. By contrast, FRFT detection exhibits a significant deterioration of BER performance at f_{ d } > 100 kHz, even if it has the advantages in conventional timefrequencyselective channel compared with coherent detection [16, 25].
In Section 5, we analyzed the characteristics of BFT detection of the multipath chirprate signal. Not only does intersymbol interference exist, but the mutual interference among different chirprate signals in the same symbol period also makes detection difficult. Nevertheless, due to the sparsity of the impulse interferences in the BFT domain, it is possible to mitigate them by signal processing. In the nonlineofsight (NLOS) communications environments, the equalization and interference cancellation is a better choice for detection.
Modulated with a fivechirprate real signal, 10Mhz bandwidth, and 1Msps symbol rate, BFT detection is simulated in the multipath environment. Two MMSEbased interference suppression algorithms are implemented according to the analysis and design presented in Section 6. The conventional LMS adaptive training process is applied to solving the optimal weight vector problem. On the other hand, we compare the BER performances at two different multipath fading environments. One is a common frequency selective fading channel, as in Fig. 8a which is the normalized channel frequency response. In Fig. 8b, we compare the BER performance of the algorithms in this channel. Auxiliary parallel interference cancellation makes the detection output close to that in flat fading. However, in this multipath environment, the performance of the MMSEDDPIC algorithm is only slightly better than that of MMSE detection, and for actual implementation, a choice based on computational complexity must be made. For a deepfading frequency selective channel as shown in Fig. 9a, the BER performances are compared in Fig. 9b. The performance without interference suppressing is unacceptable. The MMSEDDPIC algorithm does a good job, and its performance is much better than that of simple MMSE training.
Conclusions
The biorthogonal Fourier transform has been introduced to demodulate a multichirprate signal. In this paper, the detection performance of the DBFT and BFT, including the multicomponents resolution and closedform BER, are derived. Further analyses are made in the frequencyoffset and multipath environments. The small shift of compressed βdomain impulses in the high frequencyoffset channel is a remarkable and welcome output of our research. Unfortunately, the BFT of a multichirprate signal is proven to have intrasymbol and intersymbol interferences with multipath propagation or synchronization error. The theoretical result shows that the chirp modulation parameter can be adjusted to obtain the tradeoff between the time and frequency dispersion. A multipath model of BFT output is constructed, and the MMSEbased algorithm is given to suppress the interference. Aided by the decisiondirected PIC, the proposed detection performs well in the deep fading environment.
Abbreviations
 AWGN:

Additive white Gaussian noise
 BER:

Biterrorrate
 BFT:

Biorthogonal Fourier transform
 BFT^{2} :

Squarelaw BFT
 CSS:

Chirp spread spectrum
 DBFT:

Discrete BFT
 FBMC:

Filter bank multicarrier
 FRFT:

Fractional Fourier transform
 ICI:

Intercarrier interference
 ISI:

Intersymbol interference
 LFM:

Linear frequency modulation
 MCP:

Multichoice precoding
 MMSE:

Minimum mean square error
 MST:

Matched signal transform
 OFDM:

Orthogonal frequencydivision multiplexing
 PIC:

Parallel interference cancellation
 SNR:

Signaltonoiseratio
 STDFT:

Shortterm discrete Fourier transform
 TF:

Timefrequency
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Acknowledgements
The authors would like to thank the reviewers for their thorough reviews and helpful suggestions.
Funding
This work is supported in part by the National Natural Science Foundation of China (nos. 61371107 and 61571143), the Foundation of Guangxi Broadband Wireless Communication & Signal Processing Key Laboratory (no. GXKL061501), and the Foundation of Science and Technology on Communication Networks Laboratory (no. KX172600033).
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LZ is the main writer of this paper. He proposed the main idea, deduced the performance of BFT detection, completed the simulation, and analyzed the result. CYang introduced the MMSEbased algorithm in dispersive channel. CYan simulated the detection in the Doppler channel. HQ gave some important suggestions for BFT detection. All authors read and approved the final manuscript.
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Zheng, L., Yang, C., Yan, C. et al. Biorthogonal Fourier transform for multichirprate signal detection over dispersive wireless channel. J Wireless Com Network 2018, 23 (2018). https://doi.org/10.1186/s1363801810273
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DOI: https://doi.org/10.1186/s1363801810273
Keywords
 Chirprate modulation
 Biorthogonal Fourier transform
 Matched signal transform
 Dispersive channel