A joint synchronization parameter estimation in fractional Fourier transform orthogonal frequency division multiplexing underwater acoustic communication system

Abstract

In the conditions of Gaussian noise, symmetric alpha stable impulse noise, multipath propagation, Doppler effect, and so on, a joint synchronization algorithm is proposed for underwater acoustic communication system based on fractional Fourier transform orthogonal frequency division multiplexing. The algorithm realizes the orthogonal multicarrier underwater acoustic communication system synchronization based on that the symmetrical triangular linear frequency modulation signal has the energy convergence in the domain of fractional Fourier transform. In order to verify the effectiveness of the algorithm, hyperbolic frequency modulation signal and linear frequency modulation signal are used as the evaluation index, and the synchronization performance of bit error rate of them is compared by simulation and pool experiments. The simulation results show that the proposed algorithm in this article is resistant to Gaussian noise, symmetric alpha stable pulse noise, multipath propagation, and Doppler effect, compared with traditional synchronization algorithms; the synchronization error probability is low at the same signal-to-noise ratio.

Keywords

Fractional Fourier transform orthogonal frequency division multiplexing symmetric alpha stable distribution noise symmetric triangular linear frequency modulation signal joint synchronization parameter estimation

Underwater acoustic communication technology is an important technical means for the real-time acquisition, processing, and transmission of the marine environmental parameters in the marine monitoring system¹ at present. Because of the existence of the underwater acoustic channel’s space-varying characteristics and time-varying characteristics, the time of the acoustic channel not only has time delay and Doppler frequency shift but also contains various types of noise and interference. Coupled with a large number of pulse components in underwater acoustic noise, we need to use the symmetric alpha stable (SαS) stable distribution model to express. Therefore, the noise and interference with the sound source signal superposition will seriously affect the synchronization and subcarrier orthogonality of orthogonal frequency division multiplexing (OFDM) underwater acoustic communication system. How to accurately realize the synchronization parameter estimation is the key and hot research topic in OFDM^2–6 system. The main literature^7–9 gives some estimation scheme of synchronization parameters for the OFDM system. But for underwater acoustic communication, there are some problems such as the complexity of operation and the estimation accuracy. The performance of synchronization signals of hyperbolic frequency modulation (HFM) and linear frequency modulation (LFM) compared with the time domain correlation method based on time delay was studied using fractional lower order cycle fuzzy function and joint estimation of Doppler frequency shift, for SαS stable distribution noise to get better performance.^10–12 In order to further improve the performance of the synchronization parameters’ estimation in OFDM underwater acoustic communication system, the symmetrical triangular linear frequency modulation (STLFM) signals are used to obtain the higher joint estimation and synchronization performance.

The rest of the article is organized as follows. The first part introduces the synchronization parameters in underwater acoustic fractional Fourier transform (FRFT)-OFDM communication system. The second part gives the analysis and derivation of the joint synchronization algorithm. The third part gives the algorithm simulation results under different environmental noise and interference and pools the results with synchronization error rate and carrier frequency offset mean square error (MSE) as evaluation index for performance evaluation. The results are given in the last part.

Synchronization parameters

The synchronization of OFDM underwater acoustic communication system includes time (timing), sampling value, and carrier frequency synchronization.⁹

Assuming that N is the number of subcarriers in the OFDM underwater acoustic communication system, T_s is the sampling period of the transmitter, and Hypothesis $Δ f_{s}$ is the sampling frequency offset for the receiving end, then the sampling period of receiving hydrophone end is $T_{r} = N / (N / T_{s} - Δ f_{s})$ . According to the principle of FRFT-OFDM signal sampling, the sampling signal s(k) can be expressed as

\begin{array}{l} s (k) = \sum_{i = 0}^{N - 1} d_{i} A_{- α} \\ \exp {- j π {[{(\frac{k T_{r}}{N})}^{2} + {(\frac{i}{T_{s}})}^{2}] \cot α - 2 \frac{k i}{N} * \frac{T_{r}}{T_{s}} \csc α}} \end{array}

(1)

In equation (1), $A_{α} = \exp [- j π sgn (\sin α) / 4 + j α / 2] / \sqrt{| \sin α |}$ ; d_i for i is the transmission of data.

If there is relative motion between the transmitter and the receiver, the Doppler frequency shift will be generated. Assuming that $Δ f$ is the Doppler frequency shift that is included at the receiving hydrophone end, the receiving signal of hydrophone end can be represented as

\begin{array}{l} r (k + D) = \exp (j 2 π Δ f_{c} (k + D) T_{r}) \\ [s (k + D) * h (k + D)] + z (k + D) \end{array}

(2)

In equation (2), D represents the time delay, $d_{i}$ is the transmission data for the ith subcarrier, * is the convolution operation, $h (k) = \sum_{m = 1}^{M} h_{m} δ (k - τ_{m})$ represents impulse response of M multiple channel, and $z (k)$ is the underwater acoustic channel noise. During p-points sampling for each symbol and using the $l / p$ judgment, then it can be to achieve synchronization of sampled values so long as $| T_{r} - T_{s} | < (p - l) T_{r}$ .

The Doppler frequency shift for the kth subcarrier of OFDM can be represented as

F_{k} = \frac{v_{w}}{c_{w}} f_{k} \cos θ

(3)

In this equation, $f_{k}$ is the frequency of the kth subcarrier, $ν_{w}$ denotes the relative speed between the emission source and the receiving hydrophone, $c_{w}$ is the speed of underwater acoustic channel, and $θ$ is the scattering angle of underwater acoustic signal. Equation (3) shows that when the transmitting source and the receiving hydrophone have the relative speed between, the Doppler frequency shift is likely to damage the orthogonality between subcarriers, causing inter-carrier interference (ICI) and inter-symbol interference (ISI), so as to improve the bit error rate (BER). Therefore, it is needed to estimate the parameters such as time delay, Doppler frequency shift, and multipath effect comprehensively.

Joint synchronization parameter estimation

STLFM signal features

Assuming that A is the amplitude of STLFM signal, $f_{c}$ is the carrier frequency of STLFM signal, $Δ F$ represents the modulation bandwidth of STLFM signal, and $T = 2 t_{m}$ denotes the duration of the STLFM signals, the STLFM signal can be represented by equation (4)

L (t) = {\begin{matrix} A \exp {j 2 π [(f_{c} - \frac{Δ F}{2}) t + \frac{Δ F}{2 t_{m}} t^{2}]}, 0 \leq t < t_{m} \\ A \exp {j 2 π [(f_{c} + \frac{Δ F}{2}) (t - t_{m}) - \frac{Δ F}{2 t_{m}} {(t - t_{m})}^{2}]}, t_{m} \leq t < 2 t_{m} \end{matrix}

(4)

In case that the Doppler frequency shift is not too much, the frequency modulation slope of STLFM signal can be seen as the same. Using equation (4), we will get the positive and negative frequency modulation slopes of STLFM signal, represented as $u_{1} = Δ F / t_{m}$ and $u_{2} = - u_{1} = - Δ F / t_{m}$ , respectively. For underwater acoustic communication system, the parameters of STLFM signal are taken as $f_{c} = 5 kHz$ , $Δ F = 10 kHz$ , $F_{s} = 48 kHz$ , and $t_{m} = 0.125 s$ . To analyze the STLFM signal in the FRFT domain, the results are as shown in Figure 1. It can be clearly seen from Figure 1 that the STLFM signal spectrum presents energy spikes in the FRFT plane $(u, α)$ , determined by $p_{1} = - 2 arc \cot u_{1} / π$ and $p_{2} = - 2 arc \cot u_{2} / π$ . The result can be used for joint synchronization parameter estimation.

Figure 1.

FRFT domain analysis of STLFM signal.

Synchronization parameters’ estimation of FRFT-OFDM system

It can be seen obviously that in Figure 2 the STLFM signal with a length of N_L is added as the synchronization code for FRFT-OFDM underwater acoustic communication system and the transmission is achieved by an independent subcarrier in order to achieve synchronization, The period of synchronous code is represented as $T_{L} = 2 t_{m}$ . The STLFM signal is sampled, and the synchronization sequence signal of the length of $N_{L}$ is expressed as $L_{i} (k)$ . As a result, the signal from the transmitter is composed of cyclic prefix (CP), STLFM, and the information code.

Figure 2.

Structure of sent OFDM signal.

At the receiving hydrophone end, the received signal of random selection with a long of $3 N_{s}$ can be expressed as

r_{m} (k) = r_{L_{m}} (k) + r_{c p_{m}} (k) + r_{s_{m}} (k), m = 1, 2, 3

(5)

where $r_{L} (k)$ , $r_{cp} (k)$ , and $r_{s} (k)$ denote STLFM signal, CP, and information code, respectively. Setting $D_{0}$ as the symbol timing parameter, combining equations (1) and (2), and considering subcarrier position of STLFM signal, $r_{L_{1}} (k)$ can be represented as

\begin{matrix} r_{L_{1}} (k) = \exp {j 2 π Δ f_{c} [(n - 1) N_{s} + (k - D_{0})] T_{r}} \\ \times {L_{1} (k - D_{0}) A_{- α} \exp {- j π {[\frac{(k - D_{0}) T_{r}}{N}]}^{2} \cot α} * h (k - D_{0})} \end{matrix}

(6)

From the analysis of section “STLFM signal features,” we analyze the STLFM signal at the P level of FRFT

\begin{matrix} F^{p_{1}} [r_{L_{1}} (k)] \\ = F^{p_{1}} {\exp {j 2 π Δ f_{c} [(n - 1) N_{s} + (k - D_{0})] T_{r}} \\ \times {L_{1} (k - D_{0}) A_{- α} \exp {- j π {[\frac{(k - D_{0}) T_{r}}{N}]}^{2} \cot α_{1}} * h (k - D_{0})}} \\ = \exp {j 2 π Δ f_{c} [(n - 1) N_{s} - D_{0}] T_{r}} \\ \times \sum_{m = 0}^{M - 1} h_{m} A_{- α} \exp {- j π {[\frac{(D_{0} + τ_{m}) T_{s}}{N - T_{s} Δ f_{s}}]}^{2} \cot α_{1}} \\ \times F^{p_{1}} {\exp {j {(k T_{L})}^{2} {- π {[\frac{T_{s}}{(N - T_{s} Δ f_{s}) T_{L}}]}^{2} \cot α_{1}}} \\ \times \exp {j 2 π (k T_{L}) [Δ f_{c} \frac{T_{r}}{T_{L}} + \frac{(D_{0} + τ_{m}) {T_{s}}^{2}}{{(N - T_{s} Δ f_{s})}^{2}} \cot α_{1}]} L_{1} [k - (D_{0} + τ_{m})]} \end{matrix}

(7)

In equation (7), $h_{m} (k)$ represents the impulse response of the m path channel in a multipath channel, and $τ_{m}$ indicates the channel time delay of the corresponding path. Considering the effect of multipath effect, the first impulse response point is searched and set as a reference point, namely, time delay $τ_{0} = 0$ , whose position is $α_{1}$ . The $α_{1}$ expression is as follows

a_{1} = D_{0} \cos α_{1} + [Δ f_{c} \frac{N T_{s}}{(N - T_{s} Δ f_{s}) T_{L}} + \frac{D_{0} T_{s}^{2} \cot α_{1}}{{(N - T_{s} Δ f_{s})}^{2} T_{L}}] \sin α_{1}

(8)

In the same way, the second impulse response points are searched, and the location $α_{2}$ is available

a_{2} = D_{0} \cos β_{1} + [Δ f_{c} \frac{N T_{s}}{(N - T_{s} Δ f_{s}) T_{L}} + \frac{D_{0} T_{s}^{2} \cot β_{1}}{{(N - T_{s} Δ f_{s})}^{2} T_{L}} + \frac{N \csc β_{1}}{(N - T_{s} Δ f_{s}) T_{L}}] \sin β_{1}

(9)

Search the third impulse response points and remember the location $α_{3}$ as follows

\begin{array}{l} a_{3} = D_{0} \cos γ_{1} + \\ [Δ f_{c} \frac{N T_{s}}{(N - T_{s} Δ f_{s}) T_{L}} + \frac{D_{0} T_{s}^{2} \cot γ_{1}}{{(N - T_{s} Δ f_{s})}^{2} T_{L}} + \frac{(N - 1) \csc γ_{1}}{(N - T_{s} Δ f_{s}) T_{L}}] \sin γ_{1} \end{array}

(10)

Then get the solution of simultaneous equations (8), (9), and (10) as follows

{\begin{matrix} Δ f_{s}^{'} = \frac{N - ε_{1}}{T_{s}} \\ Δ f_{c}^{'} = \frac{(a_{2} \cos α_{1} - a_{1} \cos β_{1}) T_{L} ε_{1} - N \cos α_{1}}{N T_{s} \sin (β_{1} - α_{1})} \\ D_{0}^{'} = \frac{(a_{2} \sin α_{1} - a_{1} \sin β_{1}) T_{L} {ε_{1}}^{2} - N ε_{1} \sin α_{1}}{(T_{L} {ε_{1}}^{2} + T_{s}^{2}) \sin (α_{1} - β_{1})} \end{matrix}

(11)

Among them

ε_{1} = \frac{1}{T_{L}} \cdot \frac{N [\cos α_{1} \sin (β_{1} - γ_{1}) - (\cos γ_{1} - \cos β_{1}) \sin (β_{1} - α_{1})] - \cos β_{1} \sin (β_{1} - α_{1})}{(a_{2} \cos α_{1} - a_{1} \cos β_{1}) \sin (β_{1} - γ_{1}) - (a_{2} \cos γ_{1} - a_{3} \cos β_{1}) \sin (β_{1} - α_{1})}

In the same way, the FRFT signal at the STLFM level of the p₂ is analyzed, and the corresponding rotation angle is α₂, which can be obtained

{\begin{matrix} Δ f_{s}^{″} = \frac{N - ε_{2}}{T_{s}} \\ Δ f_{c}^{″} = \frac{(a'_{2} \cos α_{2} - a'_{1} \cos β_{2}) T_{L} ε_{2} - N \cos α_{2}}{N T_{s} \sin (β_{2} - α_{2})} \\ D_{0}^{″} = \frac{(a'_{2} \sin α_{2} - a'_{1} \sin β_{2}) T_{L} {ε_{2}}^{2} - N ε_{2} \sin α_{2}}{(T_{L} {ε_{2}}^{2} + {T_{2}}^{2}) \sin (α_{2} - β_{2})} \end{matrix}

(12)

Among them

ε_{2} = \frac{1}{T_{L}} \cdot \frac{N [\cos α_{2} \sin (β_{2} - γ_{2}) - (\cos γ_{2} - \cos β_{2}) \sin (β_{2} - α_{2})] - \cos β_{2} \sin (β_{2} - α_{2})}{(a_{2}^{'} \cos α_{2} - a_{1}^{'} \cos β_{2}) \sin (β_{2} - γ_{2}) - (a_{2}^{'} \cos γ_{2} - a_{3}^{'} \cos β_{2}) \sin (β_{2} - α_{2})}

The above analysis shows that the STLFM signals can be done at two FRFT in $p_{1}$ and $p_{2}$ order to achieve joint estimation of time delay, Doppler frequency shift, and multipath effect.

The simulation and experimental results’ analysis

The parameter estimation performance simulation analysis

The underwater acoustic communication simulation system we used in the experiment adopts quadrature phase shift keyin (QPSK) modulation method. Its parameters are set as follows: FRFT length 256, cyclic prefix length of 64, the sampling frequency of 48 kHz, and channel multipath number is 3. Assuming that initial distance between the emission source and receiving hydrophone is 2000 m, the relative speed of them is 20m/s, the channel noise is Gaussian. The parameters of the simulation signal are shown in Table 1.

Table 1.

Simulation signal parameters.

Simulation signal	Center frequency (Hz)	Adjustable frequency (Hz/s)	Duration (s)	Bandwidth (Hz)	Amplitude (V)
s ₁	3000	10,000	0.125	5000	1
s ₂	4000	14,000	0.125	5000	1
s ₃	5000	18,000	0.125	5000	1

Tables 2 –4 indicate the signal-to-noise ratio (SNR) in −5, 5, and 15 dB, respectively, 1000 times of the simulation results.

Table 2.

Parameter estimation results when SNR = −5 dB.

Simulation signal	Center frequency estimate mean (Hz)	Adjustable frequency estimate mean (Hz/s)	Sampling frequency offset estimate mean (kHz)	Mean carrier frequency offset estimation (Hz)	Symbol timing estimate mean (s)
s ₁	3287.9	10268.9	0.3781	0.1657	0.54 × 10⁻²
s ₂	4360.7	14255.2	0.1324	0.1843	0.69 × 10⁻²
s ₃	5240.1	18373.7	0.4954	0.2577	0.81 × 10⁻²

SNR: signal-to-noise ratio.

Table 3.

Parameter estimation results when SNR = 5 dB.

Simulation signal	Center frequency estimate mean (Hz)	Adjustable frequency estimate mean (Hz/s)	Sampling frequency offset estimate mean (kHz)	Mean carrier frequency offset estimation (Hz)	Symbol timing estimate mean (s)
$s_{1}$	3113.5	10112.3	0.0812	0.0568	0.29 × 10⁻³
$s_{2}$	4116.8	14110.7	0.0532	0.0918	0.36 × 10⁻³
$s_{3}$	5110.9	18115.6	0.1218	0.0869	0.52 × 10⁻³

SNR: signal-to-noise ratio.

Table 4.

Parameter estimation results when SNR = 15 dB.

Simulation signal	Center frequency estimate mean (Hz)	Adjustable frequency estimate mean (Hz/s)	Sampling frequency offset estimate mean (kHz)	Mean carrier frequency offset estimation (Hz)	Symbol timing estimate mean (s)
s ₁	3065.5	10043.9	0.0345	0.0097	0.24 × 10⁻⁴
s ₂	4038.9	14032.3	0.0109	0.0043	0.18 × 10⁻⁴
s ₃	5052.4	18059.4	0.0276	0.0053	0.37 × 10⁻⁴

SNR: signal-to-noise ratio.

Comparison of the performance of the simulation synchronization

In order to compare the performance of the proposed algorithm, the proposed method is used to perform the timing synchronization same as the condition with the LFM or HFM synchronization signal and the fractional lower order cyclic cross ambiguity (FCCA) function estimation, and the error rate of the estimation is analyzed and compared. The results are shown in Figure 3.

Figure 3.

Comparison of bit error rate.

Figure 3 shows that under the condition of additive white Gaussian noise channel, LFM and STLFM synchronization performance gets close; STLFM synchronization is more stable in the situation of low SNR; what is more, with Gaussian white noise and impulse noise at the same time (α = 1.25) of mixed noise, the synchronization error rate increased evidently while using HFM, indicating that the synchronization performance significantly decreased. While using FCCA, the estimation error rate can effectively be improved, and in the case of STLFM, synchronization performance has improved significantly, as the SNR is low.

The experiment in the pool

In order to verify the performance of the underwater acoustic communication system based on FRFT-OFDM, the experiments are finished in the pool at the Key Laboratory of Underwater Acoustic Communication and Marine Information Technology Ministry of Education built in Xiamen University.

In Experiment 1, two transducers are set in water with a depth of 0.5 and 5 m away, respectively. The experiment separately uses LFM signal and STLFM signal as the synchronization code and gets three groups’ effective sample data. In Experiment 2, one transducer is set in the deep water zone and the other one is in shallow water and their interval distance is 15 m; different synchronization code experiments are adopted.

By analyzing, we get the average BER from 200 experiments. It is said that 600 samples are used to get the average values. The analyzing results are shown in Table 5. It can be seen that the BER performance of the system with STLFM signal as the synchronization code has obvious advantages, and it shows better anti-noise performance.

Table 5.

Comparison of BER.

SNR (dB)	Experiment 1		Experiment 2
SNR (dB)	LFM	STLFM	LFM	STLFM
20.9	7.58 × 10⁻⁴	1.03 × 10⁻⁴	3.07 × 10⁻⁴	1.64 × 10⁻⁴
16.1	3.62 × 10⁻³	6.86 × 10⁻⁴	2.46 × 10⁻³	8.37 × 10⁻⁴
10.9	9.62 × 10⁻³	2.06 × 10⁻³	4.06 × 10⁻³	6.35 × 10⁻³
6.5	2.98 × 10⁻²	9.74 × 10⁻³	7.89 × 10⁻²	3.66 × 10⁻²
1.3	7.48 × 10⁻¹	4.39 × 10⁻²	9.34 × 10⁻¹	8.46 × 10⁻²

SNR: signal-to-noise ratio; LFM: linear frequency modulation; STLFM: symmetrical triangular linear frequency modulation.

Conclusion

This article puts forward a joint synchronization algorithm for orthogonal multicarrier underwater acoustic communication system based on STLFM signal as synchronization code. The FRFT is adopted to estimate the parameters accurately. This algorithm gets high anti-interference and synchronization performance.

The computer simulation and the pool experimental results are as follows:

The proposed algorithm can achieve the joint estimation of time delay, Doppler frequency shift, and multipath effect and has the suppression ability to many kinds of noise such as Gauss noise, multipath propagation, and Doppler effect.

For Gaussian noise, SαS noise and more complex channel of the underwater acoustic the synchronization performance of the proposed algorithm is obviously better than other methods.

Footnotes

Academic Editor: Stephen D Prior

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by NSFC projects under Grant No. 6150031264; Fujian Provincial Natural Science Foundation under Grant Nos 2013J01203, 2015J01265, and 2014H0036; Science and Technology Plan Projects of Xiamen City under Grant No. 3502Z20130005; The Plan of College Youth Outstanding Research Talents in Fujian Province (2015); and The Plan of College New Century Excellent Talents Program in Fujian Province (2016).

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