Multiple-Symbol combined differential detection for satellite-based AIS Signals
- Jingsong Hao^{1}Email author,
- Shexiang Ma^{1},
- Junfeng Wang^{1} and
- Xin Meng^{1}
https://doi.org/10.1186/s13634-015-0248-4
© Hao et al. 2015
Received: 19 January 2015
Accepted: 13 July 2015
Published: 30 July 2015
Abstract
In this paper, a multiple-symbol combined differential Viterbi decoding algorithm which is insensitive to frequency offset is proposed. According to the theories of multiple-symbol differential detection and maximum-likelihood detection, we combine the multiple-order differential information with the Viterbi algorithm. The phase shift caused by the frequency offset is estimated and compensated from the above information in the process of decoding. The simulation results show that the bit error rate (BER) of 2 bits combined differential Viterbi algorithm is below 10^{−3} when the normalized signal-to-noise ratio (NSNR) is 11 dB, and the decoding performances approach those of the coherent detection as the length of the combined differential symbols increases. The proposed method is simple and its performance remains stable under different frequency offsets.
Keywords
1 Introduction
Automatic Identification System (AIS) [1], as a new type of navigation and security ensuring system on the sea, can realize ship-to-ship and ship-to-shore station communications well. Each ship equipped with AIS transmitter periodically sends status messages in the maritime very high frequency (VHF). Each AIS receiver nearby can receive these messages and provides a map of the local maritime traffic, thus, avoiding collisions on the sea. However, the AIS system was initially developed to realize horizon communication, so it has a limited coverage range [2]. The satellite-based AIS system receives messages from a constellation of low earth orbit satellites, which extends the range of coverage and attracts attention from more and more countries [3].
The satellite-based AIS system has large frequency offset so it is hard to recover the local carrier accurately [4]. Noncoherent sequence detection, which has simple structure and good performance without accurate local carrier, is optimal for receivers. A Viterbi decoding algorithm based on Laurent decomposition, which has a performance approaches that of coherent detection at the cost of high complexity, is proposed in [5]. A sequence estimation algorithm for the differential detection of the continuous phase modulation signals, which has significant gains in bit error rate (BER) performance and with considerable resistivity to fading, is introduced in [6]. A multiple differential detection (MDD) sequence estimator, which uses a decision feedback for the demodulation of Gaussian minimum shift keying (GMSK) signals, is described in [7]. A noncoherent GMSK detector using differential phase detection combined with the soft-output Viterbi algorithm (SOVA), which overcomes the severe intersymbol interference (ISI) of GMSK signals with low B_{t}T, is presented in [8]. The schemes mentioned before can achieve good performances, but their performances remain stable only within a certain range of frequency offset. When the frequency offset exceeds the range, it needs to be estimated and compensated. An innovative receiver architecture for the satellite-based AIS, which adopts the Viterbi decoding algorithm based on Laurent decomposition in [5], is described in [2]. A highly efficient receiver, which modifies the synchronization and detection algorithms in [2] and achieves an impressive performance improvement, is proposed in [9]. Both receivers adopt noncoherent detection algorithms and achieve good performances, but they need frequency synchronization before detection. The multiple-symbol differential detection (MSDD) of M-ary phase shift keying (MPSK) signals in the presence of frequency offset is studied in [10–12], which introduce a double differential MPSK modulation to realize the robustness to both frequency and phase offsets. Their format of the transmitted signals is changed. But the modulation system of AIS signal has been defined in [1], so the double differential encoding can’t be applied to the detection for satellite-based AIS signals.
In this paper, we first introduce the baseband signal model and phase states of AIS signal. Then, a multiple-symbol combined differential detection algorithm base on the theories of multiple-symbol differential detection and maximum-likelihood detection is proposed. The phase shift caused by the frequency offset is estimated and compensated from the multiple-order combined differential information in the process of decoding. Finally, the decoding process is completed adopting Viterbi algorithm. The performance of the proposed algorithm over an AWGN channel is evaluated through computer simulation. The results show that this algorithm has good performance over AWGN channel and is insensitive to frequency offset and constant phase shift.
2 AIS baseband signal model and phase states
where E is the signal energy per information symbol, T _{ b } is the symbol period, and θ(t) is the phase of the modulating signal.
where {a _{ i }} is the information sequence, g(t) is the frequency pulse, q(t) is the phase-smoothing pulse response, and B is the 3 dB bandwidth of the Gaussian filter.
Theoretically speaking, the frequency pulse g(t) is infinite. Considering its physical realization, we truncate it to L bits.
In the phase state of GMSK modulating signals, the accumulative phase has four possible values, i.e., 0, \( \frac{\pi }{2} \), π, \( \frac{3\pi }{2} \). Furthermore, the instant phase is determined by the value of L corresponding symbols. Therefore, the number of possible phase states at time t = nT _{ b } is 4 × 2^{ L }.
3 Maximum-likelihood detection
where φ(t) = 2πf _{ d } t, f _{ d } _{,} and ϕ represent the Doppler frequency offset and the phase offset, τ is the time offset, and w(t) is zero-mean complex Gaussian noise with variance \( {\sigma}_n^2={N}_0/2 \).
where I _{0}(x) is the zeroth order modified Bessel function of the first kind. Note that |u _{ n }|^{2} is constant for all phases of GMSK signals and I _{0}(x) is a monotonically increasing function on its argument, so maximizing p(r|u) given u is equivalent to maximizing \( {\left|{\displaystyle \sum_{i=0}^{N-1}{r}_{n-i}{u}_{n-i}^{*}}\right|}^2 \). At this point, let the meaning of u _{ n }, s _{ n }, θ _{ n }, and r _{ n } be unchanged and sample the continuous signals into digital signals.
where θ _{ n − i } is the phase of the GMSK modulating signal at time (n − i)T _{ b }.
Since GMSK-modulating signals have constant envelope, the first term on the right hand side of Eq. 16 has no effect on the left hand side. So the optimum reception is equivalent to choosing \( \boldsymbol{\theta} ={\left\{{\theta}_{n-i}\right\}}_{i=0}^{N-1} \) and to maximizing \( {\displaystyle \sum_{i=1}^{N-1}\mathrm{R}\mathrm{e}\left\{{\displaystyle \sum_{m=1}^i \exp \left(-jm\varphi \right)\left[{r}_{n-i+m}{r}_{n-i}^{*} \exp \left(-j\left({\theta}_{n-i+m}-{\theta}_{n-i}\right)\right)\right]}\right\}} \). It can be readily seen that in \( {\displaystyle \sum_{i=1}^{N-1}\mathrm{R}\mathrm{e}\left\{{\displaystyle \sum_{m=1}^i \exp \left(-jm\varphi \right)\left[{r}_{n-i+m}{r}_{n-i}^{*} \exp \left(-j\left({\theta}_{n-i+m}-{\theta}_{n-i}\right)\right)\right]}\right\}} \) _{,} \( {r}_{n-i+m}{r}_{n-i}^{*} \) is the m(m = 1, …, N − 1) order differential operation of the received signal and θ _{ n − i + m } − θ _{ n − i } is the increment of the phase of modulating signal. So \( {\displaystyle \sum_{i=1}^{N-1}\mathrm{R}\mathrm{e}\left\{{\displaystyle \sum_{m=1}^i \exp \left(-jm\varphi \right)\left[{r}_{n-i+m}{r}_{n-i}^{*} \exp \left(-j\left({\theta}_{n-i+m}-{\theta}_{n-i}\right)\right)\right]}\right\}} \) is the optimum judgment formula of N symbols combined differential detection over AWGN channel.
4 Multiple-symbol combined differential detection
When N ≥ 2, the cost functions have terms containing exp(−jmφ), and φ will affect the function values for different phase states at time nT _{ b }. So, in order to realize the multiple-symbol combined differential detection, exp(−jmφ) must be compensated in Eq. 17.
4.1 Phase compensation based on combined difference
As can be seen from Eqs. 23, 24, and 25, the phase shift at time nT _{ b }, which is caused by the frequency offset in the duration of mT _{ b }, is estimated from the differential terms in the cost function without introducing any other variables or algorithms.
It can be seen from Eq. 26 that the calculation amount of the MSCDD-PCCD increases multiply with the increases of the length of the combined differential symbols. But the decoding performance gets better and better at the same time, which is shown in the simulation results. Above all, we need to consider both the decoding performance and the calculation amount when we choose the length of the combined differential symbols.
4.2 Phase compensation based on multiple-order combined difference
Comparing Eq. 31 with Eq. 26, it can be seen that the estimation of the phase shift caused by the frequency offset in Eq. 31 is more accurate. Thus, MSCDD-PCMCD has better performance, but its calculation is more complex at the same time.
4.3 Multiple-symbol combined differential Viterbi decoding
According to the cost function of the multiple-symbol combined differential detection, the decoding process adopting Viterbi algorithm can be conducted by regarding the cost function of Eq. 26 or Eq. 31 as the branch metric.
All states in the decoding process
State number | All states | |||
---|---|---|---|---|
a _{ n − 1} | a _{ n } | a _{ n + 1} | Accumulative phase | |
1 | +1 | +1 | +1 | 0 |
2 | +1 | +1 | −1 | 0 |
3 | +1 | −1 | +1 | 0 |
4 | +1 | −1 | −1 | 0 |
5 | −1 | +1 | +1 | 0 |
6 | −1 | +1 | −1 | 0 |
7 | −1 | −1 | +1 | 0 |
8 | −1 | −1 | −1 | 0 |
9 | +1 | +1 | +1 | π/2 |
10 | +1 | +1 | −1 | π/2 |
… | … | … | … | … |
28 | +1 | −1 | −1 | 3π/2 |
29 | −1 | +1 | +1 | 3π/2 |
30 | −1 | +1 | −1 | 3π/2 |
31 | −1 | −1 | +1 | 3π/2 |
32 | −1 | −1 | −1 | 3π/2 |
5 Simulation results
Verification of the proposed algorithm is carried out by comparing the corresponding simulation results in this paper, sampling the received signal at a rate which is eight times to the symbol rate (R _{ b } = 9.6 kbps). In order to improve the accuracy of our simulation, we use 1000 symbols in every decoding and take the average BER after repeating 100 times for every NSNR, let BT = 0.4 and L = 3.
6 Conclusions
In this paper, a multiple-symbol combined differential Viterbi algorithm for the detection of satellite-based AIS signals is proposed. This algorithm combines the multiple-order combined differential information with the Viterbi algorithm according to the theories of multiple-symbol differential detection and maximum-likelihood detection. The phase shift caused by the frequency offset is estimated and compensated from the above information in the process of decoding. The proposed algorithm performs well under low NSNR, and the decoding performances approach those of the coherent detection as the length of the combined differential symbols increases. Most importantly, its performance remains stable under different frequency offsets, which is attractive for the detection of signals with large Doppler frequency offsets.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China (No. 61371108), Tianjin Research Program of Application Foundation and Advanced Technology (No. 15JCQNJC01800), and Tianjin City High School Science & Technology Fund Planning Project (Nos. 20140706 and 20140707).
Authors’ Affiliations
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