Single Carrier with Frequency Domain Equalization for Synthetic Aperture Underwater Acoustic Communications

Phase-coherent underwater acoustic (UWA) communication systems typically employ multiple hydrophones in the receiver to achieve spatial diversity gain. However, small underwater platforms can only carry a single transducer which can not provide spatial diversity gain. In this paper, we propose single-carrier with frequency domain equalization (SC-FDE) for phase-coherent synthetic aperture acoustic communications in which a virtual array is generated by the relative motion between the transmitter and the receiver. This paper presents synthetic aperture acoustic communication results using SC-FDE through data collected during a lake experiment in January 2016. The performance of two receiver algorithms is analyzed and compared, including the frequency domain equalizer (FDE) and the hybrid time frequency domain equalizer (HTFDE). The distances between the transmitter and the receiver in the experiment were about 5 km. The bit error rate (BER) and output signal-to-noise ratio (SNR) performances with different receiver elements and transmission numbers were presented. After combining multiple transmissions, error-free reception using a convolution code with a data rate of 8 kbps was demonstrated.

SC-FDE is a promising low-complexity approach for mitigating intersymbol interference (ISI). The main advantage of SC-FDE is its ability to cope with long delay spread channels with single-tap equalizers. In addition, the estimated channels can be updated easily via pilot signals on a block-by-blocks. Compared to time-domain equalization, SC-FDE has lower computational complexity [11]. Compared to OFDM, SC-FDE exhibits similar or better performance with similar computational complexity while avoiding OFDM's high peak-to-average power ratio (PAPR) and sensitivity to carrier frequency offset (CFO) [11]. Therefore, SC-FDE has been proposed for IEEE 802.16e [11][12][13] and high-rate underwater acoustic communications [5][6][7][8]. In our previous work, a hybrid time frequency domain equalizer (HT-FDE) is proposed to combat the residual ISI in single carrier UWA communication systems [7]. . Each block is extended with a known PN sequence of length P to obtain blocks of size N, which is expressed as s = [x 0 , x 1 , . . . , x N−P−1 , c 0 , c 1 , . . . , c P−1 ] T , (1) where {c n } P−1 n=0 is the PN sequence of length P and (·) T represents matrix transpose. The same known PN sequence is inserted at the beginning of the transmission frame.
The known PN sequence in the previous block can be viewed as the cyclic prefix (CP) of the current transmission block. Assuming that the maximum multipath delay spread is shorter than the PN symbol's duration, the received signal r = [r 0 , r 1 , ...r N−1 ] T is treated as the circular convolution between the transmission data block and the channel impulse response (CIR). Consequently, this transmission format implements a single carrier communication that allows us to exploit the benefits of the frequency domain equalization at the receiver without having the high PAPR of OFDM. A block SC-FDE diagram for the transmitter is shown in Figure 1a. First, an information bit sequence is encoded by a 1/2 rate convolution code (the industry standard convolution code (171, 133)). We then mapped the sequence into symbols using quadrature phase shift keying (QPSK). The SC-FDE system specifications are listed in Table 1

Channel
Assume that the channel is invariant over each block duration. The CIR at block k between the transmitter and the receiver transducer in the discrete time domain is modeled as where k denotes the k-th block and L is the channel order. On the receiver side, the Doppler is usually compensated by resampling of the incoming signal [22,23]. Let r k (n) be the received baseband signal at the k-th hydrophone after resampling processing as follows: where T s is the data symbol duration; f k is the Doppler shift in k-th block, caused by relative motion between transceivers, A/D, D/A sampling, variant water current, etc.; φ k,0 is phase rotation caused by the symbol timing offset in the k-th block; and w(k, n) is additional white Gaussian noise with power σ 2 . The k-th received block is expressed as For simplicity and to avoid a loss of generality, we eliminated the block index k. The received data block is where w = [w(0), w(1), . . . , w(N − 1)] T is the noise vector, and D is a diagonal rotated-phase matrix caused by the Doppler shift, which is expressed as where is an N × N circulant matrix.

Receiver
A block diagram for the receiver is shown in Figure 1b. The SC-FDE demodulator in Figure 2 consists of a down-converter, PN remover, FFT, channel estimator, frequency domain equalizer, and inverse Fast Fourier Transform (IFFT). In the receiver, the data is converted to the frequency-domain from time-domain via FFT operation, and then frequency-domain equalization is conducted. The equalized signal is converted back to a time-domain signal. The frequency domain expression of Equation (5) is given as In summary, the FDE part consists of a rotated-phase compensation, channel estimation, and frequency domain equalization.
Channel estimation plays a critical role in underwater acoustic communications [24][25][26]. We carried out channel estimation using the known PN symbols block-by-block. The underwater channel often exhibits sparse characteristics, which cause the channel to have a large delay spread but many of the channel taps to have negligible values. We used the improved proportionate normalized least mean squares (IPNLMS) method in the receiver because it has been shown to yield a robust estimation for sparse channels [27,28]. We took this sparseness into account to apply the IPNLMS algorithm to the UWA channel estimation. The IPNLMS algorithm showed not only robust performance in non-sparse channels but also better performance than LMS in sparse channels. For the i-th block, the estimated channel impulse response vector is denoted byĥ i . In practice, we accounted for the time-varying property of the underwater acoustic channel by averaging the estimated channels from the (i − 1)-th and i-th blocks: We achieved diversity in the combination of different transmissions by maximal ratio combining (MRC) [19,27]. The basic FDE assumes the channel is invariant in one block. For channel variations in one block or channel estimation errors, there is residual ISI in the MRC output of multichannel frequency domain equalizers. We then used a single-channel adaptive time domain DFE to combat the residual ISI, as shown in Figure 3. We use the recursive least square (RLS) algorithm to update the tap coefficients of DFE.

DJK16
We start by describing the 2016 Danjiangkou (DJK16) lake experiment, in which we transmitted QPSK signals from one source to eight receivers. We conducted the DJK16 experiment in Danjiangkou Lake, Henan Province, China, in January 2016. The water depth was 45-50 m. The source was deployed at 20 m below the surface. The receiver array had eight elements that were uniformly spaced at 0.25 m apart and was deployed at depths between 18-20 m. The data were transmitted from the source using QPSK modulation with a carrier frequency of 6 kHz and a bandwidth of 4 kHz. The distances between transmitter and receivers ranged from 4.98 to 5.20 km. The longest distance between different transmitter locations was about 720 m, while the shortest was about 20 m.

Packet Design
Each packet consisted of a 200 ms-long linear frequency modulated (LFM) waveform, 100 ms zero padding period, followed by 32 SC-FDE blocks, each of which was 128 ms long. Each packet lasted 4.428 s and included 16,510 symbols. The total transmitted signal included five different packets, and its duration was about 22.14 s. We transmitted identical signals at each location. The LFM waveform at the beginning of the packet was used for synchronization and Doppler shift estimation. The coded data rate is about 2.79 kbps if we consider the synchronization LFM waveform, the additional zero-padding and the extra PN length before the first block. The structured signal was transmitted as shown in Figure 4. The detailed parameters of the communication system used in the lake experiment are shown in Table 1. The order of feedforward and feedback filters in HTFDE were both 32, respectively. We used the fractionally spaced equalizer (two samples per symbol) for the feedforward filter. The RLS forgetting factor λ in the DFE was 0.999. We corrected the residual phase offset using a second-order digital phase locked loop (DPLL) embedded in the DFE. Both K 1 and K 2 in DPLL were 0.005. The first PN sequence is used as training sequence.

Channels
Examples of channel impulse responses (CIRs) are shown as a function of geotime in the left panel of Figure 5 for the transmissions from the source to Rx# 1 during the five transmissions. The CIRs are unique because of the signals' different times and locations. A snapshot of the CIRs for each transmission at the five locations is shown in the right panel of Figure 5. The approximately 20 ms of delay spread (less than 31.5 ms) in the channels corresponds to intersymbol interference of about 80 symbols at the symbol rate of R = 1/T = 4000 symbols/s. The DJK16 channels varied slowly with time over 2.5 s and exhibits sparse characteristics. We used an IPNLMS algorithm in the analysis because it can yield robust estimates for sparse UWA channels.  The received SNRs of the five transmissions ranged from 4 dB to 11 dB as shown in Figure 6a. The SNR of the fifth transmission was the lowest, about 4 dB. The estimated SNRs of the bottom receiver element (Rx #8) were slightly higher than the SNRs for the top element (Rx #1). The estimated transmitter drifting speed (Doppler estimation) for the five transmissions is shown in Figure 6b. The drifting speed ranged from −0.32 knots to 0.05 knots, reflecting the Doppler shift from −0.64 Hz to 0.01 Hz at the center frequency of 6000 Hz.

Results
We used the uncoded bit error rate (BER) and the output SNR as the performance metrics. We investigated the impact of distributed diversity combining on the performances of BERs of Rx #1, as shown in Figure 7a    The BERs without combining were quite high (i.e., 13-22%). From one to three transmissions, the BER performance improved significantly with increasing diversity. After three transmissions (e.g., four and five transmissions), the performance improvement for the uncoded system was slight. For the MRC and FDE method, after five transmission combinations, the uncoded BER of Rx #1 was 2.36 × 10 −2 , whereas the coded BER of Rx #1 with the convolution code was 1.9 × 10 −3 . For the MRC and HTFDE method, the uncoded BER of Rx #1 was 6.6 × 10 −3 , whereas the coded BER with convolution code achieved error-free reception by five transmission combinations. Once the BER of the MRC-FDE was less than a certain level (10%), significant performance enhancement using HTFDE is achieved. To check the consistency of the results, we also plotted the performances of Rx #8 in Figure 7b. Other receivers at different depths achieved similar results.
For comparison, we also plotted the BER results of conventional multichannel equalization with 1-8 channels for the first and fifth location in Figure 8. With an 8-channel receiver in one location, the uncoded BER was about 7 × 10 −2 using MRC and FDE and the coded BER was 2 × 10 −2 . The uncoded BER using MRC and HTFDE is 3.4 × 10 −2 and the coded BER is 5.3 × 10 −3 . As shown in Figure 8, the BER improvement using the five receiver elements at one location was far smaller than the improvement using the data of Rx #1 from five transmissions. This occurred because the channel correlation between the different elements was higher than the correlation between different transmissions. A linear array consisting of eight elements was used in the lake experiment. Figure 9 shows the BER performance using up to three contiguous elements with five transmissions. When we used elements from 1 to 3, the uncoded BER of MRC-FDE decreased from 1.5 × 10 −2 to 3 × 10 −3 , whereas the uncoded BER of MRC-HTFDE decreased from 6.6 × 10 −3 to 6.1 × 10 −4 . We demonstrated that error-free using three elements combined three transmissions, especially using MRC-HTFDE. Figure 10 shows the output SNRs using different elements and transmission numbers using MRC-FDE and MRC-HTFDE methods. With the same transmission number, the output SNR improvement reduced as we used more receiver elements. The output SNRs of the three transmissions with three receiver elements were 8 dB for MRC-FDE and 10.9 dB for MRC-HTFDE. With three contiguous elements, the 0.5-m-long linear array with three transmissions achieved error-free reception. Such a three-element array is low cost and acceptable for small underwater platforms, like UUVs. Through the use of multiple elements, the BER can be reduced significantly, thus reducing the required number of transmissions, shown in Figure 10.   To illustrate the improvements using different receiver elements, we present scatterplots of equalized QPSK signals of the first packet using the IPNLMS channel estimation algorithm in Figure 11, which includes the following: (a) MRC-FDE using the 1st receiver element; (b) MRC-FDE using receiver elements 1 and 2; (c) MRC-FDE using receiver elements 1 to 3; (d) MRC-HTFDE using the 1st receiver element; (e) MRC-HTFDE using receiver elements 1 and 2; and (f) MRC-HTFDE using receiver elements 1 to 3. The top panel displays the results of using MCR-FDE; the bottom panel displays the results of using MRC-HTFDE. The output SNRs for MRC-FDE were 7.0 dB, 8.1 dB, and 8.6 dB, whereas the output SNRs for MRC-HTFDE were 8.5 dB, 11 dB, and 12.3 dB, respectively. The output SNR improvements were 1.5 dB, 3.1 dB, and 3.7 dB.  Figure 11. Scatterplot of the equalized first packet signal using the five transmissions with conventional FDE and HTFDE: (a) using the 1st element with FDE; (b) using elements 1 and 2 with FDE; (c) using elements 1 to 3 with FDE; (d) using the 1st element with HTFDE; (e) using elements 1 and 2 with HTFDE; (f) using elements 1 to 3 with HTFDE.

Conclusions
In this paper, we propose an SC-FDE scheme for synthetic aperture communications. The transmitter sent a particular set of data at different locations. Spatial and temporal diversities are obtained using multiple transmission combinations. We exploited two receiver algorithms, MRC-FDE and MRC-HTFDE, and we demonstrated error-free receptions after five transmission combinations using convolution decoding. The addition of a few receiver elements increased the output SNR and decreased the BER significantly and caused a slight increase in the cost of the system.