Blind Decision-Feedback Equalization of Shallow Water Acoustic Channels R. Weber, F. Schulz, A. Waldhorst, and J. F. B¨ohme Signal Theory Group, Ruhr-Universit a¨ t Bochum, 44780 Bochum, Germany, e-mail:
[email protected]
Summary In this paper, blind receivers for signals transmitted through shallow water are proposed. After having introduced the used modulation scheme and its favorable properties, three different blind receiver structures are studied. The first processing step in all cases is explicit non-data-aided timing recovery. Then, blind adaptive-decision feedback equalization and phase recovery is performed. The adaptive setting of the nonlinear equalizer is attained by using the constant modulus algorithm for the feed-forward section and various strategies for the feedback section. All structures have been successfully tested with measured shallow water data gathered during the ROBLINKS 1999 sea trial. The obtained results suggest that blind decision-feedback equalization is possible and may speed up convergence compared to linear CMA.
1. Introduction For the past decade, there has been a tremendous increase in research and development of underwater acoustic (UWA) communication systems. This growing interest was the response to the increasing demand for wireless underwater communications, which was initiated by a shift in applications from almost exclusively military to commercial ones. Examples for such commercial applications of UWA communications are remote control in off-shore oil industry, coastal-zone monitoring, and communication between submersibles. This development has been accompanied by an ever growing need for higher data rates to cope with the huge amount of data to be transmitted over large distances. Typical data rates range from a few kilobits per second for simple command and control tasks up to hundreds of kilobits per second for video image transmission. These desired high data rates are in contrast to the transmission conditions induced by the underwater acoustic telemetry channel, which is severely bandlimited and reverberant, thus posing many obstacles to reliable high-speed digital communications. Therefore, combating the time-varying multipath induced by the transmission channel - especially the
horizontal shallow water acoustic channel - is considered the most challenging task [10]. For a long time, incoherent modulations schemes like frequency-shift keying (FSK) [8] were the method of choice as they were seen intrinsically robust to the time and frequency spreading of the channel. Although reliable, the inefficient use of bandwidth together with the limited availability of bandwidth underwater makes incoherent systems ill-suited for high data rate applications. To increase efficiency, phase-coherent systems should be employed, which was thought to be impossible due to the time variability and dispersive multipath of the medium. It was only in the early 1990’s that Stojanovic, Catipovic, and Proakis successfully demonstrated that long range, high-rate coherent data transmission is possible by jointly optimizing an adaptive multichannel equalizer together with a phase-locked loop (PLL) [11]. However, in a time-varying environment the training of the equalizer implies the recurrent transmission of known data sequences, which especially in adverse environments reduces the effective data throughput considerably. Therefore, blind equalization methods suitable for UWA communication purposes represent an important alternative and are the topic of current research [3, 4, 9, 14].
In contrast to its linear counterpart, decision-feedback equalization (DFE) has the potential for completely opening the channel eye. One of the challenges of incorporating such DFEs into practical receivers, however, is the development of robust blind adaptive algorithms. Because the DFE uses past symbol estimates to generate new decisions, it may suffer from error propagation, and finding the desired equalizer parametrization from a blind start-up with no reliable information on the transmitted symbols being available seems a difficult task. Possibly the most successful linear blind equalization method is the constant modulus algorithm (CMA) [2]. Common problems of the CMA, however, are in the symbol rate case zeros of the channel transfer function on the unit circle and in the fractionally-spaced case common zeros of the transfer functions of the virtual channels. It is possible to overcome these problems by incorporating a feedback section. In the following, we examine three different blind DFE structures based on the CMA and study their performance by analyzing experimental data from underwater acoustic communication links. The proposed blind receiver structures are fully digital, and special emphasize is put on blind digital timing recovery to enhance the performance of the receiver. Timing recovery is necessary because like the carrier phase, symbol timing is subject to Doppler effects. The compensation could be left to the equalizer, which is able to perform this task as long as the corresponding time span of the equalizer is longer than the cumulated time offset during transmission. As we are interested in demodulation of rather long data sequences (60-400 seconds), however, and to restrict the tasks of the equalizer to inter-symbol interference (ISI) reduction, time synchronization should be better performed by a separate unit. This has to be achieved blindly because no reference signals that support timing were transmitted.
2. Signalling Scheme Minimum shift keying (MSK) [8] offers attractive properties for data transmission, making its derivative Gaussian minimum shift keying the modulation scheme of choice in the European mobile phone standard GSM. Among the prominent characteristics are that 99.5% of the signal energy are contained within a bandwidth of 1.5 times the symbol rate, and a constant envelope of the transmitted signal. At the re-
ceiver it is especially the last property, which can be exploited successfully for blind data recovery [15]. Let dn ∈ {−1, +1}, n ∈ 0 , be an independent and identically distributed sequence of binary symbols to be modulated and consider the time interval of length T associated with each symbol: t ∈ [nT, (n + 1)T ]. The continuous-phase baseband MSK communication signal in this interval is then given by k (t) = e j (φ(kT )+ 2 dk ( sMSK π
t−kT T
)) ,
(1)
where φ(kT ) is the phase at the beginning of the current symbol interval. An input symbol dk = +1 results in a phase change of +π2 during T , while an input symbol dk = −1 leads to a phase change of − π2 . It is apparent from Eq. (1) that there is an inherent phase memory making MSK a special form of full-response binary continuous-phase modulation (CPM). The signal (1) can also be re-written as an offset quadriphase-shift keying (OQPSK) signal sMSK (t) =
∞
[ak pMSK (t − 2kT ) +
k=0
+ j bk pMSK (t − (2k + 1)T )] , (2) with pulse form pMSK (t) =
cos
πt 2T
, −T ≤ t < T , otherwise
0
,
(3)
and ak
= −ak−1 d2k−2 d2k−1
bk
= −bk−1 d2k−1 d2k ,
(4)
for k ≥ 1, initialized by a0 = 1 and b0 = d0 . Equations (4) can be interpreted as a differential mapping and precoding step. Differential mapping avoids the necessity to reconstruct the absolute signal phase at the receiver, which would be impossible in a blind start-up, while precoding introduces redundancy that can be exploited. It is apparent from (2) and (4) that at even multiples of T only symbols {+1, −1} out of the finite QPSK alphabet = {+1, + j, −1, − j } are possible, while at odd multiples of T only symbols {+ j, − j } occur.
3. Receiver Architecture An overview of the proposed blind space-time processing receiver architecture is given in Fig. 1. The
x(i) MCM Alg.
+
d^n
y(n)
z(n)
ε^
−jωc kTs
e r (kTs)
Timing Synchronizer
f (n)
h RF(kTs)
Equalizer and Phase−Offset Compensator
AGC
e
^ −jΘ (n)
DPPL
e
^ j Θ(n)
−
Figure 1. General Receiver Architecture
~ y(n) b(n)
signal received at the hydrophone is sampled at a fixed rate 1/Ts and subsequently shifted to baseband using the nominal carrier frequency ωc . This leads to an error as the actual carrier frequency deviates from the nominal one due to Doppler, which has to be compensated by further processing steps. To reject additive noise outside the frequency band occupied by the transmitted signal, the baseband signal is low-pass filtered utilizing hRF (k). Subsequently, the mean signal power is normalized to one using an adaptive gain control (AGC). Then, non-data-aided timing recovery is performed, yielding an estimate ˆ of the residual timing error , which represents the misalignment between the transmitter and receiver time grids, where −0.5 ≤ ≤ 0.5 is assumed. This estimate is used to dynamically resample the signal. The input to the equalizer and phase-offset compensator is thus a signal sampled at a fraction p of the symbol interval T . The equalizer adaptively reduces ISI induced by the transmission channel. The residual time-varying carrier-phase offset is compensated using a first order digital phase-locked loop (PLL). 3.1. Timing Recovery Let the nominal oversampling factor with respect to the symbol duration be M = TTs . The received, fixedrate sampled and filtered baseband communication signal is then given by (5) r(kTs ) = ξ e (t) · e[ φ(t−T )+ωt+θ] t=kTs . The amplitude distortion ξ and the phase disturbance (t) reflect the influence of the channel, is the residual timing misalignment, ω represents a frequency offset due to Doppler effects and θ is a constant phase shift. The aim of timing recovery is to (i) estimate the timing misalignment ˆ and then (ii) r(kTs)
{r(kTs)r*[(k−M)Ts]}
2
xk
M−1 − j 2π Ml |xl+n M | e l=0
c^1,n
h pp(n)
c~1,n
−
arg(·) 2π
Figure 2. Feed-forward Timing Estimator
+
ε^ n
T
Figure 3. Equalizer Structure and Phase Recovery adjust timing. The applied feed-forward non-dataaided timing estimator is based on a modified version of the algorithm proposed in [5]. It is hang-up free and offers a constant acquisition time independent of an initial timing error. The signal [r(t)r ∗ (t − T )]2 contains a spectral component at frequency 1/T from which an estimate of may be obtained. For Ts < T2 , a completely digital implementation of the timing estimation algorithm is shown in Fig. 2. It first computes every T seconds an instantaneous estimate of the corresponding Fourier coefficient. Using planar filtering [6], this estimate is smoothed in a second processing step to reduce its variance. This estimated timing error is now used to control a post-processing unit that performs timing adjustment. In order to obtain from r(kTs ) the desired fractionally-spaced samples x(i) = r(i Tp +ˆ T ) of the input to the equalizer, where p represents the factor of oversampling with respect to T , timing adjustment is performed by digital interpolation and decimator control [1, 6]. 3.2. Blind Decision-Feedback Equalization The basic structure of the equalizer and the phase recovery unit is shown in Fig. 3. Consider the (T / p)spaced signal x(i). This signal enters an adaptive FIR filter f (n) of length L f , where the argument n highlights the fact that the filter is updated every T seconds. To obtain a compact expression for the output of the fractionally-spaced feed-forward section, consider the L f -variate vector x(n) = [x(np), . . . , x(np − L f + 1)]T ,
(6)
which consists of the sample present at time instance nT and the previous L f − 1 fractionally-spaced sam-
ples, and where (·)T denotes transposition. The symbol-spaced output of the feed-forward equalizer can then be compactly written as f (n)H x(n), where (·) H denotes Hermitian transposition. The feedback section consists of a symbol-spaced tapped-delay line b(n) of length L b , whose input are the past values
y˜ (n) = y˜ (n − 1), . . . , y˜ (n − L b )
T
.
(8)
As the optimization criterion for finding the optimal filter coefficients shall be based on the constant modulus algorithm [2], which converges independently of carrier recovery, the resulting output constellation has a phase error due to the unavoidable misadjustment between the oscillators of transmitter and receiver. Especially in underwater communications, however, severe time-varying jitter and Doppler result in a rotating signal constellation. This phase offˆ set has to be compensated using an estimate θ(n) to ˆ − j θ(n) z(n) to the create the de-rotated input y(n) = e decision device. The decision device is a nonlinear, possibly time-varying function, which generates an estimate dˆn of the transmitted symbols at its output. ˆ An estimate θ(n) of the phase error can thus be obtained by a PLL, measuring the phase difference between y(n) and dˆn . As the detected symbols are used in the feedback section, they should be related to the ˆ received signal by forward rotation ej θ(n) dˆn . 3.2.1. CMA-DFE Setting y˜ (n) = e j θ(n) dˆn , we can rewrite the output of the equalizer (8) in the following form:
x(n) H H . (9) z(n) = f (n) b (n)
y˜ (n) ˆ
:=w H (n)
A possible way to obtain decoupled update equations for the feed-forward and the feedback section is to write the CMA cost function in the overall filter vector w, 2 1 , JCMA (w) = E |z(n)|2 − γ 4
f (n + 1) = f (n)
−µ f x(n)z ∗ (n) |z(n)|2 − γ (11)
b(n + 1) = b(n)
−µb y˜ (n)z ∗ (n) |z(n)|2 − γ , (12)
(7)
The combined output of both the feed-forward and the feedback section is then given by z(n) = f (n) H x(n) + b H (n) y˜ (n) .
gradient approach, this leads to the update equations
(10)
with γ = 1 in our special case, and to calculate the gradient vector with respect to w. Using a stochastic
where µ f and µb are small positive step-size parameters. It has been shown in [7], that under certain conditions such an approach will lead to a perfect zero-forcing equalization. Unfortunately, classes of solutions exists that correspond to an all-zero setting for the feed-forward section and a single non-zero tap in the feedback section. These solutions are clearly not desired as no information about the transmitted symbols is used. Although this problem may be circumvented by imposing constraints on the forward filter [7], it is not obvious that such DFE types will perform better than their purely linear counterparts.
3.2.2. PCM-DFE Assuming for the moment that the correct symbol dn would be available at the output of the decision device and no phase offset to be present. Then, the forward error signal e f (n) = dn − f H (n)x(n) is the sum of colored noise and an interference term due to imperfect ISI removal by the forward filter. Using the feedback filter as a linear predictor by setting y˜ (n) = e f (n) in Eq. (7) results in an estimate of the forward error, eˆ f (n) = b H (n) y˜ (n), obtained from the Nb previous error signals. The output of the equalizer is then given as the desired symbol corrupted by the prediction error, z(n) = dn − (e f (n) − eˆ f (n)) ,
(13)
which leads to a reduced variance of the additive interference and thus to an improved performance compared to a purely linear equalizer. Using the constant modulus criterion to optimize both the feedforward and the feedback section, we formally obtain the same update equations (11) and (12), but with y˜ (n) = [e f (n − 1), . . . , e f (n − L b )]. It has been shown in [12] that if the forward filter provides a reasonable ISI compensation, the convergence of the feedback filter is global and the cost function has almost the same minimum as the error surface in the case of a training sequence being used.
3.2.3. P-DFE
1 0.9
It has been pointed out in [3] that the constant modulus criterion does not offer significant advantages over a quadratic cost function. Therefore, leaving the forward section unchanged, the feedback filter can be found by minimizing the exponentially weighted sum t 2 λt−n e f (n) − b H (t) y˜ (n) , (14) J (t) = n=1
with the forgetting factor λ and the feedback signal y˜ (n) = [e f (n − 1), . . . , e f (n − L b )]. The optimization is then efficiently obtained using a stabilized RLS algorithm. 4. Sea Trial Within the EC-funded project ROBLINKS, in 1999, a sea trial for underwater communication was conducted in the North Sea about 10 km off the Dutch coast. At the transmitting side, an almost omnidirectional acoustic source was deployed from the stern of a support ship and lowered to a depth of approximately 9 m. The total water depth in this shallow water area is about 20 m. The signals were received by either of two vertical arrays, consisting of 20 and 6 hydrophones, respectively. The arrays were fixed to an oceanographic platform, which also hosted all recording facilities. The 20-element array sampled the sound field from a depth of 4.4 m to 15.8 m with equally spaced sensors. The smaller array consisted of three pairs of hydrophones, separated horizontally by 15 cm at depths of approximately 7, 11, and 15 m below the sea surface. The support ship anchored at various positions with distances of 1, 2, 5, and 10 km from the platform. In addition, the ship was sailing at moderate speed the last two days to obtain measurements with a moving source. More details on the ROBLINKS experiments can be found in [13]. 5. Experimental Results To examine the performance of the proposed receiver structures we analyzed data at rate 514.32 bit/s that had been recorded while the ship moored at a distance of 2 km from the platform. The carrier frequency was 3079 Hz and approximately 100000 data bits were continuously transmitted. The signal received at hydrophone 4 - corresponding to depths
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −2
0
2
4
6 8 Delay [ms]
10
12
14
16
Figure 4. Measured Channel Impulse Response 11 m - of the planar array was processed. Prior to the information-bearing signal, a linear FM sweep of 200 ms duration spanning the frequency range from 1 to 5 kHz was transmitted to obtain initial information about the channel responses. Fig. 4 shows the result of matched-filter analysis of the received FM sweep. It is apparent, that the delay spread of the channel is in the range of 5-10 ms, giving some guidelines for choosing a proper length of the equalizer filters [8]. During time synchronization, an oversampling factor p = 2 was used and the post-processing constant α pp of the planar filtering was set to 10−3 . The measured input signal-to-noise ratio (SNR) at the hydrophone was 34.38 dB. The parameters for the different algorithms are summarized in Table 1. The results obtained with all equalizer structures are shown in Fig. 5. It is apparent that adding a feedback section leads to a faster convergence and thus to fewer bit errors. This fact is also reflected in an improved output SNR, see Table 1, which is defined as [11] 1
SNRout = 10 · log10 1 Ns
Ns
|dˆn − y(n)|2
,
(15)
n=1
with Ns , the number of received symbols. The best result is obtained using the CMA for adapting both filter sections, while there is only little improvement with the other two methods. The predictive structure (P-DFE) might perform better with the recursive Lf
Lb
µf
µb or λ
αPLL
SNRout
CMA
21
-
0.002
-
0.1
10.28
CMA-DFE
21
8
0.002
0.004
0.1
10.44
PCM-DFE
21
8
0.002
0.0009
0.1
10.35
P-DFE
21
8
0.002
0.9999
0.1
10.35
Table 1. Parameter Settings
7000
ing period? IEEE Transactions on Communications, 46(7):921–930, July 1998.
Number of Bit Errors
6000
[5] R. Mehlan, Y.-E. Chen, and H. Meyr. A fully digital feedforward MSK demodulator with joint frequency offset and symbol timing estimation for burst mode mobile radio. IEEE Transactions on Vehicular Technology, 42(4):434–443, Nov. 1993.
5000 4000 3000
1000 0 0
[6] H. Meyr, M. Moeneclaey, and S. Fechtel. Digital Communication Receivers. John Wiley & Sons, Inc., New York, 1998.
CMA CMA−DFE PCM−DFE P−DFE
2000
20
40
60
80
100
120
140
160
180
Time [s]
Figure 5. Number of Bit Errors over Time least-squares based super-exponential algorithm being used to adapt the forward section, as in this case both adaptive filters are almost decoupled [3]. 6. Conclusions Blind decision-feedback equalization based on the constant modulus criterion has been presented. Although less efficient than a jointly optimized DFE, the three proposed algorithms with independent design of the forward and feedback filters allow for a straightforward integration of blind adaptive algorithms, which are usually limited to linear filter structures, into more general nonlinear configurations. All proposed DFEs have been incorporated into a fully digital receiver. The results obtained with experimental data show that a blind feedback section leads to a reduced number of bit errors compared to a linear equalizer. References [1] F. Gardner. Interpolation in digital modems - part I: Fundamentals. IEEE Transactions on Communications, 41(3):501–507, Mar. 1993. [2] D. Godard. Self recovering equalization and carrier tracking in two-dimensional data communication systems. IEEE Transactions on Communications, 28(11):1867–1875, Nov. 1980. [3] J. Gomes and V. Barroso. Blind decision-feedback equalization of underwater acoustic channels. In Proc. Oceans 1998 MTS/IEEE Conference, pages 810–814, Nice, France, Sept. 1998. [4] J. Labat, O. Macchi, and C. Laot. Adaptive decision feedback equalization: Can you skip the train-
[7] C. B. Papadias and A. Paulraj. Decision-feedback equalization and identification of linear channels using blind algorithms of the bussgang type. In Proc. 29th Asilomar Conference on Signals, Systems and Computers, volume II, pages 335–340, Pacific Grove, CA, Oct. 30 - Nov. 1 1995. [8] J. Proakis. Digital Comunications. McGraw-Hill, Inc., Boston, third edition, 1995. [9] B. Sharif, J. Neasham, D. Thompson, O. Hinton, and A. Adams. A blind multichannel combiner for long range underwater communications. In Proc. IEEE International Conference on Acoustic, Speech and Signal Processing, ICASSP’97, pages 579–582, Munich, Germany, Apr. 1997. [10] M. Stojanovic. Recent advances in high-speed underwater acoustic communications. IEEE Journal of Oceanic Engineering, 21(2):125–136, Apr. 1996. [11] M. Stojanovic, J. Catipovic, and J. Proakis. Adaptive multichannel combining and equalization for underwater acoustic communications. Journal of the Acoustical Society of America, 94(3):1621–1631, Sept. 1993. [12] L. Tong and D. Liu. Blind predictive decisionfeedback equalization via the constant modulus algorithm. In Proc. IEEE International Conference on Acoustic, Speech and Signal Processing, ICASSP’97, pages 3901–3904, Munich, Germany, Apr. 1997. [13] M. van Gijzen, P. van Walree, D. Cano, J.-M. Passerieux, A. Waldhorst, R. Weber, and C. Maillard. The roblinks underwater acoustic communication experiments. In Proc. of the 5 th European Conference on Underwater Acoustics, ECUA 2000, pages 555–560, Lyon, France, July 2000. [14] A. Waldhorst, R. Weber, and J. F. B¨ohme. A blind multichannel DFE receiver for precoded OQPSK signal transmition in shallow water. In Proc. Oceans 2001 MTS/IEEE Conference, pages 2209–2215, Honolulu, HI, Nov. 2001. [15] R. Weber, A. Waldhorst, F. Schulz, and J. F. B¨ohme. Blind receivers for MSK signals transmitted through shallow water. In Proc. Oceans 2001 MTS/IEEE Conference, pages 2183–2190, Honolulu, HI, Nov. 2001.
