Multiuser underwater communication with space-time block codes and ...

Multiuser Underwater Communication with SpaceTime Block Codes and Acoustic Vector Sensors Huaihai Guo and Ali Abdi Emails: [email protected], [email protected] Center for Wireless Communication and Signal Processing Research Dept. of Electrical and Computer Engineering, New Jersey Institute of Technology, Newark, NJ 07102, USA Abstract—Acoustic vector sensors and particle velocity channels have recently been shown to provide new opportunities in underwater communication [1] [2]. In this paper, a multiuser vector sensor communication system is proposed using space-time block codes. Compared to multiuser spread spectrum techniques, the proposed system is particularly suitable for highly bandlimited underwater channels. This is because it allows multiple high data rate users to communicate over the same bandwidth, without compromising their rates via bandwidth spreading. Theoretical formulation and Monte Carlo simulations are provided to show the usefulness of the proposed system. Keywords—Underwater communication, acoustic communication, acoustic vector sensor, acoustic particle velocity, space-time block coding, multiuser communication.

I. INTRODUCTION The exiting trend in multiuser underwater communication is to use a spread spectrum technique, which allows multiple users to communicate via spreading codes and bandwidth expansion. Examples include code division multiple access (CDMA) systems. Bandwidth expansion is not a problem in radio frequency (RF) channels, due to the very large bandwidths of such channels. However, spectrum spreading in seriously bandlimited underwater channels reduces the data rate of each user. In this paper, a multiple access scheme is developed which does not rely on bandwidth expansion. Therefore, it can accommodate multiple high data rates users, without reducing their transmission rates. The key idea is to use space time block codes [3] [4], to communicate over acoustic particle velocity channels using vector sensors [1] [2]. The algebraic structure of space-time block codes allows for multiple access without bandwidth expansion, whereas vector sensor receivers serve as compact multichannel equalizers. The smaller delay spread of some particle velocity channels [2] helps to reduce the equalizer complexity as well. Reducing the size and complexity of the receiver is particularly important in systems which have serious size limitations. The rest of this paper is organized as follows. First the proposed general multi-input multi-output (MIMO) system over pressure and particle velocity channels is developed in Section II. Basic input/output (I/O) equations for the proposed multiuser communication system are derived in Section III. Multiuser/multichannel equalization and interference cancelation are formulated in Section IV. Section V includes

simulation set up and results for different scenarios and concluding remarks are provided in Section VI. II. THE PROPOSED MULTIUSER SYSTEM Here we explain the proposed system through a three user’s example. Extension to more users is straightforward. Consider the scenario shown in Fig. 1, where three users are transmitting data to one receiver. Each user has two pressure sensors, whereas the receiver is equipped with only one vector sensor. In this two-dimensional depth-range propagation scenario, the vector sensor measures the pressure channel p (straight dashed line), as well as the y and z components of the particle velocity channel (curved dashed lines). Extension to three dimension propagation is straightforward, where the vector sensor measures the x component of particle velocity channel as well. To make the figure easy to read, only the channels of the first transmitter of user 2, i.e., Tx21 are shown. Each user transmits its own data from the two pressure sensors using Alamouti code. It can be shown that a three channel receiver can separate up to three users [3]-[5]. Therefore, the proposed three channel vector sensor receiver in Fig. 1 can successfully recover the data of each user. Note that all the users are simultaneously sharing the same bandwidth, without spreading codes, and still can be separated at the

Figure 1. A three-user vector sensor communication system, with two pressure sensor transmitters (black dots) per user and a single vector sensor receiver (black square).

receiver. Also note that the proposed vector sensor receiver takes advantage of the vector components of acoustic field, in addition to the scalar (pressure) component. This allows the vector sensor to function as a compact multichannel receiver, as it measures the scalar and vector components, all at a single point in space [1] [2].

point orthonormal discrete Fourier transform (DFT) matrix Q 1 2π Q( p, q ) = exp(− jpq ),   where 0 ≤ p, q ≤  − 1 and  = u+v-1 . This allows us to write the I/O relations in frequency domain and in terms of frequency-transformed variables [7]

III. SYSTEM EQUATIONS

Yk ,i = Φ1k Si + Φ2k Si +1 + 0k ,i ,

For the users shown in Fig.1, Alamouti code is used [8]. Each user has two pressure transmitters. Formulas for the channel impulse response of pressure and velocity channels and vector sensor equations and communication concepts are given in [1]-[2]. In this section, we provide the system equations for single and multiple users. A. Single User System Consider a single user system, where the user is equipped with two pressure transmitter in a frequency-selective acoustic channel, and the receiver has K receive channels. The Alamouti encoder maps every pair of blocks si(u) and si+1(u) with length u into the transmission matrix X at time index i [7]-[8]:

 s (u) X=  i  si+1(u) 

−s*i +1(u)   s*i (u) 

,

(1)

where * is complex conjugation. Let yk,i and yk,i+1 denote the blocks received by the kth receive channel, which k = 1,2 …K, in tow successive slots at instants i and i+1. This gives the following I/O relationships [7]-[8]:

y k ,i = h1k ⊕ si + h2k ⊕ si +1 + nk ,i , y k ,i +1 =−h1k ⊕s*i +1 + h 2k ⊕s*i + n k ,i +1 ,

(2)

where ⊕ stands for convolution in time domain and h1k and h2k are the channel impulse responses between transmit antenna 1 and 2 to kth receive channel with the maximum channel memory v, respectively. Any channel impulse response h with length less than v will be zero-padded to length v. n is the zero mean and σ 2 variance white complex Gaussian noise vector with length u+v-1. To avoid the interblock interference and to make all the channel matrix circulant, a cyclic prefix of length v is added to each transmitted block s. Each channel impulse response can be written into a circulant toeplitz (u + v -1) × (u + v -1) matrix H, [4]

 h jk (0) 0 ... h jk (v) ⋯ h jk (1)    ⋮ ⋱ ⋱ ⋱ ⋱ ⋮    h jk (v −1) 0 h jk (0) ⋯ ⋯ h jk (v)  , H jk =  0 h jk (0) ⋯ ⋯   h jk (v) h jk (v −1)  ⋮ ⋱ ⋱ ⋱ ⋱ ⋮     h jk (v) h jk (v −1) ⋯ h jk (0)  0 0   where j = 1, 2 indicates the jth transmitter, This changes (2) to y k ,i = H1k si + H 2k si +1 + nk ,i , (3) y k ,i +1 =−H1k s*i +1 + H 2k s*i + n k ,i +1 . Now we multiply the received signal blocks y with the -

Yk ,i +1 =−Φ1k S*i +1 + Φ 2k S*i + 0 k ,i +1 .

(4)

Here Si = Qsi, 0k,i = Qnk,i, Yk,i = Qyk,i , Φjk is a diagonal matrix given by Φjk = QHjkQ†, and † denotes complex conjugate transpose. Rewriting (4) in matrix form results in:

 Yk ,i   Φ1k = Yk =  *  Y k ,i +1   Φ*2k   

Φ 2k  Si   0 k ,i   . (5)  + −Φ1*k  Si +1   0*k ,i +1 

B. Multiple User System In general, the received data at the kth receiver from mth user in an M user system, which M>1, can be written as: Yk = Λmk Sm + 0k . (6) Here m = 1, 2…M, K ≥ M and

 Φm Λmk =  m1k*  Φ 2k 

Φ m2k   −Φ1mk * 

is the Alamouti-like frequency domain channel impulse response matrix from the mth user to the kth receive channel [4]. And

 Sm   0 k ,i  Sm =  mi , 0k =  *   Si +1   0 k ,i +1     

are the transmit signal vector from the mth user and the noise vector at the kth receiver, respectively. Λmk has an Alamouti-like structure, which means that it is an orthogonal matrix. So Λ mk Λ mk † becomes a diagonal matrix. This characteristic will be used throughout the paper, to recover the signal from the noisy observations Y via a simple linear operation, as explained in Section III. Overall, the I/O equations in the multiuser system with M users and K receive channels are given by 1  Y1   Λ1    Y2   Λ12  Y= =  ⋮   ⋮    1 Y K    Λ K

Λ12 Λ 22 ⋮ Λ 2K

⋯ Λ1M  S1  ⋯ Λ 2M  S 2  ⋱ ⋮  ⋮   S M ⋯ ΛM K 

  01      +  0 2  . (7)   ⋮         0K 

For the three user system shown in Fig. 1 with the vector sensor receiver, system I/O equation can be written in frequency domain as

 Y   P1    1  Yy  =  Py  Yz   Pz1   

P2 Py2 Pz2

P3  S1   0      Py3  S 2  +  0 y  , Pz3  S3   0 z   

(8)

where the vectors Y, Yy and Yz are the received signals at the pressure, y-velocity and z-velocity channels of the vector sensor, respectively. Moreover, with m = 1, 2, 3, Pm is the

pressure channel from the mth user, Pym and Pzm are the y and z component of the velocity channel from the mth user, respectively. IV. INTERFERENCE CANCELATION AND EQUALIZATION There are two ways to recover the signal of each user: (a) Decouple the user (interference cancelation) and then apply a zero-forcing (ZF) or minimum mean square error (MMSE) equalizer to eliminate inter-symbol interference (ISI) [3]-[6]; (b) apply a joint MMSE decoupler and equalizer to overcome the noise enhancement at the decoupling stage of (a) and retrieve the signal at the same time. To illuminate the joint MMSE decoupler/equalizer, we first explain the approach where decoupling and equalization are separated. This shows how the Alamouti-like structure can be used to decouple multiple signals at the receiver. A. ZF Decoupler and MMSE Equalizer We begin with the ZF decoupler for a single user. Assume that the channel state information is completely known at the receiver. By multiplying the both sides of (5) with Λk†, because of the orthogonal structure of the Alamouti-like channel matrix Λk, one obtains 0   Si  ɶ ɶ = Λ † Y =  Ψ k (9) Y   + 0k . k k k  0 Ψ k   Si +1  Here Ψ k =| Φ1k |2 + | Φ 2 k |2 is a (u + v -1) × (u + v -1) diagonal matrix with (i, i) element equal to |Φ1k(i, i)|2+|Φ2k(i, i)|2, which is the sum of the squared ith DFT coefficients of first and ɶ is the filtered noise second channel impulse responses [7]. 0 k vector with a diagonal covariance matrix equal to diag(Ψk, Ψk). In (9), we can see how the signals Si and Si+1 of the user are decoupled from the received vector Yk using a ZF algorithm. After the ZF decoupling process, the MMSE equalizer for ɶ in (9), the mth user, by giving the decoupled signal vector Y k can be written as [3]-[6]: −1

 1  ɶ (10) Sˆ m =  Λ mk Λ mk † + I  Y γk  k  where Sˆ m includes estimations of Si and Si+1 of the mth user, γk is the signal-to-noise ratio (SNR) at the kth receiver, I is the 2(u + v -1) × 2(u + v -1) identity matrix. By multiplying Sˆ m with Q-1, the inverse DFT matrix, we obtain the original data, i.e. si and si+1 in (2), in time domain. The above ZF decouple process can be extended to multiple user system via an iterative algorithm [3]-[6]. Let us rewrite (7) as A

   1 2  ⋯ Λ Λ 1 1  Y1     Λ1 Λ 2 ⋯ Y  2 2 Y = 2 = ⋮ ⋮ ⋱  ⋮      1 Λ2 ⋯  YK   Λ K   K  C 

B   Λ1M  S1  Λ 2M  S 2  Λ3M  ⋮  M  S M Λ K   D 

  01      +  0 2  . (11)   ...         0K 

The above equation can be written in the following compact form

 Y− K   A B  S −M  =  M  YK   C D  S

  0 −K  + ,     0K 

(12)

where Y-K is the 2(K-1) size vector containing the received signal vectors, Y1,Y2,…YK-1. Similarly, S-M is the 2(M-1) signal block vector and 0-K is the 2(K-1) noise vectors. The matrices A, B, C, and D are 2(K-1)×2(M-1), 2(K-1)×2, 2×2(M1) and 2×2 channel impulse response matrices respectively. Note that to clarify the rough dimension of A, B, C, D, Y, S and 0, the length of channel output u+v-1 is omitted in above matrices size description. The decoupling matrix for the m = Mth user can be constructed as [3]-[5]

I −BD−1  (13) G M =  2( K −−1)1  .  −CA I 2   After multiplying GM with Y in (11), an equation similar to (9) can be obtained ɶ −M   R −M   (A − BD−1C)S − M   0 MY= (14) G = +  M   −1 M    ɶ M  R   (D − CA B)S   0  Note that RM, which is the received signal from the Mth user, is separated out from the other users. The matrices A-BD-1C and D-CA-1B have the Alamouti-like structure also [7]. So, by further iteration on the processed received signal vector R-M as (9)-(14), all the data vectors of other users can be recovered. According to (14), it is clear that the decoupled signal vector RM can be considered as the transmitted signal symbol ɶ M = D − CA−1B in vector SM convolved with the channel Λ M time domain. So, MMSE equalizer for R can be realized by ɶM . replacing Λ in (10) with Λ B. MMSE Joint Decoupling and Equalizer The ZF decouple process enhances the impact of noise. To avoid this, a joint MMSE solution can be developed to recover all the data symbols of all the users directly from (7) as −1 ⌢  1  S = ∆†  ∆∆† + J  Y . (15) γ   Here

 Λ11   Λ1 ∆= 2  ⋮  Λ1K 

Λ12 Λ 22 ⋮ Λ1K

⋯ Λ1M  I   ⋯ Λ 2M  I J , =  ⋮ ⋱ ⋮    ⋯ ΛM I K 

I ⋯ I  I ⋯ I , ⋮ ⋱ ⋮  I ⋯ I 

and I is the identity matrix with the size as each Λ and γ is the average SNR. Transmitted ⌢ symbols in time domain can be obtained by multiplying S in (15) with the inverse DFT matrix Q-1 with the corresponding size. The disadvantage of joint decoupling and equalization is its higher computational complexity. We have used this approach in the simulation of next section.

V. SIMULATION SET UP AND PERFORMANCE COMPARISON Monte-Carlo simulations are performed for system performance analysis. The same shallow water channel as [1] and [2] is used to simulate the proposed multiuser vector sensor system. Simulation parameters are shown in Table 1. Detailed information regarding the simulated underwater channels, such as particle velocity channel impulse responses, correlations, channel frequency responses, delay spread, can be found in [1] and [2]. The three users are vertically lined up at depths 25, 35 and 45 m below the water surface, as shown in Fig. 1. The vector sensor receiver is 63 m below the water surface. The two transmit pressure sensors of each user are vertically separated by λ, the wavelength. Each user is transmitting space-time block coded BPSK symbols with a bit rate of 2400 bits/sec. The signal vector s for each user includes equi-probable ±1 iid symbols. Alamouti’s space-time block code [8] is used in simulations. A. System Performance Fig. 2 shows the bit error rate (BER) performance of three systems: a single user system with two pressure transmitters and one pressure receiver; a three user system similar to Fig. 1, where the vector sensor receiver is replaced by a three element pressure sensor array with λ element spacing; and a three user vector sensor system of Fig. 1. According to Fig. 2, the vector sensor system has a better BER performance than the pressure only array receiver. This could be because of the correlations among the pressure channels. To investigate this, let the normalized channels in the vector sensor be defined as pɶ = (p − µ p ) σ p , pɶ y = (p y − µ py ) σ py and pɶ z = (p z − µ pz ) σ pz , where µ and σ are sample mean and standard deviation, respectively. Also let |A| denote a matrix whose elements are the absolute values of the elements of the matrix. In what follows, we calculate the average absolute value of the correlation coefficients among all the channels of the vector sensor receiver and the pressureonly array receiver, respectively:

0.84 0.14   pɶ †   1     (vector sensor  (pɶ y )†  ( pɶ pɶ y pɶ z ) =  0.84 1 0.13      receiver),  0.14 0.13   (pɶ z )†  1      pɶ 1†  0.99 0.97   1     (pressure-only  pɶ †2  ( pɶ 1 pɶ 2 pɶ 3 ) =  0.99 1 0.99      array receiver).  0.97 0.99  pɶ †  1    3 The high correlations among the elements of the pressure sensor array, 0.97 and 0.99, may explain the inferior performance of the pressure-only array. By increasing the element spacing in the pressure-only array, its BER might be decreased [1]. A statistical model for correlations in a vector sensor is developed in [9].

TABLE I SIMULATION AND CHANNEL PARAMETERS Water Depth (m) Transmitters Depth (m), User 1 Transmitters Depth (m), User 2 Transmitters Depth (m), User 3 Bottom Types Receiver Depth (m) Receiver Range (km) Carrier Frequency (kHz) Sampling Frequency (kHz) Data Rate (kbps) Nominal Sound Speed (m/s) Wavelength λ (m) Transmit sensor spacing of each user

81.158 25 35 45 Coarse silt 63 1 12 48 2.4 1500 0.125 λ

B. Impact of Imperfect Channel Estimation The BERs in Fig. 2 are obtained assuming perfect knowledge of the channel matrices. Here we study the influence of error in channel estimation. Fig. 3 shows the impact of channel estimation error on the multiuser vector sensor system. It results in a 4dB loss in SNR. C. Effect of Transmit Sensor Spacing In the previous figures the transmit element spacing was fixed at λ. Since spatial correlation exists between the transmit pressure sensors also, it is important to study the effect of transmit sensor spacing. Fig. 4 shows BERs for the three user vector sensor system with λ, 4λ and 8λ transmit sensor spacing at each user. We see that with the increase of the transmit sensor spacing, the system performance increases as well. D. Individual User Performance The BER in Fig. 2 is the multiuser system performance, obtained by averaging over the BERs of three users. In Fig. 5, however, the individual BER of each user in the vector sensor multiuser system is shown. We observe that the performance of user 2 is much worse than the other users. This could be because user 2 is located in between the other two users in Fig. 1. So, perhaps it receives more multiple access interference. E. Comparison of Separate and Joint Decoupling /Equalization Algorithms In Fig. 6 the BERs of the vector sensor receiver are shown, obtained using the separate and joint decoupling/equalization algorithms, described in Subsections IV.A and IV.B, respectively. There is a 3 dB SNR loss, possibly because of the noise enhancement by the ZF decoupler. VI. DISCUSSION AND CONCLUSION In this paper, a multiple users system for underwater channels is proposed that does not need spreading codes. Performance of a vector sensor receiver for three users is investigated in Fig. 2 to 6. The BER of the proposed multiuser space-time coded vector sensor system is close to the BER of the single user system (Fig. 2). This means that the data

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A. Abdi and H. Guo, “A new compact multichannel receiver for underwater wireless communication networks,” accepted for publication in IEEE Trans. Wireless Commun., 2008. A. Abdi, H. Guo and P. Sutthiwan, “A new vector sensor receiver for underwater acoustic communication,” in Proc. MTS/IEEE Oceans, Vancouver, BC, Canada, 2007. S. N. Diggavi, N. Al-Dhahir and A. R. Calderbak, “Algebraic properties of space-time block codes in intersymbol interference multiple-access channels,” IEEE Trans. Inform. Theory, vol. 49, pp. 2403-2414, 2003. W. M. Younis, A. H. Sayed and N. Al-Dhahir, “Efficient adaptive receivers for joint equalization and interference cancellation in multiuser space-time block-coded systems,” IEEE Trans. Signal Processing, vol. 51, pp. 2849-2862, 2003. E. G. Larsson and P. Stoica, Space-time Block Coding for Wireless Communications. Cambridge, UK: Cambridge University Press, 2003. S. Barbarossa, Multiantenna Wireless Communication Systems. Norwood, MA: Artech House, 2005. N. Al-Dhahir, “Single-carrier frequency-domain equalization for spacetime block-coded transmissions over broadband wireless channels,” in Proc. IEEE Int. Symp. Personal, Indoor, Mobile Radio Commun., San Diego, CA, 2001, pp. 143–146. S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Select. Areas Commun., vol. 16, pp.14511458, 1998. A. Abdi and H. Guo, “A correlation model for vector sensor arrays in underwater communication systems,” in Proc. MTS/IEEE Oceans, Quebec City, QC, Canada, 2008.

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symbols of the three users are successfully separated and estimated using a vector sensor. According to Fig. 2, the loss due to multiple access interference via vector sensor receiver is only about 1.5 dB. Interestingly, there is no reduction in the transmission rate of each user, as this multiuser system is not using spreading codes. This is particularly useful in highly bandwidth-constrained underwater channels. The impact of channel estimation error and transmit sensor spacing of each user are also studied. In summary, we have shown that using space-time block codes over the scalar and vector components of the acoustic field, one can have a high rate underwater multiuser system. Small size of the vector sensor receiver in the proposed system is noteworthy, as the compact vector sensor measures all particle velocity channels at a single point in space. This is important is systems which have size limitations such as unmanned underwater vehicles

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Figure 4. The impact of transmit element spacing on the multiuser vector sensor system.

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Figure 5. Individual BER of each user in the vector sensor multiuser system.

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Figure 6. Performance of the vector sensor multiuser system with ZF decoupler/MMSE equalizer and joint MMSE decoupler/equalizer.