Preequalizer Design for Spatial Multiplexing SIMO-UWB TR Systems

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 8, OCTOBER 2010

Preequalizer Design for Spatial Multiplexing SIMO-UWB TR Systems Hieu Nguyen, Zhao Zhao, Feng Zheng, Senior Member, IEEE, and Thomas Kaiser, Senior Member, IEEE

Abstract—A spatial multiplexing single-input–multiple-output (SM-SIMO) ultrawideband (UWB) communication system using the time-reversal (TR) technique is proposed in this paper. The system with only one transmit antenna, using a spatial multiplexing scheme, can transmit several independent data streams to achieve a very high data rate. To cope with the long delay spread of the UWB channel, the TR technique is adopted. TR can mitigate not only the intersymbol interference (ISI) but the multistream interference (MSI) caused by multiplexing several data streams simultaneously as well. Preequalization using the channel state information (CSI), which is already available at the transmitter in TR systems, is proposed to further eliminate the ISI and MSI. Simulation results show that the preequalization based on shortened equivalent channels can considerably improve the bit error rate (BER) performance of the system. Index Terms—Multiple-input multiple-output (MIMO), spatial multiplexing, time reversal (TR), ultrawideband (UWB).

I. I NTRODUCTION

T

HE ultrawideband (UWB) communication system, which has recently attracted great interest from both academia and industry [1]–[3], has become a promising candidate for high-data-rate and short-range communication systems. However, due to the wide-bandwidth property, UWB systems suffer from a very long delay spread by multipath effect [4]–[9]. One has to deploy RAKE receivers with an impractical amount of fingers to combat the intersymbol interference (ISI). The time-reversal (TR) technique, which originated from underwater acoustics and ultrasonic [10], [11], has shown its potential in dealing with the ISI problems in UWB [12], [13]. In a TR system, the time-reversed channel impulse response (CIR) is used as a filter at the transmitter side. This effort leads to a very narrow focus of power at the receiver at one specific time instant and one specific space position if the CIRs between any two communication pairs at different locations are well decorrelated. In other words, a TR-UWB prefiltering system possesses a feature of space-time domain focusing. Space-time focusing is also beneficial in the multipleinput–multiple-output (MIMO) spatial multiplexing scheme [14]–[16]. Several studies have applied the TR technique for multiple-antenna beamforming systems [17]–[22]. In [17], Manuscript received October 19, 2009; revised February 12, 2010 and May 28, 2010; accepted July 14, 2010. Date of publication August 5, 2010; date of current version October 20, 2010. The review of this paper was coordinated by Dr. C. Cozzo. The authors are with the Institute of Communication Technology, Leibniz University of Hannover, 30167 Hannover, Germany (e-mail: hieu.nguyen@ ikt.uni-hannover.de; [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2010.2064345

[20], and [22], the same data are beamformed by a TR filter. In [18] and [19], a joint zero-forcing (ZF) and TR preequalizer is designed to minimize ISI and maximize received power at the intended receiver. The potential of a UWB system using a spatial multiplexing scheme is considered in [23], where the matching filter plays the role of a passive TR filter and the maximum-likelihood (ML) detector is deployed to deal with the MSI but neglects the ISI. The SM-MIMO-UWB system using TR is introduced in [24] with the capability of transmit antenna selection. In this paper, TR is proven to achieve transmit diversity gain for SM-MIMO systems. When the number of transmit antennas is small, the received signal is not well focused in the space-time domain. Therefore, the induced MSI and ISI severely deteriorate the performance. The technique for the further mitigation of MSI and ISI is necessary. In this paper, the cascade of a TR filter and preequalizer is proposed to cope with both MSI and ISI in spatial multiplexing MIMO as a design for a low-complexity receiver. The shortened equivalent channels are used to design the appropriate preequalizer. With the help of the TR and preequalizer, several data streams can be transmitted over only one antenna. This system, which is called TR SM-SIMO-UWB, works like the system using virtual transmit antennas to convey data streams simultaneously. If power constraints at the transmitter are imposed, two power-constrained preequalizers, namely, ZF and minimum mean square error (MMSE), are proposed. The simulation results in Section IV show that the cascading of the TR and preequalizer is a promising technique to combat the ISI and MSI for the SM-SIMO-UWB system. II. S YSTEM D ESCRIPTION Let us consider an impulse UWB system using binary pulse amplitude modulation (BPAM) with a proper UWB monopulse. The transmitted signal is x(t) =

+∞ Eb dk p(t − kT )

(1)

k=−∞

where dk = {±1} is the transmit data, Eb is the bit energy, p(t) is the desired pulse shape, and T denotes the symbol duration. The pulse-shaped signal is transmitted over a multipath channel with its CIR being h(t). In this paper, the CIR is modeled by a tap delay line model with Ls taps: h(t) =

Ls

αl δ(t − τl )

l=1

where αl is the amplitude, and τl is the delay of the lth tap.

0018-9545/$26.00 © 2010 IEEE

(2)

NGUYEN et al.: PREEQUALIZER DESIGN FOR SPATIAL MULTIPLEXING SIMO-UWB TR SYSTEMS

The received signal is y(t) = x(t) ⊗ h(t) + n(t)

(3)

where n(t) is the additive noise, and ⊗ denotes convolution operation. At the receiver side, we assume perfect synchronization, and the received signal is sampled with a sampling rate of 1/T .

The TR technique is proposed to combat the ISI problem [20], [21]. In the TR technique, the temporal reverse of the CIR h(−t) is deployed at the transmitter side as a prematched filter to generate the encoded data c(t) = h(−t) ⊗ x(t). Thereby, the received signal can be expressed as y(t) = h(t) ⊗ [h(−t) ⊗ x(t)] + n(t) = R(t) ⊗ x(t) + n(t)

(4)

where R(t) = h(t) ⊗ h(−t) is the autocorrelation of the CIR. As aforementioned, the power of the time-reversed signal is focused within a very narrow time duration. Therefore, the time-domain equalization can be greatly simplified. B. MIMO Multiplexing for TR-UWB In the MIMO spatial multiplexing scheme, the MIMO channel degrees of freedom are exploited to achieve the high data rate without expanding the bandwidth or adopting a high order of constellation map. Basically, the UWB MIMO channel capacity can be increased proportional to the number of antennas. However, the SM scheme, without adopting complicated codings, has a limitation in performance caused by MSI. Let us consider an SM-MIMO system with M transmit and N receive antennas. There are M × N multipath channel realizations between the transmitter and the receiver. We assume that the maximum length of each channel realization is L. The CIR between transmit antenna j and receive antenna i is L l=1

which is based on the original CIR matrix reversed in time and transposed in space. Then, the CIR matrix of the equivalent MIMO time-reversed channel can be denoted as ⎛˘ ˘ 1,2 (t) · · · h ˘ 1,N (t) ⎞ h1,1 (t) h ⎟ ⎜h ˘ ˘ ˘ ⎜ 2,1 (t) h2,2 (t) . . . h2,N (t) ⎟ ˘ (8) H(t) =⎜ ⎟ . . . .. .. .. .. ⎠ ⎝ . ˘ N,2 (t) · · · h ˘ N,N (t) ˘ N,1 (t) h h with each entry of the matrix being calculated by

A. TR for UWB

hi,j (t) =

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αli,jδ t−τli,j ,

i = 1, . . . , N, j = 1, . . . , M. (5)

We can arrange these channels in a matrix form as follows: ⎛ h (t) h (t) · · · h (t) ⎞ 1,1

1,2

⎜ h2,1 (t) h2,2 (t) H(t) = ⎜ .. .. ⎝ . . hN,1 (t) hN,2 (t)

··· .. . ···

1,M

h2,M (t) ⎟ ⎟. .. ⎠ . hN,M (t)

(6)

Suppose a TR filter for the MIMO system is applied at the transmitter with the filtering matrix given by ⎛ h (−t) h (−t) · · · h (−t) ⎞ 1,1 2,1 N,1 ⎜ h1,2 (−t) h2,2 (−t) · · · hN,2 (−t) ⎟ ⎟ HT R (t) = ⎜ .. .. .. .. ⎠ ⎝ . . . . h1,M (−t)

h2,M (−t) · · ·

hN,M (−t)

(7)

˘ i,j (t) = h

M

hi,k (t) ⊗ hj,k (−t),

i, j = 1, . . . , N.

(9)

k=1

˘ Note that the matrix of equivalent MIMO channel H(t) is a square matrix, with its main diagonal entries being the summation of the autocorrelation of the original CIRs and its other entries being the summation of the cross correlation of the original CIRs. C. Reducing to SM-SIMO-TR-UWB Two significant remarks can be drawn from the matrix of ˘ the MIMO time-reversed channel H(t) in (8). First, the maximum number of independent data streams that the system can transmit simultaneously is N , which is the number of the receive antennas. This differs from the conventional narrowband MIMO spatial multiplexing scheme where min(N, M ) is the maximal multiplexing gain if the channel matrix is of full rank [16]. Second, the central taps of the diagonal TR-MIMO composite channel has already captured most of the energy of the delay spread, making detection based on these taps quite robust against the fading of the spread taps. Moreover, the TR in SM-MIMO-UWB can exploit up to M orders of transmit diversity. In other words, more transmit antennas will provide more degrees of transmit diversity, making the link even more robust. By the aforementioned analysis, we propose a reduced structure of the time-reversed MIMO UWB system with only one transmit antenna, namely, SM-SIMO-TR-UWB. It sacrifices some degrees of transmit diversity but can still achieve the same multiplexing gain and maintain a reliable link. It is meaningful for substantially reducing the transceiver complexity and cutting the cost. The performance of the proposed SM-SIMOTR-UWB is evaluated by the simulation in Section IV. A comparison is made between the SM-SIMO-TR-UWB and SMMIMO-TR-UWB. III. P REEQUALIZATION As introduced in Section II, the time-reversed UWB MIMO system possesses good ISI oppression capability and good MSI mitigation ability, but neither ISI nor MSI can be completely eliminated because the equivalent composite MIMO channel is neither an ideal diagonal matrix in the spatial domain nor a δ-like function in the temporal domain. We propose to use a linear preequalizer cascaded with the TR filter to deal with

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TABLE II KEY PARAMETERS OF THE CHANNEL MODELS

Fig. 1. Block diagram of the TR-MIMO-UWB system. TABLE I POWER DISTRIBUTION OF EQUIVALENT AND ORIGINAL CHANNELS

the remaining ISI and MSI. The preequalizer is designed based on the equivalent CIR of the composite channel. The block diagram of the system is shown in Fig. 1.

see that the channel CM3 has the best compression property at 90% energy with nearly half of the channel length shortened. Let us discretize the CIR of the equivalent composite channel of the system. Suppose that the maximum length of each channel is L taps. For a given threshold of the collected energy level (in percentage), choose Ns taps (Ns L) of the equivalent channel, and calculate the appropriate parameters for the linear preequalizer. We call this new CIR of the equivalent channel the ˆ s. shortened CIR and denote it as H Consider a block of received signals at time instants k, k + 1, . . . , k + K − 1, respectively, which can be represented as ˆ Lx + n y=H

(10)

where A. Shortening to Equivalent MIMO TR Channel Regarding the equivalent MIMO time-reversed channel observed from the preequalizer side, we note that each entry of ˘ the matrix H(t) in [8] contains many taps that charge highly on the cost of an equalizer if the equalizer is designed based on all of the taps. Considering the fact that the CIR of a TR system is actually compressed in the time domain, we do not need to ˘ Table I shows, use the full CSI of the equivalent channel H. compared with the original no-time-reversed channel, how the power of the equivalent channel distributes across the channel taps. It is obtained based on the average of 20 realizations of the simulated standard UWB channel models CM1–CM4 [8], where the sampling interval is chosen to be 0.125 ns. The parameters of the channel models used in the simulations are shown in Table II, where Λ, λ, Γ, γ, σ1 , σ2 , and σx are the cluster arrival rate, ray arrival rate, cluster decay factor, ray decay factor, standard deviation of the cluster lognormal fading term, standard deviation of the ray lognormal fading term, and standard deviation of the lognormal shadowing term for the total multipath realization, respectively [8]. In Table I, we can see that most energy of the equivalent channels falls into taps near the peak instant of the autocorrelation. Compared to the original channels, the compression of TR is extremely evident at the threshold of 50% of the total energy. The channel lengths of CM1, CM2, and CM4 have been shortened by more than tenfold. Thus, the computational burden for the channel equalizer can be greatly reduced. In the following, we consider the number of taps that capture 90% of the energy of equivalent channels, which will vary from 1% to 12%, depending on the channel model. From Table I, we can

1 T x [k−L+1], xT [k−L+2], . . . , xT [k] . . . x=√ M

T . . . , xT [k+K −1]

T y = yT [k], yT [k+1], . . . , yT [k+K −1]

T n = nT [k], nT [k+1], . . . , nT [k+K −1] ⎛ ⎞ ˆ ˆ H[L−1] ··· H[0] 0 ··· 0 ⎜ ˆ ˆ 0 H[L−1] ··· H[0] ··· 0 ⎟ ⎟ ˆL=⎜ H ⎜ ⎟ . . . . . . . . . . . . ⎝ . . . . . . ⎠ ˆ ˆ 0 ··· 0 H[L−1] · · · H[0] ⎛˘ ⎞ ˘ 1,2 [k] · · · h ˘ 1,N [k] h1,1 [k] h ⎜h ˘ 2,1 [k] h ˘ 2,2 [k] · · · h ˘ 2,N [k] ⎟ ⎟ ˆ =⎜ H[k] ⎜ ⎟. .. .. .. .. ⎝ ⎠ . . . . ˘ ˘ ˘ hN,1 [k] hN,2 [k] · · · hN,N [k] In the aforementioned equation, x[k] = [x1 [k], . . . , xN [k]]T is the input symbol in data streams at time instant k, and y[k] = [y1 [k], . . . , yN [k]]T is the received signal at time instant k, n[k] = [n1 [k], . . . , nN [k]]T is receiver noise (assumed to be ˆ ˆ ˆ − 1]} Gaussian) at time instant k, and {H[0], H[1], . . . , H[L are the channel matrices characterizing the channel between the input x[k] and the output y[k] (it is the discrete-time version of √ the MIMO channel [8]). The scale factor 1/ M in x keeps the total transmit power per antenna at the same power as the SISO case for fair comparison. Notice that we have abused the notation to some degree. For example, x and x[k] are different vectors, i.e., x is the stacking of x[k] for different values of k.


A similar notational convention is used in the remainder of this section. Due to the temporal-focusing property of the TR channel, the effective length of the equivalent composite TR channel can be reduced to Ns , centering around the peak tap. Suppose Ns is an odd number. Let the tap index for the peak tap be kp , ˆ p − ks ], H[k ˆ p− and define ks = (Ns − 1)/2. Therefore, {H[k ˆ p ], . . . , H[k ˆ p + ks − 1], H[k ˆ p + ks ]} will be ks + 1], . . . , H[k selected as the CIR matrices of the shortened equivalent channels. We then reindex the taps of the CIR of the shortened equivalent channels to be 0, 1, . . . , Ns exactly in the original order. By further slight abuse of notation, we denote the corˆ ˆ ˆ s − 1]} responding MIMO channel as {H[0], H[1], . . . , H[N again. Then, (10) can be approximated by ˆ s xs + n y≈H

It is easy to find the preequalizer matrix G in (12) as follows (denoted as GZF for the ZF case): −1 ˆs ˆH ˆ† = α H ˆ HH H (13) GZF = αH s s s where † and (•)H denote the Moore–Penrose pseudoinverse and the conjugate transpose of a matrix, respectively. The coefficient α is introduced for the power constraint of the transmit signal, which is determined by the following equation:

¯ x2 = Etx (14) E HG˜ 2 or

¯ ˆ † 2 ˜ = Etx Hs x E αH

(15)

2

(11)

where ⎛

⎞ ˆ s −1] ˆ H[N ··· H[0] 0 ··· 0 ⎜ ˆ ˆ s −1] · · · H[0] ··· 0 ⎟ 0 H[N ⎟ ˆ s =⎜ H ⎜ . .. ⎟ .. .. .. .. .. ⎝ . . . . . ⎠ ˆ ˆ s −1] · · · H[0] 0 ··· 0 H[N 1 T xs = √ x [k−Ns +1], xT [k−Ns +2], . . . , xT [k] . . . M

T xT [k+K −1] .

˜= The preequalizer is applied to the block data stream x [xT [k], xT [k + 1], . . . , xT [k + K − 1]]T . Therefore, the preequalizer is characterized by a matrix G of dimension N (K + Ns − 1) × N K, which is used before the TR filter to manipu˜ . The system model can be rewritten as late the data stream x ˆ s G˜ ˜=H x+n y

3801

(12)

˜ T [k + 1], . . . , y ˜ T [k + K − 1]]T , and y[k] ˜ = [˜ where y yT [k], y is the receiver output of the TR system when the preequalizer G is applied at the transmitter side. Note that the preequalizer ˜ instead of block data xs . will be applied to block data x The covariance matrices of the transmitted signal and noise ˜ T ] = σx2 IN K and E[nnT ] = σn2 IN K , are characterized by E[˜ xx respectively, where IN K is an N K × N K identity matrix. The design criterion for the equalizing filter G can be either the ZF or MMSE. They are analyzed in the following sections.

where • 22 denotes the Euclidean norm of a vector. Equation (14) indicates that the total power of the transmitted signal at the point of transmit antennas for all the processing schemes ¯ is the TR filter in the format is fixed to be Etx . The matrix H of block data processing, which is an M K × N (K + Ns − 1) block Toeplitz matrix defined by ⎛ H[0] ¯ ¯ s −1] · · · H[N 0 ¯ ¯ s −1] ··· H[N ⎜ 0 H[0] ¯ =⎜ . H .. .. .. ⎝ . . . . . ¯ 0 ··· 0 H[0]

··· ··· .. .

0 0 .. .

⎞

⎞ ⎟ ⎟ ⎠

¯ s −1] · · · H[N (16)

where ⎛ h [k] 1,1 ⎜ h1,2 [k] ¯ H[k] =⎜ .. ⎝ .

h2,1 [k] h2,2 [k] .. .

··· ··· .. .

hN,1 [k] hN,2 [k] .. .

h1,M [k]

h2,M [k]

···

hN,M [k]

The power constraint equation follows as H 2 † H ¯ ˆ† ˆ ¯ ασx tr Hs H HHs = Etx

⎟ ⎟. ⎠

(17)

where tr(•) is the trace of a matrix. The scale factor of the power constraint is E 1 tx . α = 2 H (18) σx ˆ †s ¯ HH ¯H ˆ †s tr H H

B. ZF Preequalizer The preequalization with the ZF criterion guarantees that the received data are identical to the transmitted data in the absence of receiver noises. This simple equalizer, however, is sensitive to receiver noises when the channel suffers deep fading. In this case, the values of some entries of the equalizer matrix, which is derived from the inverse of the CIR matrix, are very large, causing the transmit power amplifier to be inefficient. To deal with this problem, the power constraint for the transmit signal (after the preequalizer and TR filter) is introduced below.

C. MMSE Preequalizer The MMSE preequalizer is designed to minimize the mean square error (MSE) between the receive and transmit data. The equalizer matrix is a solution of the following optimization problem:

˜ 22 x−y arg min E ˜ G

¯ x2 = Etx . s.t. E HG˜ 2

(19)

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The MSE equation can be rewritten as

ˆ s G˜ ˜ 22 =E ˜ x−H E ˜ x−y x −n22 ˆ s G)H (IN K − H ˆ s G) +σ 2 . =σx2 tr (IN K − H n (20) The power constraint equation gives ¯ H HG) ¯ σx2 tr(GH H − Etx = 0.

(21)

Construct the Lagrangian function with Lagrangian multiplier λ ∈ R+ , i.e.,

¯ x2 − Etx ˜ 22 + λ E HG˜ L = E ˜ x−y 2 ˆ s G)H (IN K − H ˆ s G) + σ 2 = σx2 tr (IN K − H n 2 H ¯H ¯ (22) + λ σx tr(G H HG) − Etx . Setting the first derivative of the Lagrangian function to zero, we obtain the solution for the MMSE preequalizer as −1 ¯ HH ˆ s + λH ¯ ˆH ˆ HH (23) GMMSE = H H s s where λ is the solution of the following equation: −1 ¯ HH ¯ H ˆ s + λH ¯ ˆ HH ˆs ˆ HH tr H H s s −1 Etx H ˆ H ¯ H ¯ ¯ ˆ × Hs Hs + λH H = 2 . H σx

(24)

When the transmit power is large, the solution λ to (24) will be very small. In this case, the MMSE preequalizer matrix is close to the ZF case. In general, a conceptual TR transceiver consists of a channel estimator, a reconfigurable finite-impulse response filter, and an interpolator. The complexity and cost of these parts can be lowered by various simplified implementations with some loss in performance [25]. From (13) and (23), we can see that the computational complexity of both ZF and MMSE equalizers is dominated by matrix inversion and multiplication, while the complexity of both matrix inversion and multiplication are of order O(nc0 ) for n0 × n0 matrices with 2 < c ≤ 3 [26], [27]. In our proposed equalizers, the relevant matrices are of dimension N (K + Ns − 1) × N (K + Ns − 1), where N and K are much smaller than that required by the traditional equalizer without TR. Therefore, the approach of shortening the equivalent channel length Ns in the TR approach can substantially reduce the complexity of the preequalizer. IV. N UMERICAL R ESULTS Simulations are conducted to verify the performance of the SIMO/MIMO-TR-UWB system with the proposed preequalizers. In the simulations, the data symbols are modulated in the BPAM format. The monocycle UWB waveform is 2 t−tc 2 t − tc (25) e−2π( w ) p(t) = 1 − 4π w where w is a parameter corresponding to the pulsewidth, and tc is a time shifting of the pulse. In our simulations, w = 1 ns, and

Fig. 2. BER performance of the SM-MIMO-UWB system versus the number of transmit antennas. Here, SNR = 12 dB with four receive antennas.

tc = w/2. The symbol duration T is 1.5 ns, corresponding to a data rate of 667 Mb/s per data stream. We assume that the signal is transmitted over UWB channels and perfectly synchronized at the receiver. In this paper, the IEEE 802.15.3a CM1-CM4 channel models [8] are used in the simulations. The average bit error rate (BER) is evaluated over 100 channel realizations for each simulation. Fig. 2 illustrates the transmit diversity capability of an SMMIMO-TR-UWB system for different channel models. At the same SNR = 12 dB for the same number of receive antennas, when the number of transmit antennas increases, the average BER decreases. It can be seen that transmit diversity has been achieved. However, the increasing rate becomes small when the number of transmit antennas is high. For high-speed data transmission and a small number of transmit antennas, it is necessary to deploy the equalizer for ISI and MSI mitigation. Fig. 3 compares the performance between SM-SIMO-TRUWB and SM-MIMO-TR-UWB. The channel model CM3 and the ZF equalizer are used. We can observe performance degradation if we reduce the number of transmit antennas while maintaining the spatial multiplexing order as four. However, deploying more transmit antennas will increase the cost and complexity of the TR system. Therefore, in the following, we will concentrate on investigating the performance of the proposed SM-SIMO-TR-UWB system. The BER performance of the SM-SIMO-TR-UWB systems with and without the preequalizer for the CM1–CM4 channels are shown in Figs. 4–7, respectively. The ZF preequalizer without power constraint is designed based on the shortened equivalent channels. The length of shortened channels is chosen from Table I according to the criterion that the power threshold is 90% of the total power. In the simulation, N (= 2, 3, and 4, respectively) independent data streams are transmitted over a single transmit antenna. The performance of the SM-SIMO system severely degrades if the equalizer is not used. The greater the number of data streams, the poorer the BER performance of the system becomes. This is because the involved MSI increases with the number of data streams. From Figs. 4–7, it can be seen that the proposed preequalizer


Fig. 3. BER performance of the SM-SIMO-UWB and SM-MIMO-UWB with the ZF equalizer for the CM3 channel.

Fig. 4. BER performance of SM-SIMO-UWB with and without the ZF equalizer for the CM1 channel.

can significantly improve the performance of the systems. The error floor of the average BER is reduced by almost one order. For example, when the number of data streams N = 4, the BER performance for the CM3 channel decreases from BER = 3 × 10−2 for the case of no equalizer to BER = 9 × 10−4 for the case of using the ZF equalizer. If the number of data streams is small, e.g., N = 2, the BER error floor of the system with the ZF preequalizer is broken up for channels CM2 and CM3. Fig. 8 shows the dependence of average BER on the length of the shortened channel used in the preequalizer. The BER performance is plotted with respect to the ratio between the length of the shortened channel Ns and the full channel length L. The SNR is fixed at 12 dB. When the length of the shortened channel increases, the BER first drastically decreases and then approaches some points of saturation. At the saturation point, if the shortened channel length increases further, the BER decreases slightly. Simulation results show that the equivalent channel CM3 possesses a better compression property than other channels CM1, CM2, and CM4. These results agree with this fact, as illustrated in Table I.

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Fig. 8. BER performance of the SM-SIMO-UWB system versus the length of the shortened channel used in the equalizer.

SIMO-UWB with only one transmit antenna to convey several independent data streams. The TR SM-SIMO-UWB system works like a system using virtual multiple transmit antennas for communications. In this paper, the spatial correlation among the different channels has not been considered. Therefore, the results are generally valid when the receive antennas are separated sufficiently far. In the SM-UWB-SIMO-TR systems considered in this paper, the multiple receive antennas can be the antennas from different users. Therefore, a separation of several tens of centimeters among the antennas is quite normal. On the other hand, it is shown in [28] and [29] that the system performance of UWB-MIMO systems will not suffer too much degradation if the antennas are separated from each other by more than 10 cm. Finally, we would like to point out that it is interesting to combine the design of the TR prefilter and the preequalizer into a one-step design since both designs use the same information. It is not yet clear whether the integrated design can reduce the complexity in the system implementation. However, how to exploit the nice property of the composite TR channel remains an issue in the integrated design. ACKNOWLEDGMENT The authors would like to thank the two anonymous reviewers for their constructive comments. R EFERENCES

Fig. 9. BER performance of SM-SIMO-UWB with ZF/MMSE equalizers for the CM3 channel (dashed line: ZF; solid line: MMSE).

When there is a power constraint at the transmitter side, the performance of two preequalizers, i.e., power constraint ZF and MMSE preequalizers, is compared. The result is illustrated in Fig. 9. As can be seen, the performance of the MMSE preequalizer is slightly better than that of the ZF preequalizer. V. C ONCLUSION Spatial multiplexing MIMO-UWB using TR yields the multipath and spatial diversity to combat the ISI and the MSI. However, a great number of transmit antennas require a lot of expensive hardware. Thus, the SM-SIMO-TR-UWB system is of great interest in practice. In this paper, ZF and MMSE preequalizers have been proposed to reduce the effects of ISI and MSI on the performance of the SM-SIMO-TR-UWB and SMMIMO-TR-UWB systems. The preequalizer has been designed based on the shortened equivalent channels to reduce the complexity of systems. The performance of system considerably improved when the preequalizer has been applied. The cascade of the proposed preequalizer and TR filter enables the TR SM-

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Hieu Nguyen received the Ph.D. degree in electronics engineering from Chungbuk National University, Cheonju, Korea, in 2007. He is currently a Scientific Researcher with the Institute of Communications Technology, Faculty of Electrical Engineering and Computer Science, Leibniz University of Hannover, Hannover, Germany. His major research interests are signal processing for communications, multicarrier and multiple-antenna systems, and ultrawideband communications.

Zhao Zhao received the B.Sc. and M.Sc. degrees in electrical engineering from Shanghai Jiaotong University, Shanghai, China, in 2004 and 2007, respectively. He is currently working toward the Ph.D. degree with the Leibniz University of Hannover, Hannover, Germany. Since 2007, he has been a Scientific Researcher with the Institute of Communications Technology, Faculty of Electrical Engineering and Computer Science, Leibniz University of Hannover. His research interests are in communication theory and radio resource management specifically applied to two-tier cellular networks.

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Feng Zheng (SM’06) received the B.Sc. and M.Sc. degrees in electrical engineering from Xidian University, Xi’an, China, in 1984 and 1987, respectively, and the Ph.D. degree in automatic control from the Beijing University of Aeronautics and Astronautics, Beijing, China, in 1993. Previously, he held an Alexander von Humboldt Research Fellowship with the Gerhard Mercator University of Duisburg, Duisburg, Germany, as well as research positions with Tsinghua University, Beijing, the National University of Singapore, Singapore, and the University of Limerick, Limerick, Ireland. From 1995 to 1998, he was with the Center for Space Science and Applied Research, Chinese Academy of Sciences, Beijing, as an Associate Professor. Since July 2007, he has been with the Institute of Communications Technology, Faculty of Electrical Engineering and Computer Science, Leibniz University of Hannover, Hannover, Germany, as a Wissenschaftlicher Beirat. He is the author of around 70 refereed journal/conference papers and a book entitled Ultra Wideband Systems With MIMO (Wiley, 2010, coauthored with Prof. T. Kaiser). His research interests include ultra-wideband-multiple-input– multiple-output wireless communications, signal processing, and systems and control theory. Dr. Zheng is a corecipient of several awards, including the Third National Natural Science Award in 1999 from the Chinese Government, the First Science and Technology Achievement Award in 1997 from the State Education Commission of China, and the Best Paper Award in 1994 from the Society of Instrument and Control Engineering (SICE) of Japan at the 33rd SICE Annual Conference, Tokyo, Japan.

Thomas Kaiser (M’98–SM’04) received the Diploma degree in electrical engineering from Ruhr-University Bochum, Bochum, Germany, in 1991 and the Ph.D. (with distinction) and German Habilitation degrees in electrical engineering from Gerhard Mercator University, Duisburg, Germany in 1995 and 2000, respectively. From 1995 to 1996, he spent a research leave with the University of Southern California, Los Angeles, which was grant-aided by the German Academic Exchange Service. From April 2000 to March 2001, he was the Head of the Department of Communication Systems, Gerhard Mercator University, and from April 2001 to March 2002, he was the Head of the Department of Wireless Chips and Systems, Fraunhofer Institute of Microelectronic Circuits and Systems, Duisburg. From April 2002 to July 2006, he was Coleader of the Smart Antenna Research Team, University of Duisburg-Essen, Duisburg. In the summer of 2005, he joined the Smart Antenna Research Group, Stanford University, Stanford, CA, and in the winter of 2007, he joined the Department of Electrical Engineering, Princeton University, Princeton, NJ, both as a Visiting Professor. Currently, he chairs the Institute of Communication Technology, Leibniz University of Hannover, Hannover, Germany, and is the Founder and Chief Executive Officer of mimoOn GmbH, which is a start-up company developing software for the next generation of cellular communication systems. He is the author of more than 100 papers in international journals and conference proceedings and a book entitled Ultra Wideband Systems With MIMO (Wiley, 2010). He has been a Guest Editor of several special issues and is a Coeditor of four books on multiantenna and ultrawideband (UWB) systems. He serves on the editorial board of the EURASIP Journal of Applied Signal Processing. He is involved in several national and international projects and has given a couple of keynote speeches on multiantenna and UWB systems. His research interest lies in applied signal processing with emphasis on future wireless communication and localization systems. Dr. Kaiser is Member-at-Large of the Board of Governors of the IEEE Signal Processing Society. He was the founding Editor-in-Chief of the e-letter of the IEEE Signal Processing Society. He has been the Chair or a Cochair of a number of special sessions on multiantenna implementation issues and is a technical program committee member of multiple conferences. He was the General Chair of the IEEE International Conference on UltraWideBand in 2008 and the General Chair of the International Conference on Cognitive Radio Oriented Wireless Networks and Communications in 2009.