MMSE Pre-distortion Scheme for Multiuser UWB - Semantic Scholar

Wireless Pers Commun (2013) 70:1307–1319 DOI 10.1007/s11277-012-0748-6

MMSE Pre-distortion Scheme for Multiuser UWB Bruno A. Angélico · Paul Jean E. Jeszensky · Taufik Abrão

Published online: 10 July 2012 © Springer Science+Business Media, LLC. 2012

Abstract The paper proposes a new formulation of a minimum mean square error predistortion scheme for multiple-input single-output impulsive ultra-wideband in multiple user environment. For the performance evaluation, scenarios CM1 and CM3 of the IEEE 802.15.3a channel model are considered. Results show that such a scheme has a good trade-off between intersymbol interference and multiple access interference reduction and signal-to-noise ratio preservation, performing better than time reversal and zero forcing pre-distortion schemes. Keywords

MISO · MMSE · Pre-distortion · Time reversal · UWB

1 Introduction Due to the very large bandwidth of impulsive ultra-wideband (I-UWB) systems, the channel impulse response is characterized by many resolvable paths. In order to effectively capture the energy spread over the multipath components, the transmit based time reversal technique (sometimes called pre-Rake) has been investigated [4,7,11,13,16]. In baseband time reversal, the channel impulse response (CIR) is estimated from a probe signal, and the data is convolved with the complex conjugate time reversed version of the estimated CIR (namely, time reversal coefficients) prior to transmission.

B. A. Angélico Federal University of Technology—Paraná (UTFPR), Cornélio Procópio, Brazil e-mail: [email protected] P. J. E. Jeszensky Department of Telecommunications and Control Engineering, Escola Politécnica of the University of São Paulo, São Paulo, Brazil e-mail: [email protected] T. Abrão (B) Department of Electrical Engineering, State University of Londrina, Londrina, Brazil e-mail: [email protected]

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When the transmission rate is moderated, multiple-input single-output (MISO) multiuser time reversal (TR) I-UWB is shown to be efficient [5,10,11]. However, for high transmission rates, the residual intersymbol interference (ISI) and multiple access interference (MAI) will still degrade the system performance [1]. In this work, a minimum mean square error (MMSE) pre-distortion scheme is proposed for MISO multiuser I-UWB scenario with very dense multipath, including errors in the CIR estimation. There are scarce contributions in the literature treating the MMSE criterion on the transmission (precoding) side combined to the multiuser and multipath channels, specially in the UWB context. For instance, in [8], authors propose to implement the linear MMSE criteria both at Tx end and Rx end of a multiuser MIMO system. Therein, the MMSE based block diagonalization (BD) algorithm is deployed to decompose the multiple-user MIMO downlink channel into multiple parallel independent single-user MIMO channels and solve all separate SU-MIMO decoding algorithm at the receiver. The proposed algorithm brought a negligible complexity increase compared with the zero forcing (ZF) based BD algorithms but outperforms them. Same principle in combining block diagonalization with a minimum mean square error vector precoding for achieving further gain in performance with minimal computational overhead is deployed in [12]. Specifically for I-UWB with very dense multipath condition, in [14] a MMSE scheme is cascaded with a time reversal (TR) pre-filter in a singe-user scenario. In [3] a comparison among MMSE, zero-forcing (ZF) and TR in an UWB single-user context is carried out. However, to the best of authors’ knowledge, there are no existing literature results developing MMSE formulation for a multi-user MISO/MIMO I-UWB in dense multipath channels.

2 Channel and System Model The IEEE 802.15.3a DS-UWB [6] standard, as well as the IEEE 802.15.4a UWB standard [9], assume a square-root raised-cosine (RRC) pulses that requires a carrier. Based on this, the schemes presented in this paper are performed in baseband considering a RRC pulse shape. A discrete-time complex baseband version of the IEEE 802.15.3a model is used. Two scenarios are considered: CM1 and CM3. The kth discrete-time complex baseband channel impulse response (CIR) with sampling interval T and length L C is obtained by [14]    h k [m] = gT (t) ∗ h k (t) ∗ g R (t)

mT

,

(1)

where T is the reciprocal of the symbol rate, ∗ denotes convolution, g R (t) is matched to the  pulse gT (t), which has a square-root raised-cosine shape, and h k (t) is a baseband version of the CIR described in the IEEE 802.15.3a channel model. The parameter T is used for the pulse generation, but the effective symbol rate is controlled by the space between consecutive symbols, Ts = κ T , where κ is an integer. According to [15], the multipath components are independent across antennas, but the shadowing terms χk are correlated. Therefore, the CIRs are normalized before inserting the shadowing effect, which for the four antennas case has the following correlation matrix [15]: ⎤ ⎡ 1 0.86 0.54 0.25 0.86 1 0.86 0.54 ⎦. R x = ⎣ 0.54 0.86 1 0.54 0.25

123

0.54

0.86

1

(2)

MMSE Pre-distortion Scheme for Multiuser UWB

1309

Fig. 1 MISO multiuser transmission with pre-distortion, Nt antennas and U users

Due to the large number of resolvable paths, the CIR is truncated according to the BER analysis according to Angélico et al. [2]. The maximum relative delay among CIRs is fixed at 2.5 ns. CIR estimation errors model is based on [4]. A sequence of N P probe pulses with repetition period longer than the maximum effective delay spread of the channel, τef , is transmitted from the receiver to the transmitter side, and the N P CIRs received on each antenna are coherently averaged. Defining the signal-to-noise ratio per antenna as S N R = E b /N0 , where N0 /2 is the double-sided power spectral density of the additive white Gaussian noise (AWGN), and E b the mean transmitted bit energy per antenna, and assuming a static channel during the frame period, the estimation noise on the th resolvable path of the kth antenna is a complex Gaussian random variable with variance of the in-phase and quadrature components given by N0 /(2 N P ). Figure 1 illustrates the considered system model. The h k,u [m] term represents the mth CIR coefficient on the kth antenna of the uth user. yu is the uth received signal. With BPSK modulation, symbols bui ∈ {±1}, the uth kth transmitted signal is given by  ∞  su,k [m] = E bu,k (3) biu γu,k [m − iκ], i=−∞

where γu,k [m] is the pre-filter, with m = 0, 1, . . . , L γ − 1, and L γ being the length of the pre-filter. Assuming perfect synchronism, the matched filter (MF) output sampled at the effective symbol rate, 1/Ts , is

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yu [n] =

∞ 

biu xu [n − i] +

U ∞  

q

bi f u,q [n − i] + z u [n] ,

(4)

i=−∞ q=1 q=u

i=−∞

with z u [n] being the equivalent additive Gaussian noise (AWGN) at the MF output, and xu [n] and f u,q [n] being obtained sampling xu [m] =



E bu,k

Nt 

γk,u [m] ∗ h k,u [m],

(5)

k=1

and f u,q [m] =



E bu,k

Nt 

γk,q [m] ∗ h k,u [m],

(6)

k=1

respectively, by a factor κ = Ts /T . Equation (4) can be rewritten as yu [n] = bnu xu [0] +

∞ 

u bn−i xu [i] +

U ∞  

q

bn−i f u,q [i] + z u [n].

(7)

i=−∞ q=1 q=u

i=−∞ i=0

The first term represents the desired signal, while the second and the third are the ISI and the MAI terms, respectively. Equation (7) can be written in a vector matrix form as follows yu,n = b u,n xu +

U 

 bq,n fu,q + z u,n ,

(8)

q=1 q=u u u where bu,n = [bn+L , . . . bnu , . . . , bn−L ] has p = L γ + L C − 1 elements, and (.) γ −1 C +1 stands for matrix transposition operator. An estimative of the received signal after MF is given by

˜u + y˜u,n = b u,n x

U 

 ˜ bq,n fu,q + z u,n

q=1 q=u

⎡˜ ⎤ fu,1 ⎢ .. ⎥ ⎥

⎢ ⎢ . ⎥    ⎢ = b1,n . . . bu,n . . . bU,n · ⎢ x˜ u ⎥ ⎥ + z u,n ⎢ . ⎥ . ⎣ . ⎦ f˜u,U Defining y˜ n = [ y˜1,n y˜2,n . . . y˜U,n ], it follows that ⎡ x˜ 1 f˜2,1 · · · ⎢ f˜1,2 x˜ 2 · · · 

⎢   y˜ n = b .. . . 1,n b2,n . . . bU,n · ⎢ ..    ⎣ . . .  bc ˜f1,U f˜2,U · · ·   Ac

123

⎤ f˜U,1 f˜U,2 ⎥ ⎥  .. ⎥ +zn . . ⎦ x˜ U 

(9)

(10)

MMSE Pre-distortion Scheme for Multiuser UWB

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Note that Ac is given by ⎡ ˜ ˜ 2γ 1 H1γ 1 H ⎢H ˜ 1γ 2 H ˜ 2γ 2 ⎢ Ac = ⎢ . .. ⎣ .. . ˜ 2γ U ˜ 1γ U H H

˜ Uγ 1 ⎤ ··· H ˜ Uγ 2 ⎥ ··· H ⎥ .. ⎥ , .. . . ⎦ ˜ Uγ U ··· H

(11)

where γ u = [γγ 1,u , . . . , γ k,u , . . . , γ Nt ,u ] concatenates the pre-filter coefficients of all Tx ˜ u γu , f˜u,q = H ˜ u γq , with u  = q, and antennas of the uth user. In addition, x˜ u = H   h˜ u0 ⎢ ⎢ ⎢   ⎢ ⎢ h˜ u1 ⎢ ⎢ ⎢ .. ⎢ ⎢ . ⎢ ⎢  ⎢ ⎢ ˜ L C −1  ⎢ hu ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ .. ⎢ ⎢ . ⎢ ⎣ 0 ⎡

˜u = H

0   h˜ u0   h˜ u1 .. .   h˜ uL C −1 .. . ···

0 .. .

··· .. .

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ .. ⎥  . ⎥ 0 ⎥   ⎥ ⎥ .. ⎥ 0 . ⎥ h˜ u ⎥   ⎥ ⎥ ⎥ h˜ u1 ⎥ ⎥ ⎥ .. .. ⎥ ⎥ . .   ⎥ ⎦ L −1 C  0 h˜ u

(12)



 is a circulant matrix with h˜ u = h˜ 1,u [] . . . h˜ Nt ,u [] . 2.1 TR Pre-distortion In the time reversal scheme, the pre-filter coefficients on the kth antenna of the uth user, T R = [γ T R [0] . . . γ T R [L − 1]], are obtained as γ k,u γ k,u k,u  ∗ TR γk,u [m] = Ck,u h˜ k,u [−m] ,

(13)

where the constant Ck,u depends on the power allocation scheme. In this paper, Ck,u is set to be equal for all the antenna elements and users, i.e.,   Nt (14) Ck,u = C =   2 ,   Nt  ˜ hk,u  k=1

2

with h˜ k,u = [h˜ k,u [0] . . . h˜ k,u [L γ − 1]] being the vector of estimated taps for each antenna element of the uth user. Note that the total energy of the normalized coefficients from all Nt antennas is equal to Nt . 2.2 ZF Pre-distortion With the above formulation, a zero-forcing (ZF) scheme can be directly obtained finding γ 1 , . . . , γ U , such that

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⎤ dδ 0 · · · 0 . ⎥ ⎢ ⎢ 0 dδ · · · .. ⎥ ⎥ = Dδ , Ac = ⎢ ⎥ ⎢ . . . ⎣ .. .. . . 0 ⎦ 0 · · · 0 dδ ⎡

(15)

with dδ = [0 . . . 1 . . . 0] and 1 at the Lγ th position. Note, however, that Eq. (15) is equivalent to ⎡ ˜ H1γ 1 ⎢H ˜ ⎢ 2γ 1 ⎢ . ⎣ .. ˜ Uγ 1 H

˜ 1γ 2 H ˜ 2γ 2 H .. . ˜ Uγ 2 H

··· ··· .. . ···

˜ 1γ U ⎤ H ˜ 2γ U ⎥ H ⎥ ˜ c c = Dδ , .. ⎥ = H . ⎦ ˜ Uγ U H

(16)

⎡

⎤ ˜1 H

 . ⎥ ˜c = ⎢ where H ⎣ .. ⎦, with dimension (U p × q), and  c = γ 1 . . . γ U , with dimension ˜U H (q × U ), being p = L c + L γ − 1 and q = Nt L γ . Hence, the concatenated matrix of coefficients in the ZF pre-filter is given by tmp

c

˜ c† Dδ , =H

(17)

tmp

tmp

tmp tmp such that  c = [ γ tmp . . . γ U ]. The superscript tmp in  c stands for tempo1 γ2 rary, because Eq. (17) has to be normalized in order to obtain a constrained total power Ptot ≤ U Nt , i.e., tmp  

ZF ZF c ZF = U N  ZF = γ γ . . . γ   , t c 1 2 U  tmp 2  c 

(18)

F

where · F is the Frobenius norm. 2.3 MMSE Pre-distortion The MMSE criterion minimizes the expectation of the square norm of the error signal.  ˜ The modification in (16) cannot be used, because b c Hc  c + zn is not equivalent to the line  received vector considering all users, y˜ n , in (10). However, a new formulation for the MMSE pre-distortion is proposed, where y˜ n is described as ⎡

 ˜ ˜ b 1,n H1 b2,n H1 ⎢ b H  ˜ ˜2 ⎢ 1,n 2 b2,n H y˜ n = ⎢ . . ⎢ .. .. ⎣  ˜ ˜ b 1,n HU b2,n HU

123

⎤  H ˜1 ⎡γ ⎤ · · · bU,n 1  H ˜2 ⎥ ⎥ γ · · · bU,n ⎥⎢ 2 ⎥ ⎥⎢ ˜ + zn = B ⎢ . . c H c γ c + zn , .. ⎥⎣ . ⎥ .. . . ⎦ ⎦  H ˜U γU · · · bU,n

(19)

MMSE Pre-distortion Scheme for Multiuser UWB

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2 2 ˜ with B c (U × U p ) and H c (U p × U q) given as ⎤ ⎡   b1,n · · · bU,n 01×U · p ··· 01×U · p ⎥ ⎢ .. .. ⎥ ⎢  . · · · bU,n . ⎥ ⎢ 01×U · p b  1,n Bc = ⎢ ⎥, .. .. .. ⎥ ⎢ . . ⎣ . 01×U · p ⎦  01×U · p ··· 01×U · p b 1,n · · · bU,n ⎤ ⎡ ˜ H1 0 p×q · · · 0 p×q ⎢ .. ⎥ ⎢ 0 p×q H ˜ 1 ··· . ⎥ ⎥ ⎢ ⎥ ⎢ . .. .. ⎢ .. . . 0 p×q ⎥ ⎥ ⎢ ⎢ ˜1 ⎥ ⎥ ⎢ 0 p×q · · · 0 p×q H ⎥ ⎢ ˜ ⎢ H2 0 p×q · · · 0 p×q ⎥ ⎥ ⎢ .. ⎥ ⎢ ˜ 2 ··· ⎥ ⎢ 0 p×q H . ⎥ ⎢ ⎥ ⎢ .. . . ˜ .. .. 0 Hc = ⎢ . ⎥, p×q ⎥ ⎢ ⎥ ⎢0 ˜ ⎢ p×q · · · 0 p×q H2 ⎥ ⎥ ⎢ . ⎥ ⎢ .. ⎥ ⎢ ⎥ ⎢ ˜ ⎢ HU 0 p×q · · · 0 p×q ⎥ ⎥ ⎢ .. ⎥ ⎢ ˜U ··· ⎥ ⎢ 0 p×q H . ⎥ ⎢ ⎥ ⎢ . . . .. .. 0 ⎦ ⎣ .. p×q ˜ 0 p×q · · · 0 p×q HU

and

⎤ γ1 ⎢γ2 ⎥ ⎥ ⎢ γc = ⎢ . ⎥ ⎣ .. ⎦ γU

(20)

(21)

⎡

(22)

represents the concatenated vector of the pre-filter coefficients for all users. For the MMSE derivation, the total transmitted power by all users is considered to be Ptot = U Nt . Assuming that the signal received by each user is multiplied by a scalar ζ , the main objective of the MMSE consists in minimizing

(23) J M M S E = E bn − ζ y˜ n 22 ,

 subjected to γγ c 22 = U Nt . The vector bn is given by bn = bn1 bn2 . . . bnU , and represents the symbol vectors transmitted by all U users at the instant n. As shown in Appendix A, the MMSE concatenated vector is given by γ MMSE = c

opt  γ¯ c  . U Nt   opt  γ¯ c 

(24)

2

where

 opt ˜c+ ˜ cH H γ¯ c = H

1 SN R

−1

H

˜ c dn , H

(25)

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Table 1 Simulation parameters

Specification

Parameters

Roll of factor

α = 0.3

Rb ≈ 250 Mbps

κ = 4, T = 1 ns

CIR length for CM1 CIR length for CM1

L C = 35; L γ = L C L C = 60; L γ = L C

CIR estimation

N P = 100

N t = 2, CM1

−1

10

−2

−2

10

10

−3

BER

BER

N t = 2, CM3

−1

10

10

−4

−3

10

−4

10

10

TR ZF MMSE

−5

10

TR ZF MMSE

−5

10

−6

−6

10

0

5

10

15

10

20

0

5

SNR [dB]

10

15

20

SNR [dB]

Fig. 2 BER performance for U = 3, Nt = 2

and dn =

 

dn1

 

dn2



  , . . . dU n

(26)

with the following line sub-vectors with dimension (1 × U p) 

dn1



= [(0 · · · 1 · · · 0)(0 · · · 0 · · · 0) · · · (0 · · · 0 · · · 0)];    p entries

 2  dn = [(0 · · · 0 · · · 0)(0 · · · 1 · · · 0) · · · (0 · · · 0 · · · 0)];  U  dn = [(0 · · · 0 · · · 0)(0 · · · 0 · · · 0) · · · (0 · · · 1 · · · 0)].

(27)

It is important to observe that, despite the channel matrix and the vector dn have a large dimension, the computational complexity can be reduced, since there are many zeros in fixed position in this structure. The same is valid to Dδ in the ZF scheme. 3 Simulation Configuration and Results The Monte Carlo simulation (MCS) method with at least 50 errors per point. Channel state is assumed to be static during the frame transmission. Two, three and four antenna elements are adopted. In order to reduce complexity without compromise the system performance significantly, the CIR was truncated according to a BER criterion. Simulation specification and parameters are summarized in Table 1, as shown in [2]. BER results as a function of signal-to-noise ratio (S N R) in dB are presented in Figs. 2, 3, 4.

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MMSE Pre-distortion Scheme for Multiuser UWB

BER

10 10 10 10 10

N t = 3, CM1

−1

N t = 3, CM3

−1

10

−2

−2

10

BER

10

1315

−3

−4

−3

10

−4

10

TR ZF MMSE

−5

TR ZF MMSE

−5

10

−6

−6

0

5

10

15

10

20

0

5

SNR [dB]

10

15

20

SNR [dB]

10

−1

10

−2

10

−3

10

−4

10

−5

10

−6

N t = 4, CM1

BER

BER

Fig. 3 BER performance for U = 3, Nt = 3

TR ZF MMSE 0

5

10

15

SNR [dB]

20

10

−1

10

−2

10

−3

10

−4

10

−5

10

−6

N t = 4, CM3

TR ZF MMSE 0

5

10

15

20

SNR [dB]

Fig. 4 BER performance for U = 3, Nt = 4

It can be observed that the ZF and MMSE schemes are better than the TR one. When Nt < U , the results are not satisfactory at all. In the case of Nt > U , the MMSE scheme has a good performance specially for high SNRs, characterizing the main contribution of this paper. If Nt = U , both the ZF and MMSE schemes have similar performance at high SNRs, but latter performs better at low SNR. As for the ZF scheme, mainly when Nt > U , and despite ISI and MAI are well combated, the transmit power is penalized for doing so. This effect is the dual of noise enhancement in receive ZF schemes. It is known that the MMSE scheme is sensitive to channel estimation errors. In order to confirm this statement, the BER performances of the TR and MMSE schemes for U = Nt = 3 were obtained considering perfect channel estimation. The results are shown in Fig. 5. Comparing this results with those on Fig. 3, the MMSE performance with perfect CIR estimation does not present a BER saturation tendency within the considered SNR range; as a consequence, the overall performance is improved.

4 Conclusions In the present work, a new MMSE formulation for a multi-user MISO UWB system has been proposed. Performance results comparing with those obtained for the well-known time

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N t = 3, CM1

−1

10

−2

10 −2

BER

10

BER

N t = 3, CM3

10 −1

−3

10

−4

10 −3 10 −4

10

TR MMSE

−5

10

TR MMSE

10 −5

−6

10 −6

10

0

5

10

15

20

0

5

SNR [dB]

10

15

20

SNR [dB]

Fig. 5 BER performance for U = 3, Nt = 3, considering perfect CIR estimation

reversal (TR) and zero-forcing (ZF) schemes have demonstrated improved performance for the MMSE scheme if the number of transmitted antennas is larger than the number of active users, specially when accurate channel estimates are available. Such scheme can possibly be a promising solution for applications under no fast varying fading channels, which have a good computational capacity at the transmitter side, but require a very low-complexity receiver.

Appendix A: MMSE Criterion Equation (23) can be rewritten as

J M M S E = E ( bn − ζ y˜ n ) H ( bn − ζ y˜ n )







 H H ˜ ˜ ˜ ˜ − b − ζ y + ζ y b E ζ y E b E ζ . = E b y n n n n n n n n The first term in (28) is given by





E b n bn = U.

For the second term







(28)

(29)



˜  ˜ n = E b E b n ζy n ζ Bc H c γ c + E ζ bn zn





 ˜ ˜ c ζγγ c , γ c = dnH = E b n Bc H c ζγ

where dn , with dimension (1 × U 2 p), is given by       2   , dn . . . dU dn = dn1 n

(30)

(31)

with the following line subvectors with dimension (1 × U p)  1  dn = [(0 · · · 1 · · · 0)(0 · · · 0 · · · 0) · · · (0 · · · 0 · · · 0)];    p entries

 2  dn = [(0 · · · 0 · · · 0)(0 · · · 1 · · · 0) · · · (0 · · · 0 · · · 0)];  U  dn = [(0 · · · 0 · · · 0)(0 · · · 0 · · · 0) · · · (0 · · · 1 · · · 0)].

123

(32)

MMSE Pre-distortion Scheme for Multiuser UWB

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The third term is given by,





˜ cH Bc bn + E znH ζ bn γ cH H E ζ y˜ nH bn = E ζγ ˜ cH E [Bc bn ] = ζγγ cH H ˜ cH dn . = ζγγ cH H

(33)

Finally, the fourth term can be expanded as  H  

˜ ˜ ζ B E ζ y˜ nH y˜ n ζ = E ζ B H γ + ζ z H γ + ζ z c c n c c n c c



˜ cH E Bc B ˜ γ c + |ζ |2 E znH zn = ζγγ cH H c H c ζγ ˜ cH H ˜ c ζγγ c + U |ζ |2 σz2 . = ζγγ cH H

(34)

Therefore, ˜ c ζγγ c − ζγγ cH H ˜ cH dn + ζγγ cH H ˜ cH H ˜ c ζγγ c + U |ζ |2 σz2 . J M M S E = U − dnH

(35)

 2 Making γ¯ c = ζγγ c , results in γ¯ c 2 = |ζ |2 γγ c 22 = |ζ |2 U Nt and, consequently, |ζ |2 = U1Nt γ¯ cH γ¯ c . Hence, the last term is (35) is written as U |ζ |2 σz2 = where S N R = in

Nt σz2

σz2 H 1 γ¯ c γ¯ c = γ¯ H γ¯ c , Nt SN R c

(36)

is the signal-to-noise ratio per antenna. Substituting (36) in (35), it results

˜ cγ¯ c − γ¯ cH H ˜ cH dn + γ¯ cH H ˜ cH H ˜ cγ¯ c + J M M S E = U − dnH

1 γ¯ H γ¯ . SN R c c

(37)

The derivative of (37) regarding the conjugate of γ c is ∂ JM M S E ˜ cγ¯ c + 1 γ¯ c . ˜ cH dn + H ˜ cH H = −H ∗ γ ∂γ c SN R Making the result equal to zero, it turns that   ˜ cH H ˜ cH dn , ˜c+ 1 H γ¯ c = H SN R

(38)

(39)

and, opt γ¯ c

Observing that |ζ |2 =



˜c+ ˜ cH H = H

1 H U Nt γ¯ c γ¯ c ,

1 SN R

−1

H

˜ c dn . H

(40)

if follows that

 1   opt  ζ opt = √ γ¯ c  , 2 U Nt

(41)

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and, hence, the MMSE concatenated vector is given by  γ¯ c γ¯ c  . ⇒ γ MMSE = U Nt  c opt  opt  ζ γ¯  opt

= γ MMSE c

opt

c

(42)

2

Note that ζ is just a auxiliary variable that do not need to be implemented at the receiver.

References 1. Angélico, B. A., Burt, P. M.Sc., Jeszensky, P. J. E., Hodgkiss, W. S., & Abrão, T. (2008). Performance of MISO time reversal ultra-wideband over an 802.15.3a channel model. In 10th IEEE international symposium on spread spectrum techniques and applications (ISSSTA’08) (pp. 577–581). 2. Angélico, B. A., Jeszensky, P. E. J., & Abrão, T. (2010). Miso time reversal and constrained least squares pre-equalizer ultra-wideband systems. In 11th IEEE international symposium on spread spectrum techniques and applications. 3. Angélico, B. A., Jeszensky, P. J. E., & Abrão, T. (2011). Pre-distortion schemes for MISO single-user ultra-wideband systems. In MILCOM 2011—military communications conference (pp. 513–518). 4. Cao, W., Nallanathan, A., & Chai, C. C. (2007). Performance analysis of prerake DS UWB multiple access system under imperfect channel estimation. IEEE Transactions on Wireless Communications, 6(11), 3892–3896. 5. Chen, M., & Li, X. (2004). Transmitter-based channel equalization and MUI suppression for UWB systems. In Proceedings of the international conference modern problems of radio engineering, telecommunications and computer science (pp. 501–504). doi:10.1109/TCSET.2004.1366044. 6. Fisher, R., Kohno, R., McLaughlin, M., & Welbourn, M. (2005). DS-UWB physical layer submission to IEEE 802.15 task group 3a (Doc. Number P802.15-04/0137r4). IEEE P802.15. 7. Guo, N., Sadler, B. M., & Qiu, R. C. (2007). Reduced-complexity UWB time-reversal techniques and experimental results. IEEE Transactions on Wireless Communications, 6(12), 4221–4226. 8. Huayu, Z., Rong, Z., & Qin, Z. (2010). Low complexity mmse precoding and decoding for multiuser mimo. In 2nd international conference on future computer and communication (ICFCC) (Vol. 3, pp. V3–282 –V3–285). 9. IEEE 802.15.4a-2007. (2007). Part 15.4: wireless medium access control (MAC) and physical layer (PHY) specifications for low-rate wireless personal area networks (WPANs); Amendment 1: Add Alternate PHYs. 10. Liu, Z., Tian, Y., & Yang, C. (2008). A preprocessing algorithm of ultra-wideband signal for space-time focusing transmission. In 9th international conference on signal processing, ICSP’08. (pp. 1896–1899). doi:10.1109/ICOSP.2008.4697512. 11. Nguyen, H. T., Kovács, I. Z., & Eggers, P. C. F. (2006). A time reversal transmission approach for multiuser UWB communications. IEEE Transactions on Antennas and Propagation, 54(11), 3216–3224. 12. Park, J., Lee, B., & Shim, B. (2012). A mmse vector precoding with block diagonalization for multiuser mimo downlink. Communications, IEEE Transactions On, 60, –2569577. 13. Qiu, R. C., Zhou, C., Guo, N., & Zhang, J. Q. (2006). Time reversal with MISO for ultrawideband communications: Experimental results. IEEE Antennas and Wireless Propagation Letters, 5(1), 269–273. 14. Torabi, E., Mietzner, J., & Schober, R. (2008). Pre-equalization for pre-rake MISO DS-UWB systems. In IEEE international conference on communications (ICC ’08) (pp. 4861–4866). 15. Zhiwei, L., Xiaoming, P., Png, K. B., & Chin, F. (2007). Kronecker modelling for correlated shadowing in UWB MIMO channels. In IEEE wireless communications and networking conference (WCNC 2007) (pp. 1583–1587). 16. Zhou, C., Guo, N., Sadler, B. M., & Qiu, R. C. (2007). Performance study on time reversed impulse MIMO for UWB communications based on measured spatial UWB channels. In IEEE military communications conference (MILCOM 2007) (pp. 1–6).

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Author Biographies Bruno A. Angélico received the B.S. in Electrical Engineering from State University of Londrina in 2003 and M.Sc. in Electrical Engineering from Escola Politécnica of the University of São Paulo (EP-USP) in 2005. He was a visiting scholar at the California Institute for Telecommunications and Information Technology (Calit2), University of California, San Diego in 2007–2008. He is currently an Assistant Professor at UTFPR—Federal Technological University of Paraná. PR, Brazil. His research interests include ultra wideband, MIMO systems, spread spectrum, and multi-user detection.

Paul Jean E. Jeszensky received the B.S., M.S. and Ph.D., all in Electrical Engineering from EPUSP-Escola Politécnica of University of São Paulo (Brazil), in 1972, 1981, and 1989, respectively. Since 1990 he has been with EPUSP where he is Full Professor and Researcher in Communication Systems. He was visiting professor at UPC-Universitat Politécnica de Catalunya, Barcelona (Spain) in 1995 and at TUBTechnical University of Budapest (Hungary) in 2001. He is author of the book Sistemas Telefˆonicos (in Portuguese), Editora Manole, 2003 and his current research interests include CDMA systems, multi-user detection, code sequences analysis and related topics.

Taufik Abrão received the B.S., M.Sc. and Ph.D., all in Electrical Engineering from EPUSP—Escola Politécnica of University São Paulo, Brazil, in 1992, 1996, and 2001, respectively. He is currently an Associate Professor at the Electrical Engineering Department of UEL—State University of Londrina (Brazil). From 2007 to 2008, he was a visiting professor at TSC/UPC—Department of Signal Theory and Communications, Universitat Politécnica de Catalunya, Barcelona, Spain. His research interests include multi-user detection, MC-CDMA and MIMO systems, heuristic and optimization aspects of multiple access systems and networking. He is author or co-author of more than 80 research papers published in specialized periodicals and key conferences in the area of wireless communication and networking.

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