2012 Comparison of multiple-input single-output single ...

Universidade de São Paulo

2012

Comparison of multiple-input single-output

single-user ultra-wideband systems with pre-

distortion TRANSACTIONS ON EMERGING TELECOMMUNICATIONS TECHNOLOGIES, HOBOKEN, v. 23,

n. 3, supl. 1, Part 3, pp. 240-253, APR, 2012http://www.producao.usp.br/handle/BDPI/36821

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TRANSACTIONS ON EMERGING TELECOMMUNICATIONS TECHNOLOGIESTrans. Emerging Tel. Tech. 2012; 23:240–253

Published online 5 Decemeber 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/ett.1528

RESEARCH ARTICLE

Comparison of multiple-input single-outputsingle-user ultra-wideband systems withpre-distortionB. A. Angélico1, P. J. E. Jeszensky2, P. M. S. Burt2, W. S. Hodgkiss3 and T. Abrão4*1 Federal Technology University of Paraná (UTFPR), Cornélio Procópio, Brazil2 Department of Telecommunications and Control Engineering, Escola Politécnica, University of São Paulo, São Paulo, Brazil3 Department of Electrical and Computer Engineering, University of California–San Diego, San Diego, CA, USA4 Department of Electrical Engineering, State University of Londrina, Londrina, Paraná, Brazil

ABSTRACT

This paper presents a performance analysis of a baseband multiple-input single-output ultra-wideband system overscenarios CM1 and CM3 of the IEEE 802.15.3a channel model, incorporating four different schemes of pre-distortion:time reversal, zero-forcing pre-equaliser, constrained least squares pre-equaliser, and minimum mean square errorpre-equaliser. For the third case, a simple solution based on the steepest-descent (gradient) algorithm is adopted and com-pared with theoretical results. The channel estimations at the transmitter are assumed to be truncated and noisy. Resultsshow that the constrained least squares algorithm has a good trade-off between intersymbol interference reduction andsignal-to-noise ratio preservation, providing a performance comparable to the minimum mean square error method butwith lower computational complexity. Copyright © 2011 John Wiley & Sons, Ltd.

*Correspondence

T. Abrão, Department of Electrical Engineering, State University of Londrina, Rod. Celso Garcia Cid-PR445, s/n, CampusUniversitário, 86051-970, PO Box 6001, Londrina, Paraná, Brazil.E-mail: [email protected]

Received 11 October 2010; Revised 20 September 2011; Accepted 22 September 2011

1. INTRODUCTION

Ultra-wideband (UWB) is an emerging technology thatemploys ultra-short pulses to transmit information, result-ing in a very large bandwidth. Because of its attractivecharacteristics, such as very high data rates, low prob-ability of interception and good time domain resolutionallowing location and tracking applications at centimetrelevel, UWB has been considered as a promising solutionfor short-distance high-data-rate communications, such aswireless personal area networks.

Ultra-wideband channel is characterised by a densemultipath environment. For the energy spread over the mul-tipath components to be effectively captured, the transmit-based time reversal (TR) technique (sometimes calledpre-Rake) has been investigated [1–6]. In baseband TR,the channel impulse response (CIR) is estimated from aprobe signal, and the data are convolved with the com-plex conjugate time-reversed version of the estimated CIR(namely, TR coefficients) prior to transmission. This tech-nique is based on the channel reciprocity, which was

experimentally verified in [1] for a particular UWB envi-ronment. TR-based UWB transmission can provide inter-symbol interference (ISI) mitigation by reducing the delayspread of the channel and also co-channel interferencerejection by focusing the signal on the point of interest.

However, for high transmission rates, the residual ISIwill still degrade the system performance because theequivalent CIR after TR* is not a delta function. In order tohandle such impairments, we can employ a receiver-basedchannel equalisation scheme with fewer taps than that usedwithout TR [7–9].

In wireless communications, it is sometimes desiredto keep portable devices as simple and as power effi-cient as possible. From this standpoint, a receiver-basedequaliser may aggregate an undesired complexity to the

*The term equivalent CIR refers to the convolution between the TR

coefficients and the original CIR. In a multiple-input single-output

(MISO) system, it is the summation of the resultant convolutions from

each transmit antenna element.

240 Copyright © 2011 John Wiley & Sons, Ltd.

B. A. Angélico et al.

receiver in a downlink scenario. Hence, some transmitter-based equalisation can be applied to mitigate ISI, with-out adding complexity to the receiver. In [10], the authorscompared the performance of pure TR with a zero-forcing(ZF) pre-equalised system, for fixed wireless access chan-nels, and also proposed a new joint ZF and TR scheme.In [11], the authors proposed additional spatial and fre-quency filters to the ZF and TR pre-filters over anIEEE 802.11n channel model. In [12], two novel mini-mum mean square error (MMSE)-based symbol-level pre-equalisation for MISO direct-sequence UWB (DS-UWB)systems in cascade with pre-Rake combining are pro-posed and shown to achieve a good bit error rate (BER)performance.

In this work, a performance comparison of a down-link MISO UWB system with four different schemes ofpre-distortion is presented. An objective function modeledas a constrained least squares (CLS) problem is consid-ered. Imperfect channel estimation, no coding and perfectdata synchronisation are considered. The transmission isassumed to be from an access point with relatively goodcomputational capacity to a lower complexity device withhardware constraints. When the number of transmit anten-nas is Nt > 1, it is assumed that the small-scale fad-ing components across antennas are independent, but theshadowing factors are correlated, according to the methodin [13].

Overall, despite many papers in the field of impulseUWB considering a carrier-free pulse, the IEEE 802.15.3aDS-UWB [14] standard, as well as the IEEE 802.15.4aUWB standard [15], assumes a square-root raised-cosine(RRC) pulse that requires a carrier. All the schemes pre-sented in this paper are performed in baseband consideringan RRC pulse shape.

The rest of the paper has the following organisation.Section 2 describes the UWB channel and system model.Section 3 considers the derivation of pre-filter coefficients,whereas Section 4 presents a complexity analysis, andSection 5 considers the signal-to-interference-plus-noiserate (SINR) analysis. Section 6 shows the simulationparameters and results, whereas Section 7 points out themain conclusions.

2. CHANNEL AND SYSTEM MODEL

2.1. Channel model

A discrete-time complex baseband version of the IEEE802.15.3a model is used [16, 17], where multipath com-ponents arrive in clusters. Cluster and ray arrivals withineach cluster are Poisson distributed with rateƒ and � > ƒ,respectively. The arrival times of the `1th cluster and the`2th ray within the `1th cluster are denoted by �`1 and�`1;`2 . The multipath gain ˇ`1;`2 is described by a log-normal distribution, and its phase assumes only 0 or �with equal probability. A channel realisation at kth antenna

Table I. Channel parameters for CM1 and CM3 of IEEE802.15.3a model.

Parameters CM1 CM3

ƒ (1/ns) 0.0233 0.0667� (1/ns) 2.5 2.1�1 (dB) 3.39 3.39�2 (dB) 3.39 3.39�x (dB) 3 3

consists of

h0

k .t/D �k

L1�1X`1D0

L2�1X`2D0

ˇk`1;`2ı�t � �k`1

� �k`1;`2

�(1)

where ı .�/ is the Dirac delta function and �k D

10.��=20/wk is the log-normal shadowing associated withthe Gaussian random variable (r.v.) wk , with �� D 3 dBbeing the standard deviation of �k [16]. Two scenariosare considered: CM1 and CM3. Table I summarises thechannel parameters [16].

The terms �1 and �2 are respectively the standard devi-ation of cluster log-normal fading (ˇk

`1;`2for a fixed `2)

and ray log-normal fading (ˇk`1;`2

for a fixed `1).The kth discrete-time complex baseband CIR with sam-

pling interval T and length L is obtained by [12]

hk Œm�D gT .t/ � h00

k .t/ � gR .t/ˇ̌̌mT

(2)

where T is the reciprocal of the symbol rate, � denotesconvolution, gR .t/ is matched to the pulse gT .t/, whichhas an RRC shape, and h

00

k.t/ is a baseband version of

h0

k.t/. Here, parameter T is used for the pulse generation,

but the effective symbol rate is controlled by the intervalbetween consecutive symbols, Ts D �T , where � is aninteger.† According to [13], the multipath components areindependent across antennas, but the shadowing terms �kare correlated. Therefore, the CIRs are normalised beforeinserting the shadowing effect, which for the three antennacases has the following correlation matrix [13]

R� D

24 1 0:86 0:54

0:86 1 0:86

0:54 0:86 1

35 (3)

For two transmitter antennas, R� is a 2 � 2 matrix with0:86 in the secondary diagonal. Considering �k ;�j , thecorrelation coefficient between the log-normal variables�k , �j , and wk ;wj , the correlation coefficient between

†In fact, if there was no time interval (multiple of T ) between

consecutive symbols, 1=T would be the effective symbol rate.

Trans. Emerging Tel. Tech. 23:240–253 (2012) © 2011 John Wiley & Sons, Ltd.DOI: 10.1002/ett

241


the Gaussian variables wk , wj , related to �k , �j , it wasshown in [13] that

wk ;wj D1

2�2�ln

��e�2�

�2 � 1

��k ;�j C 1

�(4)

with D ln .10/ =2.Hence, from wk ;wj , a Gaussian correlated vector of

size Nt is obtained. Finally, a log-normal correlated vectorof size Nt is obtained according to �k D 10.��=20/wk .

The discrete-time CIR with resolution T , length L andcorrelated shadowing is given by

hk Œm�D

L�1X`D0

˛k` ı Œm� `� (5)

2.2. Channel estimation

Because of the large number of resolvable paths, the CIRon each antenna is truncated to obtain the TR coefficients.The criterion for this truncation is illustrated in Figure 1.The normalised power delay profile does not take intoaccount the interval (zero samples) before the first signif-icant path. Mainly in the scenario CM3, there might exista relative delay between the first significant path on eachantenna element. However, such a delay must be reinsertedin the estimated channels in order to properly combine thecomponents from each antenna. As the original channelmodel does not consider multiple antennas, the maximumrelative delay among CIRs was fixed at 2:505 ns, whichcorresponds to five times the channel resolution.

Moreover, the TR coefficients are obtained, consideringestimate errors. The method for generating CIR estimationerrors is based on [4]. Assuming a time duplex divisionsystem, a sequence of NP probe pulses with a repetitionperiod longer than the maximum effective delay spreadof the channel, �ef, is transmitted from the receiver to the

0 20 40 60 80 100 120 140 160 180−60

−50

−40

−30

−20

−10

0

Time [ns]

Nor

mal

ized

PD

P [d

B] CM3

CM1

without initial delay

without initial delay

67 ns23 ns

Figure 1. Normalised power delay profile (PDP) for CM1 andCM3, not considering the zero samples before the first signifi-cant path. A �20-dB criterion is chosen for the CIR truncation

on each antenna.

transmitter side. Assuming perfect synchronisation, theNPCIR realisations estimated on each antenna are coherentlyaveraged. If the additive white Gaussian noise (AWGN)double-sided power spectral density per antenna is givenby N0=2, the signal-to-noise ratio (SNR) per antenna isdefined as

SNRDEkb

N0(6)

where EkbD Eb=Nt is the mean bit energy per antenna,

considering Nt antennas. Assuming a static channel dur-ing the frame period, the estimated coefficients on the kth

antenna,nQ̨k`

oL�1`D0

, are represented as

Q̨k` D1

NP

NPXnD1

Q̨k` .n/D ˛k` C e

k` ; (7)

with ek`

being a complex Gaussian r.v. that represents thenoise of the imperfect channel estimation on the `th resolv-able path, with variance of the in-phase and quadraturecomponents given byN0=2NP [4]. The estimated discrete-time CIR after truncation is defined as Qhk Œm�, with lengthLC. It is important to note that channel estimation errorsare not explicitly shown in the results. However, in allcases, the pre-filter coefficients are obtained, taking intoaccount the estimated CIR, which is noisy and truncated.

2.3. System model

The discrete-time model considered is shown in Figure 2.�k represents the sequence of pre-filter coefficients on thekth antenna, and zŒm� is the sampled AWGN. Signalsand systems are represented by their complex basebandequivalents. For antipodal binary signalling with symbolsbi 2 f˙1g and Nt antennas, the signal to be transmittedon the kth antenna element is represented as

sk Œm�D

qEkb

1XiD�1

bi �k Œm� i�� (8)

with �k Œm� representing the pre-filter on the kth antenna.Considering perfect synchronisation, the output of thereceive matched filter (MF), resampled at the rate 1=Ts , is

y Œn�D

1XiD�1

bi x Œn� i �C z Œn� (9)

where z Œn� is the discrete-time AWGN at the output ofthe MF and x Œn� denotes the equivalent CIR, which isobtained from

x Œm�D

qEkb

NtXkD1

.�k � hk/ Œm� (10)

242 Trans. Emerging Tel. Tech. 23:240–253 (2012) © 2011 John Wiley & Sons, Ltd.DOI: 10.1002/ett


Sync.

Signal

Output

Input

Data

Data

DownsamplingSymbol rate

Figure 2. Equivalent discrete-time model. ISI represents the intersymbol interference, yn D y.nTs/ and zn D z.nTs/.

downsampling it by a factor of � D Ts=T . AssumingEkbD 1, the variance of the AWGN per antenna, zk Œm�,

k D 1; � � � ; Nt , is given by �2kD 1=SNR (the same for

each antenna).

3. PRE-FILTER COEFFICIENTS

In all the schemes considered, the length of the pre-filteron each antenna is set as LPF 6 LC. An estimate ofx Œm� from Equation (10) in a matrix-vector notation canbe obtained as

QxD QH” (11)

where ” D

�”0>� � ��”LPF�1

�>�>is a .Nt LPF/ � 1

vector with ”` D��1Œ`� � � � �Nt Œ`�

>, with f�g> meaningtransposition; QH is a p � q block Toeplitz matrix, wherep D LCCLPF � 1 and q DNtLPF, given by

QHD

266666666666666666664

�Qh0�>

0> � � � 0>�Qh1�> �

Qh0�> : : :

:::

:::�Qh1�> : : : 0>�

QhLC�1�> :::

: : :�Qh0�>

0>�QhLC�1

�> �Qh1�>

:::: : :

: : ::::

0> � � � 0>�QhLC�1

�>

377777777777777777775

(12)whose first Nt columns are padded with LPF � 1 null vec-

tors, 0>, with length Nt and Qh` DhQh1Œ`� � � � QhNt Œ`�

i>.

After obtaining ”, we reshaped it into a matrix of Ntrows and LPF columns, whose kth row represents thepre-filter coefficients on the kth antenna element, ”k DŒ�k Œ0�; � � � ; �k ŒLPF � 1��.

3.1. Time reversal pre-filter

Time reversal coefficients on the kth antenna, ”k D ”TRkD�

�TRkŒ0� � � � �TR

kŒLPF � 1�

, are obtained as

�TRk Œm�D Ck

�Qhk Œ�m�

��(13)

where the constant Ck depends on the power allocationscheme.‡ In this paper, Ck is set to be equal for all theantenna elements, that is,

Ck D C D

vuuuutNt

NtPkD1

�� Qhk��22

(14)

with Qhk DhQhk Œ0� � � �

Qhk ŒLPF � 1�i

being the vector of

estimated taps for each antenna element. Note that the totalenergy of the normalised coefficients from all Nt antennasis equal to Nt . It is possible to see that the vector of TRcoefficients for all Nt antennas, ” D ”TR, is given by the.LPF/th line of the matrix QH in Equation (12), multipliedby the normalisation factor Ck .

3.2. Zero-forcing pre-equaliser

The ZF pre-equaliser attempts to cancel the ISI. Here, theZF coefficients for all Nt antennas, ” D ”ZF, are obtained,considering that

QH”ZF D dv (15)

where dv D Œ0; � � � ; 0; 1; 0; � � � ; 0�> has length p, with1 at the vth position. v and are specified below. The ZFsolution is given by

” D ”ZF D QH� dv (16)

‡For a causal representation, hTRk Œm� D Ck

�Qhk ŒLPF �m� 1�

��

should have been assumed, but it does not change the theoretical results

because we are further considering that the index 0th represents the

information timing.


243


with QH� being the pseudo-inverse of QH. If q > p andrank . QH/ D p, there are many solutions. One partic-

ular solution is QH� D QHH�QH QHH

��1, which min-

imises�� QH”ZF � dv

��22

. In this case, v is selected in

order to maximise because it determines the power

of the received signal [10]. Assuming��”ZF

��22D Nt , it

follows that��”ZF��2 DNt

) j j2 dv�QH QHH

��1QH QHH

�QH QHH

��1dv DNt

) j j2��QH QHH

��1�v;v

DNt (17)

Hence, v is chosen so that

��QH QHH

��1�v;v

is mini-

mum. Note that the condition Nt > 2 must be satisfiedfor a ZF solution. If Nt D 2, LPF cannot be smaller thanLC.

On the other hand, if p > q and rank . QH/D q, a ZF solu-tion cannot be guaranteed. However, v can be fixed as v DLPF, for example, and an approximated solution (denoted

by ZF0) that minimises�� QH” � dv

��22

can be obtained as

” D QH� dv , where QH� D�QHH QH

��1QHH . This solu-

tion has to be normalised in order to generate a power-

constrained solution given by ”ZF0 D

qNt=k”k

22 ”.

3.3. Constrained least squarespre-equaliser

Because Qhk is available at the transmitter, let Qhk D C Qhk bea normalised version of Qhk . Thus, a normalised estimationof xŒm� in a matrix-vector can be obtained as

QxD QH” (18)

where QH is given by Equation (12), substituting Qh`D

Ck

hQh1Œ`� � � � QhNt Œ`�

i>for Qh` D

hQh1Œ`� � � � QhNt Œ`�

i>.

Considering a power constraint k”k22 6 Nt , the idea ofthe CLS equaliser is to make Qx be as close as possible tothe vector d D Œ0 � � � Nt � � � 0�

> with length p and Nt atposition LPF. Defining JCLS D kQx� dk22, we can obtain” D ”CLS as

min”JCLS

s:t: k”k22 6Nt(19)

In [18], a solution for the CLS problem using thesingular-value decomposition (SVD) of QH and Lagrangemultipliers is presented and named least squares minimi-sation over a sphere. The matrix QH is decomposed, suchthat UH QHV D † QH, where V D

�v1 v2 � � � vq

.q � q/

and U D�u1 u2 � � � up

.p � p/ are unitary matrices,

whereas † QH is the matrix (not necessarily square) whose

diagonal elements are the singular values of QH. With thistransformation,

��UH�QH” � d

��22D��UH

�QHVVH” � d

��22

D��UH QHVVH” �UHd

��22

D��† QH Q” � Qd��22 (20)

where QdD UHd, Q” D VH” and f�gH the Hermitian oper-

ator. Note that k Q”k22 D��VH”

��22D k”k22. Therefore, the

following Lagrange problem is obtained

L . Q”; �/D�† QH Q” �

Qd�H �

† QH Q” �Qd�C �

�Q”H Q” �Nt

�(21)

with � being the Lagrange multiplier. With L . Q”; �/ beingdifferentiated with respect to Q”� and the resulting gradientbeing set to zero, it follows that

@L . Q”; �/@ Q”�

D 0 )†>QH

�† QH Q” �

Qd�C � Q” D 0

)��I C†>QH

† QH

�Q” D†>QH

Qd (22)

Hence, the CLS filter coefficients are given by

Q” D

"�1

�C �21

Qd1�2

�C �22

Qd2 � � ��r

�C �2r

Qdr

#>(23)

with the following constraint

rXiD1

�i

�C �2i

!2 ˇ̌̌Qdi

ˇ̌̌2DNt (24)

where r is the rank of the matrix QH. Consequently, ifrPiD1

ˇ̌̌Qdi

ˇ̌̌2=�2i >Nt , then

” D ”CLS D

rXiD1

�i Qdi

��C �2i

!vi

��

rXiD1

�i

��C �2i

!2 ˇ̌̌Qdi

ˇ̌̌2DNt (25)

else,

” D ”CLS D

rXiD1

Qdi

�i

!vi (26)

The term �� can be found using, for instance, the bisec-tion method. In order to find an iterative solution forthe CLS problem, we firstly considered the unconstrained



problem. The gradient of J is rJ D �2 QHH

e, whereeD .d� QH”CLS/ is the error vector. In the steepest-descentalgorithm, the coefficients on the .i C 1/th iteration are

updated as ”CLS.i C 1/D ”CLS.i/C��QHH

e.i/�

, where

� is the convergence factor. This algorithm comes out witha solution that is unconstrained. Therefore, Algorithm 3.3is proposed to generate an energy-constrained vector ”CLS.

The coefficients are normalised right after the updatingprocess within each iteration, and the error and gradientcomputations are performed with respect to those nor-malised coefficients. Figure 3 presents the convergence ofthe proposed CLS algorithm. The convergence factor � ischosen as in the unconstrained gradient algorithm, that is,0 < � < 2=�R

max, where �Rmax represents the maximum

eigenvalue of the matrix R D QHH QH. Note that 15 to 30

iterations are enough for achieving convergence.

3.4. Minimum mean square errorpre-equaliser

Another possible power-constrained pre-equalisation isbased on [12], where the filter coefficients are obtained byconsidering the MMSE criterion applied previously to a TRfilter. In this paper, the MMSE filter will not be used in cas-cade with a TR filter but directly, like in the CLS scheme.Considering b D

�bnCLPF�1 � � � bn � � � bn�LCC1

, a vec-

tor of information bits with LCCLPF�1 components and

bn at the .LPF/th position, the estimated received signalafter MF can be written as

Qyn D bH QxC zn D bH QH”C zn (27)Equation (27) is a non-causal representation, but this

does not change the results. As considered in [12], thereceived signal is multiplied by a constant in order to helpthe minimisation procedure. The design goal of the MMSE

criterion is to minimise JMMSE D Ehjbn � Qynj

2i, sub-

jected to k”k22 DNt . JMMSE is given by

JMMSE DEhjbn � Qynj

2iD E

�.bn � Qyn/

� .bn � Qyn/

DEh1� bn Qyn �

� Qy�nbnC j j2 j Qynj

2i

DEh1� bn .bH QH”C zn/

� �.�H QHHbC z�n/bnC j j2j Qynj

2i

(28)

The term Ehj Qynj

2i

in Equation (28) is given by

EŒj Qynj2�DE

h��H QHHbC z�n

� �b> QH”C zn

�iD”H QHH QH”C �2z (29)

Hence,

JMMSE D 1C j j2 �2z � �”Hhn � hHn ”

Cj j2”H QHH QH” (30)

where hn D QHHEŒbnb� D QHHdn and dn is the vectorwhose elements are zero, except for the .LPF/th element,which is equal to 1. Defining N” D ” and noting thatj j2 D .1=Nt / N”

H N”, we automatically insert the powerconstraint in JMMSE [12], resulting in

JMMSE D 1C�2zNtN”H N”�N”Hhn�hHn N”CN”

H QHH QH” (31)

0 10 20 30 40 50

100

ITER0 10 20 30 40 50

100

ITER

Figure 3. Convergence of the constrained least squares algorithm.


245


With JMMSE being differentiated with respect to N”� andbeing set to zero,

@JMMSE

@ N”�D

1

SNRN” � hnC QHH QH” D 0 (32)

)

�QHH QHC I

1

SNR

�N” D hn

Therefore,

N”opt D

�QHh QHC I

1

SNR

��1hn (33)

where I is an identity matrix. Finally,

” D ”MMSE DN”opt

opt) ”MMSE D

pNt

N”opt�� N”opt

��2

(34)

The term is just a constant that does not need to beimplemented at the receiver side.

Observing Equation (34), one can conclude that if

SNR!1; N”opt !�QHH QH

��1hn D

�QHH QH

��1QHHdn,

and, consequently, after normalising the power, ”MMSE!

”ZF0

. On the other hand, if SNR ! 0; N”opt ! C Ihn D

C QHHdn, which represents the .LPF/th line of QH inEquation (12) multiplied by a constant C , and, therefore,

after normalising the power, ”MMSE! ””TR

.

3.5. Channel impulse response comparison

Figure 4 shows examples of equivalent CIRs, consideringthe schemes TR, ZF, MMSE and CLS, with LPF D LC,but here LC is obtained by considering a truncation crite-rion of �30 dB (Figure 1); NP is set to NP D 100. Thehigher SNR is obtained with TR (it represents an MF), butthe residual ISI is also higher than the other schemes. TheZF pre-equaliser eliminates the ISI but results in a rela-tively low power at the receiver, which can be interpretedas the dual problem of noise amplification when ZF is usedat the receiver side. For the SNR conditions considered, thepeak of the equivalent CIR with MMSE is a little lowerthan the case with CLS, but the MMSE scheme results in asomewhat better ISI mitigation.

4. COMPLEXITY ANALYSIS

Table II presents a comparative analysis of the computa-tional complexity for ZF, CLS and MMSE schemes, wherep D .LCCLPF � 1/ and q D .NtLPF/. Complex multi-plication (or division), complex addition (or subtraction),square root extraction and comparison are considered assingle operations. All multiplications and additions areassumed to be complex operations. For the CLS solutionbased on SVD, only the operations needed for the com-putation of SVD are taken into account, which is the mostcostly part. For a matrix inversion of dimensionN �N , theGaussian elimination method is considered, which requiresa total of approximately 2N 3=3 operations.

5. SIGNAL-TO-INTERFERENCE-PLUS-NOISE RATIO ANALYSIS

Equation (9) can be rewritten as

yn D

1XiD�1

bn�ixiCzn D bn x0„ƒ‚…Signal

C

1XiD�1i¤0

bn�ixi

„ ƒ‚ …ISI

C zn„ƒ‚…Noise

(35)where yn D y.nTs/; zn D z.nTs/ and xi D x.iTs/.Note that the discrete-time sequence that represents thesampled noise after MF, zn, is still AWGN and with vari-ance (or power) �2z D Nt �

2kD Nt=SNR. The decision

variable is V D<fyng, where <f�g represents the real-partoperator. If the information symbols are independent andidentically distributed with bn D ˙1, the instantaneousSINR conditioned on the j th set of channel realisations canbe obtained as [9]

SINRj D<fx

j0 g2

1PiD�1i¤0

<nxji

o2C �2

(36)

where �2 D �2z =2 D Nt=2 SNR represents the variance ofthe in-phase and quadrature components of zn.

If the residual ISI in Equation (36) is assumed to beGaussian distributed, the BER conditioned to the j th setof channel realisations can be written as

BERj DQ�p

SINRj�

(37)

where Q.x/ D 1=p2� �

R1x e�y

2=2 dy. ConsideringJ sets of channel realisations, the average BER can becomputed as

BERD1

J

J�1XjD0

BERj (38)

6. SIMULATION CONFIGURATIONAND RESULTS

Performance results are obtained by considering MonteCarlo simulation and the semi-analytical approach (THEO)shown in Section 5. Two and three antenna elements areadopted. The transmission rate is set to Rb D 499 Mbps(� D 4) and Rb D 665:3 Mbps (� D 3), and the RRCpulse is generated by considering ˛ D 0:3 and T D 501 ps.Pre-filter coefficients, as well as the CIR, are assumed tobe static during the frame duration Tf, which is consideredsufficiently long. The CIRs are randomly chosen amongthe 100 realisations proposed in [16]. NP D 100 probe



60 80 100 120 140

10−2

100 TR

Amostras

60 80 100 120 140

10−2

100 ZF

Amostras

60 80 100 120 140

10−2

100 MMSE

Amostras

Am

plitu

de

60 80 100 120 140

10−2

100 CLS Proposed

Amostras

20 40 60 80 100 120 140

10−2

100 CLS SVD

Samples

(a)

20 40 60 80 100 120 140

10−2

100 TR

Amostras

20 40 60 80 100 120 140

10−2

100 ZF

Amostras

20 40 60 80 100 120 140

10−2

100 MMSE

Amostras

Am

plitu

de

20 40 60 80 100 120 140

10−2

100 CLS Proposed

Amostras

20 40 60 80 100 120 140

10−2

100 CLS SVD

Samples

(b)

50 100 150 200 250 300 350

10−2

100 TR

Amostras

50 100 150 200 250 300 350

10−2

100 ZF

Amostras

50 100 150 200 250 300 350

10−2

100 MMSE

Amostras

Am

plitu

de

50 100 150 200 250 300 350

10−2

100 CLS Proposed

Amostras

50 100 150 200 250 300 350 400

10−2

100 CLS SVD

Samples(c)

100 150 200 250 300 350

10−2

100 TR

Amostras

100 150 200 250 300 350

10−2

100 ZF

Amostras

150 200 250 300 350

10−2

100 MMSE

Amostras

Am

plitu

de

150 200 250 300 350

10−2

100 CLS Proposed

Amostras

50 100 150 200 250 300 350 400

10−2

100 CLS SVD

Samples(d)

Figure 4. Comparison among equivalent channel impulse responses considering time reversal (TR), zero forcing (ZF), minimum meansquare error (MMSE) and constrained least squares (CLS), for CM1 and CM3 and NP D 100. Ppk represents the peak of the equiv-alent channel impulse response, whereas PR is the ratio between Ppk and the total power. SVD, singular-value decomposition; SNR,

signal-to-noise ratio.

Table II. Computational complexity analysis.

Scheme Operations needed

ZF 2p3=3C .2q� 1/p2C .2p� 1/.pqC q/C 2qC 1MMSE 2q3=3C .2p� 1/ .q2C q/C .2q� 1/ .1C q/C 3qC 3CLS (SVD) 9 � q3C 4 � p2qC 8 � pq2

CLS (Algorithm 3.3) ITER � .4 � pqC 4 � q� pC 2/

Note that pD LCC LPF � 1 and qD NtLPF.ZF, zero forcing; MMSE, minimum mean square error; CLS, constrained least squares; SVD, singular-value decomposition.


247


transmissions are considered for testing the channel inorder to obtain the CIR estimations at the transmitter.

Figures 5 and 6 show the TR, CLS and MMSE analyti-cal BER performances regarding the number of pre-filtercoefficients. The CLS scheme is implemented with themodified steepest-descent algorithm considering ITER D

15. For Nt D 2, SNR D 15 dB is set, whereas for Nt D 3,SNR D 12 dB is set. It is possible to see that in all curvesthe performances become flat for LPF � 0:7LC withNt D 3 and LPF � 0:8LC with Nt D 2.

Bit error rate results as a function of SNR in decibels arepresented in Figures 7 and 8 for rates Rb D 665:3 Mbps

15 20 25 30 35 40 4510

−6

10−5

10−4

10−3

10−2

10−1

BE

R

BE

RB

ER

BE

R

TR

MMSE

CLS

15 20 25 30 35 40 4510

−6

10−5

10−4

10−3

10−2

10−1

TR

MMSECLS

15 20 25 30 35 40 4510

−6

10−5

10−4

10−3

10−2

10−1

TR

MMSE

CLS

15 20 25 30 35 40 4510

−6

10−5

10−4

10−3

10−2

10−1

TR

MMSECLS

Figure 5. Bit error rate (BER) as a function of LPF considering CM1. TR, time reversal; MMSE, minimum mean square error; CLS,constrained least squares; SNR, signal-to-noise ratio.

40 50 60 70 80 90 100 110 120 130

BE

R TR

MMSE

CLS

40 50 60 70 80 90 100 110 120 130

BE

R

TR

MMSE CLS

40 50 60 70 80 90 100 110 120 130

BE

R

TR

MMSE

CLS

40 50 60 70 80 90 100 110 120 130

10−6

10−5

10−4

10−3

10−2

10−1

10−6

10−5

10−4

10−3

10−2

10−1

10−6

10−5

10−4

10−3

10−2

10−1

10−6

10−5

10−4

10−3

10−2

10−1

BE

R TR

MMSE

CLS

Figure 6. Bit error rate (BER) as a function of LPF considering CM3. TR, time reversal; MMSE, minimum mean square error; CLS,constrained least squares; SNR, signal-to-noise ratio.



and Rb D 499 Mbps, respectively. In both figures, itwas considered that LPF � 0:7LC with Nt D 3 andLPF � 0:8LC with Nt D 2 for TR, CLS and MMSE,whereas for ZF, LPF D LC (ZF0 is not considered). Con-sidering the criterion of �20 dB for the CIR truncation,LC D 46 in CM1 and LC D 134 .C0 to 5/ in CM3. Notethat the CLS performance with ITER D 15 is superior tothe other techniques for SNR > 9 dB. For the low-SNRregion, TR and MMSE perform better than CLS. The ZFscheme, even for LPF D LC, has not a satisfactory per-formance with Nt D 2 antennas, but with Nt D 3, it isbetter than TR for high SNRs. Moreover, ZF, MMSE andCLS are less sensitive than TR regarding an increase in thetransmission rate from 499 to 665 Mbps.

Table III shows a numerical analysis of the computa-tional complexity considering the configurations presentedin Figures 7 and 8. For the ZF scheme, LPF D LC, and

for the CLS with SVD, only the SVD complexity wastaken into account. Observe that the method CLS (SVD)is more complex than the other techniques, and ZF has acomplexity of the same order as that of MMSE. It is alsoworth noticing that the CLS method based on the mod-ified steepest-descent algorithm results in a complexitysubstantially lower than that of the other schemes.

Considering a higher CIR estimation error condition,Figure 9 shows the BER performance for Rb D 499Mbps,Nt D 2 antennas and NP D 20. It can be seen that CLSkeeps performing better than MMSE. It is possible to seethe superiority of CLS in relation to MMSE, as observed inthe previous lower CIR estimation error case (NP D 100),specially for higher SNRs.

In order to help us understand why CLS performs bet-ter than MMSE under the conditions considered, Figure 10presents BER versus SNR results considering CM1, LPF D

0 2 4 6 8 10 12 14 16 1810−6

10−5

10−4

10−3

10−2

10−1

BE

R

CM 1

0 2 4 6 8 10 12 14 16 1810−6

10−5

10−4

10−3

10−2

10−1

BE

RCM 3

THEO TRMCS TRTHEO MMSE

MCS MMSETHEO CLSMCS CLSTHEO ZF

THEO TRMCS TRTHEO MMSE

MCS MMSETHEO CLSMCS CLSTHEO ZF

Figure 7. Performance of bit error rate (BER) versus signal-to-noise ratio (SNR) for Rb D 665:3 Mbps and NP D 100 probe pulses forchannel impulse response estimation. MCS, Monte Carlo simulation; TR, time reversal; MMSE, minimum mean square error; CLS,

constrained least squares; ZF, zero forcing.

0 2 4 6 8 10 12 14 16 1810−6

10−5

10−4

10−3

10−2

10−1

BE

R

CM 1

0 2 4 6 8 10 12 14 16 1810−6

10−5

10−4

10−3

10−2

10−1

BE

R

CM 3

THEO TRMCS TRTHEO MMSEMCS MMSETHEO CLSMCS CLSTHEO ZF

THEO TRMCS TRTHEO MMSEMCS MMSETHEO CLSMCS CLSTHEO ZF

Figure 8. Performance of bit error rate (BER) versus signal-to-noise ratio (SNR) for Rb D 499 Mbps and NP D 100 probe pulses forchannel impulse response estimation. MCS, Monte Carlo simulation; TR, time reversal; MMSE, minimum mean square error; CLS,

constrained least squares; ZF, zero forcing.


249


Table III. Numerical computational complexity analysis.

Number of operations

Configuration Scheme

ZF MMSE CLS (SVD) CLS (mod. grad.)

CM1, Nt D 2 2:3342� 106 1:1697� 106 9:1030� 106 3:6565� 105

CM1, Nt D 3 2:6112� 106 2:0663� 106 1:6179� 107 4:5326� 105

CM3, Nt D 2 6:1390� 107 3:0111� 107 2:3682� 108 3:1961� 106

CM3, Nt D 3 6:8684� 107 5:3298� 107 4:2041� 108 3:9619� 106

ZF, zero forcing; MMSE, minimum mean square error; CLS, constrained least squares; SVD, singular-value decomposition.

0 2 4 6 8 10 12 14 16 1810−6

10−5

10−4

10−3

10−2

10−1

BE

R

CM 1

0 2 4 6 8 10 12 14 16 1810

−6

10−5

10−4

10−3

10−2

10−1

BE

R

CM 3

THEO TR

MCS TR

THEO MMSE

MCS MMSE

THEO CLS

MCS CLS

THEO ZF

THEO TR

MCS TR

THEO MMSE

MCS MMSE

THEO CLS

MCS CLS

THEO ZF

Figure 9. Performance of bit error rate (BER) versus signal-to-noise ratio (SNR) for Rb D 499 Mbps, Nt D 2 antennas and NP D 20probe pulses for channel impulse response estimation. MCS, Monte Carlo simulation; TR, time reversal; MMSE, minimum mean

square error; CLS, constrained least squares; ZF, zero forcing.

LC, noiseless CIR estimation at the transmitter and aCIR truncation criterion of �30 dB (Figure 1), whichcorresponds to 40 ns, approximately, rather than 23 ns(�20-dB criterion). Under these conditions, the perfor-mance of MMSE is similar to CLS for 12 < SNR < 16 dB

and slightly better than CLS for 0 < SNR < 12 dB. There-fore, it is possible to conclude that the CLS scheme is morerobust than the MMSE one regarding errors on the CIRestimation (noise and truncation). This observation sug-gests that the robustness of CLS and MMSE in relation to

0 2 4 6 8 10 12 14 16 1810−6

10−5

10−4

10−3

10−2

10−1

BE

R

CM 1

0 2 4 6 8 10 12 14 16 1810−6

10−5

10−4

10−3

10−2

10−1

BE

R

CM 1

THEO CLSMCS CLSTHEO MMSEMCS MMSE

THEO CLSMCS CLSTHEO MMSEMCS MMSE

Figure 10. Bit error rate (BER) as a function of signal-to-noise ratio (SNR) in CM1, with noiseless estimation and truncation ofthe channel impulse response in 40 ns. MCS, Monte Carlo simulation; MMSE, minimum mean square error; CLS, constrained

least squares.



CIR estimation errors depends on how the respective pre-filter coefficients ”CLS and ”MMSE deviate from their idealnoiseless values ”0CLS and ”0MMSE, respectively, obtainedconsidering perfect knowledge of the CIR at the transmit-ter side. In their turn, these deviations can be measured in

each case by �2ED E

h��” � ”0��2i =LPF.

In order to test this dependency, we estimated �2E

forCLS and MMSE, considering different values of SNR(Eb=N0) and the number of probe transmissions NP. Athousand noisy CIR generations were considered for eachsimulated point. The results are presented in Figures 11 and12. It can be seen that CLS performs better than MMSEfor higher Eb=N0 and/or higher NP. Moreover, for higherEb=N0 and lower values of NP, CLS is quite better thanMMSE. These results are consistent with the BER per-formance in Figures 7 to 9 and therefore corroborate thedependency of the BER on �2

E.

7. CONCLUSIONSThis paper presented a performance analysis for a single-user MISO UWB system incorporating four different pre-distortion schemes: TR, ZF, MMSE and CLS. Resultsshowed that CLS has a BER performance comparable tothat of the MMSE and better than that of TR and ZF. Forinstance, when Nt D 2 antennas, the performance of ZFis not satisfactory because of its power inefficiency, andits performance for Nt D 3 antennas is better than that ofTR, considering high SNRs. Furthermore, when TR is con-sidered for high transmission rates, there is a residual ISIdegrading the system’s performance, which suggests somepost-equalisation scheme at the receiver side.

Under the conditions considered in this paper, it is pos-sible to conclude that the CLS scheme is more robustthan the MMSE one regarding errors on the CIR estima-tion. Besides, the CLS method using the modified gradi-

03

69

1215

18 10 20 30 40 50 60 70 80 90 1000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

03

69

1215

18 10 20 30 40 50 60 70 80 90 1000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04CLS MMSE

Figure 11. Constrained least squares (CLS) and minimum mean square error (MMSE) sensitivity to channel impulse response errorsin CM1, Nt D 2 and Rb D 499 Mbps.

03

69

1215

18 10 20 30 40 50 60 70 80 90 1000

0.005

0.01

0.015

0.02

03

69

1215

18 10 20 30 40 50 60 70 80 90 1000

0.005

0.01

0.015

0.02CLSMMSE

Figure 12. Constrained least squares (CLS) and minimum mean square error (MMSE) sensitivity to channel impulse response errorsin CM3, Nt D 2 and Rb D 499 Mbps.


251


ent algorithm has a lower complexity compared with ZFand MMSE. Such a scheme could be a good solution forthe downlink of high-data-rate applications having goodcomputational capacities at the access point and requiringlow-complexity receivers, in no fast varying channels.

ACKNOWLEDGEMENTS

This work was partially funded by the Brazilian agencyThe National Council for Scientific Technological Devel-opment (CNPq, research grant process #303426/2009-8).The authors would like to express gratitude to the AssociateEditor and anonymous reviewers for their invaluably con-structive suggestions and helpful comments, which haveenhanced the readability and quality of the manuscript.

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3. Guo N, Sadler BM, Qiu RC. Reduced-complexity UWBtime-reversal techniques and experimental results. IEEETransactions on Wireless Communications 2007; 6(12):4221–4226.

4. Cao W, Nallanathan A, Chai CC. Performance analy-sis of pre-rake DS UWB multiple access system underimperfect channel estimation. IEEE Transactions onWireless Communications 2007; 6(11): 3892–3896.

5. Zhou C, Guo N, Sadler BM, Qiu RC. Performance studyon time reversed impulse MIMO for UWB communica-tions based on measured spatial UWB channels. In IEEEMilitary Communications Conference (MILCOM 2007),2007; 1–6.

6. Kaiser T, Zheng F. Ultra Wideband Systems with MIMO.John Wiley and Sons: Chichester, UK, 2010.

7. Strohmer T, Emami M, Hansen J, Papanicolaou G,Paulraj AJ. Application of time-reversal with MMSEequalizer to UWB communications, In IEEE GlobalTelecommunications Conference (GLOBECOM ‘04),Vol. 5, 2004; 3123–3127.

8. Song HC, Hodgkiss WS, Kuperman WA, StevensonM, Akal T. Improvement of time-reversal communica-tions using adaptive channel equalizers. IEEE Journalof Oceanic Engineering 2006; 31(2): 487–496.

9. Angélico BA, Burt PMS, Jeszensky PJE, HodgkissWS, Abrão T. Improvement of MISO single-user timereversal ultra-wideband using a DFE channel equal-izer, In 10th IEEE International Symposium on Spread

Spectrum Techniques and Applications (ISSSTA‘08),2008; 582–586.

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AUTHORS’ BIOGRAPHIES

B. A. Angélico received the BS degree in ElectricalEngineering from the State University of Londrina in2003 and the MS and and PhD degrees in ElectricalEngineering from Escola Politécnica of the Universityof São Paulo (EP-USP) in 2005 and 2010, respectively.He was a visiting scholar at the California Institute forTelecommunications and Information Technology (Calit2),University of California, San Diego in 2007-2008. He iscurrently an Assistant Professor at the Federal Universityof Technology-Paraná. His research interests include ultra-wideband, multiple-input, multiple-output systems, spreadspectrum and multi-user detection.



P. J. E. Jeszensky received the BS, MS and PhD degrees,all in Electrical Engineering from Escola Politécnicaof University of São Paulo (EPUSP, Brazil), in 1972,1981 and 1989, respectively. Since 1990, he has beenwith EPUSP where he is a Full Professor and aResearcher in Communication Systems. He was a visitingprofessor at Universitat Politécnica de Catalunya (UPC),Barcelona (Spain) in 1995 and at Technical Univer-sity of Budapest (TUB, Hungary) in 2001. He is theauthor of the book Sistemas Telefônicos (in Portuguese),Editora Manole, 2003, and his current research interestsinclude code division multiple access systems, multi-userdetection, code sequences analysis and related topics.

P. M. S. Burt received bachelor, master and doctoraldegrees in Electrical Engineering from the University ofSão Paulo (Escola Politécnica), São Paulo, Brazil, wherehe has been since 1992 and where he is currently anAssociate Professor. From 1984 to 1992, he worked in thetelecommunications industry as a development engineer. In2003, he spent a sabbatical year at the Institut National desTélécommunications, Evry, France. His research interestsinclude adaptive signal processing, telecommunicationsapplications and processing of acoustic signals.

W. S. Hodgkiss received the BSEE degree from BucknellUniversity, Lewisburg, PA, in 1972, and the MS andPhD degrees in Electrical Engineering from Duke Univer-sity, Durham, NC, in 1973 and 1975, respectively. From1975 to 1977, he worked with the Naval Ocean Systems

Center, San Diego, CA. From 1977 to 1978, he was afaculty member in the Electrical Engineering Department,Bucknell University. Since 1978, he has been a memberof the faculty of the Scripps Institution of Oceanography,University of California, San Diego, and on the staffof the Marine Physical Laboratory. Currently, he is theDeputy Director, Scientific Affairs, Scripps Institution ofOceanography. His present research interests are in theareas of signal processing, propagation modeling andenvironmental inversions with applications of these tounderwater acoustics and electromagnetic wave propaga-tion. Dr Hodgkiss is a Fellow of the Acoustical Societyof America.

T. Abrão received the BS, MSc and PhD degrees, allin Electrical Engineering from Escola Politécnica ofUniversity São Paulo (EPUSP), Brazil, in 1992, 1996and 2001, respectively. He is currently an AdjunctProfessor at the Electrical Engineering Department ofUEL-State University of Londrina (Brazil). Currently(2007/2-2008/1), he is 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, multi-carrier code division multiple access and multiple-input multiple-output systems, heuristic and optimizationaspects of direct-sequence code division multiple accesssystems and 4G systems. He is author or co-author ofmore than 80 research papers published in specializedperiodicals and symposiums.


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