Approaching the matched filter bound with block transmission techniques

TRANSACTIONS ON EMERGING TELECOMMUNICATIONS TECHNOLOGIESTrans. Emerg. Telecomms. Techs. 2012; 23:76–85

Published online 14 November 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/ett.1513

RESEARCH ARTICLE

Approaching the matched filter bound with blocktransmission techniquesFábio Silva1,4*, Rui Dinis1,2, Nuno Souto1,3 and Paulo Montezuma2,4

1 Instituto de Telecomunicações, Lisboa, Portugal2 DEE, FCT-UNL, Portugal3 ISCTE, Lisboa, Portugal4 UNINOVA, Caparica, Portugal

ABSTRACT

Block transmission techniques, with appropriate cyclic prefixes and employing frequency-domain equalisation techniques,have been shown to be suitable for high data rate transmission over severely time-dispersive channels. The most pop-ular techniques based on this concept are orthogonal frequency division multiplexing (OFDM) and single-carrier withfrequency-domain equalisation (SC-FDE).

In this paper, we consider OFDM and SC-FDE transmission schemes and study the impact of the number of multipathcomponents and the diversity order on their performance. It is shown that when we have a high number of separablemultipath components, the asymptotic performance of both schemes approaches the matched filter bound, even withoutdiversity. When we have diversity, the performance approaches the matched filter bound faster, with a small number ofseparable multipath components. It was also observed that the SC-FDE has an overall performance advantage over theOFDM option, especially when employing the iterative block decision feedback equaliser with turbo equalisation and forhigh code rates. Copyright © 2011 John Wiley & Sons, Ltd.

KEY WORDS

matched filter bound; SC-FDE; OFDM; IB-DFE: Iterative Block Decision Feedback Equaliser; turbo equalisation; diversity

*Correspondence

Fábio Silva, Instituto de Telecomunicações, Lisboa, Portugal.E-mail: [email protected]

Received 2 May 2011; Revised 13 August 2011; Accepted 17 August 2011

1. INTRODUCTION

It is widely recognised that bock transmissions com-bined with frequency-domain detection schemes are suit-able for broadband wireless systems. Among these, themost popular schemes are orthogonal frequency divisionmultiplexing (OFDM) modulations [1] and single-carrierwith frequency-domain equalisation (SC-FDE) modula-tions [2]. In both cases, we employ frequency-domainequalisation (FDE) at the receiver, whose implementa-tion can be very efficient because the discrete Fouriertransform (DFT)/inverse DFT operations can be realisedusing the fast Fourier transform algorithm. Moreover, thereceiver complexity is almost independent of the channelimpulse response, making them suitable for severely time-dispersive channels [3, 4]. Because of the lower envelopefluctuations of the transmitted signals, SC-FDE schemesare especially appropriate for the uplink transmission

(i.e. the transmission from the mobile terminal to thebase station), and OFDM schemes are preferable for thedownlink transmission (i.e. the transmission from the basestation to the mobile terminal) because of lower signalprocessing requirements at the receivers [3, 4].

The conventional receiver for SC-FDE schemes is a lin-ear FDE. However, it is known that nonlinear equalisersoutperform linear equalisers [5]. Iterative block deci-sion feedback equaliser (IB-DFE: Iterative Block Deci-sion Feedback Equaliser) [6] is a promising iterative FDEtechnique for SC-FDE that was first proposed in [7] andextended to diversity scenarios [8] and layered space–timeschemes [9]. These receivers can be regarded as low-complexity turbo FDE schemes [10, 11], where the chan-nel decoder is not involved in the feedback. True turboFDE schemes can also be designed based on the IB-DFE: Iterative Block Decision Feedback Equaliser concept[12, 13]. It was observed that the asymptotic performance

76 Copyright © 2011 John Wiley & Sons, Ltd.

F. Silva et al. Approaching the MFB with block transmission techniques

of IB-DFE: Iterative Block Decision Feedback Equaliserschemes can be sometimes very close to the matched filterbound (MFB), but in other cases, it is relatively far from it[8]. However, it is not clear under which circumstances wecan expect performances close to the MFB.

In this paper, we study the impact of the number of mul-tipath components and the diversity order in the asymptoticperformance of SC-FDE and OFDM schemes and investi-gate under which conditions the asymptotic can be close tothe MFB. This paper is organised as follows: OFDM andSC-FDE receivers are described in Section 2. Analyticalexpressions for the MFB, when we have multipath prop-agation and diversity, are presented in Section 3. A set ofperformance results is presented in Section 4, and Section 5is concerned with the conclusions of this paper.

2. SYSTEM DESCRIPTION

In this paper, we consider wireless systems employingOFDM and SC-FDE block transmission techniques. In aconventional OFDM scheme, the time-domain block isfsnInD 0; 1; : : : ; N�1g D IDFT fSk I kD 0; 1; : : : ; N�1g,with Sk denoting the frequency-domain data symbols tobe transmitted, associated to the kth subcarrier, whichare selected from a given constellation (e.g. a quadraturephase-shift keying (QPSK) constellation). On the otherhand, for a SC-FDE scheme, the time-domain symbolsfsnIn D 0; 1; : : : ; N � 1g are directly selected fromthe constellation. For both block transmission schemes,a cyclic prefix, with length longer than the overall chan-nel impulse response, is appended leading to the signalfsCPn In D �NG ; : : : ; N � 1g, which is transmitted overa time-dispersive channel.

The transmission structures for both OFDM and SC-FDE schemes are depicted in Figures 1(a) and 1(b),respectively.

We consider receivers with NRx diversity branches forboth schemes. The signal associated to the l th branch issampled, and the cyclic prefix is removed leading to the

time-domain block fy.l/n InD 0; 1; : : : ; N�1g. The corres-ponding frequency-domain block, obtained after an appro-

priate size-N DFT operation, is fY .l/kI k D 0; 1; : : : ;

N � 1g, where Y .l/k

can be written as

Y.l/kD SkH

.l/kCN

.l/k

(1)

withH .l/k

denoting the overall channel frequency responsebetween the transmit antenna and the l th receive antennafor the kth frequency, Sk denoting the frequency-domain

of the transmitted block andN .l/k

denoting the correspond-ing channel noise.

2.1. Orthogonal frequency divisionmultiplexing receiver structure

Figure 2 presents a maximal-ratio combining [14] diver-sity scheme, which is implemented for each subcarrier k.The received sample for the l th receive antenna and the kthsubcarrier is given by Equation (1). The analog-to-digitalconversion, serial/parallel conversion and demodulationoperations are implicit in the block Rx.

The equalised samples f QSk I k D 0; 1; : : : ; N � 1g are

QSk D

NRxXlD1

F.l/kY.l/k

(2)

where fF .l/kI k D 0; 1; : : : ; N � 1g is the set of FDE

coefficients related to the l th diversity branch, denoted by

F.l/kD

H.l/�k

NRxPl 0D1

ˇ̌̌H.l 0/k

ˇ̌̌2 (3)

Applying Equations (1) and (3) to Equation (2), thecorresponding equalised samples can then be given by

QSk D Sk C

NRxPlD1

H.l/�k

NRxPl 0D1

ˇ̌̌H.l 0/k

ˇ̌̌2N .l/k (4)

2.2. Iterative block decision feedbackequaliser receiver structure

Although a linear FDE leads to good performance forOFDM schemes, the performance of SC-FDE can beimproved if the linear FDE is replaced by an IB-DFE:Iterative Block Decision Feedback Equaliser [7]. Thereceiver structure is depicted in Figure 3 [6, 8]. Theanalog-to-digital conversion, the serial/parallel conver-sion and the demodulation operation are implicit in theblock Rx.

In this case, for the i th iteration, the frequency-domain

block at the output of the equaliser is f QS .i/kI k D

0; 1; : : : ; N � 1g, with

QS.i/kD

NRxXlD1

F.l;i/k

Y.l/k�B

.i/kOS.i�1/k

(5)

where fF .i/kI k D 0; 1; : : : ; N � 1g are the feedforward

coefficients and fB.i/kI k D 0; 1; : : : ; N � 1g are the feed-

back coefficients. f OS .i�1/k

I k D 0; 1; : : : ; N�1g is the DFT

of the block fOs.i�1/n InD 0; 1; : : : ; N�1g, with Osn denotingthe ‘hard decision’ of sn from the previous FDE itera-tion. It can be shown that the optimum coefficients Bk and

DOI: 10.1002/ett77Trans. Emerg. Telecomms. Techs. 23:76–85 (2012) © 2011 John Wiley & Sons, Ltd.

Approaching the MFB with block transmission techniques F. Silva et al.

Map.

{ }n

sInput bitsdata

IDFT

{ }k

SInsertCP

Channel

{ }CP

ns

Noise

{ }n

yRemove

CP

(a)

Map.InsertCP

Channel

{ }n

s { }CP

nsInput bitsdata

Noise

RemoveCP

{ }n

y

(b)

Figure 1. (a) Orthogonal frequency division multiplexing transmission chain; (b) Single-carrier with frequency-domain equalisationtransmission chain.

ˆ{ }k

SDecision Device

DFT X

DFT X

( ){ }RxNkY

(1){ }ny

( ){ }RxNny

(1){ }kY

(1){ }kF

( ){ }RxNkF

{ }k

S

Rx

Rx

Figure 2. Orthogonal frequency division multiplexing receiver structure with an NRx -branch space diversity.

Fk that maximise the overall instantaneous signal-to-noiseratio (SNR), associated to the samples QSk , are [8]

B.i/kD �

0@NRxXlD1

F.l;i/k

H.l/k� 1

1A (6)

and

F.l;i/kD

�H.l/�k

˛C�1�

��.i�1/

�2�NRxPl 0D1

ˇ̌̌H.l 0/k

ˇ̌̌2 (7)

respectively, where � denotes the so called correlation

factor, ˛ D EŒjN.l/kj2�=EŒjSk j

2� (which is common toall data blocks and diversity branches) and � selected toguarantee that

1

N

N�1XkD0

NRxXlD1

F.l;i/k

H.l/kD 1 (8)

As we can see from Equations (6) and (7), the correlationfactor �.i�1/ is a key parameter for the good performanceof IB-DFE: Iterative Block Decision Feedback Equaliserreceivers because it gives a blockwise reliability measure

of the estimates employed in the feedback loop (associatedto the previous iteration), which reduces error propagationproblems. The correlation factor �.i�1/ is defined as

�.i�1/ DEŒOs

.i�1/n s�n �

EŒjsnj2�DEŒ OS

.i�1/k

S�k�

EŒjSk j2�

(9)

where the block fOs.i�1/n InD 0; 1; : : : ; N � 1g denotes thedata estimates associated to the previous iteration, that isthe hard decisions associated to the time-domain block atthe output of the FDE, fQs.i/n InD 0; 1; : : : ; N � 1g = IDFT

f QS.i/kI k D 0; 1; : : : ; N � 1g.

It should be noted that Equation (5) can be written as

QS.i/kD

NRxXlD1

F.l;i/k

Y.l/k�B

0.i/kS.i�1/k;Block (10)

where B0.i/kDB

.i/k=�.i�1/ and S

.i�1/k;Block D �

.i�1/ OS.i�1/k

(as stated before, �.i�1/ can be considered as the block-

wise reliability of the estimates f OS .i�1/k

I k D 0; 1; : : : ;

N � 1g).For the IB-DFE: Iterative Block Decision Feedback

Equaliser performance to be improved, it is possible

78DOI: 10.1002/ett

Trans. Emerg. Telecomms. Techs. 23:76–85 (2012) © 2011 John Wiley & Sons, Ltd.


Figure 3. Iterative block decision feedback equaliser receiver structure with an NRx -branch space diversity.

to use ‘soft decisions’, s.i/n , instead of ‘hard deci-

sions’, Os.i/n . Under these conditions, the ‘blockwiseaverage’ is substituted by ‘symbol averages’ [13]. This can

be carried out by using fS.i�1/k;Symbol I k D 0; 1; : : : ; N �

1g= DFT fs.i�1/n;Symbol

In D 0; 1; : : : ; N � 1g instead of

fS.i�1/k;Block I k D 0; 1; : : : ; N � 1g= DFT fs.i�1/

n;BlockIn D

0; 1; : : : ; N � 1g, where s.i�1/n;Symbol

denotes the averagesymbol values conditioned to the FDE output from pre-

vious iteration, Qs.i�1/n . To simplify the notation, we will

replace s.i�1/n;Symbol

by s.i�1/n in the following equations.For QPSK constellations, the conditional expectations

associated to the data symbols for the i th iteration are givenby

s.i/n D tanh

LI.i/n

2

!Cj tanh

LQ.i/n

2

!D �In Os

InCj�

Qn Os

Qn

(11)with the loglikelihood ratios (LLRs) of the ‘in-phase bit’

and the ‘quadrature bit’, associated to sIn and sQn , respec-

tively, given by LI.i/n D 2

�2i

QsI.i/n and LQ.i/n D 2

�2i

QsQ.i/n ,

respectively, with

�2i D1

2EŒjsn � Qs

.i/n j

2��1

2N

N�1XnD0

jOs.i/n � Qs

.i/n j

2 (12)

where the signs of LIn and LQn define the hard decisions

OsIn D˙1 and OsQn D˙1, respectively. In Equation (11), �Inand �Qn denote the reliabilities related to the ‘in-phase bit’and the ‘quadrature bit’ of the nth symbol and are given by

�I.i/n DEŒsIn Os

In �=EŒjs

In j2�D

ˇ̌̌ˇ̌tanh

LI.i/n

2

!ˇ̌̌ˇ̌ (13)

and

�Q.i/n DEŒs

Qn Os

Qn �=EŒjs

Qn j2�D

ˇ̌̌ˇ̌tanh

LQ.i/n

2

!ˇ̌̌ˇ̌ (14)

respectively. Therefore, the correlation coefficientemployed in the feedforward coefficients will be given by

�.i/ D1

2N

N�1XnD0

.�I.i/n C �

Q.i/n / (15)

Obviously, for the first iteration, we have �I.0/n D

�Q.0/n D 0 and, consequently, sn D 0.The receiver that employs ‘blockwise reliabilities’ is

referred as IB-DFE: Iterative Block Decision FeedbackEqualiser with ‘hard decisions’, whereas the receiver thatemploys ‘symbol reliabilities’ is referred as IB-DFE: Iter-ative Block Decision Feedback Equaliser with ‘soft deci-sions’. The feedforward coefficients used in both typesof IB-DFE: Iterative Block Decision Feedback Equaliserreceivers are still given by Equation (7), but the feedbackloop of the IB-DFE: Iterative Block Decision FeedbackEqualiser with ‘hard decisions’ uses the estimated datablock, weighted by a reliability coefficient common to theentire block, whereas for IB-DFE: Iterative Block Deci-sion Feedback Equaliser with ‘soft decisions’, the feedbackloop uses a different reliability coefficient for each symbol(changes from bit to bit).

The IB-DFE: Iterative Block Decision FeedbackEqualiser receiver can be implemented in two differentways, depending whether we involve the channel decodingoutput in the feedback loop or not. In the first case, the IB-DFE: Iterative Block Decision Feedback Equaliser can beregarded as a turbo equaliser implemented in the frequencydomain, and therefore, we will denote it as ‘Turbo IB-DFE:Iterative Block Decision Feedback Equaliser’. In the sec-ond case, we do not perform the channel decoding in thefeedback loop, and this receiver can be regarded as a lowcomplexity turbo equaliser implemented in the frequencydomain. Because this is not a true ‘turbo’ scheme, wewill call it ‘conventional IB-DFE: Iterative Block DecisionFeedback Equaliser’.

Thus, the main difference between ‘conventional IB-DFE: Iterative Block Decision Feedback Equaliser’and ‘Turbo IB-DFE: Iterative Block Decision FeedbackEqualiser’ is in the decision device: in the first case, the



decision device is a symbol-by-symbol soft-decision (forQPSK constellation, this corresponds to the hyperbolic tan-gent, as in Equation (11)); for the turbo IB-DFE: Itera-tive Block Decision Feedback Equaliser, a soft-in, soft-out(SISO) channel decoder is employed in the feedback loop.The SISO block can be implemented as defined in [15] andprovides the LLRs of both the ‘information bits’ and the‘coded bits’. The input of the SISO block are LLRs of the

‘coded bits’ at the FDE output, given by LI.i/n and LQ.i/n .For uncoded scenarios, it only makes sense to employ

conventional IB-DFE: Iterative Block Decision FeedbackEqualiser schemes. In coded scenarios, we could stillemploy a ‘conventional IB-DFE: Iterative Block DecisionFeedback Equaliser’ and perform the channel decodingprocedure after all the iterations of the IB-DFE: Itera-tive Block Decision Feedback Equaliser. However, becausethe gains associated to the iterations are very low atlow-to-moderate SNR values, it is preferable to involvethe channel decoder in the feedback loop, that is to usethe ‘Turbo IB-DFE: Iterative Block Decision FeedbackEqualiser’. Because we are trying to approach the MFBperformance, we always employ a Turbo IB-DFE: IterativeBlock Decision Feedback Equaliser in the coded case.

3. MATCHED FILTER BOUND

The MFB characterises the bit error rate (BER) perfor-mance of a receiver operating in fading channels, on thebasis of the assumption that a single pulse is transmit-ted on an environment without intersymbol interference(ISI). Therefore, the MFB represents the best possible errorperformance for a given receiver.

The MFB performance with NRx-order space diversityis defined as

Pb;MFB DE

264Q

0B@vuuut2Eb

N0

1

N

N�1XkD0

NRxXlD1

ˇ̌̌H.l/k

ˇ̌̌21CA375(16)

where the expectation is over the set of channel realisations

(it is assumed that EŒjH .l/kj2 D 1�).

For the particular case in which a single ray is transmit-ted between the transmitter and each receiver antenna, thechannel exhibits Rayleigh flat fading and the performanceis given by [5]

Pb;Ray D

�1��

2

�NRx NRx�1XlD0�

NRx � 1C l

l

��1C�

2

�l(17)

with

�D

vuuut EbN0

1CEbN0

(18)

However, for the general case in which different rayswith different powers are considered, the calculation ofthe MFB is more complex. The analytical expressionsfor obtaining the MFB in uncoded scenarios when wehave multipath propagation and diversity are presented inAppendix A.

Because for the case with channel coding (in this casewith convolutional codes) it is very difficult to obtain ana-lytical BER expressions, even for an ideal additive whiteGaussian noise channel, the MFB needs to be computedby simulation.

4. PERFORMANCE RESULTS

In this section, we present a set of performance resultsconcerning the impact of the number of multipath com-ponents and the diversity on the performance of OFDMand SC-FDE receivers as well as the correspondent MFB.In both cases, we consider blocks with N D 512 ‘use-ful’ data symbols, plus an appropriate cyclic prefix. It wasobserved that the performance is independent of the num-ber of symbols N of each transmitted block. The mod-ulation symbols are selected from a QPSK constellationunder a Gray mapping rule. Unless otherwise stated, thechannel is characterised by a uniform power delay profile(PDP), with U D U1 D � � � D UNRx equal-power symbol-spaced multipath components and uncorrelated Rayleighfading for all diversity branches. However, we would liketo point out that we have a similar behavior for other chan-nels such as exponential PDP. The major difference was onthe higher number of multipath components needed to havesimilar results to those of the uniform PDP. This is due tothe fact that the number of relevant multipath componentsis lower for the exponential PDP because the last ones havemuch lower power. We assumed linear power amplificationat the transmitter and perfect synchronisation and channelestimation at the receiver.

Our performance results are expressed as function ofEb=N0, where N0 is the one-sided power spectral densityof the noise and Eb is the energy of the transmitted bits(i.e. the degradation due to the useless power spent on thecyclic prefix is not included).

Figure 4 shows the typical behavior of the BER, foran IB-DFE: Iterative Block Decision Feedback Equaliser,without channel coding, and a channel with U D 16 sep-arable multipath components for each diversity branch.The SC-FDE employs an IB-DFE: Iterative Block Deci-sion Feedback Equaliser receiver with four iterations andthe particular case with a single iteration that correspondsto a linear FDE. Clearly, there is a significant perfor-mance improvement with the subsequent iterations, and the

80DOI: 10.1002/ett



−2 0 2 4 6 8 10 12 14 16

10-1

10-2

10-3

10-4

BE

RIter. 1(SC-FDE)Iter. 2Iter. 4MFB

− − − − :_______ :− ⋅ − ⋅ − :⋅ ⋅ ⋅ ⋅ ⋅ ⋅ :

Diversity order:: N Rx = 1: N Rx = 2

100

Eb/N

0(dB)

Figure 4. Bit error rate performance of an iterative block deci-sion feedback equaliser in the uncoded case.

asymptotic BER performance becomes closer to the MFB.We did not include the OFDM results because the uncodedOFDM performance is very poor because OFDM schemesare severely affected by deep-faded subcarriers.

Let us now evaluate the impact of channel coding. Forboth modulation schemes, we employed a channel encoderbased on a 64-state, 1/2-rate convolutional code with thepolynomials generators 1 C D2 C D3 C D5 C D6 and1CDCD2CD3CD6. We also considered punctured ver-sions of this code (with rates 2/3 and 3/4), with the purposeof increase the user bit rate while maintaining the grossbit rate [16]. The coded bits are interleaved before beingmapped into the constellation points and distributed by thesymbols of the block.

We start by presenting the BER results of a codedtransmission considering both a SC-FDE (with a TurboIB-DFE: Iterative Block Decision Feedback Equaliserreceiver), and OFDM schemes with the same chan-nel encoder, and a channel with U D 8 relevantseparable multipath components for each diversity branch.Figures 5–7 consider 1/2-rate, 2/3-rate and 3/4-rate,respectively. It can be seen from Figure 5 that with a rate-1/2 convolutional code, the OFDM modulation has slightlybetter performance than SC-FDE with a linear FDE (cor-responding to the IB-DFE: Iterative Block Decision Feed-back Equaliser’s first iteration). However, for the followingiterations, Turbo IB-DFE: Iterative Block Decision Feed-back Equaliser outperforms OFDM. Figures 6 and 7 showthe results obtained with the 2/3-rate and 3/4-rate con-volutional code, respectively. It is clear that for highercode rates, the SC modulation has better performance thanOFDM, even with only a single iteration.

Next, we present the required values of Eb=N0 for aBER D 10�4 for the SC-FDE and OFDM, as well asthe corresponding MFB (Figure 8–10). These values areexpressed as a function of the number of multipath com-ponents U . It can be observed that SC-FDE has an overallperformance advantage over the OFDM, especially whenemploying the IB-DFE: Iterative Block Decision Feedback

− − − − :_______ :− ⋅ − ⋅ − :⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ :

Turbo IB-DFE, Iter. 1 (SC-FDE)Turbo IB-DFE, Iter. 4Coded OFDMMFB


−2 0 2 4 6 8

10-1

10-2

10-3

10-4

BE

R

Eb/N

0(dB)

100

Figure 5. Bit error rate performance for a rate-1/2 code.

Turbo IB-DFE, Iter. 1(SC-FDE)Turbo IB-DFE, Iter. 4Coded OFDMMFB

− − − − :_______ :− ⋅ − ⋅ − :⋅ ⋅ ⋅ ⋅ ⋅ ⋅ :


−2 0 2 4 6 8 10 12

10-1

10-2

10-3

10-4

BE

R

100

Eb/N

0(dB)


−2 0 2 4 6 8 10 12 14 16 18

Turbo IB-DFE, Iter. 1(SC-FDE)Turbo IB-DFE, Iter. 4Coded OFDMMFB

− − − − :_______ :− ⋅ − ⋅ − :⋅ ⋅ ⋅ ⋅ ⋅ ⋅ :


10-1

10-2

10-3

10-4

BE

R

Eb/N

0(dB)

100


Equaliser with turbo equalisation and/or diversity. There-fore, by using SC modulation with turbo equalisation andchannels with a high number of multipath components, wecan be very close to the MFB after a few iterations (natu-rally, for U D 1, the BER is identical to the MFB, althoughthe performance is very poor because this corresponds to



0 20 40 60 80 100

−2

0

2

4

6

8

10

12

14

Eb/N

0(dB

) fo

r B

ER

=10

−4

− − − − :_______ :− ⋅ − ⋅ − :⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ :



U

Figure 8. Required Eb=N0 to achieve BER D 10�4 for the rate-1/2 convolutional code, as function of the number of multipath

components.

0 20 40 60 80 100

0

2

4

6

8

10

12

14

16

Eb/N

0(dB

) fo

r B

ER

=10

−4

− − − − :_______ :− ⋅ − ⋅ − :⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ :



U


components.

a flat fading channel). It is also possible to observe thatthe improvements with the iterations are higher withoutdiversity. This is also the case where a higher number ofmultipath components are required to allow performancesclose to the MFB (about U D 70).

Finally, we might ask what happens for different fad-ing models. To answer this question, we considered thecase where we have Nakagami fading with factor � oneach multipath component (clearly, � D 1 correspondsto the Rayleigh case and � D C1 corresponds to casewhere there is no fading on the different multipath compo-nents). Figure 11 presents the required values of Eb=N0for BER=10�4 as a function of the number of multipathcomponents, concerning the MFB and an IB-DFE: IterativeBlock Decision Feedback Equaliser with four iterations.

It should be pointed out that the performance degra-dation is due to two main factors: the fading effects andthe residual ISI. The fading effects in each ray (and, con-sequently, on the overall received signal) decrease as weincrease the Nakagami factor �; they also reduce when weincrease the number of components because of multipath

0 20 40 60 80 1000

5

10

15

20

Eb/N

0(dB

) fo

r B

ER

=10

−4

− − − − :_______ :− ⋅ − ⋅ − :⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ :



U


components.

0 5 10 15 20 25 308

10

12

14

16

18

20

22

N Ray

Iter. 4MFB

1416

_______ :− ⋅ − ⋅ − :

Figure 11. Required Eb=N0 to achieve BER D 10�4 at thematched filter bound and at the fourth iteration of the itera-tive block decision feedback equaliser, for an uncoded scenariowithout diversity and with a Nakagami channel model with

factor �.

effects. That is why the MFB is better for larger values of� and/or larger number of components. The IB-DFE: Iter-ative Block Decision Feedback Equaliser is a very efficientequaliser, being able to reduce significantly the residual ISIeffects, especially for large number of components. Natu-rally, the overall performance will be the result of thesecombined effects. For Rayleigh fading channels, the fad-ing effects are very strong and have a higher impact on theperformance than the residual ISI effects when we havea small number of multipath components. Therefore, theperformance improves steadily as we increase the numberof components. When we have smaller fading effects oneach ray (as in Nakagami channels with � > 1), the degra-dation due to residual ISI becomes more relevant, espe-cially when we have a small number of components (butmore than one). This leads to the somewhat unexpected IB-DFE: Iterative Block Decision Feedback Equaliser behav-ior of Figure 11 where there is a slight degradation as weincrease the number of components up to a value, after

82DOI: 10.1002/ett



−1 0 1 2 3 4 5 6

− − − − :_______ :− ⋅ − ⋅ − :⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ :

Linear DFETurbo IB-DFE, Iter. 4Coded OFDMMFB

2 Rays8 Rays32 Rays

10-1

10-2

10-3

10-4

BE

R100

Eb/N

0(dB)

Figure 12. Bit error rate performance of orthogonal frequencydivision multiplexing and single-carrier with frequency-domainequalisation, for U D 2, 8 and 32, and a Nakagami channel

with �D 4.which we have the steadily improvement with the numberof components.

With respect to the coded case, the impact of the factor� on the performance is negligible for a large number ofmultipath components. Figure 12 shows the performanceof OFDM and SC-FDE (with both a linear FDE and a turboFDE with four iterations), as well as the MFB, for 2, 8 and32 multipath components. Clearly, the best performance isachieved for the turbo FDE and the worse performance forthe linear FDE, with the performance of OFDM schemessomewhere in between. The difference between the MFBand the achieved performance is higher for a moderatenumber of propagation components (around 8).

5. CONCLUSIONS

In this paper, we studied the impact of the number of multi-path components and the diversity order on the asymptoticperformance of OFDM and SC-FDE, in different channelcoding schemes.

It was shown that the number of relevant separablemultipath components is a fundamental element that influ-ences the performance of both schemes and, in theIB-DFE: Iterative Block Decision Feedback Equaliser’scase, the gains associated to the iterations. Our resultsdemonstrate that SC-FDE has an overall performanceadvantage over OFDM, especially when employing the IB-DFE: Iterative Block Decision Feedback Equaliser, espe-cially when there is a high number of separable multipathcomponents because it allows a performance very close tothe MFB, even without diversity. When we have diversitythe performance approaches MFB faster, even for a smallnumber of separable multipath components.

APPENDIX A: ANALYTICALCOMPUTATION OF THE MATCHEDFILTER BOUND

In this appendix, we present an analytical approach toobtain the MFB without channel coding (for the coded

case, we do not have closed formulas for the BER; there-fore, the MFB is computed by simulation).

We derive the MFB using an approach similar to [17].Let us consider the case of a transmission over a multi-path Rayleigh fading channel withNRx diversity branches,where all branches can have different fading powers orcan be correlated. Assuming a discrete multipath channelfor each diversity branch l , composed of Ul discrete taps,where the magnitude of each tap i has a mean square valueof �2

i;l, the channel response, at time t , to a pulse applied

at t � � can be modeled as

cl .�; t/D

UlXiD1

˛i ;l .t/ ı�� i ;l

�; l D 1 : : : NRx

(19)with ˛i ;l .t/ being a zero-mean complex Gaussian randomprocess, �i ;l the respective delay (assumed constant) andı.t/ the Dirac function. For the derivation of the MFB, weassume a transmission of one pulse s � g.t/, where s is asymbol of a QPSK constellation and g.t/ is the impulseresponse of the transmit filter.

Assuming a slowly time-varying channel, the sum of thesampled outputs, from the matched filters of the diversitybranches, can be written as

y .t D t0/Ds �

NRxXlD1

UlXiD1

UlXi 0D1

˛i ;l˛�i 0;l

R��i ;l � �i 0;l

�C

NRxXlD1

l (20)

where l represents additive white Gaussian noise sampleswith power spectral density N0 and R .�/ is the autocor-relation function of the transmit filter. The instantaneousSNR is given by SNR D 2Eb

N0�, where Eb denotes the

average bit energy and � is defined as

� D

NRxXlD1

UlXiD1

UlXi 0D1

˛i ;l˛�i 0;lR

��i ;l � �i 0;l

�D zH†z

(21)In (21), z represents aUtotal�1 (withUtotal D

PNRxlD1

Ul )

vector containing the random variables ˛i ;l , and zH

denotes the conjugate transpose of z. † is a Utotal � UtotalHermitian matrix constructed as

† D

266666664

R1 � � � 0

:::: : :

:::

0 � � � RNRx

377777775

(22)



where Rl is a matrix associated to the l th diversity branch,defined as

Rl D

264

R .0/ � � � R��Ul ;l � �1;l

�:::

: : ::::

R��1;l � �Ul ;l

�� R .0/

375(23)

For a QPSK constellation, the instantaneous BER can bewritten as

Pb .�/D1

2erfc

sEb

N0�

!(24)

where erfc(x) is the complementary error function. Toobtain the probability density function (PDF) of �, wewill write � as a sum of uncorrelated random variableswith known PDFs. Denoting ‰ as the covariance matrixof z (‰ D Cov Œz�), which is Hermitian and positive-semidefinite, it is possible to decompose ‰ into ‰ DQQH . In fact, if we apply the Cholesky decomposition,Q will be a lower triangular matrix. Moreover, using thismatrix, we can define a new vector z0 D Q�1z, whosecomponents will be uncorrelated unit-variance complexGaussian variables and � becomes

� D z0HQH†Qz0 D z0H†0z0 (25)

with

†0 DQH†QDˆƒˆH (26)

whereƒ is a diagonal matrix whose elements are the eigen-values i (i D 1; : : : ; Utotal) of †0 and ˆ is a matrixwhose columns are the orthogonal eigenvectors of †0.The decomposition of †0 in Equation (26) is possiblebecause of its Hermitian property. We can then rewriteEquation (25) as

� D z0HˆƒˆH z0 D z00Hƒz00 DUtotalXiD1

iˇ̌z00i

ˇ̌2 (27)

where we have defined two more vectors, z00H D z0Hˆand z00 D ˆH z0, whose components are still uncorre-lated unit-variance complex Gaussian variables. Accordingto Equation (27), � can be expressed as a sum of inde-pendent random variables with exponential distributionswhose characteristic function is

Ene�j��

oD

UtotalYiD1

1

1C ji�(28)

If there are U 0 distinct eigenvalues, each with a multi-plicity of qi , iD 1 : : : U 0, we can apply the inverse Fouriertransform to Equation (28) and obtain the PDF of � as

p .�/D

U 0XiD1

qiXmD1

Ai ;m

qii .qi �m/Š .m� 1/Š

�m�1e� ��i

(29)

with

Ai ;m D

2666664@qi�m

@sqi�m

0BBBBB@

U 0Yj D 1

j ¤ i

1�1C sj

�qj

1CCCCCA

3777775sD� 1

�i

(30)It is easy to verify that the average BER can be obtained

as

Pbav D

Z C1�1

Pb .�/ p .�/ d�DU 0XiD1

qiXmD1

Ai ;m

qi�mi .qi �m/Š�

1��i

2

m�

m�1XrD0

�m� 1C r

r

��1C�i

2

rwhere

�i D

vuut EsN0i

1C EsN0i

(31)

ACKNOWLEDGEMENTS

This work was partially supported by Fundação para aCiência e Tecnologia (pluriannual funding and ADCODprojects).

REFERENCES

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AUTHORS’ BIOGRAPHIES

Fábio Silva received his MS degree from Faculdade deCiências e Tecnologia da Universidade Nova de Lisboa(FCT-UNL), Portugal, in 2010. During that year, he wasa member of the research center Institute for Systems andRobotics (ISR) at Instituto Superior Técnico (IST).

He is currently working towards his PhD degree andas a researcher at Instituto de Telecomunicações (IT),Lisbon, Portugal.

He has been involved in research projects in thebroadband wireless communications area. His mainresearch interests include modulation, equalisation andchannel coding.

Rui Dinis received his PhD degree from Instituto SuperiorTécnico (IST), Technical University of Lisbon, Portugal, in2001. From 2001 to 2008, he was a professor at IST. Since2008, he is teaching at Faculdade de Ciências e Tecnologiada Universidade Nova de Lisboa (FCT-UNL).

He was a researcher at Centro de Análise e Processa-mento de Sinais (CAPS at IST) from 1992 to 2005; from2005 to 2008, he was researcher at Instituto de Sistemase Robótica (ISR at IST); in 2009, he joined the researchcenter Instituto de Telecomunicações (IT).

He has been involved in several research projects inthe broadband wireless communications area. His mainresearch interests include modulation, equalisation andchannel coding.

Nuno Souto graduated in aerospace engineering (avionicsbranch) in 2000 at Instituto Superior Técnico, Lisbon, Por-tugal. From November 2000 to January 2002, he workedas a researcher in the field of automatic speech recognitionat Instituto de Engenharia e Sistemas de Computadores,Lisbon, Portugal. He finished his PhD at Instituto SuperiorTécnico in 2006.

He currently works as an assistant professor at InstitutoUniversitário de Lisboa (ISCTE-IUL) and as a researcherat Instituto de Telecomunicações (IT), Lisbon, Portugal.

He participated in several European research projects.His research interests include wideband CDMA systems,OFDM, channel coding, modulation, channel estimationand MIMO systems.

Paulo Montezuma received his PhD degree from Fac-uldade de Ciências e Tecnologia da Universidade Novade Lisboa (FCT-UNL) in 2007. Since 2001, he is teach-ing at the Department of Electrical Engineering (DEE) ofFCT. He was a researcher at Centro de Análise e Proces-samento de Sinais/Instituto Superior Técnico (CAPS/IST)from 1996 to 2005. In 2001, he joined the Uninova-CTSresearch centre.

He has been involved in several research projects inTelecommunications area, both national and European. Hismain research interests include modulation, channel cod-ing, signal design for wireless communications, nonlineareffects and spread spectrum systems.


Approaching the matched filter bound with block transmission techniques

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