· ict318362emphatic date: 3/4/2014 enhanced multicarriertechniquesfor professional ad-hoc...

ICT 318362 EMPhAtiC Date: 3/4/2014

Enhanced Multicarrier Techniques for Professional Ad-Hocand Cell-Based Communications

(EMPhAtiC)

Document Number D4.2

MIMO channel estimation and data detectionContractual date of delivery to the CEC: 28/02/2014Actual date of delivery to the CEC: 07/04/2014Project Number and Acronym: 318362 EMPhAtiCEditor: Eleftherios Kofidis (CTI), Christos

Mavrokefalidis (CTI)Authors: Christos Mavrokefalidis (CTI), Eleftherios

Kofidis (CTI), Athanasios Rontogiannis(CTI), Antonios Beikos (CTI), MarkkuRenfors (TUT), Juha Yli-Kaakinen (TUT),Jerome Louveaux (UCL)

Participants: CTI, TUT, UCLWorkpackage: WP4Security: Public(PU)Nature: ReportVersion: 1.2Total Number of Pages: 66

Abstract:This report summarizes the results from research efforts within WP4 (Task 4.2) on the problemsof channel estimation and equalization/detection for MIMO FB-MC systems. Both spatial multi-plexing and space-time diversity schemes are studied. The focus is on doubly dispersive channels,corresponding to propagation conditions of high time- and frequency-selectivity. The studied topicsinclude optimal preamble-based channel estimation, efficient linear and decision-feedback adaptiveequalization algorithms, and turbo equalization for space-time bit-interleaved coded modulation.Extensive simulation results are reported that confirm the applicability of the proposed techniquesin realistic environments. Important remaining questions are pointed out and possible future re-search directions to address them are proposed.

Keywords: BCJR, BICM, channel equalization, channel estimation, coded modulation, CP-OFDM, DFE, FB-MC, FBMC/OQAM, FC-FB, FMT, iterative receivers, LTE, PMR, pream-ble, RLS, scattered pilots, turbo equalization

ICT-EMPhAtiC Deliverable D4.2 1/66


Document Revision HistoryVersion Date Author(s) Summary of main changes0.1 26.02.2014 C. Mavrokefalidis First draft of Chapter 4

0.2 18.03.2014 E. Kofidis Chapter 3

0.3 18.03.2014 M. Renfors and J.Yli-Kaakinen

Chapter 5

0.4 18.03.2014 J. Louveaux Chapter 6

0.5 23.03.2014 M. Renfors and J.Yli-Kaakinen

Updating some figures in Chapter 5

0.6 26.03.2014 E. Kofidis Revising Chapter 4

0.7 27.03.2014 J. Louveaux Revising Chapter 6

1.0 30.03.2014 C. Mavrokefalidis Homogenizing the notation; first draft ofChapters 1, 2, and 7

1.1 31.03.2014 E. Kofidis Revising Chapters 1, 2, and 7

1.2 03.04.2014 E. Kofidis Incorporating comments of other authors



Table of Contents

1 Introduction 4

2 Background 6

3 Preamble-based MIMO-FBMC/OQAM Channel Estimation 83.1 Optimal complex-valued preambles . . . . . . . . . . . . . . . . . . . . . . . 93.2 Optimal real-valued preambles . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 What about MIMO-OFDM? . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 Simulation example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Adaptive Equalization of Doubly Dispersive Channels in MIMO-FBMC/OQAMSystems 164.1 Adaptive decision feedback equalization . . . . . . . . . . . . . . . . . . . . . 17

4.1.1 BLAST RLS-based DFE equalization . . . . . . . . . . . . . . . . . . 174.1.2 Channel estimation-based initialization of the DFE filters . . . . . . . . 264.1.3 Using pilots scattered throughout the frame . . . . . . . . . . . . . . 32

5 Fast-convolution Implementation of Linear Equalization Based MultiantennaDetection Schemes 395.1 Frequency sampling based multi-tap subcarrier equalizer approach . . . . . . . 395.2 Embedded subcarrier equalization in FC-FB . . . . . . . . . . . . . . . . . . . 395.3 Simulation-based performance evaluation . . . . . . . . . . . . . . . . . . . . 415.4 Complexity evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6 Iterative Receivers for Bit-interleaved Coded MIMO-FBMC/OQAM 466.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.2 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.3 Iterative receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.3.1 Virtual complex equalizer . . . . . . . . . . . . . . . . . . . . . . . . 496.3.2 Full real model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.4.1 Iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.4.2 Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.4.3 Comparison of the correlation matrix approximation . . . . . . . . . . 566.4.4 Advantage of the full model . . . . . . . . . . . . . . . . . . . . . . . 57

7 Conclusions and Future Research 60

8 References 62



1. IntroductionUsing multiple antennas at the transmitter and/or the receiver side has been proven to bean enabling technology for providing increased reliability and capacity of data transmissions,without sacrificing the (especially, nowadays) scarce available bandwidth. This is achievedthrough the exploitation of the spatial dimension along with the time and frequency dimensionsof the signal [1]. The capacity of the transmission can be increased by assigning a differentstream of data to each transmit antenna, leading to the so-called spatial multiplexing (SM)gains and associated transmission schemes. On the other hand, a carefully coded stream of datacan be shared by all transmit antennas, offering diversity gains via the associated space-timecoding (STC) schemes.

Modern wireless communication systems are expected to accommodate numerous applica-tions and services that have quite different requirements in terms of data rate and reliability.These applications range from classical voice and data communications found in cellular-basedsystems to more elaborate ones such as the Professional Mobile Radio (PMR), utilized by publicsafety organizations [2]. The efficient use of the spectrum in these contexts requires a high levelof flexibility as far as spectrum access is concerned and the ability that different transmissionsystems operate in dense neighboring bands that are not necessarily contiguous. Multicarriermodulation (MCM) is capable of providing the desired flexibility by separating the availablespectrum into subchannels whose bandwidth and center frequency can be configured appropri-ately and assigned on demand to each different wireless system. In this context, the increasedrequirements for high data transmission rate and reliability point to the need of combiningMCM with the multiple antenna (MIMO) technology.

The result of this combination, namely MIMO using orthogonal frequency division multi-plexing (OFDM), has demonstrated a great potential in current wireless communications, andit has therefore been a central part of 4G wireless standards recommendations [3]. However, thewell known limitations of OFDM, with the bad spectral containment and the bandwidth/powerloss induced by the use of guard intervals being the most important ones, carry over to MIMO-OFDM systems as well, making them rather inadequate for the highly demanding – in flexibility,reliability, and spectral efficiency – networks of the future generation, particularly those of anad hoc nature. Filter bank-based multicarrier (FB-MC) systems have shown the potentialto overcome these limitations [4]. Two important examples are FB-MC using offset QAM(FBMC/OQAM) [5] and Filtered MultiTone (FMT) [6]. Both schemes are able to offer highspectral containment, with the former offering the possibility for maximum spectral efficiency,at a much lower computational complexity. FBMC/OQAM, however, is significantly more chal-lenging (due to its intrinsic interference effect) in designing basic receiver operations, such aschannel estimation / equalization and detection of data streams. Solutions to these challengesare reported in the EMPhAtiC deliverables D3.1 and D3.2, for single-antenna systems. In MIMOsystems, the problems are even more interesting, in view of the additional (spatial) interferencecoming from the use of multiple antennas [4].

The EMPhAtiC deliverable D4.1 focused on efficient detection techniques for MIMO FBMC/OQAM, including space-time-frequency coding. The feasibility of using compact antenna arraysin a PMR scenario was also investigated. The aim of the present document is to report workdone within WP4, Task 4.2 on MIMO-FBMC/OQAM channel estimation, equalization (withemphasis on doubly selective channels), and efficient detection algorithms relying on the turboequalization and coded modulation ideas. Notably, embedding linear adaptive equalization inthe fast convolution filter bank (FC-FB) structure is studied. The realistic scenario of a highlyfrequency-(and time-) selective channel is considered throughout. This results in subchannels



of nonnegligible frequency selectivity, challenging earlier solution approaches that relied on afrequency flat subchannel model. It is worth noting that such a scenario is far more realisticin a broadband PMR environment, characterized by high data rates and high mobility. Thelatter requirement in turn advocates for higher subcarrier spacing and hence higher frequencyselectivity inside subchannels.

The rest of the document is organized as follows. Background information, including thedefinition of the MIMO FB-MC system under study, basic assumptions, and notation used inthe following, is presented in Chapter 2. Preamble-based estimation of the MIMO channelimpulse response is studied in Chapter 3. This work extends related work presented in D3.1 forsingle-antenna systems. Again, no assumption is made on the channel frequency selectivity, andoptimal (in the mean squared error (MSE) sense) preambles of the shortest possible durationare designed. The same problem is addressed for MIMO-OFDM as well. The simulation resultsreported therein show the efficacy of the estimation methods and preambles, most notablydemonstrating the absence of the MSE error floor typically appearing in FBMC/OQAM channelestimation [7]. Chapter 4 is devoted to adaptive equalization of doubly dispersive MIMOchannels. The algorithm presented stems from the equivalence of the Bell Labs Layered Space-Time (BLAST) scheme for optimal input recovery in a MIMO system with the generalizeddecision feedback equalization (DFE) structure. Both the BLAST ordering adaptation and theupdate of the equalizer filters are performed via an efficient, numerically robust recursive leastsquares (RLS) algorithm, able to perform satisfactorily in a highly time-/frequency-selectiveenvironment. Important improvements on the basic form of the algorithm are brought by achannel estimate-based initialization and tracking, resulting in a considerable reduction of therequired training overhead. The algorithm is extensively evaluated for both FBMC/OQAM andFMT, as well as OFDM systems. In Chapter 5, linear adaptive equalization is studied in thecontext of the FC-FB structure developed in D2.1, with emphasis on FBMC/OQAMmodulation.Minimum MSE (MMSE) frequency sampling based equalization is studied and embedded inthe FC-FB model. The effectiveness of the proposed scheme is successfully demonstrated inboth low and high mobility scenarios, also for harsh propagation scenarios typical in PMR.The challenging task of STC in MIMO-FBMC/OQAM systems is revisited in Chapter 6. Theapproach that is followed therein combines the ingredients that work well in FBMC/OQAM,such as the linear MIMO equalizer and the interference cancellation, through the adoption of abit-interleaved coded structure with turbo equalization. This structure allows to benefit fromdiversity with a lot of flexibility in the choice of the code, and still acceptable complexity. Theassociated computational burden is shown to be effectively reduced via alternative approximationtechniques. Chapter 7 reports conclusions and summarizes some directions for future research.



2. BackgroundHere, the basic input-output relations for a MIMO FB-MC system with NT transmit and NRreceive antennas are recalled (Fig. 2-1). The output signal of the synthesis filter bank (SFB)at the tth transmit antenna and at time l is given by [5]

st(l) =M−1∑k=0

∑n

dtk,ngk,n(l), (2.1)

where, for FBMC/OQAM, dtk,n are T/2-spaced (with T being the symbol period) real OQAMsymbols (for t = 1, 2, . . . , NT), generated by a complex to real OQAM modulator and

gk,n(l) = g(l − nM2

)e 2πMk

(l−Lg−1

2

)eφk,n , (2.2)

with g being a real symmetric prototype filter impulse response of length Lg = MK andunit energy. M is the (even) number of subcarriers, K is the overlapping factor and φk,n =(k+n)π2−knπ [5]. Finally, the pair (k, n) corresponds to a frequency-time point with subcarrierindex k and time index n.

The signal st is transmitted through doubly (i.e., time- and frequency-) dispersive channelswith impulse responses hjt(l, p), between the tth transmit and jth receive antenna at discretetime l, where p = 0, 1, . . . , Lh − 1. Lh is the length of the channel impulse responses (CIR),which is assumed the same (without loss of generality) for all pairs of transmit-receive antennas.Additive white Gaussian noise (AWGN) with zero mean and spatial covariance σ2INR is assumedto be present at the receiver front-end.

The received signal at the output of the kth subchannel of the analysis filter bank (AFB)of the jth receive antenna at time n is written as

yjk,n =NT∑t=1

M−1∑m=0

Lc−1∑p=0

cjtkm(n, p)dtm,n−p + ηjk,n, (2.3)

where cjtkm(n, p) is the impulse response of the subchannel from the mth input of the SFB atthe tth transmit antenna to the kth output of the AFB at the jth receive antenna at time n,and p = 0, 1, . . . , Lc − 1 with Lc =

⌈2Lg+Lh−2

M/2

⌉. This is the convolution of gm,n, hjt, and gk,n,

SFB

...

SFB

d1

dNT

NRxNT

channel

AFB

...

AFB

y1

yNR

+

+

w1

wNR

Figure 2-1: The MIMO FB-MC system.



downsampled by M2 [8]. Moreover, ηjk,n is the noise term (of zero mean and variance σ2) at the

output of the AFB. More information on its statistics will be given shortly.Collecting the received signals at subchannel k of all receive antennas yields the following

input-output relation:

yk,n =M−1∑m=0

Lc−1∑p=0

Ckm(n, p)dm,n−p + ηk,n, (2.4)

where

yk,n =[y1k,n y2

k,n · · · yNRk,n

]T,

dm,n =[d1m,n d2

m,n · · · dNTm,n

]T,

ηk,n =[η1k,n η2

k,n · · · ηNRk,n

]Tand Ckm(n, p) is the NR×NT matrix of cjtkm(n, p)’s. As for the noise term ηk,n, it is known tobe also Gaussian, with zero mean. Since the noise at the AFB inputs is spatially uncorrelated,the same holds at the outputs of the AFBs, that is, E{ηj1k,n(ηj2k,n−ζ)∗} = 0, for j1 6= j2 and forall ζ. The noise temporal correlation, at a single antenna and a single subcarrier k, is givenby [9, 7]

rηk(ζ) = E{ηjk,n(ηjk,n−ζ)∗} = σ2[−(−1)k]ζ∑l

g(l − nM2

)g(l − (n− ζ)M2

)(2.5)

For prototype filters g that are well-localized in time, it can be assumed that rηk(ζ) ≈ 0 forζ ≥ 2 and based on (2.5) it can be shown that rηk(0) = σ2 and rηk(±1) = ∓(−1)kγσ2, wherethe notation γ = ∑Lg−1

l=M2g (l) g

(l − M

2

)is adopted from [7].

For prototype functions that are well-localized in frequency, it can be assumed that inter-ference exists only between neighboring subcarriers. Thus, (2.4) can be written as

yk,n ≈k+1∑

m=k−1

Lc−1∑p=0

Ckm(n, p)dl,n−p + ηk,n (2.6)

In view of the above expression for the noise correlation, one can see that E{ηk,n(ηk,n)H} =σ2INR , E{ηk,n(ηk,n+1)H} = σ2(−1)kγINR and E{ηk,n(ηk,n−1)H} = −σ2

η(−1)kγINR .Notation. Lower and upper case bold letters will be used to denote vectors and matrices, respec-tively. Superscripts T and H will respectively signify transposition and conjugate (Hermitian)transposition. The Moore-Penrose pseudoinverse of a matrix A is denoted by A†. Complexconjugation will be denoted by ∗. 0r×s will be the all zeros matrix of dimensions r × s. Is isthe sth-order identity matrix. The r × 1 vector of all ones will be denoted by 1r. Dimensionswill be omitted whenever understood from the context. The diagonal matrix with the entries ofthe vector x on its main diagonal will be denoted by diag(x). E{·}, <{·} and ={·} stand forexpectation, real and imaginary part, respectively. � is the Hadamard (element-wise) product.The Euclidean norm will be denoted by ‖ · ‖. will denote the imaginary unit.



3. Preamble-based MIMO-FBMC/OQAM Channel Estima-tion

Let hi,j be the CIR of the channel from the transmit antenna j to the receive antenna i andassume (without loss of generality) that all NTNR channels have the same length, Lh. Thischapter is devoted to developing an estimation procedure for such a system, with trainingbased on a preamble and with no assumption made on the channel frequency selectivity (asit has been the case in previous related works [10, 8]). Moreover, optimal preambles of theshortest possible duration – namely consisting of one pilot FBMC symbol accompanied by atleast one guard symbol – are designed. These results extend those of D3.1 from the single- tothe multiple-antenna setup.

It is not difficult to extend the input/output equation developed in D3.1 to the presentcontext. Indeed, one can write

yi =NT∑j=1

Γ(dj)hi,j + ηi, i = 1, 2, . . . , NR, (3.1)

where yi contains the AFB outputs at the ith receive antenna and similarly for the noise signalsηi, and Γ(dj) is an M ×Lh matrix defined as in D3.1, with dj being the FBMC symbol inputto the SFB of the jth transmit antenna. Recall that the latter matrix is a priori known, as itonly depends on the filter bank parameters and the FBMC input. The above can be writtenmore compactly as

yi =[

Γ(d1) Γ(d2) · · · Γ(dNT)]

︸︷︷︸Γ

hi,1

hi,2

...

hi,NT

︸︷︷︸hi,·

+ηi

= Γhi,· + ηi (3.2)

Collecting the above for all receive antennas results in

y =

Γ 0 · · · 0

0 Γ · · · 0... ... . . . ...

0 0 · · · Γ

︸︷︷︸

¯Γ

h1,·

h2,·

...

hNR,·

︸︷︷︸

h

+η

= ¯Γh+ η (3.3)

Assuming, as usual, that the noise signals at different receive antennas are uncorrelated witheach other (i.e., spatially white noise), the covariance matrix of the noise term in the latterequation has also a block diagonal structure, and hence it is sufficient to address the estimationproblem separately for each receive antenna, as suggested by (3.2). As for the noise covariance



in the latter problem, it is obviously the same with the corresponding single-input single-outputproblem (see D3.1), namely

Cηi = σ2B,

with B being a circulant matrix depending on the amount of correlation with adjacent sub-carriers (see D3.1). It is then obvious that the Gauss-Markov estimate for (3.2) will be givenby

hi,· = (˜Γ

H ˜Γ)−1 ˜ΓHyi, (3.4)

with the notation · having the meaning of a pre-whitening transformation, that is, multiplicationby a Hermitian square root of B. The above presumes that the matrix Γ is full column rankand hence tall, that is,

M ≥ NTLh (3.5)

3.1 Optimal complex-valued preambles

Making the assumption in (3.5) and applying the optimality condition for the matrix ˜Γ, namelythat its columns should be orthogonal for the preambles dj to be MSE-optimal, results in theconclusion that each of them should be of the type

dj =√√√√ ENTλmj

fmj , (3.6)

where fmj and λmj are the corresponding column of the M ×M normalized DFT matrix Fand eigenvalue of B, respectively. E denotes the total transmit energy budget (at the SFBoutput), which in this context can be written as

NT∑j=1

(dj)HBdj ≤ E ,

and it is assumed that it is uniformly distributed to the NT transmit antennas. In additionto (3.6), one must take care not to choose the samem for two antennas, as this would result in arank deficient matrix Γ. The optimal value formj can be found in the way described in D3.1. Infact, recalling the orthogonality conditions from that deliverable and looking at the structure ofthe matrices involved therein, one can easily see that oncemj has been determined, the possiblevalues forms, s 6= j, exclude the set {mj, ((mj+1))M , ((mj+2))M , . . . , ((mj+Lh−1))M}, where((·))M denotes circular ordering in the set {1, 2, . . . ,M}. This implies that, as also suggestedby (3.5), the more transmit antennas and CIR taps there are, the fewer degrees of freedom areavailable in designing the corresponding preambles. This limitation can be mitigated if longer(than one pilot FBMC symbol) preambles are considered. Such designs, where the duration ofthe preamble is proportional to the number of transmit antennas, have been previously studied(cf. [10, 7]), yet only for the flat subchannel case. For the latter simpler scenario, MSE-optimalsingle-symbol preambles (of the sparse type) for MIMO-FBMC/OQAM were developed in [8](see also [7]) and later adopted in [11].

3.2 Optimal real-valued preambles

It is of interest to specialize the above in the particular but highly realistic case of 2 transmitantennas and real-valued pilot symbols. One can then obtain an optimal design directly (i.e.,



without search) from

d1 =√E2f 1 =

√E

2M 1M (3.7)

d2 =√E2f M

2 +1 =√E

2M [(−1)p]M−1p=0 , (3.8)

where the fact that λ1 = λM2 +1 = 1 was also used. Note that this choice respects the

constraint on the column indexes stated above, in the sense that 1 + M2 is permissible in view

of the assumption (3.5).

3.3 What about MIMO-OFDM?

It is of interest to also look at the corresponding problem in MIMO-OFDM systems using a CP.Eq. (3.2) takes then the form

yi =[X1F Lh

X2F Lh· · · XNTF Lh

]hi,· + ηi

= Fhi,· + ηi (3.9)

whereXj = diag(xj),

with xj being the pilot OFDM symbol at the jth transmit antenna, and F Lh=√MF (:, 1 : Lh)

denotes the M × Lh submatrix of the (unnormalized) DFT matrix consisting of its first Lhcolumns. Notice that the noise term, ηi, is white with covariance σ2IM . Again, a necessarycondition for (3.9) to have a solution is given in (3.5). With F being of full column rank, theLS estimate of hi,· is given by

hi,· = F †yi (3.10)

and the resulting MSE is minimized subject to a constraint on the transmit energy,∑NTj=1 ‖xj‖2 ≤

E , when the matrix F above has orthogonal columns. This implies that

F HLh

(Xj)HXrF Lh= F H

Lhdiag((xj)∗ � xr)F Lh

∝ δj−rILh , (3.11)

for all antenna indexes j, r, where δj−r is the Kronecker delta. Obviously, the condition aboveis satisfied for j 6= r if (xj)∗ � xr = 0, which holds true when the nonzero entries of xj arelocated at the positions where xr has zero entries. This represents a common (in standards)transmission scenario in which only one of the antennas transmits at each of the pilot subcarrierswith the rest of the antennas transmitting nulls. Assuming then (without loss of generality)that

(xj)p+1 6= 0, only at p = mNT + j − 1, (3.12)with m = 0, 1, . . . , M

NT− 1, where it was assumed (for the sake of simplicity) that NT divides

M . Then the (k + 1, l + 1)th entry of the matrix in (3.11) with j = r becomes

M−1∑p=0

epk2πM |(xj)p+1|2e−pl

2πM = e(j−1)(k−l) 2π

M

MNT−1∑

m=0|(xj)mNT+j|2e

m 2πM/NT

(k−l),

which is equal to zero for any k 6= l if all nonzero entries of xj have the same modulus squared,say µj. If this is the case, the above equals M

NTµj for k = l.



Summarizing: MSE-optimal single-symbol preambles for MIMO-OFDM are built by placingequipowered pilots of maximum modulus at the positions dictated by (3.12). Note that thesepreambles belong to the so-called frequency division multiplexing (FDM) pilots, tested forMIMO-FBMC/OQAM in [8].

3.4 Simulation example

Fig. 3-1 shows two examples of the application of the above methods in a 2 × 2 system, forchannels of (a) medium and (b) high frequency selectivity, represented here by ITU-VehA and3GPP-HT models, respectively. The normalized MSE (NMSE) is plotted as a function of thesignal to noise ratio (SNR). Filter banks with M = 128 and K = 3 designed as in [12] wereemployed. A subcarrier spacing of 15 kHz was assumed. The CP duration was chosen as M

8and M

4 in (a) and (b), respectively. In addition to the preambles designed above (as in (3.7),(3.8) for FBMC/OQAM), pseudo-randomly chosen preambles were also tested. Observe thatin all cases the proposed method avoids error floors altogether. The superiority of the optimalpreamble is apparent in both modulation systems. The significant performance deterioration ofthe proposed FBMC method when using sub-optimal preambles for a 3GPP-HT channel can beattributed to the fact that in such scenarios an ill-conditioned matrix results for ˜Γ. This holdstrue for CP-OFDM (and the matrix F) as well but to a smaller extent. Observe the error floorexhibited by CP-OFDM in the (b) case, where CP is shorter than the channel length, resultingin residual interference. Fig. 3-2 depicts the modulated optimal preambles for both multicarrierschemes in a 2× 2 system as the one described above. Notice the impulse-like appearance ofthe signals for FBMC/OQAM. Observe than the “impulses" appear in the two antennas witha time difference of M

2 = 64 samples. The entries (moduli) of the MSE matrices for OFDM

((FHF)−1) and FBMC/OQAM ((˜ΓH ˜Γ)−1) are shown in Figs. 3-3(a) and (b), respectively, for

a VehA channel. As expected they are both diagonal. Observe the shape of the diagonal inthe FBMC/OQAM case. It shows that the channels for the links emanating from the secondantenna are not estimated as well as those of the first antenna. This should be expected inview of the forced suboptimality of the corresponding pilot symbol, d2 (cf. eqs. (3.7), (3.8)).Analogous conclusions were drawn for methods relying on the flat subchannel model in [10, 7].The corresponding matrix for a 3GPP-HT channel is plotted in Fig. 3-4.



−5 0 5 10 15 20 25 30 35 40 45−60

−50

−40

−30

−20

−10

0

SNR (dB)

NM

SE

(dB

)2 x2, M=128, K=3, ITU−VehA (L

h=14)

CP−OFDM (Random)CP−OFDM (Optimal)FBMC/OQAM (Random)FBMC/OQAM (Optimal)

(a)

−5 0 5 10 15 20 25 30 35 40 45−50

−40

−30

−20

−10

0

10

20

30

SNR (dB)

NM

SE

(dB

)

2 x2, M=128, K=3, 3GPP−HT (Lh=43)

CP−OFDM (Random)CP−OFDM (Optimal)FBMC/OQAM (Random)FBMC/OQAM (Optimal)

(b)

Figure 3-1: NMSE performance of the proposed method as compared to (time domain) CP-OFDM for a 2 × 2 system involving (a) ITU-VehA and (b) 3GPP-HT channels. Filter bankswith M = 128, K = 3 were employed.



0 20 40 60 80 100 120 140−3

−2

−1

0

1

2

3

Sample

CP−OFDM (Antenna 1)

0 20 40 60 80 100 120 140−3

−2

−1

0

1

2

3

Sample

CP−OFDM (Antenna 2)

0 100 200 300 400−20

−10

0

10

20

Sample

FBMC/OQAM (Antenna 1)

0 100 200 300 400−20

−10

0

10

20

Sample

FBMC/OQAM (Antenna 2)

imaginary partreal part

Figure 3-2: Modulated CP-OFDM and FBMC/OQAM preambles at the two transmit antennasin the experiment of Fig. 3-1.



05

1015

2025

0

5

10

15

20

25

0

0.5

1

1.5

2

2.5

3

3.5

4

x 10−3

column indexrow index

(a)

05

1015

2025

05

1015

2025

0

0.5

1

1.5

2

2.5

x 10−3

column indexrow index

(b)

Figure 3-3: As in Fig. 3-1. Absolute values of the entries of the MSE matrix for (a) CP-OFDMand (b) FBMC/OQAM for ITU-VehA channels.



010

2030

4050

6070

80

0

20

40

60

80

0

1

2

3

4

5

6

x 10−3

column index

FBMC/OQAM (3GPP−HT)

row index

Figure 3-4: As in Fig. 3-3(b), for 3GPP-HT channels.



4. Adaptive Equalization of Doubly Dispersive Channels inMIMO-FBMC/OQAM Systems

In this chapter, adaptive equalization in an FBMC/OQAM system with multiple antennasat both the transmitter and the receiver sides is studied, for doubly dispersive channels.FBMC/OQAM equalization has been studied in the past, for both single- and multiple-antennasystems. In the former case, both adaptive (using the LMS algorithm) and non-adaptive schemeswere reported (e.g., [13, 14]). The literature for MIMO-FBMC/OQAM equalization has beenmainly restricted to non-adaptive solutions (e.g., [15, 16]). Adaptive equalization for MIMO-FBMC/OQAM has so far only been studied in [17, 18].

In high-mobility applications, like in the PMR systems of interest in the present work, analgorithm with increased adaptation abilities is necessary, to be able to cope with the fast varia-tion of the propagation environment. Important features needed include: (a) Fast convergenceof the equalizer filters during the initial training phase. This translates to using shorter trainingsequences, in order not to severely harm the bandwidth efficiency of the system. (b) Support ofthe tracking task of the algorithm by as sparse as possible pilots throughout the frame payload.These are a necessity (as it will be seen in the sequel) in highly time-varying environments.(c) Low computational complexity and numerical stability. The latter is often an issue in fastconverging algorithms of the RLS type.

To help reduce the overhead in scattered pilots and achieve a good performance at anaffordable cost, a decision feedback equalizer (DFE) is adopted here. In fact, as previouslyshown [19], a DFE structure naturally emerges1 in a MIMO equalizer when a BLAST orderingof the recovered streams is imposed. Such an algorithm, relying on efficient RLS recursionsto update both the DFE filters and the optimum ordering, is presented here, adapted to theFBMC/OQAM modulation. Examples of an extensive study of its convergence behavior and itsdependence on the selection of its parameters (filter lengths and delays) are reported. The effectof the noise color (at the AFB output) as well as that of the contribution of adjacent subcarriersare also discussed. It is argued that it is a widely linear type of equalization algorithm, as alsosupported by simulation examples.

Examples of convergence behavior show that a rather long preamble is required for training.In order to keep this as short as possible, a channel estimate-based initialization of the algorithmis adopted and demonstrated to be very effective in this respect. Specifically, the quantitiesthat need to be initialized are identified and expressed as functions of the CIR. This allowsthe use of a short preamble for initial channel estimation and training of the equalizer. Forthe sake of the comparison, analogous techniques are implemented for both the MIMO-OFDMBLAST DFE algorithm and a DFE algorithm extending to MIMO the LMS-based DFE algo-rithm previously developed for single-antenna FBMC/OQAM [20]. Simulation results reportingthe mean squared symbol estimation error (MSE) evolution in time are provided, which demon-strate the advantages that can be expected from such a training approach. Moving on to thedata transmission phase, appropriately placed (in accordance with an LTE pilot configuration)scattered pilots are used to support the equalizer ability to track time variations. These pilotsmay be used either for providing training information directly to the equalization algorithms orto aid channel tracking and subsequent re-initialization of the equalizer filters similarly with thepreamble. These options are experimentally evaluated for channels of medium and high time-and frequency-selectivity. The comparative study includes both conventional and BLAST RLSDFEs and encompasses CP-OFDM and FMT in addition to the FBMC/OQAM modulation

1Via the equivalence between V-BLAST and Generalized DFE (GDFE) [19].



Filter update

and ordering

DFE1

DFE2

DFE

......

Stage 1

Stage 2

Stage NT

...NT

SFB1

...

SFBNT

AFB1

AFBNR

NRxNT

... Equalizer

(k-th subchannel)

...

Channel

Figure 4-1: The per-subchannel DFE structure.

scheme.For the simulations, a 2 × 2 LTE-like broadband system was considered, corresponding

to a bandwidth of 1.4 MHz at a sampling frequency of 1.92 MHz. This translates to M =128 subcarriers with a subcarrier spacing of ∆f = 15 kHz. The carrier frequency was setto 400 MHz. The ITU-VehA and 3GPP-HT channel models were tested, as representativeexamples of channels with medium and high frequency selectivity, respectively. QPSK symbolswere transmitted. In all the experiments, the FBMC/OQAM filter banks employed the prototypefilter designed in [12], with overlapping factor K = 3.

4.1 Adaptive decision feedback equalization

In the following, the problem of per-subchannel adaptive decision feedback equalization will beinvestigated. The general model of the system and the adopted equalizer structure are givenin Fig. 4-1. The equalizer is applied on the T/2-rate signal described by (2.6).

4.1.1 BLAST RLS-based DFE equalization

The equalization algorithm developed in [17] is our starting point. In [17], it was assumedthat there is no interference between adjacent subchannels and that the noise is temporallywhite. Moreover, the algorithm was only tested with a long training preamble, in rather slowlyfading channels. Here, the equalization performance of this algorithm is studied in a high-mobility scenario. First, the required training sequence length is minimized by supporting theinitialization of the equalizer with the aid of a channel estimate. Additional training informationis supplied in the payload in the form of scattered pilots. The general pattern of these pilotsfollows the Cell-specific Reference Signal (CRS) layout as defined for antenna ports 0, 1 in theLTE standard [21]. Of course, an appropriate adaptation of this pilot format to FBMC/OQAMwas implemented, based on the well known idea of help (auxiliary) pilots. Dealing with thenoise color at the equalizer input and the possible interference from neighboring subchannels isalso discussed in this subsection.

For each subcarrier k, the equalization algorithm consists of NT stages, i.e., as many as thetransmit antennas. At stage p, the output of the DFE is given by [17]

dk,n(op) = <{fHk,n(p)yk,n + bH

k,n(p)dk,n(p)}, (4.1)

where fk,n(p) is the KfNR× 1 feedforward filter and bk,n(p) the (KbNT + p− 1)× 1 feedback



filter at the pth stage. Moreover,

yk,n =[yTk,n−Kf+1 yT

k,n−Kf+2 · · · yTk,n

]T, (4.2)

dk,n(p) =[dTk,n−Kb dT

k,n−Kb+1 · · · dTk,n−1 dk,n(o1) dk,n(o2) · · · dk,n(op−1)

]T

In the previous equations, the detection of data symbols is performed based on a BLAST-typeordering, where the streams with the higher post-detection SNR, which leads to lower detectionerrors, are first selected for detection. This ordering is optimal in terms of SNR. In the following,op denotes the index of the data stream which is detected at stage p and time n (among thosethat have not been detected yet). Due to the <{·} operation, eq. (4.1) can be written as [17]

dk,n(op) =[={fk,n(p)

}T<{fk,n(p)

}T] =

{yk,n

}<{yk,n

}+ <{bk,n(p)}T dk,n(p)

=[={fk,n(p)

}T<{fk,n(p)

}T<{bk,n(p)}T

]︸︷︷︸

wTk,n

(p)

={yk,n

}<{yk,n

}dk,n(p)

︸︷︷︸

xk,n(p)

= wTk,n(p)xk,n(p) (4.3)

The DFE algorithm is applied in a per-subchannel fashion. For each subchannel, there are NTstages and at each stage a DFE is employed, which is efficiently updated via an RLS recursion.The input symbols are detected using BLAST ordering, determined by the maximization of thepost detection SNR criterion. RLS is also used to adaptively update the optimum ordering. Inthe feedback filters, not only past detected symbols are used but also those detected at earlierstages of the current time. The algorithm is computationally efficient and numerically robust,being of a square-root RLS type [22]. The latter is implemented by updating a Cholesky factorof the DFE input autocorrelation matrix, namely

Rk,n(p) = Chol{Φk,n(p)},

whereΦk,n(p) =

n∑q=0

λn−qxk,q(p)xHk,q(p), (4.4)

with 0 < λ < 1 denoting the RLS forgetting factor. Applying this factorization in the LSnormal equations, one can write the LS equalizer weight vector for the rth stream as

w(r)k,n(p) = R−1

k,n(p)v(r)k,n(p),

wherev

(r)k,n(p) = R−T

k,n (p)z(r)k,n(p),

andz

(r)k,n(p) =

n∑q=0

λn−qxk,q(p)drk,q (4.5)



is the corresponding cross-correlation vector. The BLAST-optimal equalizer will then be givenby wk,n(p) ≡ w(op)

k,n (p). The algorithm steps are outlined below. More details can be foundin [17].

Initialization: For p = 1, 2, . . . , NT, op = p, v(p)k,0 = 0, Ek,0(p) = 1. For r = 1, 2, . . . , NT,

v(r)k,0(1) = 0. Qk,0 = 0. R−1

k,0(1) = δ−1/2I, where δ is a small positive constant.For n = 1, 2, . . .

(1) Compute tk,n(1) = R−Tk,n−1(1)xk,n(1), and do1k,n = dec

[do1k,n

], where do1

k,n = wTk,n−1(1)xk,n(1).

(2) Find a sequence of Givens rotations T k,n(1) such that

T k,n(1)

−λ−1/2tk,n(1)

1

=

0

αk,n(1)

(3) Time-update the inverse Cholesky factor:

T k,n(1)

λ−1/2R−Tk,n−1(1)

0T

=

R−Tk,n (1)

X

,where X denotes “don’t care" elements.

(4) For p = 2, 3, . . . , NT

(a) Order-update tk,n(p):

tk,n(p) =

tk,n(p− 1)dop−1k,n

−dop−1k,n√

Ek,n−1(p−1)

(b) Compute decisions from d

opk,n = wT

k,n−1(p)xk,n(p), dopk,n = dec[dopk,n

].

(5) Time-update matrix Qk,n ≡∑nl=0 λ

n−ldk,ldTk,l:

Qk,n = λQk,n−1 + dk,ndTk,n (4.6)

(6) For r = 2, 3, . . . , NT

(a) Time-update v(r)k,n(1):

T k,n(1)

λ1/2v(r)k,n−1(1)

drk,n

=

v(r)k,n(1)

?

(b) E (r)k,n(1) =

[Qk,n

]r,r−∥∥∥v(r)

k,n(1)∥∥∥2

(7) Set Ek,n(1) = minr E (r)k,n(1) and let vk,n(1) be the corresponding v(r)

k,n(1).



0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Channel tap index

Mag

nitu

de

A subchannel impulse response for Veh−A.

(a) VehA channel model

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Channel tap index

Mag

nitu

de

A subchannel impulse response for 3GPP−HT

(b) 3GPP-HT channel model

Figure 4-2: Subchannel impulse response for M = 128, K = 3, ∆f = 15 KHz.

(8) For p = 2, 3, . . . , NT

(a) For r ∈ Op(n) = {1, 2, . . . , NT}\{o1, o2, . . . , op−1}

(i) Order-update v(r)k,n(p):

v(r)k,n(p) =

v(r)k,n(p− 1)

[Qk,n]op−1,r−vT

k,n(p−1)v(r)k,n

(p−1)√Ek,n(p−1)

(ii) E (r)k,n(p) = E (r)

k,n(p − 1) −∣∣∣[v(r)

k,n(p)]

end

∣∣∣2, where [v(r)k,n(p)

]end

is the last elementof v(r)

k,n(p).

(b) Set Ek,n(p) = minr E (r)k,n(p) and let vk,n(p) be the corresponding v(r)

k,n(p).

In the training mode, symbols dk,n are provided as pilots. When working in the decision-directed mode, drk,n are computed using the optimum equalizer and ordering found at theprevious time, n− 1, as specified in Steps 1 and 4(b) above.

4.1.1.1 Setting the filter lengths and the equalization delay

The filter lengths Kf , Kb and the equalization delay2 ∆ are important parameters and theirdetermination in the literature is based on ad-hoc rules, exhaustive search or sub-optimumiterative techniques. Here, we follow the ad-hoc rule that determines Kf as a multiple (usually3-5 times) of the “anticausal" part of the CIR and Kb as the length of its “causal" part. Theterms“anticausal" and “causal" refer to those parts of the CIR that are found at the left-hand andright-hand sides of its strongest tap, respectively. In Figs. 4.2(a), 4.2(b), the impulse response ofan arbitrary subchannel is shown for the ITU-VehA and 3GPP-HT channel models. From thesefigures and according to the previous rule, the feedforward and feedback filter lengths shouldbe set to 2 or 3 for both channel models. As for the delay, after exhaustive experimentation,

2Meaning that the current decision of the equalizer, at subchannel k and time n, corresponds to the inputsymbol at the same subcarrier and time n−∆.



it has been set to ∆ = 1.3 Figs. 4.3(a), 4.3(b) show the MSE learning curves (based on atraining sequence) for these two channel models. Results for four different configurations of

0 50 100 150 200 250 300−25

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0

Iterations

MS

E (

dB)

SNR=21

KfxKb=1x1KfxKb=2x2 − ∆=1KfxKb=3x3 − ∆=1KfxKb=3x3 − ∆=2

(a) ITU-VehA channel model

0 50 100 150 200 250 300−25

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−5

0

Iterations

MS

E (

dB)

SNR=21

KfxKb=1x1KfxKb=2x2 − ∆=1KfxKb=3x3 − ∆=1KfxKb=3x3 − ∆=2


Figure 4-3: MSE learning curves (all-training mode) for M = 128, K = 3, ∆f = 15 KHz.Time-invariant channels.

the equalizer parameters are depicted, at SNR=21 dB. It can be observed that (a) a single-tapequalizer (with ∆ = 0) is not adequate, (b) the delay parameter plays an important role, and(c) the selected values for the filter lengths lead to an acceptable performance in terms ofsteady-state MSE.

4.1.1.2 Impact of noise color

As mentioned previously, the noise sequence at each AFB output is temporally correlated. InFigs. 4.4(a), 4.4(b), the noise autocorrelation sequences at the AFB input and an arbitraryAFB output are plotted, for M = 128, K = 3. As observed (and expected from eq. (2.5)), the

−20 −15 −10 −5 0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1AFB input

Abs

olut

e va

lue

of c

orre

latio

n

Time lag

(a) Input to the AFB

−20 −15 −10 −5 0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1AFB output

Abs

olut

e va

lue

of c

orre

latio

n

Time lag

(b) Output of the AFB

Figure 4-4: Autocorrelation noise sequences at the input and output of the AFB (M = 128,K = 3).

3More generally, the delay can be different in each subchannel. A single delay for all subchannels was chosenhere, for the sake of the simplicity and because the main characteristics of the subchannel impulse responsepreviously shown do not vary much from one subcarrier to another.



white noise at the input of the AFB becomes colored at its output. However, noise correlationdecreases rapidly as the time lag increases.

It should be noted that the BLAST RLS algorithm has been originally designed under theassumption that the noise sequence is uncorrelated in time. In order to evaluate the impactof the above correlation on the equalization performance, two approaches are examined. First,a whitening filter is employed. Second, a decorrelating matrix is used, which is equal to theinverse square root factor of the noise autocorrelation matrix.

Specifically, identical whitening filters are used for each antenna/subchannel pair, designedas feedforward prediction-error filters [23]. For receive antenna j and subchannel k, the input-output relation of such a filter is expressed as

vjk,n = ηjk,n −N∑i=1

u∗i ηjk,n−i, (4.7)

where ui are the taps of the Nth-order prediction-error filter. The filter taps ui are computedas the solution of a Wiener-Hopf system of equations, constructed using the autocorrelationsequence of the filter input (such as the one in Fig. 4.4(a)).

As for the decorrelating matrix, the noise term of interest in subchannel k is

ηk,n =[ηTk,n−Kf+1 ηT

k,n−Kf+2 · · · ηTk,n

]T

and its correlation matrix can be written as

C η = σ2

INR (−1)kγINR 0 · · · 0

−(−1)kγINR INR (−1)kγINR · · · 0

0 −(−1)kγINR INR · · · 0... ... . . . . . . ...

0 0 · · · −(−1)kγINR INR

Let its eigenvalue decomposition be C η = V ΛV H. Then its inverse square root can becomputed as Λ−1/2V H. This matrix is applied on the signal given in (4.2).

In Fig. 4-5, MSE learning curves (all-training case) of the algorithm are plotted, when awhitening filter is employed (at the output of the AFB), for three different prediction filterorders. The SNR is again set to 21 dB. The parameters for the DFE equalizer are chosen asKf = 3, Kb = 3, ∆ = 1 and the forgetting factor λ of the RLS algorithm is set to 1. Theparameters of the time-invariant channel model are as before. It is observed that the applicationof the whitening filter actually makes the MSE performance worse. This is due to the fact thatthe overall subchannel impulse response is now increased in length and the equalizer filters areno longer of the appropriate size. Improvement in performance can be achieved by increasingthe size of the equalizer filters, at the cost of higher complexity (and slower convergence).Finally, in Fig. 4-6, the MSE performance is presented when the decorrelating matrix (describedabove) is used for noise whitening. As observed, no gain in performance is achieved by thisapproach. In view of these and other results not reported here, and in order to minimize thecomputational burden, no noise prewhitening will be employed henceforth.



0 10 20 30 40 50 60 70 80−20

−15

−10

−5

0

5

10

Iterations

MS

E (

dB)

Order=0Order=2Order=5

Figure 4-5: MSE learning curves (all-training mode) for three different orders of the whiteningfilter for the ITU-VehA channel model.

4.1.1.3 Interference from adjacent subchannels

In FMBC/OQAM systems, the signal received at subchannel k is interfered by the signals ofneighboring subchannels, especially the adjacent ones at positions k± 1 (see, e.g., (2.6)). Thisinterference may be taken into account when designing a linear equalizer or the feedforwardfilter of a DFE, as in [24, 15, 25]. An alternative approach is to employ two additional feedbackfilters in the DFE structure, one for each of the two adjacent subcarriers, as proposed in [26]for example.

So far, it was assumed that a single feedback filter is utilized per subchannel k in theBLAST RLS DFE algorithm. In order to assess the impact of using three feedback filters,the DFE structure described in Section 4.1.1, is extended with the feedback filters bk−1,n andbk+1,n. Moreover, it is assumed that, for each time n, the equalization is performed seriallyfrom the first subchannel to the last one. In particular, at time n and subchannel k, all threefeedback filters are fed with decisions concerning the times up to n− 1, and additionally bk−1,nis fed with the current (at time n) decisions for subchannel k − 1, assumed to be available.The input of bk,n is as described above. It is noted that bk+1,n is not fed with current symbolsas they are not yet detected.4

In Fig. 4-7, the MSE learning curves (in an all-training mode) are plotted for a DFE withKf = 3 taps in its feedforward filter, Kb = 3 taps in each of its three feedback filters,equalization delay equal to ∆ = 1, and λ = 1. The time-invariant Veh-A channel model isassumed. As observed, at practical SNR values, the use of three feedback filters has a smallimpact on steady-state performance, as opposed to what is happening at quite high SNRs. Thisshould be expected in view of the well known fact that intrinsic interference is “hidden" behindnoise and only prevails at weak noise regimes. In view of the above results, and for the sake ofthe simplicity, only DFEs with a single feedback filter will be considered in the sequel.

4A simple approach to transforming this serial processing of the subchannels to a parallel one would be toonly consider past symbols.



0 10 20 30 40 50 60 70 80−18

−16

−14

−12

−10

−8

−6

−4

−2

0

2

Iterations

MS

E (

dB)

Noise covariance−based whiteningWithout whitening

Figure 4-6: MSE learning curves (all-training mode) with and without correlation matrix whiten-ing for the ITU-VehA channel.

4.1.1.4 On widely linear processing

The widely linear (WL) approach [27] is known to be well suited in FBMC/OQAM equalizationin view of the non-circular nature of OQAM. This has been studied, for example, in [14, 28]among others. The output of a WL DFE is computed as

dk,n(op) = fHk,n(p)yk,n + fH

k,n(p)y∗k,n + bHk,n(p)dk,n(p), (4.8)

that is, both the signal yk,n and its complex conjugate are processed. In [24, 14], WL MMSEequalization for single-antenna FBMC/OQAM was studied. A central conclusion there was thatthe two filters in the feedforward part are conjugate to each other (that is, f = f ∗) and thefeedback filter is, effectively, real. This is also true in the WL version of the RLS algorithmderived in [29] (although the latter work does not really address an equalization problem). Onthe other hand, the formulation of the FBMC/OQAM equalization algorithms (including theBLAST DFE) basically follows Tu’s principle [30], namely, that since the input to be recoveredis real-valued, no degrees of freedom should be spent to recover the imaginary part as well.

One can easily show that these two approaches, namely the WL (see (4.8)) and that putforward by Tu (see (4.1)), can be seen as equivalent (as also alluded to in [14]), in the sensethat the same kind of processing is taking place in both of them. Thus, (4.1) can be rewritten



0 50 100 150−14

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0

2

4

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MS

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Single FB filterThree FB filters

(a) SNR=16 dB

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0

2

Iterations

MS

E (

dB)


(b) SNR=21 dB

0 50 100 150−35

−30

−25

−20

−15

−10

−5

0

5

Iterations

MS

E (

dB)


(c) SNR=43 dB

Figure 4-7: MSE learning curves when either a single or three feedback filters are used perstage. ITU-VehA time invariant channels were assumed.

as5:

d = <{wH

1 y + bHd}

= wT1,RyR +wT

1,IyI + bTRd

=[wT

1,R wT1,I

] yR

yI

+ bTRd

=[uT

1 uT2

] yR

yI

+ bTRd (4.9)

5Here, the notation is simplified for convenience. However, the correspondence of the terms with thosein (4.1) is obvious.



0 5 10 15 2010

−4

10−3

10−2

10−1

100

Eb/N

0 (dB)

BE

R

NormalMethod of TuWidely−Linear

Figure 4-8: BER curves for the ITU-VehA channel model (for Kf = 2, Kb = 2, ∆ = 1, λ = 1).

Similarly, one can write (4.8) as:

d = <{aH

1 y + aH2 y∗ + bHd

}=

= aT1,RyR + aT

1,IyI + aT2,RyR − aT

2,IyI + bTRd

=[aT

1,R + aT2,R aT

1,I − aT2,I

] yR

yI

+ bTRd

=[vT

1 vT2

] yR

yI

+ bTRd (4.10)

In these equations, the subscripts (·)R and (·)I denote the real and imaginary parts of thecorresponding quantities, respectively. It can be seen that the output of the equalizer is similarlycomputed in the two cases. The equivalence of the WL and Tu’s approaches has also beenverified via simulations with the BLAST RLS DFE algorithm. Such an example is given inFig. 4-8, where the resulting bit error rate (BER) is plotted as a function of the SNR. Thestraightforward implementation, i.e., when neither of the WL and Tu’s approaches is used andno account is taken of the real-valued nature of the transmitted data, was also tested for thesake of the comparison. The training sequence consisted of 50 OFDM symbols, sufficientlymany to guarantee the convergence of the algorithm in each case. It is observed that theTu’s and WL versions of the algorithm perform identically and outperform the straightforwardversion of the equalizer.

4.1.2 Channel estimation-based initialization of the DFE filters

In Figs. 4.9(a), 4.9(b), MSE learning curves are plotted for the BLAST RLS DFE, for varioustraining sequence lengths, at SNR=21 dB. It is observed that in order to achieve a good



0 50 100 150 200 250 300−25

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0

5

Iterations

MS

E (

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Training=280Training=140Training=50Training=24Training=12Training=6Training=2

(a) ITU-VehA channel model

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−20

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−5

0

5

Iterations

MS

E (

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Training=280Training=140Training=50Training=24Training=12Training=6Training=2


Figure 4-9: MSE curves for various training sequence lengths.

performance, a high number of multicarrier symbols should be assigned to the initial trainingphase (e.g., 50 symbols). One way around this problem is to initialize the filters of the DFEbased on channel estimates, which, in principle, can be acquired with considerably less training.

4.1.2.1 Initialization of the BLAST RLS DFE algorithm

The initialization of the BLAST RLS DFE feedforward and feedback filters will be describedbased on the calculation of the mean values of three key quantities of the associated square-rootRLS algorithm, namely R−1

k,n(1), v(r)k,n(1), and the matrix Qk,n (see (4.6)). Φk,n(p) and z(r)

k,n(p)will be replaced by their expected values, Φk,n(p) = E{Φk,n(p)} and zrk,n(p) = E{zrk,n(p)},respectively. The expectation is meant here with respect to the symbols dk,n and the noise ηk,n.In the following, by making the common assumption that the DFE makes correct decisions, itis shown how Φk,n(p) and z(r)

k,n(p) can be computed if knowledge of the channel is available.To this end, using (2.6), yk,n can be written as

yk,n =k+1∑

m=k−1Ckmsm,n + ηk,n, (4.11)



where Ckm is a block Toeplitz matrix with first block row equal to

[ Ckm(Lc) Ckl(Lc − 1) · · · Ckm(0) 0 · · · 0 ] (4.12)

andsm,n =

[dTm,n−Kf+1−Lc dT

m,n−Kf+2−Lc · · · dTm,n

]T, (4.13)

ηk,n =[ηTk,n−Kf+1 ηT

k,n−Kf+2 · · · ηTk,n

]T(4.14)

Using the fact that dm,n’s are real numbers, the real and imaginary parts of yk,n can be expressedas

<{yk,n} =k+1∑

m=k−1<{Ckm}sm,n + <{ηk,n} (4.15)

and={yk,n} =

k+1∑m=k−1

={Ckm}sm,n + ={ηk,n}, (4.16)

respectively. Then, the input autocorrelation matrix at the first stage of the DFE (i.e., p = 1)can be written as

Φk,n(1) = E{Φk,n(1)}

=n∑q=0

λn−q E{xk,q(1)xHk,q(1)}

=n∑q=0

λn−q E{

={yk,q}

<{yk,q}

dk,q(1)

[={yT

k,q} <{yTk,q} d

Tk,q(1)

]}

=n∑q=0

λn−q

E{={yk,q}={yT

k,q}} E{={yk,q}<{yTk,q}} E{={yk,q}d

Tk,q(1)}

E{<{yk,q}={yTk,q}} E{<{yk,q}<{yT

k,q}} E{<{yk,q}dTk,q(1)}

E{dk,q(1)={yTk,q}} E{dk,q(1)<{yT

k,q}} E{dk,q(1)dTk,q(1)}

≡ κ(n)Θ (4.17)

where κ(n) = ∑nq=0 λ

n−q = 1−λn+1

1−λ ,

E{={yk,q}={yTk,q}} = σ2

d

k+1∑m=k−1

={Ckm}={CTkm}+CηII , (4.18)

E{={yk,m}<{yTk,m}} = σ2

d

k+1∑l=k−1

={Ckm}<{CTkm}+CηIR , (4.19)

E{={yk,q}dTk,q(1)} = σ2

d={Ckk}S, (4.20)

E{<{yk,q}<{yTk,q}} = σ2

d

k+1∑m=k−1

<{Ckq}<{CTkq}+CηRR , (4.21)

E{<{yk,q}dTk,q(1)} = σ2

d<{Ckk}S, (4.22)

E{dk,q(1)dTk,q(1)} = σ2

dINTKb , (4.23)



σ2d is the symbol power and S = E{sk,qd

Tk,q(1)} is a matrix of zeros with the exception of a

diagonal (starting at row (Kf + Lc − 1−Kb + ∆)NT + 1) which is filled with ones. Use wasmade of the assumption of i.i.d. transmitted data. Moreover,6 CηII = E{={ηk,n}={ηT

k,n}},CηIR = E{={ηk,n}<{ηT

k,n}} and CηRR = E{<{ηk,n}<{ηTk,n}}. The remaining block entries

of Θ in (4.17) are the transpose versions of the corresponding terms in the above equations.The first-stage cross correlation vector can be written as

z(r)k,n(1) = E{z(r)

k,n(1)}

=n∑q=0

λn−q E{xk,q(1)drk,q}

=n∑q=0

λn−q E{

={yk,q}

<{yk,q}

dk,q(1)

drk,q}

=n∑q=0

λn−q

E{={yk,q}drk,q}

E{<{yk,q}drk,q}

E{dk,q(1)drk,q}

≡ θκ(n) (4.24)

whereE{={yk,q}drk,q} = σ2

s={Ckk}er, (4.25)

E{<{yk,q}drk,q} = σ2s<{Ckk} er, (4.26)

andE{dk,q(1)drk,q} = 0NTKb×1, (4.27)

where er = E{sk,qdrk,q} is a vector filled with zeros except for the position (Kf + Lc − 1 −∆)NT + r, with r = 1, 2, . . . , NT. Finally,

Qk,n = E{Qk,n} = σ2dκ(n)INT (4.28)

Based on (4.17) and (4.24), the initialization of the first-stage Cholesky factor Rk,n(1) canbe written as

Rk,n(1) = Chol{Φk,n(1)} =√κ(n)Chol{Θ} ≡

√κ(n)ΘChol (4.29)

Hence, the first-stage inverse of the Cholesky factor and the temporary equalizers are initializedas

R−1k,n(1) = 1√

κ(n)Θ−1

Chol (4.30)

andv

(r)k,n(1) = R

−Tk,n(1)z(r)

k,n(1) = κ(n)√κ(n)

Θ−TCholθ =

√κ(n)Θ−T

Cholθ, (4.31)

respectively.6CηIR is assumed zero in the following.



4.1.2.2 Initialization of the RLS DFE algorithm

FBMC/OQAM adaptive equalization has been mainly studied for single-antenna systems [14,20]. In these works, a DFE was proposed in which the feedback filter operates on past decisionsonly and the LMS algorithm was used for filter adaptation. In the present work, that DFEstructure was extended to the MIMO context and the faster in terms of convergence RLSalgorithm was adopted. Specifically, assuming that the feedforward and feedback filer lengthsare KfNR and KbNT, respectively, the straightforward RLS algorithm is used as in [31].

Initialization of the RLS MIMO DFE filters based on channel estimation is also possible,in a way similar to that presented previously for the BLAST RLS DFE. This approach, again,reduces the length of the training sequence required in the initial phase of the algorithm.

4.1.2.3 The CP-OFDM case

Channel estimation-based initialization of the equalizer could be also applied to CP-OFDM, asdetailed below. Assume that a sufficiently long CP is inserted and hence the frequency selectivechannels are transformed into a set of frequency flat subchannels. In a MIMO-OFDM system,this means that for each subcarrier, there is a frequency flat NR ×NT MIMO channel betweenthe transmitter and the receiver. In this case, the input-output relation for the kth subchannelcan be written as

yk,n = Hk,nsk,n + ηk,n, (4.32)

where yk,n is a NR×1 received signal vector at time n, sk,n is the NT×1 complex input symbolvector (with symbol power equal to σ2

s), Hk,n is the flat MIMO channel frequency responsematrix and ηk,n is a noise vector of zero mean and covariance equal to σ2INR .

If the BLAST RLS algorithm is applied in MIMO-OFDM, the initialization of the samequantities is required as previously. Now, the length of the feedforward filters at all stages isNR, having yk,n as common input, while the feedback filter input at each stage is the vector ofsymbols detected at the previous stages [32]. Hence, following a similar procedure as the oneabove, the mean autocorrelation matrix at the first stage of BLAST RLS is given by

Φk,n(1) = κ(n)(σ2sHk,nH

Hk,n + σ2INR

)(4.33)

and the cross-correlation vectors z(r)k,n(1) are given by

z(r)k,n(1) = σ2

sHk,n(:, r)κ(n), (4.34)

where Hk,n(:, r) is the rth column of Hk,n. Once the above quantities have been calculated,the associated Cholesky factor as well as the temporary equalizers and the matrix Q are givenas previously.

4.1.2.4 Simulation results

The reported simulation results aim at a) evaluating the performance of the previously de-scribed channel estimation-based equalization schemes and b) assessing the improvement inperformance achieved in high mobility scenarios, when, except for initial training, scatteredpilots are also exploited by the algorithms. In Figs. 4.10(a), 4.10(b), MSE learning curves (all-training case) for the two channel models are shown. The performances of the BLAST RLS andRLS algorithms are compared with and without initialization, both with perfect and estimatedchannel state information (CSI). For channel estimation-based initialization, two FBMC symbols



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Figure 4-10: MSE curves for assessing channel based initialization of equalizer filters and Kf =3, Kb = 3, λ = 1, ∆ = 1.

are used for estimating the impulse response of the MIMO channels according to the methodpresented in [8], which employs a MSE-optimal sparse preamble. As observed from these fig-ures, the initialization of the filters, both with perfect and estimated CSI, leads to the rapidconvergence of the MSE to the desired steady state, meaning that only two FBMC symbolswould be sufficient for the initial channel estimation-based initialization of the algorithms.

In Figs. 4.11(a), 4.11(b), BER results are given for the two algorithms corresponding tothe three initialization schemes described for Figs. 4.10(a) and 4.10(b). In the no initializationcase, 6 FBMC symbols are used for training the equalizer directly. For the other two cases,2 symbols are used for channel estimation and equalizer initialization and 4 for training theequalizer after its initialization. Thus, the same training sequence length is used in all cases. Itis observed that channel-based initialization offers a tremendous improvement as compared tono initialization. Finally, comparing the initialization performance based on the estimated CSIwith perfect CSI, it is observed that these are closer for ITU-VehA channels. This is attributedto the fact that the channel estimator used is based on the assumption that the subchannelsare frequency flat, which is closer to being valid for this channel model.



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(b) 3GPP-HT channel model (Kf = 3, Kb = 3, λ = 1, ∆ = 1)

Figure 4-11: BER curves for assessing channel-based initialization of DFE equalizer filters inthe RLS and BLAST DFE algorithms (FBMC/OQAM).

Figs. 4.12(a), 4.12(b) compare FBMC/OQAM and CP-OFDM systems. The simulationparameters are as in Figs. 4.11(a) and 4.11(b). For CP-OFDM, the CP length was set toM4 = 32 (and the associated SNR loss was taken into account). With the assumption thatthis CP duration is sufficient to remove ISI and ICI, the equalizer’s filter lengths were set toKf = Kb = 1. The training sequence length was 3 OFDM symbols, of which the first oneis used for channel estimation whenever applicable. One can see that CP-OFDM also benefitsfrom the channel-based initialization of the equalizer filters. At practical (low and moderate)SNR values, it is outperformed by FBMC/OQAM. However, at high SNRs, where the residualintrinsic interference in FBMC/OQAM prevails over noise, the situation is reversed.

4.1.3 Using pilots scattered throughout the frame

In high mobility scenarios, the algorithm has to be able to track the time variations in the datatransmission mode. In order to support the operation of the BLAST RLS algorithm, pilots thatare scattered throughout the frame were also used for refreshing the channel estimate and use



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BLAST−RLS − no initBLAST−RLS − init − estimated channelBLAST−RLS − init − perfect channelCP−OFDM − no initCP−OFDM − init − estimated channelCP−OFDM − init − perfect channel

(a) ITU-VehA channel model (Kf = 2, Kb = 2, λ = 1, ∆ = 1)

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(b) 3GPP-HT channel model (Kf = 3, Kb = 3, λ = 1, ∆ = 1)

Figure 4-12: BER curves for assessing channel-based initialization of DFE equalizer filters inthe BLAST DFE algorithm (FBMC/OQAM vs. CP-OFDM).

it to re-initialize the equalizer filters.The pilot pattern follows the LTE CRS format and is defined on the basis of a 12 × 14

frequency-time grid as shown in Fig. 4-13 for a two transmit antennas system. The P’s and0’s denote the positions where a pilot or a zero value is sent, respectively. The other positionsare filled with data symbols. This pattern is used by the first antenna. The second antennauses the same pattern with P’s and 0’s interchanged. This way when a pilot symbol is sentby one antenna, there is no interference by the other antenna, thus making channel estimationeasier at the specific time-frequency position. This pilot pattern has been originally designed forCP-OFDM. For an FBMC/OQAM system, the corresponding grid is of size 12× 28. To copewith the interference intrinsic in FMBC/OQAM, help pilots are employed, which control theinterference to the pilots and zeros. In Fig. 4-14, the FBMC/OQAM pilot layout is presented.The help pilot positions are designated by H’s.

In addition to the scattered pilots, a preamble of 3 OFDM symbols is used for the initialtraining of the equalizer. If, additionally, channel estimation is used in this initial phase, thenthe first OFDM symbol is used for channel estimation and the remaining two for training of the



0

P

P

0

0

P

P

0

Time

Fre

qu

en

cy

0

P

P

0

0

P

P

0

Figure 4-13: LTE-based scattered pilot pattern, based on the CRS frame format, for the firstantenna of a 2×X system.

0

P

P

0

0

P

P

0

Time

Fre

qu

en

cy

0

P

P

0

0

P

P

0

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

Figure 4-14: LTE-based scattered pilot pattern updated with help pilots for application in aFBMC/OQAM system.

equalizer filters. First, the algorithm is tested for ITU-VehA channels, at two mobile speeds,50 and 150 km/h, with Kf = Kb = 2 and ∆ = 1. The corresponding forgetting factor valuesare λ = 0.985 and λ = 0.975. The resulting MSE learning curves are shown in Figs. 4.15(a),4.15(b). It is seen that the periodic re-initialization (denoted by ‘reinit’ in the figures) of theequalizer based on channel estimates obtained by exploiting the scattered pilots is necessary forthe algorithm to have an acceptable performance in a mobile environment. The same conclusioncan be drawn from Figs. 4.15(c), 4.15(d), showing the BER results.

For the 3GPP-HT channel model, the MSE learning curves at SNR=18 dB are presentedin Fig. 4-16. When perfect CSI is available at the scattered pilot positions, results are againfavorable as previously. However, the performance with estimated channel response is now seento severely degrade with time. It must be noted, of course, that the help pilot idea relies on theassumption of a locally flat CFR, which is far from being accurate in such a highly frequencyselective channel. But what is even more important here is that the number of pilots tonesat a given time is not large enough to allow estimating such a long impulse response. A way



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(a) MSE curves for 50 km/h and SNR=18 dB.

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(d) BER curves for 150 km/h (λ = 0.975).

Figure 4-15: Assessing the impact of scattered pilots on the performance of the BLAST RLSDFE algorithm (for ITU-VehA channels, with Kf = Kb = 2 and ∆ = 1).

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BLAST−RLS − Init − No reinit − Estimated channelsBLAST−RLS − Init − No reinit − Perfect channelsBLAST−RLS − Init − Reinit − Estimated channelsBLAST−RLS − init − Reinit − Perfect channels

Figure 4-16: MSE learning curves for the 3GPP-HT channel model with parameters as inFig. 4-15, at 50 km/h.



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(b) MSE curves for 50 km/h and SNR=21 dB.

Figure 4-17: MSE performance of the BLAST RLS algorithm, for a sparse channel of lengthLh = 36 with only 3 nonzero coefficients.

around this problem can be to exploit the highly sparse structure of this channel, which impliesthat the number of (significant) unknown taps is much smaller than the channel length, hencerequiring far less training information for their estimation. To this end, one should employsome sparsity-aware channel estimation technique, as in the EMPhAtiC deliverable D3.1. Oneof these techniques, which was successfully tested in D3.1, is the so-called Orthogonal MatchingPursuit (OMP) algorithm. To clearly demonstrate the ability of such a technique to overcomethe problem, an example of a strictly sparse channel (generated as a sample-spaced version ofa 3GPP-HT channel) was tested in the above scenarios and the results are shown in Figs. 4-17.It is clearly seen that exploiting the CIR sparsity can solve the problem and offer performanceclose to that with PCI. Further research is required in this direction, to successfully address thisproblem with other channel models as well, most importantly for the more realistic scenariosinvolving compressible or group sparse CIRs.

Of course, should the system parameters are such that the number of pilots is not muchsmaller than the CIR length, the direct (not sparsity-aware) channel estimation approach canprovide satisfactory results. Such an example is given in Fig. 4-18, where the 3GPP-HT channelis again considered at the same bandwidth as before, but with a larger number of subcarriers,M = 256, and narrower subcarrier spacing, ∆f = 7.5 kHz.

It is of interest to also compare with other FB-MC schemes. FMT was selected for sucha comparison, in view of its long history and large number of applications (including PMRsystems) [6]. No channel estimation is used here. A long preamble of 30 OFDM symbolsis assumed, to guarantee convergence of all equalizers. Scattered pilots were transmittedand used to directly adapt the equalizers, with no channel estimation being involved. BERresults are plotted in Fig. 4-19, for FBMC/OQAM, FMT, and CP-OFDM, for zero mobility.The corresponding results for mobile speeds of 20 and 50 km/h are shown in Fig. 4-20. InFMT, a square root raised cosine pulse of length 12M was used as a prototype filter, a quitecommon choice. Observe that these filters are much longer than those (of length 3M) usedin FBMC/OQAM. The CP in CP-OFDM was M/4 samples long, which is shorter than theCIR order (Lh = 43). FMT was tested both without CP and with a CP of the same lengthas in CP-OFDM. A pilot configuration similar to that used previously was assumed. However,since no channel estimation is performed, both antennas transmit pilot symbols at the positionspreviously occupied by zeros. The 3GPP-HT channel model was adopted. It is observed that



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Figure 4-18: MSE performance of the BLAST RLS algorithm for the 3GPP-HT channel; M =256, K = 3, ∆f = 7.5 kHz, no mobility.

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Figure 4-19: BER performance of the BLAST RLS algorithm using scattered pilots withoutchannel estimation. 3GPP-HT channels, at zero mobility; M = 128, K = 3, ∆f = 15 kHz.For FBMC/OQAM and FMT: Kf = Kb = 3, ∆ = 1, λ = 1.



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(b) At 50 km/h.

Figure 4-20: As in Fig. 4-19, with mobile speeds of 20 km/h (λ = 1) and 50 km/h (λ = 0.98).

the gain from using the scattered pilots only for training the equalizer filters is small for all threesystems at these speeds. In all three scenarios, CP-OFDM performs better than FMBC/OQAMat lower SNR values while it gets worse at higher values of the SNR. Their difference becomesnegligible in the higher mobility scenario. The performance deterioration of CP-OFDM at higherSNRs can be attributed to the inadequacy of the CP employed here to avoid inter-symbol (ISI)and inter-carrier interference (ICI), particularly at a mobile environment. These effects becomemore prevalent when noise is weaker. FMT is seen to be outperformed by both FBMC/OQAMand CP-OFDM in this experiment. Only for a time invariant channel (see Fig. 4-19) it attainsthe CP-OFDM performance when a CP is also employed. For the mobile scenarios, one canmake here analogous comments with those concerning the effect of the short (or no) CP inCP-OFDM above.



5. Fast-convolution Implementation of Linear EqualizationBased Multiantenna Detection Schemes

Linear minimum mean-square error (LMMSE) detection is one of the most basic approachesfor detection in multiantenna systems. In spatial multiplexing (SM) MIMO-OFDM systems,the subchannels are flat fading and the LMMSE solution is reached basically through channelmatrix inversion for each subcarrier. The subcarrier signals from different receiver antennas arecombined through complex weights, i.e., single-tap subcarrier equalizers. In FB-MC systems,the flat-fading subcarrier model is not strictly valid, and the system optimization may actuallylead to significant frequency selectivity at the subcarrier level. This leads to the need for multi-tap subcarrier equalizers for LMMSE detection. In the broadband PMR (B-PMR) context,the Hilly Terrain channel model with LTE-like FBMC/OQAM waveform parametrization is oneexample where the subchannels become highly frequency-selective.

5.1 Frequency sampling based multi-tap subcarrier equalizer approach

Frequency sampling based subcarrier equalizer design was originally developed for single-antennaFBMC/OQAM systems [33] and single-carrier systems [34], but it has also been extended toSIMO and spatial multiplexing MIMO FBMC/OQAM systems in [16]. Fig. 5-1 shows thefrequency sampling based equalizer structure for the 1 × 2 SIMO case. Even though theequalizer adaptation is based on the frequency sampled model, the subcarrier equalizers areimplemented as multi-tap complex FIR filters. Actually, the scheme is very flexible, and thesame design principle and the same subcarrier equalizer structure can be used in all FB-MCsystems and SC systems with frequency-domain equalization in which significant frequencyselectivity appears at the subcarrier/subband level. The key idea is to find the linear MMSEsolution, in the same way as in MIMO-OFDM, in a number of frequency points within eachsubband. Then the multi-tap subcarrier equalizers are designed to reach the target frequencyresponse at those frequencies.

One of the core ideas of EMPhAtiC is to utilize the fast-convolution filter bank (FC-FB)model for implementing different filter bank based SC and MC waveforms in a unified andflexible manner, as described in the EMPhAtiC deliverable D2.1 and [35]. The idea of embed-ded equalization was introduced for the single-antenna case in the deliverable D3.1. In thissection, we show how the frequency sampled subcarrier equalizer approach can be effectivelyembedded to the fast-convolution processing structure also in the SIMO and MIMO cases, whilemaintaining the multimode capabilities of the FC-FB scheme.

5.2 Embedded subcarrier equalization in FC-FB

The idea of FC-FB is to use multirate fast-convolution processing for implementing the sub-channel filters of analysis and synthesis filter banks [35]. Overlap-save processing is utilized toapproximate well linear convolution by the structure that implements a cyclic convolution bynature. In this structure, the subband widths, center frequencies, and transition band shapescan be tuned independently of each others. This gives also the possibility to process differentwaveforms in different subbands simultaneously. In the basic scheme, the weight coefficients aredesigned to optimize the passband and stopband frequency response characteristics, primarilytargeting at minimizing the inband interference of the transmission link and maximizing stop-band attenuation to minimize out-of-band interference coupling. However, the same weights



RF front-end

&AD

C

RF front-end

&AD

C

10CFIR2x –

oversampled

M-subchannelanalysis FB

OQ

AM

post-processing

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pledM

-subchannelanalysis FB

( )1H z1kCFIR

11MCFIR −

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1RN

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Evaluation of MRC-weighted target values

Frequency-sampling design of CFIR equalizers

,( )qk iH ,( )

qk iC

[ ]qkw i

+

1[ ]Ma l−

[ ]ka l

1ˆ [ ]Ma l−

ˆ [ ]ka l

0[ ]a l 0ˆ [ ]a l

1[ ]r m

[ ]RNky n[ ]RNr m

1[ ]ky n 1[ ]kz n

[ ]kv n

[ ]RNkz n

1[ ]Mv n−

0[ ]v n

Figure 5-1: SIMO system with multi-tap subcarrier equalizers based on the frequency samplingapproach.

N-point FFT

L -point IFFT

Select LS

bins

Blocks of N input samples with 2NO sample overlap

Block of LS output samples

w1,1,1,i

N-point FFT

RX antenna 2 RX antenna 1

L -point IFFT

Select LS

bins

Stream 2 Stream 1

w2,1,1,iw1,2,1,i w2,2,1,i

Figure 5-2: 2 × 2 spatial multiplexing MIMO receiver utilizing the FC-FB structure with em-bedded equalization.



can be used also for implementing the subcarrier equalizers, as explained in the EMPhAtiCdeliverable D3.1.

Fig. 5-2 shows the FC-FB structure with embedded subcarrier equalizers in the 2×2 MIMOcase, with two transmitted data streams. Here the frequency sampling based equalizer coef-ficients are directly combined with the weights of the basic FC-FB design. The weights areimplemented separately for each signal path from one of the antennas to an output stream, andIFFT processing is implemented after combining the antenna signals. This embedded equal-izer structure is independent of the transmitted waveform, subcarrier bandwidth and centerfrequency. The frequency domain weights are computed from the channel frequency responsesand the pre-designed basic weight mask of each subcarrier. The weight for each frequency binis obtained as the product of the weight coming from the basic weight mask and the linearequalizer weights computed from the channel frequency responses:

wq,p,k,i = w(FB)k,i w

(eq)q,p,k,i (5.1)

Here q is the index of the receiver antenna, p is the index of the data stream, k is the subcar-rier/subband index and i is the IFFT bin index. We can see that the frequency sampling basedsubcarrier equalizer can be embedded in a natural way to the FC analysis filter bank. Withzero-forcing criterion, the equalizer weights would be obtained for each frequency bin from thepseudo-inverse of the channel matrix Hk,i at the corresponding bin,

W(ZF)k,i = (HH

k,iHk,i)−1HHk,i, (5.2)

where the superscript H stands for Hermitian (complex conjugate transpose). With MSEcriterion, the corresponding expression is

W(MSE)k,i = (HH

k,iHk,i + ρINT)−1HHk,i, (5.3)

where ρ is the noise-to-signal power ratio and INT is an identity matrix of dimension NT, thenumber of transmit antennas.

Some clarification concerning the linear receiver structure used in different antenna config-urations is appropriate in this context. In single-input single-output and SIMO configurations,it is possible to utilize the frequency-domain equalizer to closely approximate the classical,optimal receiver structure where the fractionally-spaced equalizer implements the matched fil-ter, matched both to the transmitted pulse shape and the channel response, along with linearchannel equalizer [34]. However in the SM MIMO configuration, each antenna chain receivesmultiple data streams affected by different channel responses and it is not possible to matchthe receiver processing for multiple channel responses at the same time. Therefore, our receiverstructure consists of the pulse shape matched filter, implemented by the basic weight mask ofthe FC-FB structure, and the MSE-criterion based weights obtained from eq. (5.3). It can benoted that the zero-forcing solution would completely cancel the interference from other datastreams and also from adjacent subcarriers in case of FBMC/OQAM waveform. However, theMSE solution results in lower MSE in the equalized signal, by balancing the effects of imperfectequalization and noise enhancement.

5.3 Simulation-based performance evaluation

We consider the 5 MHz LTE-like scenario with FBMC/OQAM waveform, 300 active subcarriersand subcarrier spacing of 15 kHz [36]. Results are shown for the 2 × 2 spatial multiplexing



0 5 10 15 20 25 3010-4

10-3

10-2

10-1

100

Eb/No in dB

Unc

oded

BE

R

2x2 MIMO, VehA channel with PCI, Uncoded BER performance

1-tap equalizer 200 km/hEmbedded equalizer 200km/h1-tap equalizer 0 km/hEmbedded equalizer 0km/h

Figure 5-3: LMMSE detection performance for 2×2 spatial multiplexing MIMO with VehicularA channel and perfect channel knowledge in the receiver.

MIMO configuration using Vehicular A (VehA) and Hilly Terrain (HT) channel models with0 km/h and 200 km/h mobilities and the carrier frequency of 450 MHz. In the simulations,independent instances of the corresponding channel model are used for the four propagationpaths of the 2× 2 MIMO configuration. In addition to the embedded equalizer, also the single-tap subcarrier equalization model (using the same equalizer weights in FC-FB implementationfor all bins of a subcarrier) is included in the comparisons. Figs. 5-3 and 5-4 show the resultsassuming perfect channel knowledge in the receiver. While the single-tap equalizer modelworks quite well with the Vehicular A channel, its performance with Hilly Terrain channel issignificantly degraded. The embedded equalizer is able to handle very well also the highlyfrequency-selective subchannel case of the HT channel. The equalizer weights are constantover each FFT processing block, which introduces some performance degradation with highmobility. Figs. 5-5 and 5-6 give results with pilot based channel estimation. The usedpilot structure is shown in Fig. 5-7 and it is based on the auxiliary pilot model [37] to controlintrinsic interference of the pilot symbols. The channel estimation based results should be takenas preliminary; careful optimization of the pilot interpolation methods is expected to improvethe results.

5.4 Complexity evaluation

The FC-FB structure has potential for reduced computational complexity, in terms of multipli-cation and addition rates, in comparison to the commonly used polyphase filter bank structure[35]. In the following, we assume that the number of transmitted streams in a SM MIMOsystem is equal to NT. We assume the EMPhAtiC demonstrator parametrization for FC-FBreceiver with FFT length of N = 512L/2 = 4096, IFFT length of L = 16 and overlap factorof (L− LS)/L = 6/16, where LS = 10 is the number of useful output samples (half-symbols)



0 5 10 15 20 25 3010-4

10-3

10-2

10-1

100

Eb/No in dB

Unc

oded

BE

R

2x2 MIMO, HT channel with PCI, Uncoded BER performance

1-tap equalizer 200 km/h1-tap equalizer 0 km/hEmbedded equalizer 200km/hEmbedded equalizer 0km/h

Figure 5-4: LMMSE detection performance for 2 × 2 spatial multiplexing MIMO with HillyTerrain channel and perfect channel knowledge in the receiver.

0 5 10 15 20 25 3010-4

10-3

10-2

10-1

100

Eb/No in dB

Unc

oded

BE

R

2x2 MIMO, Estimated VehA channel, Uncoded BER performance

200 km/h0km/h

Figure 5-5: LMMSE detection performance for 2×2 spatial multiplexing MIMO with VehicularA channel and pilot based channel estimation.



0 5 10 15 20 25 3010-3

10-2

10-1

100

Eb/No in dB

Unc

oded

BE

R

2x2 MIMO, Estimated HT channel, Uncoded BER performance

200 km/h0km/h

Figure 5-6: LMMSE detection performance for 2 × 2 spatial multiplexing MIMO with HillyTerrain channel and pilot based channel estimation.

Time

Sub

carr

ier

d d d d d d d d d d d P1 d P2 d

d P1 d P2 d d d d d d d d d d d






d P1 d P2 d d d d d d d d d d d d d d d d d d d d d d P1 d P2 d



d P1 d P2 d d d d d d d d d d d FFT block 1 FFT block 2 FFT block 3

P1: Pilots for antenna 1

P2: Pilots for antenna 2

Figure 5-7: Pilot structure used for 2x2 spatial multiplexing MIMO.



per IFFT processing block. In the structure of Fig. 5-2, long FFT is implemented blockwise foreach receiver antenna. The short IFFTs are implemented for each output stream. We assumethat the FFT and IFFT lengths are powers of two and that the implementation uses the splitradix algorithm taking B(log2(B)− 3) + 4 real multiplications for length-B FFT or IFFT.

Then in the FC-FB receiver implementation, the following elements are needed in the ex-ample case: (i) NR FFTs of length N = 4096, (ii) MusedNT IFFTs of length L = 16, and(iii) MusedNTNR sets of weight coefficients each containing L− 1 = 15 non-trivial coefficients.There is a need for using arbitrary complex coefficients as weights, instead of fixed basic weightcoefficients which, depending on the specific parametrization and design, could also take realvalues. From each processing block, MusedNTLS/2 data symbols are detected. The needednumber of real multiplications per detected symbol is about 53. This compares favorably withthe multiplication rate of the polyphase implementation with 3-tap subcarrier equalizers [16, 38],which takes about 65 real multiplications per detected symbol with the polyphase filter bankoverlap factor of K = 3, or about 72 multiplications per detected symbol with polyphase filterbank overlap factor of K = 4. The arithmetic complexity of the FC-FB based implementationdepends greatly on the overlap factor. As another example, if the IFFT length is 24, the FFTlength is 6144, and the overlap factor is 6/24, then the multiplication rate is reduced to 39 realmultiplications per detected symbol.

The above calculations include only the multiplications in the data symbol path. Additionalcomputations are needed for channel estimation and calculation of the equalizer weight coef-ficients. It was assumed that the weights are recalculated for each IFFT bin for each FFTprocessing block, which naturally leads to relatively high complexity. To reduce the complexity,methods based on interpolation over each set of weight coefficients and between FFT processingblocks should be developed.



6. Iterative Receivers for Bit-interleaved Coded MIMO-FBMC/OQAM

6.1 Introduction

The focus of the previous developments has been on spatial multiplexing receivers. The ap-plication of space-time coding to FBMC/OQAM systems has been shown to be an even morechallenging task. For instance, the direct application of the simple Alamouti scheme does notsucceed due to the interference structure of the FBMC/OQAM modulation. Several solutionshave been proposed. Receivers based on interference cancellation were studied in [39] thatare able to provide acceptable performance but with generally much higher complexity thanin OFDM. Alternative coding schemes, such as the Block-Alamouti scheme, have also beenproposed [40].

In this chapter, we propose another solution that allows to reach the large diversity gains ofspace-time coding with an FBMC/OQAM transmission. The idea is to combine all the ingredi-ents that work well in FBMC/OQAM, such as the linear MIMO equalizer and the interferencecancellation by using a bit-interleaved coded structure with turbo equalization [41]. The mainadvantage of this structure is that it decouples the choice of the coding and the modulationoperation. Hence simple linear MIMO equalizers can be used to treat the specific interference ofFBMC/OQAM, and the coding can be chosen independently in order to offer diversity. Iterativereceivers based on turbo-equalization have been shown to approach optimal MAP performancefor this type of transmitter schemes in OFDM and single-carrier (SC) modulations. The objec-tive of this chapter is to adapt the receivers for an FBMC/OQAM system taking into accountits intrinsic interference effect, evaluate the performance, and examine whether such a schemeis effectively able to offer the expected diversity.

The scenario considered here is a point-to-point multi-antenna transmission with the samenumber of antennas at the transmitter and the receiver. In a PMR scenario, it is most likelyto be limited to 2 antennas at each side. No channel state information is available at thetransmitter, so no resource allocation is performed and space-time coding is used to offer somediversity.

6.2 System model

The transmitter structure is based on space-time bit interleaved coded modulation (STBICM) asdescribed in [41, 42]. It is depicted in Fig. 6-1. The information bits are organized in frames ofsize Lf . A frame of information bits ua, a = 0, 1, . . . , Lf −1 is first encoded by a convolutionalencoder, then interleaved by a random permutation. The frame of interleaved coded bits is thensplit into NT subblocks sent to the NT transmit antennas. Each of these subblocks is further

Convolutional

encoderInterleaver Demux

FBMC

FBMC

uα

dk,i

Figure 6-1: Transmission scheme: space-time bit-interleaved MIMO-FBMC/OQAM.



mapped into QPSK symbols and split into M subcarriers using FBMC/OQAM modulation.The system model used here takes as inputs the real data entries corresponding to the real

and imaginary part of the QPSK symbols. The real information symbol to be transmitted onantenna i at subcarrier k and at time instant n is here denoted by dk,i(n), i = 1, 2, . . . , NT,k = 0, 1, . . . ,M − 1. These symbols are entering the FBMC/OQAM modulator at the rate2/T , where T is the FBMC symbol duration. For simplicity, we assume that the real symbolsare normalized at dk,i(n) = ±1. So the information symbols are zero mean with varianceσ2d = 1. The real information symbols for the NT transmit antennas are grouped into a vector

denoted by dk(n) for subcarrier k and time n:

dk(n) =[dk,1(n) dk,2(n) · · · dk,NT(n)

]T(6.1)

We assume that the transmitter has no knowledge about the channel state information, sothe signals sent to the different antennas have identical QPSK mappings and uniform powers.We consider a MIMO frequency selective fading channel modeled by tapped-delay lines. Thevariances of the various taps are characterized by a power delay profile. We assume a quasi-static Rayleigh fading so that the channel is assumed to be constant over the duration of aframe.

At the receiver side, we denote by yk,j(n) the output of the FBMC/OQAM demodulationon subcarrier k of antenna j at time n. These received symbols for all NR receive antennasare also regrouped in a vector denoted by yk(n) for subcarrier k and time n. In the caseof mildly selective channels, it can be assumed that the channel is approximately flat insideeach subcarrier. Taking into account the particular interference structure resulting from theFBMC/OQAM modulation, the transmission model can be written as [5]

yk(n) = Hkdk(n) + Hk

k+1∑k′=k−1

n+3∑n′=n−3

tk,k′(n− n′)dk′(n′) + ηk(n), (6.2)

where Hk is the NR ×NT flat fading channel matrix for subcarrier k. tk,k′(n− n′) representsthe transmultiplexer response of the filterbank (tk,k(0) is assumed to be equal to 0 as thesymbol of interest does no create any component on the imaginary part). It is restricted intime to 3 symbols before and after the current symbols and the other coefficients are assumednegligible. Finally, ηk(n) is the vector of noise samples at the different receivers. The additivenoise is assumed to be white Gaussian.

6.3 Iterative receiver

An iterative receiver is used, based on the well-known turbo principle. It is an extension of theturbo demodulation presented in [43], also applied in [44, 45] to MIMO systems. In this case,we extend the application to the MIMO-FBMC/OQAM transmitter, taking into account theinterference structure of FBMC/OQAM.

For simplicity, we assume perfect channel knowledge and perfect synchronization at thereceiver. The receiver is depicted in Fig. 6-2. It is made of two soft-in/soft-out (SISO) stagesexchanging soft information in the form of log likelihood ratios (LLRs). The first stage is asoft demodulator using the received symbols yk(n) as inputs and taking into account the softinformation from the second stage on the real information symbols dk(n). It has to mitigateinter symbol interference (ISI), inter carrier interference (ICI) as well as co-antenna interference(MAI) in order to produce a proper estimate of the information symbols, which can then be



FBMC

FBMC

Space-equ

Subc 0

Space-equ

Subc M-1

P/S deint.

decoder ûα

S/P int.

yk,i

Lα(d)

Le(d)

Le(dk,i)

Lα(dk,i)

Figure 6-2: Iterative receiver structure.

used as soft information for the second stage. The second stage is a classical SISO binarydecoder taking into account the soft information from the first stage on the coded bits as wellas the code structure (but not the received symbols themselves as they are treated throughthe demodulator in the first stage). It is implemented based on the BCJR algorithm. Theinterleaver is used in order to be able to assume independence between the local symbols ineach stage and therefore split the implementation. As mentioned, the second stage (SISObinary decoder) is classical and does not need further detail. On the other hand, the first stageis not straightforward to implement.

The optimal implementation would require to compute the a posteriori probabilities of theinformation symbols, P (dk,i(n)|yk(n)), based on the received observations, taking into accountthe soft information from the second stage provided as a priori probabilities on these informationsymbols, Pa(dk,i(n)). This can be rewritten in synthetic form as

P (dk,i(n)|yk(n)) =∑

dk′,i′ (n′)P (yk(n)|dk−1,1(n−3), . . . , dk+1,NT(n+3))

∏Pa(dk′,i′(n′)) (6.3)

up to a factor 1/P (yk(n)), which is constant for all hypotheses. The summation has to be madeover all possible values of the information symbols dk−1,1(n− 3) to dk+1,NT(n + 3) except forthe symbol of interest. This corresponds to a complexity on the order of 221NT , which is clearlyintractable. This could also be performed using the BCJR algorithm but that still requires anintractable complexity due to the number of terms involved in the channel model (6.2).

A more realistic implementation is to use a linear space-time equalizer to produce an estimateof each symbol dk,i(n). This estimate can then be used as soft information to transfer to thesecond stage (after deinterleaving). This equalizer structure is an extension of results reportedin [43, 46, 47, 44, 45]. In the case of MIMO-FBMC/OQAM however, there are several furtherchoices to be made and particular attention need to be paid to the way interference cancelationis managed. First, it is necessary to decide which received symbols yk′,j(n′) are going to be usedfor computing the estimate of a given information symbol dk,i(n). Assuming a mildly selectivechannel, and similarly to what is done in OFDM, only yk(n) is used here for the computation ofthe estimates dk(n). This restricts the size of the model and allows for a simple computationof the equalizer. It could however be extended to a multi-tap equalizer, taking into account amore complete model, also valid for more frequency selective channels. Then, the criterion forcomputing the equalizer has to be chosen. Several solutions are presented and discussed below.

In all cases, the model makes use of the symbol means and variances obtained on the basisof the soft information from the second stage, used as a-priori probabilities Pa(dk,i(n)) fromthe previous stage. The mean of each information symbol can be computed as

dk,i(n) = E {dk,i(n)} = 2Pa(dk,i(n))− 1, (6.4)



and the variance is given by

vk,i(n) = E{d2k,i(n)

}− d2

k,i(n)= 1− d2

k,i(n) (6.5)

since it was assumed that the symbols are mapped to ±1. Note that, as it is always done inthese types of equalizers, the mean and variance coming from the a-priori information on thecurrent symbol dk,i(n) are never used for the computation of its own estimate dk,i(n). The twoproposed versions of the equalizers are described below.

6.3.1 Virtual complex equalizer

The first option to compute the equalizer is to equalize the information symbols dk,i(n) as ifthey were complex symbols. It comes down to rewrite the model as if the imaginary interferenceterm in (6.2) were corresponding to the imaginary part of the information symbol. Hence anew virtual complex symbol is defined as

d(c)k,i(n) = dk,i(n) +

k+1∑k′=k−1

n+3∑n′=n−3

tk,k′(n− n′)dk′,i(n′) (6.6)

taking into account the interference of the FBMC/OQAM modulation. With this new virtualcomplex symbol, the channel model is simply given by

yk(n) = Hkd(c)k (n) + ηk(n) (6.7)

It is important to note that this model assumes that the channel is approximately flat insidethe subcarrier. The soft information (mean and variance) can also be rewritten for these newvirtual symbols assuming independence between the real information symbols:

d(c)k,i(n) = dk,i(n) +

k+1∑k′=k−1

n+3∑n′=n−3

tk,k′(n− n′)dk′,i(n′) (6.8)

v(c)k,i(n) = vk,i(n) +

k+1∑k′=k−1

n+3∑n′=n−3

t2k,k′(n− n′)vk′,i(n′) (6.9)

The objective of the equalizer is now to equalize these virtual complex symbols while takinginto account the soft information. Based on the a-priori soft information, and the simplemodel (6.7), the MMSE equalizer for computing the estimate of the virtual complex symbold

(c)k,i(n) is given by

d(c)k,i(n) = gH

k,i(n)[yk(n)−Hkd(c)

k,i(n)], (6.10)

whered(c)k,i(n) = [d(c)

k,1(n) · · · d(c)k,i−1(n) 0 d

(c)k,i+1(n) · · · d(c)

k,NT(n)]T, (6.11)

withgk,i(n) = σ2

d

[HkRi

dd,k(n)HHk + σ2INR

]−1Hkei (6.12)

andRidd,k(n) = diag

{v

(c)k,1(n), . . . , v(c)

k,i−1(n), σ2d, v

(c)k,i+1(n), . . . , v(c)

k,NT(n)

}, (6.13)



where INR is the identity matrix of size NR×NR, and ei is a selection vector of size NT×1 withall zeros except for a unity at position i. The equalization first performs an interference cancel-lation, based on the soft symbols dk′,i′(n′). Inserting the expression (6.11) of soft informationon the virtual symbols, the estimate can be rewritten as

d(c)k,i(n) = gH

k,i(n)yk(n)−Hkdk,i(n)− Hk

k+1∑k′=k−1

n+3∑n′=n−3

tk,k′(n− n′)dk′,i(n′) , (6.14)

where similarly,

dk,i(n) = [dk,1(n) · · · dk,i−1(n) 0 dk,i+1(n) · · · dk,NT(n)]T (6.15)

It should be noted that the symbol of interest dk,i(n) is obviously not canceled, hence thedefinition of a specific vector dk,i(n), different for each symbol, where the corresponding softinformation is set to zero. In addition, the expression (6.14) does not cancel the ISI and ICIfrom the same antenna, i.e. all terms in dk′,i(n′), for fixed i, that appear in the second termof (6.10). This is because the equalizer takes these terms as part of the virtual symbol andhence as useful information that needs to be estimated. However since the imaginary part isnot really useful in this case, it is also beneficial to perform additional interference cancelationto remove this interference as much as possible. So the equalizer can finally be written as

dk,i(n) = <

gHk,i(n)

yk(n)−Hkdk,i(n)− Hk

k+1∑k′=k−1

n+3∑n′=n−3

tk,k′(n− n′)dk′(n′) , (6.16)

where the only difference with (6.14) is the use of the vectors

dk′(n′) = [dk′,1(n′) · · · dk′,NT(n′)]T (6.17)

denoting the full vectors of soft information (where no symbol has been set to zero). All in all,only the terms based on dk,i(n) are not canceled. The interference cancelation part is followedby a combination vector (6.12) or space-equalizer across the antennas computed according tothe MMSE criterion. Similarly, the knowledge of the soft variance of the symbol of interest isnot taken into account in the computation of the correlation matrix Ri

dd,k(n). Here it shouldbe noted that due to the structure of the virtual symbol, each variance term of the correlationmatrix Ri

dd,k(n) is actually a sort of local average of the neighboring soft variances. Afterthis combination, the real part should be extracted to recover the real information symboldk,i(n). The advantage of this virtual complex formulation is that the equalizer automaticallytries to recover the correct phase of the virtual complex symbol and hence concentrates theISI and ICI interference terms on the imaginary part. They are then removed when taking thereal part. The disadvantage is that it still tries to do that even when the interference can becanceled based on the soft information about the neighboring symbols. So when the reliabilityabout the other symbols increases with the iterations, it cannot take full advantage of thissoft information. The co-antenna interference is also handled by this model and it takes intoaccount both contributions from the current symbols dk(n) as well as the contributions fromneighboring symbols dk′(n′) through the imaginary terms. Note that the soft variances of theseneighboring symbols is taken into account through the definition of the variance of the virtualsymbols (6.9).

Due to the difference of means and variances for each symbol, the linear combination vectorgk,i(n) is different for each symbol and hence the implementation of (6.16) requires a matrixinversion for each information symbol in the frame at each iteration. In order to decrease thecomplexity, several approximations can be considered.



Fixed correlation matrix: The strongest approximation is to use a fixed value of the corre-lation matrix

Ridd,k(n) = σ2

dINT = INT , (6.18)

which is equivalent to ignore the soft information about the symbol variances and assume themaximum variance (1 in this case since we assume ±1 symbols) for all symbols. In this case,the combination vector gk,i(n) is the same for all instants n and does not change throughiterations. It may of course still differ across the subcarriers k because of the channel frequencyselectivity. So NTM inversions are necessary in total. This approximation can be expectedto perform suboptimally as it only takes into account the mean part of the soft informationand not the variance, so the soft-equalizer cannot adapt itself to the increased reliability acrossiterations.

Average correlation matrix: In order to keep a reduced complexity while taking into accountthe soft information, an efficient approximation that has been used in turbo-demodulationschemes [46] corresponds to taking the mean a-priori variance across all symbols

v = 1NTMLn

NT∑i=1

M−1∑k=0

Ln−1∑n=0

vik(n) (6.19)

This seems even more relevant here as the virtual variances are already a local average ofneighboring symbols. The correlation matrix to be used in the computation of the combinationvector becomes

Ridd,k(n) = diag

{v, . . . , v, σ2

d, v, . . . , v}

(6.20)

Once again this combination vector becomes identical for all instants n but changes throughiterations. So NTM inversions are necessary per iteration.

Based on the symbol estimates (6.16), the extrinsic LLRs on the coded bits can be computedand the sent to the next stage as soft information. This computation based on a simplemodel [43]. The estimate dk,i(n) is assumed to be the output of an equivalent AWGN channelof the form

dk,i(n) = µk,i(n)dk,i(n) + θk,i(n), (6.21)

where θk,i(n) is a zero mean Gaussian variable with variance ν2k,i(n). The parameters µk,i(n)

and ν2k,i(n) can be computed as a function of the vector gk,i(n):

µk,i(n) = gHk,i(n)Hkei and ν2

k,i(n) = µk,i(n)σ2d − µ2

k,i(n)σ2d (6.22)

Then, the symbol extrinsic probabilities can be approximated as

Pe(dk,i(n)) ≈ P (dk,i(n)|dk,i(n)) = 1√2πν2

k,i(n)exp

−|dk,i(n)− µk,i(n)dk,i(n)|2ν2k,i(n)

(6.23)

6.3.2 Full real model

In this section, we describe a second option for implementing the space-time equalizer. Thismethod does not rely on the virtual complex representation to ensure that the interference iskept on the imaginary part of the estimate but instead precisely models the whole structureof this interference and uses the complete real model of the transmission. This allows to usethe soft information on these various symbols to a larger extent in the optimization of the



equalizer and could therefore potentially provide even better results when the reliability of thesoft information increases through iterations. In this case, we limit ourselves again to usingyk(n) only for the computation of the estimates dk(n) at subcarrier k and time instant n,which corresponds to a per-subcarrier equalizer with a single tap in the time domain (but withcombinations across the antennas). The model could however easily be extended to a multiple-tap equalizer in the case of more frequency selective channels by extending the observationinterval.

The channel impulse response is assumed to be known perfectly. Due to the selectivityof the filter bank, only adjacent subcarriers can interfere with the subcarrier of interest. Theoverlapping factor of the considered filter bank is equal to K = 4. So it is also assumed thatthe span of the filter bank compounded with the channel is limited to 3 instants before and3 instants after the symbol of interest, which is the case in most practical situations, evenunder highly frequency selective channels. The relevant information symbols are stacked intothe vector

dk(n) =[dTk−1(n− 3) · · · dT

k−1(n+ 3) dTk (n− 3) · · · dT

k (n+ 3) dTk+1(n− 3) · · · dT

k+1(n+ 3)]T

Based on the knowledge of the channel and the filter bank, the overall model of the receivedsymbols at subcarrier k and time instant n can be written as

yk(n) = Ckdk(n) + ηk(n), (6.24)

where the matrix Ck takes into account the combination of the SFB, the channel and the AFB.If the channel is only mildly selective, it corresponds to the model described above in (6.2) andthe matrix has the following structure

Ck =[tk,k−1(3)Hk · · · tk,k−1(−3)Hk tk,k(3)Hk · · · tk,k(−3)Hk

tk,k+1(3)Hk · · · tk,k+1(−3)Hk

]+ [0 · · ·0 Hk 0 · · ·0] (6.25)

The model is however not restricted to the quasi-flat case and can handle more frequencyselective channels. In order to easily take into account the real model of the informationsymbols, the received symbols are further split into real and imaginary parts (remembering thatall elements of dk(n) are purely real):

yk(n) =

<{yk(n)}

={yk(n)}

(6.26)

=

<{Ck}={Ck}

dk(n) +

<{ηk(n)}

={ηk(n)}

(6.27)

= Hkdk(n) + ηk(n) (6.28)

Based on this model, the MMSE space-time equalizer for information symbol dk,i(n), takinginto account the soft information (6.4) and (6.5) is given by

dk,i(n) = gk,i(n)T[yk(n)− Hk

˜dk,i(n)], (6.29)



where

˜dk,i(n) = [dk−1,1(n− 3) · · · dk,i−1(n) 0 dk,i+1(n) · · · dk+1,NT(n+ 3)]T (6.30)

contains the soft averages of all relevant symbols except the symbol of interest which is setto zero, where the combination vector, or spatial equalizer, is found according to the MMSEcriterion (in the real domain)

gk,i(n) = σ2d

[HkRi

dd,k(n)HTk + σ2

2 I2NR

]−1

Hkei, (6.31)

where

Ridd,k(n) = diag

{vk−1,1(n− 3), . . . , vk,i−1(n), σ2

d, vk,i+1(n), . . . , vk+1,NT(n+ 3)}

(6.32)

similarly contains all the soft variances of the relevant symbols except for the symbol of interest,which is set to σ2

d. The vector ei is the selection vector of size 21NT × 1 with all zeros exceptfor a unity at the position corresponding to dk,i(n). The equalization has the same structure asin the virtual complex case with an interference cancellation followed by a combination acrossantennas. All elements are real here however. The computation of this equalizer requires amatrix inversion for each symbol at each iteration. Note that it is the inversion of a 2NR×2NRreal matrix instead of the inversion of an NR × NR complex matrix in the case of the virtualcomplex model. Once again, an approximation can be considered to reduce the complexity.Using a fixed correlation matrix is not relevant here as it would provide similar results to thevirtual complex case. The approximation of the average correlation matrix can be considered.As previously, it corresponds to averaging the soft variance across all the symbols and using

Ridd,k(n) = diag

{v, . . . , v, σ2

d, v, . . . , v}, (6.33)

which is constant over the different instants n. It is also constant across the subcarriers but thecomputation of the combination vector (6.31) requires a different inversion for each subcarrieranyway, unless the channel is completely flat across all subcarriers.

A similar method can be used to compute the LLRs of the coded bits as a function of theestimates. The same model (6.21) is considered for the estimates, with parameters

µk,i(n) = gHk,i(n)Hkei and ν2

k,i(n) = µk,i(n)σ2d − µ2

k,i(n)σ2d (6.34)

6.4 Simulation results

In this section, results of the iterative receiver are presented and compared for different con-figurations. The parameters are detailed in Table 6.4. The convolutional encoder at the inputof the transmitter is the rate 1/2 code with octal representation [238, 358]. We consider anFBMC/OQAM transmission with M = 128 subcarriers, with the prototype filter designed asin [12], with an overlapping factor of K = 4. There are NT = 2 antennas at the transmitterand NR = 2 antennas at the receiver. Each frame is made of 12 FBMC symbols (on bothantennas) so the overall frame length is Nf = 3072. We consider a multiple taps Rayleighfading channel with various configurations of the power delay profile, corresponding to a flatchannel, Ped-A, and Ped-B channel models as well as intermediate situations. All simulationsare performed by averaging the results over 1600 channel realizations.



Number of subcarriers 128

Carrier frequency 400 MHz

Sample frequency 1.92 MHz

Frame duration 800 µs

Number of FBMC symb per frame 12

Constellation size QPSK

Number of antennas 2x2

Prototype filter PHYDYAS

Overlapping factor 4

Propagation channel model Ped-A & Ped-B

Table 6-1: Simulation parameters.

6.4.1 Iterations

Fig. 6-3 shows the evolution of the BER of the decoded bits as a function of the number ofiterations for the virtual complex equalizer for a multi-tap channel with 2 dominant taps spacedby 3 samples. The power delay profile of the channel is given by [1 0 0 0.1]. It appears thatthe performance is already quite close to the last iteration after around 4 iterations. Whileadditional iterations still provide some improvement, the results after 4 iterations are alreadysignificant of what the receiver can achieve. For this reason, we limit ourselves to 4 iterationsfor the remainder of this section. The figure also shows the large gain of diversity that can beachieved by the receiver.

6.4.2 Diversity

In this subsection, we analyze the diversity that can be achieved by the method as a functionof the number of taps of the channel. The virtual complex equalizer is considered here onceagain. Fig. 6-4 presents the BER after the 1st and 4th iterations for 1-(flat fading), 2- and 3-tapchannels, respectively. Note that the 2-tap channel corresponds to the Ped-A model and has apower delay profile of [1 0 0 0.1], while the 3-tap channel corresponds to the Ped-B model andhas a power delay profile of [1 0 0.25 0 0.1]. The increased diversity for increasing multipathis clearly observed, already after the first iteration. It is difficult, and out of the scope of thisresearch to provide precise diversity computations, especially as SNR range investigated here islimited. Roughly, it can be observed that the 1st iteration benefits from a diversity a bit belowNRκ, where κ is the number of taps (although taps with smaller power contribute less to thediversity due to the limited SNR range). After convergence, the situation is a bit different. Thetypical "waterfall" effect of this type of iterative receiver can be observed for the flat channel:there is some gain for low SNR but from 7 dB and on, it converges to the same slope as whatis obtained after 1 iteration, corresponding to a diversity of 2. For higher number of taps, thereseems to be a larger performance gain and potentially a larger diversity, but the same effectwaterfall effect could probably be observed at higher SNRs. In conclusion, even though it isdifficult to identify quantitatively the diversity gain obtained by the iterative receiver, the results



3 4 5 6 7 8 9

10−5

10−4

10−3

10−2

SNR

BE

R

1st iter2nd iter3rd iter4th iter5th iter8th iter

Figure 6-3: Evolution of the BER with number of iterations, for a 2-tap channel, and using avirtual complex equalizer.

3 4 5 6 7 8 9

10−5

10−4

10−3

10−2

SNR

BE

R

flat − 1st itflat − 4th it2tap − 1st it2tap − 4th it3tap − 1st it3tap − 4th it

Figure 6-4: BER of the virtual complex equalizer for various channels, showing the diversityevolution.



3 4 5 6 7 8 9

10−5

10−4

10−3

10−2

SNR

BE

R

1st it

LocalR − 4th it

AvgR − 4th it

FixedR − 4th it

Figure 6-5: Comparison of correlation matrix approximations for the virtual complex equalizer ona Ped-A channel. The 1st iteration and the 4th iteration are presented for the 3 approximations.

clearly show the efficiency of the method to benefit from the diversity available in the multipletaps, with a reasonable complexity.

As can be seen in the figure, the performance is already quite good and difficult to evaluateprecisely for the 3-tap channel (due to the limited number of realizations). So we will focusmainly on the 1-tap and 2-tap channel for our performance comparison below.

6.4.3 Comparison of the correlation matrix approximation

We compare here the effects of the various approximations proposed in Section 6.3.1 for thecomputation of the correlation matrix Ri

dd,k(n) for the virtual complex equalizer. Fig. 6-5shows the BER performance of the various approximations for a Ped-A channel (2 taps). Theconsidered approximations are the following

• ’LocalR’ denotes the correlation matrix computed by averaging the variance of neighboringsymbols and corresponding to the more precise model (6.13);

• ’FixedR’ denotes the fixed correlation matrix (6.18) corresponding to ignoring the infor-mation on the soft variance of the symbols;

• ’AvgR’ denotes the correlation matrix averaged over all symbols of the frame (6.20) andproviding a reduced complexity implementation.

The first iteration is the same in all cases as no soft information is available yet. After 4iterations, the ’FixedR’ aproximation does not perform as well, which confirms the advantage intaking the soft variance of the symbols into account in the computation of the space equalizer. Itappears that the ’AvgR’ approximation is really efficient and its performance is almost identicalto the ’LocalR’ model for a reduced complexity. This behaviour was already observed earlierfor these types of receivers [44] and is even easier to understand in this case as the variancesof the virtual symbol are already some kind of averaged values over the neighboring symbols.



3 4 5 6 7 8 9

10−3

10−2

SNR

BE

R

1st it

LocalR − 4th it

AvgR − 4th it

FixedR − 4th it

Figure 6-6: Comparison of correlation matrix approximations for the virtual complex equalizeron a flat fading channel.

Fig. 6-6 provides the same comparison for the flat fading channel. The performance is notas good as before due to the lack of diversity but the same general conclusions hold here aswell: the ’FixedR’ approximation incurs a loss in performance while the ’AvgR’ approximationprovides almost identical result as the initial model.

6.4.4 Advantage of the full model

In this subsection, we compare the two proposed methods: the virtual complex equalizer andthe equalizer based on the full real model of Section 6.3.2. Based on the results of the previoussection, we focus here on the ’AvgR’ approximation of the correlation matrix for the virtualcomplex equalizer, using the average of the variance across all symbols of the frame. For thefull model, we do not use any approximation as one of the objectives of the model is to beable to take the complete soft information in account when computing the space equalizer.Fig. 6-7 shows the results for the Ped-A channel. The full model appears to provide a smallperformance improvement from the first iteration already. This improvement does not gethigher with iterations however, so it does not seem that the full model benefits so much fromusing the soft information precisely.

Another objective of the full real model is to be able to take into account a more frequencyselective channel. Hence for comparison purposes, the results are investigated for an increasedfrequency selectivity of the channel. Note that it is not relevant to simply increase the numberof taps of the channels to consider more frequency selective channels, as this also increases thediversity and therefore improves the performance of the receiver. For this reason, the channelconsidered here is still a 2-tap channels, but with an increased delay between the taps. The delayis increased to 8 samples in this case, and the power of the second tap is slightly increased toobtain a higher frequency selectivity. Fig. 6-8 compares the results of both methods (once again,the ’AvgR’ approximation is used here for the virtual complex equalizer) after 4 iterations, for a2-tap channel with increased delay, providing increased frequency selectivity. The performanceis generally better than in the first case thanks to the increased power of the second tap. The



3 4 5 6 7 8 9

10−5

10−4

10−3

10−2

SNR

BE

R

Virt − 1st itVirt − 4th itFull − 1st itFull − 4th it

Figure 6-7: Comparison of the virtual complex equalizer and of the full real model-basedequalizer on a Ped-A channel.

3 4 5 6 7 8 9

10−5

10−4

10−3

10−2

SNR

BE

R

Virt − 1st itVirt − 4th itFull − 1st itFull − 4th it

Figure 6-8: Comparison of the two methods for a more frequency selective channel (’AvgR’assumption).



virtual complex equalizer does not seem to suffer that much from the frequency selectivityhowever. At higher SNR, the virtual equalizer even seems to get better than the full model for4 iterations although with the number of realizations used here, the difference is still within theconfidence interval and might not be really significant.



7. Conclusions and Future ResearchThis document reported results of research efforts made in the framework of WP4, Task 4.2,on “MIMO channel estimation and data detection." Studied topics include channel estimation,adaptive equalization of doubly dispersive channels (in both a general and a FC-based FBMCMIMO system), and STC based on turbo equalization and bit-interleaved coded modulation.

The problem of MIMO-FBMC/OQAM channel estimation was addressed in its preamble-based version, using preambles of the shortest possible duration. The latter feature is ofa high practical importance in highly time-varying environments such as the ones found inPMR applications. The main contribution lies on the derivation of MSE-optimal preamblesfor estimating MIMO CIRs of high frequency selectivity. The problem was also addressed forMIMO-OFDM systems, resulting in optimal single-symbol preambles. The training designsdeveloped here were demonstrated to considerably improve upon the estimation performanceof pseudo-random preamble signals. Further research is needed to extend these designs topreambles of a longer duration, necessary to support the estimation of even more frequencyselective channels.

An adaptive BLAST DFE structure, enjoying low complexity, fast convergence and numericalstability, was developed and extensively studied as an effective means of equalizing MIMO-FBMC/OQAM channels of high time- and frequency-selectivity. Channel estimation was calledto assist in further reducing the requirements for training input, with very favorable results. AnLTE-compliant pilot configuration was implemented and tested for adapting the equalizer inthe payload of the frame. It was observed, however, that this might be insufficient for providingreliable estimates of long CIRs, such as those following the 3GPP-HT channel model. This isbecause the density of the pilots in the frequency direction is not sufficiently high for providinga sufficient number of channel measurements. Possible ways around this are to employ otherdenser pilot formats or to take the sparsity characteristics of the CIR into account. Somepreliminary experimental results were provided to support these alternatives. Future researchis needed to complete this study. It must be noted that MIMO-FMT was also included inthe MC schemes under study, along with MIMO-OFDM. FMT was shown to be outperformedby FBMC/OQAM, especially in high mobility cases, and demonstrated to necessitate the useof a guard interval (CP) for improving its performance. Means of improving the pilot-assistedalgorithm’s performance for FBMC/OQAM so as to better compete with CP-OFDM at all SNRvalues need to be investigated.

Adaptive linear equalization, of the frequency sampling type, was studied in the contextof the FC-FB structure of D2.1. One of the important features of this equalization schemeis that the equalizer is effectively embedded in the FB structure itself, resulting in an elegantand computationally efficient implementation. Simulation results were provided to demonstratethe applicability of this equalizer in both medium and highly frequency selective channels, athigh mobile speeds. It is noted that channel estimation was also required here for setting theequalizer filters on the basis of scattered pilots. The computational requirements of this schemewere analyzed and favorably compared to those of the classical polyphase structure. To furtherreduce the complexity, methods based on interpolation over each set of weight coefficients andbetween FFT processing blocks should be developed.

Finally, the challenging task of STC in MIMO-FBMC/OQAM systems was re-visited. Theadopted approach is based on the well known turbo principle. A two-stage iterative receiver wasproposed. Its first stage is a space-time equalizer, which mitigates the interference and providessoft information to the second stage, where a classical soft-input soft-output binary decoder isemployed. Two versions were studied. In the first one, the symbols are considered as complex,



with the imaginary part corresponding to the intrinsic interference of FMBC/OQAM. The initialversion of this equalizer requires a matrix inversion at each iteration and for each symbol. Toreduce the complexity, a number of approximations were studied and tested. The secondversion stems from the will to further exploit the soft information fed back by the second stage,especially when the reliability of the decoded symbols is increased. It relies on a strictly real-domain formulation. In the simulations, a small number of iterations (3 or 4) was observed tobe sufficient for achieving acceptable performance and benefit from the available diversity. Theimpact of the aforementioned approximations was also discussed. The analysis was restricted toa single-tap equalizer. Its extension to a multi-tap equalizer is expected to significantly improvethe performance, especially in highly frequency selective channels. Extending the experimentalstudy to more realistic propagation conditions, involving doubly selective channels, remains asa subject of future research.



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Glossary and Definitions

Acronym MeaningAFB Analysis Filter Bank

AWGN Additive White Gaussian Noise

BCJR Bahl-Cocke-Jelinek-Raviv algorithm

BER Bit Error Rate

BICM Bit-Interleaved Coded Modulation

BLAST Bell Labs Layered Space Time

B-PMR Broadband PMR

CFR Channel Frequency Response

CIR Channel Impulse Response

CP Cyclic Prefix

CRS Cell-specific Reference Signal

CSI Channel State Information

DFE Decision Feedback Equalizer

FB-MC Filter Bank-based MultiCarrier

FBMC/OQAM Filter bank-based multicarrier using OQAM

FC-FB Fast Convolution-based Filter Bank

FIR Finite Impulse Response

FMT Filtered MultiTone

ICI Inter-Carrier Interference

(I)FFT (Inverse) Fast Fourier Transform

ISI Inter-Symbol Interference

ITU International Telecommunication Union

LLR Log Likelihood Ratio

LMMSE Linear Minimum Mean Squared Error

LMS Least Mean Squares

LS Least Squares

LTE Long Term Evolution

MAP Maximum A Posteriori

MC Multi Carrier

MCM Multi Carrier Modulation

MIMO Multiple Input Multiple Output

(M)MSE (Minimum) Mean Squared Error

NMSE Normalized MSE

OFDM Orthogonal Frequency Division Multiplexing

OMP Orthogonal Matching Pursuit



OQAM Offset Quadrature Amplitude Modulation

PCI Perfect Channel Information

PMR Professional (or Private) Mobile Radio

RLS Recursive Least Squares

SC Single Carrier

SFB Synthesis Filter Bank

SIMO Single Input Multiple Output

SISO Soft Input Soft Output

SM Spatial Multiplexing

SNR Signal to Noise Ratio

STBICM Space-Time Bit-Interleaved Coded Modulation

STC Space-Time Coding

3GPP 3rd-Generation Partnership Project

V-BLAST Vertical-Bell Laboratories Layered Space-Time

ZF Zero Forcing


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