05345737G-2010

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 3, MARCH 2010 1269

Overspread Digital Transmission Over WirelessLinear Time-Varying MIMO Systems

Pedro M. Marques and Sílvio A. Abrantes, Member, IEEE

Abstract—Accompanying the journey for higher spectral ef-ficiency in mobile wireless communications, this paper studiesmultiple-input–multiple-output (MIMO) transmission in a unifiedframework that consolidates the phenomena responsible for fre-quency selectivity and time selectivity, i.e., delay overspreadingand Doppler overspreading, respectively. Starting with a basebanddescription of the linear time-varying MIMO system, a novelapproach using continuous, discrete, and hybrid linear operatorsleverages the process of channel orthonormalization and modeldiscretization. Notably, this process leads to an optimal time-varying semiorthonormal matrix matched filter (the ORTHO-TS-MMF). Monte Carlo simulations test the error performance fortypical wireless MIMO realizations. In particular, they unveil thatcombining the ORTHO-TS-MMF with optimal linear detectiontakes advantage of both delay and Doppler diversities, signifi-cantly reducing the symbol error probability.

Index Terms—Delay- and Doppler-dispersive wireless channels,linear operator orthogonalization, maximum-likelihood sequenceestimation (MLSE), multiple-input–multiple-output (MIMO) sys-tems, noise whitening, nonstationary multivariate decompositions,spatial and delay-Doppler diversity, time-varying matrix matchedfilters (MMFs).

I. INTRODUCTION

THE TYPICAL delay spread of the wireless mobile com-munications channel in common urban or suburban prop-

agation environments is much smaller than its coherence time.This means that the channel may be viewed as almost timeinvariant during consecutive time frames spanning less thanthe coherence time. Since the coherence time is on the orderof milliseconds, one approach for designing a digital commu-nications receiver is to first consider the channel as interval-wise invariant, devise an equalizer to the received signal (e.g., alinear or decision-feedback equalizer), and then use an adaptivetechnique (e.g., least mean square, recursive least squares, orKalman) to track the channel’s variations and update the equal-izer parameters [1]–[5]. This procedure finds its most important

Manuscript received April 14, 2009; revised August 13, 2009 andNovember 7, 2009. First published December 4, 2009; current version pub-lished March 19, 2010. This work was supported in part by the Ministérioda Ciência e Ensino Superior, Fundação para a Ciência e Tecnologia, Lisbon,Portugal, under Grant SFRH/BD/17131/2004. The review of this paper wascoordinated by Dr. C. Ling.

P. M. Marques was with the Department of Electrical and Computer Engi-neering, Faculty of Engineering, University of Porto, 4200-465 Porto, Portugal.He is now with EFACEC Sistemas de Electrónica, S.A., 4471-907 Maia,Portugal (e-mail: [email protected]).

S. A. Abrantes is with the Department of Electrical and Computer Engi-neering, Faculty of Engineering, University of Porto, 4200-465 Porto, Portugal(e-mail: [email protected]).

Digital Object Identifier 10.1109/TVT.2009.2037640

application in frequency-selective slowly fading channels and,under mild conditions, may be replaced by blind or decision-directed techniques [6]. There are, however, several drawbacksin the adaptive process: 1) It is not a smooth process, as abruptchanges between consecutive intervals may occur [2]. 2) Nopilot-aided transmission is exploited. 3) It may be unstable un-der certain response variations [7]. 4) It does not take advantageof the time variations in the channel to improve the receiverperformance. Introducing the “vertical Bell Labs layered space-time” (V-BLAST) [8] architecture into the picture also has itsown problems [9], [10].

It has been shown that the RAKE receiver is inefficientwhen the channel is time varying [11], and attempts have beenmade to correct the problem using techniques such as time-frequency (canonical) representations and the basis-expansionmodel [12]–[15]. In [16], maximum-likelihood sequence esti-mation (MLSE) and maximum a posteriori probability equal-ization approaches are researched in the context of doublyselective channels. These developments encourage the studyof wireless channels in a more general setting. Instead ofdevising a receiver that matches the current channel conditionsand then using an adaptive algorithm to track the variationsin an interval-by-interval basis, one that matches both channeland variability conditions across adaptation intervals should beobtained. The primary barrier for such a study is the analysis oflinear time-varying (LTV) filters in a system-wide perspective,which is much more involved than analyzing conventionallinear time-invariant systems. This paper will try to shed somelight onto this subject by dwelling on practical aspects ofmultiple-input–multiple-output (MIMO) transmission and re-ception. No constrictive assumptions will be presumed fromthe start, except for that of accurate channel estimation at thereceiver [17]–[20].

The wireless channel between each pair of antennas will beconsidered linear and time varying, with maximum Dopplershift fD and RMS delay spread στ . In addition, depending onthe coherence time (Δt)c ≈ (2πfD)−1, coherence bandwidth(Δf)c ≈ (2πστ )−1, signal bandwidth W , and signaling inter-val T , it can be of two types, i.e., underspread or overspread.The typical wideband channel (e.g., CDMA2000 or W-CDMA)is overspread in delay (στ � 1/W → W � (Δf)c) and,hence, frequency selective yet slowly fading (T � (Δt)c).On the contrary, to capitalize on capacity, some multicar-rier systems (e.g., orthogonal frequency-division multiplex-ing) with very low carrier separations (e.g., not uncommonlylower than 10 kHz, viz., Third-Generation Partnership Project(3GPP) long-term evolution (LTE) evolved Multicast Broadcast

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1270 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 3, MARCH 2010

Multimedia Service and 1 kHz in Digital Audio Broadcastingmay be frequency flat (W < (Δf)c→στ < T/2π) and, hence,delay underspread; however, their signaling rate may be too lowto permit a slow-fading behavior. These narrowband subchan-nels are possibly fast fading; hence, Doppler overspread (T >(Δt)c→fD > W/2π), which means that, since intercarrier in-terference could be introduced, the induced Doppler spreadmight not be negligible from the receiver perspective. Channelssuch as those invoked self-dictate the need for a thoroughanalysis of transmission overspreading in the context of MIMOtechnology.

This paper is organized as follows: Section II introduces thecontinuous baseband input/output model of the time-varyingMIMO channel in both time and frequency domains and de-velops some considerations about the design of the input andoutput of a digital MIMO system. The method for discretiz-ing the baseband input/output model by means of channel-extracted basis functions is presented in Section III, where anorthonormalized version of the matrix matched filter (MMF) forthe time-varying MIMO channel is also proposed. Section IVdiscusses noise whitening under time-varying conditions, andSection V derives three MMSE-based linear detectors for theequivalent discrete input/output model. Section VI resorts toMonte Carlo simulations to assess the error performance of thesemiorthonormal MMF with linear detection, and Section VIIsummarizes this paper.

II. INPUT/OUTPUT MODEL

For a multipath wireless MIMO channel, the LTV im-pulse response between the jth input and the ith output isgiven by [21]

cij(τ, t) =Nij(t)∑k=1

α(ij)k (t)δ

(τ − τ

(ij)k (t)

)(1)

accounting for the attenuations α(ij)k (t) and delays τ

(ij)k (t)

of Nij(t) multipath replicas. The time-varying parameters of(1) must be interpreted as sample functions of statisticallyindependent stochastic processes.

A. Baseband MIMO Channel Model

In a typical wireless application, the input will be limitedto a passband [fc − B/2, fc + B/2] of bandwidth B = 2Wand center frequency fc so that one may define the base-band response as c ij(τ, t) = 2cij(τ, t)e−j2πfcτ . Denoting thenumber of inputs by nT , the baseband signal from the jthinput by sj(t), and the additive noise at the ith output byni(t), the baseband signal at the ith output is given by thesuperposition

ri(t) =12

nT∑j=1

∫τ

c ij(τ, t)sj(t − τ)dτ + ni(t). (2)

In a more compact vector-matrix notation

r(t) =12

∫τ

C(τ, t)s(t − τ) dτ + n(t)

=12C(τ, t)⊗

τs(t) + n(t) (3)

where one defines the column vector-valued signals r(t) =[ri(t)]nR×1, n(t) = [ni(t)]nR×1, and s(t) = [sj(t)]nT ×1, andthe matrix-valued channel response C(τ, t) = [c ij(τ, t)]nR×nT

(with nR being the number of outputs). The notation ⊗x

refers to a MIMO convolution in the x variable. The noiseterm n(t) = 2n(t)e−j2πfct is modeled as a zero-mean wide-sense stationary vector random process with autocorrelationRnn(τ) = N0δ(τ)I and power density spectrum Snn(f) =2Snn(f + fc) = N0I.

Defining the 2-D Fourier transform of C(τ, t) as C(f, ν),the noiseless baseband model in the frequency domain will begiven by

u(f) =12C(f − ν, ν)⊗

νs(f) (4)

which, due to the convolution in the variable ν, explains thechannel’s Doppler spread.

B. Baseband MIMO Input Design

A vector-valued sequence {ak} at a symbol rate fs andsignaling interval Ts = 1/fs usually lies at the input of aMIMO system. It can be conveyed to the receiver by a vector-valued signal x(t) with bandwidth fs/2 and satisfying

x s(t) = x(t)∑

k

δ(t − kTs) =∑

k

δ(t − kTs)ak. (5)

Accordingly, a low-pass pulse shaping filter G(f) may beapplied before transmission without the risk of losing anyinformation required to reconstruct the samples. The MIMOoutput will be given by

r(t) =∑

k

H(t − kTs, t)ak + n(t) (6)

where H(τ, t) is the full channel response, i.e.,

H(τ, t) =12

∫α

C(α, t)G(τ − α) dα. (7)

C. Baseband MIMO Output Design

Given the discrete input of transmission model (6), it makessense to search for a model with a discrete output that lacks norelevant information for the decision process. For the most part,one’s interest is a discrete input/output model after matchedfiltering of the received signal, and therefore, presampling is nota requirement. Moreover, avoiding MIMO output presamplingwill also avoid working at a higher sampling rate than the

MARQUES AND ABRANTES: OVERSPREAD DIGITAL TRANSMISSION OVER WIRELESS LTV MIMO SYSTEM 1271

signaling rate, which will simplify derivations. Starting with thecontinuous model in (6), the discrete model will be found byresorting to 1) the theory of general LTV operators (see [22]–[24] for some insight) and 2) the classical method of MLSEextended to a time-varying environment.

III. DISCRETIZING THE

MULTIPLE-INPUT–MULTIPLE-OUTPUT

INPUT/OUTPUT MODEL

It was found that the discrete-input model of the MIMOchannel is given by

r(t) =u(t) + n(t) = H(τ, t)⊗τ

a l + n(t)

=∑

l

H(t − lTs, t)a l + n(t) (8)

where the matrix-valued functions H(t − lTs, t) are withdrawnfrom parallel lines of equation t = τ + lTs (henceforward to beknown as the domains of transmission) in support of the graphof H(τ, t). When the channel is time varying and particularlyunder fast-fading conditions, a significantly different H(t −lTs, t) is used every lTs seconds.

To simplify the treatment of time-varying MIMO filters, itwill be important to circumvent the intricate convolution nota-tion. (In LTI systems, one simply takes the Fourier transform,which does not help in the LTV case.) One possibility is to useoperator theory to describe the entire MIMO system. In fact, theexpression for u(t) can be rewritten as

u(t) = H(τ, t)⊗τ

x s(t) =∫τ

H(τ, t)x s(t − τ) dτ

=τ=t−t′

∫t′

�

H(t, t′)x s(t′) dt′ (9)

where�

H(t, t′) = H(t − t′, t) ⇔ H(t, t′) =�

H(t′, t′ − t) (10)

and�

H denotes a two-sided-continuous time-varying matrix

operator (or a matrix of operators).�

H can be interpreted asa matrix of continuous-time matrices (the coefficients), andx s(t′) can be interpreted as a vector of continuous-time vectors(functions). In fact, standard matrix notation and algebraicconventions may be used without limitations; one just has toremind (10) and acknowledge that continuous operator multi-plication is an infinite sum of infinitesimal parts (the integral).

The product of operators�

A and�

B is defined as

�

C =�

A�

B Δ=�

C(t, t′) =∫t′′

�

A(t, t′′)�

B(t′′, t′) dt′′ (11)

and the Hermitian adjoint product is given by

�

C =�

AH �

B Δ=�

C(t, t′) =∫t′′

�

AH

(t′′, t)�

B(t′′, t′) dt′′. (12)

The continuous input/output model is thus rewritten as

r = u + n =�

Hx s + n. (13)

Moreover, one can also introduce a one-sided-discrete (hy-brid) matrix operator and reassemble (8) as

r(t) =∑

l

H(t − lTs, t)a l + n(t)

Δ=∑

l

H(t, lTs)a l + n(t) Δ= r = Ha + n (14)

where the dot above the operator indicates that it is discrete onone of its sides (implicit from the construction) and continuouson the other, and the dot above the vector indicates that it actsas a vector of discrete-time vectors. Similarly, two upper dotswill denote a two-sided-discrete matrix operator.

A. Defining a Basis for MIMO Response Expansion

The foremost objective is to find an approximation u(t) foru(t) as a function of generic orthonormal matrix-valued basisfunctions Φ(t − lTs, t) and vector-valued coefficients b l

u=Φb Δ= u(t)=∑

l

Φ(t − lTs, t)b l =Φ(τ, t)⊗τ

b l (15)

such that the following semiunitarity condition is satisfied:

ΦHΦ = I Δ=

∫t

ΦH(t − kTs, t)Φ(t − lTs, t) dt = δk−lI.

(16)

A finite-energy requirement on u(t) is set as

‖u‖2 = uHu Δ= ‖u(t)‖2 =∫t

uH(t)u(t) dt < ∞. (17)

By minimizing the energy in the difference between u andu, the standard minimum squared error criterion determines theoptimal vector b as follows:

bopt = arg minb

‖u − u‖2 = arg minb

‖u − Φb‖2. (18)

Let uo be the orthogonal projection of u onto the range of Φ,R(Φ), and write

bopt = arg minb

‖u − uo + uo − Φb‖2

[u − uo] ∈ R⊥(Φ) = N (ΦH

); [uo − Φb] ∈ R(Φ) (19)

where N (ΦH

) is the null space of the Hermitian adjoint ΦH

,and ⊥ denotes the orthogonal complement. Hence, applying the


Pythagorean theorem, it follows that

bopt = arg minb

‖uo − Φb‖2 (20)

which leads to the conclusion that setting u = Φb = uo min-imizes the energy in the vector-valued error function. Lettingu = uo + [u − uo], it is clear that

[u − uo] ∈N (ΦH

) → ΦH

[u − uo] = 0

Δ=∫t

ΦH(t − lTs, t) [u(t) − uo(t)] dt = 0 (21)

and consequently

ΦHΦbopt = Φ

Hu → bopt = Φ

Hu = u Φ

Δ=b(opt)l =

∫t

ΦH(t − lTs, t)u(t) dt. (22)

As expected, the optimal coefficients are obtained by orthog-onally projecting u(t) onto the basis functions Φ(t − lTs, t)or, equivalently, onto R(Φ). The optimal coefficients are least-square solutions of Φb = u.

The additional sufficient restriction

ΦΦH

=�

I Δ=∑

l

Φ(t − lTs, t)ΦH(t′ − lTs, t′) = δ(t′ − t)I

(23)

renders Φ as a full-orthonormal (i.e., full-unitary) matrix oper-ator and guarantees an approximation almost everywhere (a.e.)between u(t) and u(t)

‖u − u‖2 Δ=∫t

(u(t) − u(t))H (u(t) − u(t)) dt = 0

→ u(t) = u(t) a.e. (24)

but it will nevertheless be shown in Section III-D that (23) isunnecessary for optimal detection.

The optimal coefficients bopt can be rewritten in terms of theoriginal coefficients a (the sequence of transmitted symbols) asfollows:

bopt = ΦHu = Φ

HHa = H Φa

Δ=b(opt)k =

∑l

H Φk,la l =∑

l

HΦl,kak−l (25)

where H Φ = ΦHH denotes the projection of H onto Φ or

their correlation

H Φk,l =∫t

ΦH

(t, kTs)H(t, lTs) dt. (26)

In turn, using (23), H can be expressed in terms of Φ andH Φ as

H = ΦH ΦΔ= H(t, lTs) =

∑k

Φ(t, kTs)H Φk,l. (27)

The first equality in (27) resembles the familiar QR fac-torization from linear algebra with matrices replaced by linearoperators, except that (to be causal) H Φ is necessarily a lower

triangular operator, i.e., H Φk,l = 0 ∀k < l. To determine Φ,it shall be assumed (with no loss of generality) that the firstand last symbols to be transmitted are a1 and aL, respectively,such that

H(t, lTs) =L∑

k≥l

Φ(t, kTs)H Φk,l. (28)

The concept behind the factorization is a backward block-wise Gram–Schmidt semiorthonormalization procedure. AfteraL is received and H(t, LTs) is known, the Lth normal basismatrix is computed as

H(t, LTs) = Φ(t, LTs)H ΦL,L, H ΦL,L =∥∥∥H(t, LTs)

∥∥∥Φ(t, LTs) = H(t, LTs)

∥∥∥H(t, LTs)∥∥∥−1

(29)

where ‖H(t, LTs)‖ = [HHH]1/2

LL is given by

∥∥∥H(t, LTs)∥∥∥ =

⎡⎣∫t

HH

(t, LTs)H(t, LTs) dt

⎤⎦1/2

. (30)

The subsequent basis matrices Φ(t, (L − m)Ts), 1 ≤ m <L are found by projecting onto the previous space, computingthe difference, and normalizing the outcome, i.e.,

Φ (t, (L − m)Ts) =

⎡⎣H (t, (L − m)Ts) −L∑

i=L−m+1

Φ(t, iTs)

·∫t

ΦH

(t, iTs)H (t, (L − m)Ts) dt

⎤⎦H−1

ΦL−m,L−m (31)

where

H ΦL−m,L−m =

∥∥∥∥∥∥H (t, (L − m)Ts) −L∑

i=L−m+1

Φ(t, iTs)

·∫t

ΦH

(t, iTs)H (t, (L − m)Ts) dt

∥∥∥∥∥∥ (32)

and the remaining (below diagonal) entries are obtained from

H Φi,L−m =∫t

ΦH

(t, iTs)H (t, (L − m)Ts) dt (33)

for L − m < i ≤ L.This process builds an orthonormal basis for the received

signal and guarantees a causal H Φ. The main drawback forachieving causality is that the orthonormalization runs back-ward from H(t, LTs), which is something that may not befeasible if L is too large. An alternative approach is to keep L


Fig. 1. Discrete MIMO model after projecting the output onto R(Φ).

sufficiently low and perform the orthonormalization for everyL vector symbols.

From the channel decomposition, the MIMO noiseless outputis given by

u = Ha = ΦH Φa = Φbopt = uo (34)

indicating that the system Φb = u is, in fact, consistent. (Φ hasu(t) in its range.) Expressing the input/output model as

r =u + n = Φbopt + n

Δ= r(t) =∑

l

Φ(t − LTs, t)b(opt)l + n(t) (35)

and applying the adjoint operator on the left yield the discretemodel

r Φ = ΦHr=Φ

Hu+Φ

Hn= bopt+Φ

Hn

Δ= rΦk =∫t

ΦH(t−kTs, t)r(t)dt=b(opt)k +nΦk. (36)

One should notice that applying the adjoint operator isequivalent to filtering with ΦH(−τ, t − τ) and then samplingat t = kTs because

rΦk =∫τ

ΦH(−τ, kTs − τ)r(kTs − τ) dτ

= ΦH(−τ, t − τ)⊗τ

r(t)]

t=kTs

. (37)

As shown in Fig. 1, after cascading the channel with ΦH

,the discrete output is determined from the discrete input byapplying the two-sided-discrete operator H Φ (or a discretematrix filter HΦl,k).

The operator Φ does not span the entire noise space of the

MIMO output, because the condition ΦΦH

=�

I is not met. Asa result, some information is lost after the projection of r(t)onto R(Φ). It will now be determined whether this informationis relevant from an optimal estimation perspective.

B. Time-Varying MLSE (TV-MLSE)

Let Ψ be an ideal full-orthonormal matrix operator that satis-

fies both conditions ΨHΨ = I and ΨΨ

H=

�

I . As previouslydiscussed, this operator spans the entire space of the MIMOnoisy output r(t). Since its estimate is r = Ψr Ψ, where r Ψ =

ΨHr, then it is equal to r(t) at least a.e., that is, r = ΨΨ

Hr a.e.=

r. Developing this, an equivalent discrete representation thatlosslessly describes r(t) is found by orthogonally projectingr(t) onto the range of Ψ. That is

r Ψ = ΨHr = u Ψ + n Ψ

Δ= rΨk = uΨk + nΨk (38)

where, generically, xΨk =∫

t ΨH(t − kTs, t)x(t) dt. The dis-

cretized noise is space-time white

R n Ψn Ψ=

12E

[n ΨnH

Ψ

]= N0I

Δ=RnΨknΨl

(k − l) = N0δk−lI (39)

and normally distributed

nΨk ∼ N(E[nΨk],RnΨk

nΨk

)≡ N (0, N0I)

fn Ψ(n Ψ) =

[∏k

(2πN0)−nR

]e−(2N0)

−1nH

Ψn Ψ . (40)

The probability density function in (40) lies at the root ofthe maximum-likelihood (ML) detection criterion of the inputvector-valued sequence of symbols. If the symbols of the inputset are equally likely to be transmitted, then the classical Bayes’theorem establishes that

aML = arg maxa

f(r Ψ | u Ψ) = arg mina

‖r Ψ − u Ψ‖2. (41)

C. TS-MMF

Expanding the norm in (41) and using r Ψ − u Ψ = ΨH

(r −u), one arrives at

aML = arg mina

[−2Re{aH y} + ‖Ha‖2

](42)


Fig. 2. Discrete MIMO model after matched filtering with the TS-MMF.

where y = HHr is the output information required for ML

decision. Again

y = HHr Δ= y(kTs) = HH(−τ, t − τ)⊗

τr(t)

]t=kTs

(43)

yielding HH(−τ, t − τ) as the MMF for the time-varyingMIMO channel. This is the optimal predetection filter. It isnoncausal and nonpredictive and, in effect, translates to the

adjoint operator�

HH

(t′, t) in the operator domain; therefore, theoptimal receive filter operator is always the Hermitian adjointoperator of the channel operator. HH(−τ, t − τ) is merely

required to satisfy HH(−τ, kTs − τ) = HH

(kTs − τ, kTs) sothat it can be matched to the domains of transmission. Infact, the receiver is correlating with each matrix-valued func-tion H(t − lTs, t) as

y(kTs) =∑

l

⎡⎣∫t

HH(t − kTs, t)H(t − lTs, t)dt

⎤⎦a l

+∫t

HH(t − kTs, t)n(t) dt (44)

or, more simply, as

y(kTs) =∑

l

S ((k − l)Ts, kTs)a l + nHk (45)

i.e., y = Sa + n H, where S = H H = HHH is the global

system operator, and

S(kTs, lTs) =∫t

HH

(t, kTs)H(t, lTs) dt. (46)

Contrary to the LTI scenario, S(kTs, lTs) is nonstationary, be-cause it depends on both the difference k − l and each particulark; hence, it is not Toeplitz. The same is true for the noiseautocorrelation, i.e.,

R n Hn H= N0S

Δ= RnHknHl

(k − l, k)

= N0S ((k − l)Ts, kTs) . (47)

One may also check that ‖u‖2 = aH Sa ≥ 0, i.e.,∫t

uH(t)u(t) dt =∑

k

aHk

∑l

S(kTs, lTs)a l ≥ 0 (48)

meaning that S is nonnegative definite Hermitian and, hence, avalid covariance operator. In addition, since r H is a sufficientstatistic, the ML criterion is given by

aML = arg mina

(r H − Sa)H S−1

(r H − Sa). (49)

The presence of the middle factor S−1

is caused by the lackof semiorthonormality of the channel operator H; therefore,removing the first is an extra motivation for orthogonalizing theTS-MMF.

Writing MMF(τ, t) = HH(−τ, t − τ), it is clear that thereis an implicit linear transformation

(τ, t) → (−τ, t − τ) ⇔(

τt

)=

(−1 0−1 1

)(τ ′

t′

)(50)

where (τ ′, t′) are the old coordinates. This reveals that thematched filter has the form of a π/4-skewed mirroring ofthe full channel response H(τ, t). In other words, H(τ, t)experiences a vertical (time) shear of unit slope, followed bya horizontal (delay) reflection. In light of this, the matchedfilter will be distinguished with the name time-shear MMF (TS-MMF). The system diagram after filtering with the TS-MMF isshown in Fig. 2.

The TS-MMF is always implementable, except in a veryatypical scenario, where (Δt)c < Ts < τmax, because H(τ, t)will effectively change before it can be measured. In all othersituations, the TS-MMF may be determined by channel estima-tion or sending pilots. Whenever the channel is fast fading suchthat τmax < (Δt)c < Ts, a candidate for filling estimation gapsis interpolation [25], [26].

D. Optimality of the Discrete MIMO Model in R(Φ)

At this point, one no longer needs the operator Ψ, whichis introduced in Section III-B. The next step is to return tothe semiorthonormal channel operator Φ from Section III-Aand try to express the ML criterion within R(Φ), checkingfor its equivalence. The projection of the noise n(t) onto the

space R(Φ) is denoted by n Φ = ΦHn so that the MIMO


Fig. 3. Decomposition of the TS-MMF and global system response into discrete and semiorthonormal filters.

input/output relation after the channel decomposition H =ΦH Φ is equal to r = Ha + n = Φ(H Φa + n Φ) + ε, where

εΔ= ε(t) is the noise part out of R(Φ). In addition

ε =n − Φn Φ = r − Φr Φ

ΦH

ε =0 → ε ∈ N (ΦH

) = R⊥(Φ). (51)

From these considerations, the ML criterion may be re-written as

aML = arg mina

∥∥∥(r − Φr Φ) + Φ(r Φ − H Φa)∥∥∥2

= arg mina

‖r Φ − H Φa‖2 (52)

because (r Φ − H Φa)HΦ

Hε = 0, and ‖r − Φr Φ‖ is indepen-

dent of a. The ML estimates aML are entirely based on thediscretized MIMO model of Fig. 1, and one should notice

the exclusive use of the semiorthonormality condition ΦHΦ =

I. All the information required for optimal detection is stillavailable after projecting the MIMO output by means of the

operator ΦH

extracted from the full channel response H(τ, t);hence, one can confidently rely on the time-varying causaldiscrete model

r Φ = H Φa + n ΦΔ= rΦk =

k∑l=k−D

H Φk,la l + nΦk. (53)

Not only is the model attractively discrete and causal, but thenoise is also space-time white: R n Φn Φ

= N0I. Capitalizing

on this, (52) has the form

aML = arg mina

∑k

∥∥∥∥∥rΦk −k∑

l=k−D

H Φk,la l

∥∥∥∥∥2

(54)

from which one may easily derive a detector based on theViterbi algorithm. This would not be possible without a semi-orthonormal MMF.

E. Decomposing the TS-MMF Into Discrete andSemiorthonormal Factors

In addition to decomposing the channel operator as in (27), itis also relevant to seek an equivalent expression that factorizesthe TS-MMF HH(−τ, t − τ). With operator conventions

HH

= HH

ΦΦH Δ= H

H(t, lTs) =

∑k

HH

Φk,lΦH

(t, kTs).

(55)

Manipulating, (37) and (43) yield

y(kTs) =∑

l

HH

Φl,kr Φl = HHΦ−l,k−l ⊗

lrΦk

=HHΦ−l,r−l ⊗

αΦH(−τ, t − τ)⊗

τr(t)

]t=kTs

(56)

revealing a convolution between a discrete MMF HHΦ−l,r−l and

a continuous semiorthonormalized MMF ΦH(−τ, t − τ) (fromthis point onward to be designated ORTHO-TS-MMF).

Moreover, it is straightforward to show that the global systemresponse can also be decomposed into discrete and orthogonalterms. From (27) and (46), one finds

S = HH

ΦH ΦΔ= S(kTs, lTs) =

∑r

HH

Φr,kH Φr,l (57)

which, after some manipulations, yields

S ((k − l)Ts, kTs) =∑

r

HHΦr−k,rHΦr−l,r

l→k−l⇒ S(lTs, kTs) =HHΦ−r,k−r ⊗

rHΦl,k (58)

which is an expression that resembles the continuous correla-tion function S(τ, t) = HH(−α, t − α) ⊗α H(τ, t). The roleof the discrete MMF HH

Φ−r,k−r is to match the receiver toHΦl,k. The system diagram pertaining to these decompositionsis shown in Fig. 3.


IV. NOISE WHITENING

The uncorrelated noise samples at the ORTHO-TS-MMFoutput represent a clear advantage, as they induce a reductionin detection complexity. In the following, the possibility oforthonormalizing the TS-MMF using a fully discrete time-varying matrix filter will be addressed.

Since the noise autocorrelation at the output of the TS-MMFis given by Rn Hn H

= N0S, one may question the existence of

a filter Υ that, when applied at the TS-MMF output, renders thenoise white, i.e.,

R nHΥH n

HΥH= N0ΥH

H

ΦH ΦΥH

= N0I. (59)

If such a fully discrete filter indeed exists, it should satisfy thecondition

(HΥH

)HHΥH

= (H ΦΥH

)HH ΦΥH

= ΘHΘ = I

where ΘH

= ΥHH

ΦΦH

. It is known, nonetheless, that the

solution of the minimum Frobenius norm of the system HH

=H

H

ΦΦH

is obtained from the Moore–Penrose pseudoinverse of

H Φ, i.e., H†Φ; therefore, one may test the choice of a filter that

satisfies

ΘH

= ΥHH

=Υ=H

†H

Φ

H†HΦ H

H= H

†HΦ H

H

ΦΦH

. (60)

To make things clear, H Φ is expanded using its singular

value decomposition (SVD) H Φ = UDVH

, where UHU =

VHV = I, and R(U) = R(H Φ); R(V) = R(H

H

Φ). The

pseudoinverse is decomposed as H†Φ = VD

−1U

H, and (60)

is rewritten in the form

ΘH

= UD−H

VHVD

HU

HΦ

H= UU

HΦ

H. (61)

Consequently, a sufficient condition for whitening is UUH

=I, which is satisfied if U is square and nonsingular; hence, H Φ

should be a full row-rank operator. The full row-rank formula

H†Φ = H

H

Φ

(H ΦH

H

Φ

)−1

→ H ΦH†Φ = I

shows that the pseudoinverse is a right inverse of H Φ. In other

words, Υ = H†HΦ whitens the noise at the TS-MMF output, i.e.,

R n Θn Θ= R n Φn Φ

= N0I.

It should be kept in mind that H Φ is the band-based sparseL factor resulting from a block-wise QL operator factorization.Its dimensions are nT NL × nT L, where L is the number oftransmitted vector-valued symbols, and NL is the number of in-dependent matrix-valued functions H(t − lTs, t). NL is equalto L with high probability. From the statistical independence ofa set of NL domain functions H(t − lTs, t), one assesses thestatistical independence of the matrix coefficients within each

column of H Φ, and hence, whitening can be deemed possiblewith high probability.

V. LINEAR DETECTION FOR OVERSPREAD

MULTIPLE-INPUT–MULTIPLE-OUTPUT CHANNEL

Optimal detection of the sequence of vector samples a isaccomplished by employing the Viterbi algorithm extended toa MIMO setup, as it yields an ML estimate of a. As usual,the penalty incurred by optimal detection is computationalcomplexity, and in the MIMO case, it may become overwhelm-ing. When high detection complexity becomes an issue, time-varying linear filtering is an alternative method that warrantslinear complexity scaling, and therefore, it may be used instead.This section presents the optimal linear detector for the over-spread MIMO channel: the unconstrained linear detector. Theidea is to extend standard linear estimation (see [28]–[31]) totime-varying channel operators.

A. Unconstrained Linear Detection

The optimal linear detector can be derived by processingan entire sequence of vector samples rΦ0 · · · rΦL (i.e., r Φ).

This means that all the information available in H Φ will beused for detection, improving the error performance. The linearestimator of a is a = Dr Φ, and the error induced by theestimator is the sum of two independent entities, i.e., the symboldistortion and the residual noise, i.e.,

e = a − a = a − DH Φa︸︷︷︸a distortion

− Dn Φ︸︷︷︸n residual

. (62)

The error is minimized when it is orthogonal to the matchedfilter’s output, i.e.,

E[erH

Φ

]= 0 → Dopt = arg min

D

E[eH e

]. (63)

Expanding the error leads to E[arHΦ

] = DoptE[r ΦrHΦ

],which means that two statistical expectations need to be

determined: 1) E[arHΦ

] = 2R aaHH

Φ, and 2) E[r ΦrHΦ

] =

2H ΦR aaHH

Φ + 2R n Φn Φ, which, since R n Φn Φ

= N0I,

simplifies to R r Φr Φ= H ΦR aaH

H

Φ + N0I. Equating, one

obtains

R aaHH

Φ = Dopt

(H ΦR aaH

H

Φ + N0I)

which, after applying the generalized inverse (or Gaussianelimination), yields the general expression

Dopt = R aaHH

Φ

(H ΦR aaH

H

Φ + N0I)†

. (64)

For independent zero-mean symbols, (64) simplifies to

R aa = (Pa/2)I → Dopt = HH

Φ

(H ΦH

H

Φ +1γt

I)†

(65)


where, again, γt = Pa/σ2n Φ

. The minimum average error en-

ergy is given by

E[eH e]min = 2tr{

(I − DoptH Φ)R aa

}= 2tr

{N0DoptH

†HΦ

}(66)

where H†HΦ is the Moore–Penrose pseudoinverse of H

H

Φ.

VI. ERROR PERFORMANCE OF OPTIMAL TIME-VARYING

SEMIORTHONORMAL MATRIX MATCHED FILTER

WITH LINEAR DETECTION

This section will study the error performance of the ORTHO-TS-MMF when combined with the unconstrained linear detec-tor. The goal is to infer how well the receiver compensates (andpossibly benefits from) the delay-Doppler overspreading of theMIMO channel. To accomplish this, one shall engage in Monte-Carlo-based numerical simulation for several wireless MIMOchannel realizations.

A. Continuous MIMO Channel Model

The continuous passband channel model was given in (1). Itsbaseband version is

cij(τ, t) = 2Nij(t)∑n=1

α(ij)n (t)ejϕ

(ij)n (t)δ

(τ − τ (ij)

n (t))

(67)

where the multipath delays and associated phases are τ(ij)n (t) =

τ(ij)n − (ω(ij)

n /ωc)t, and ϕ(ij)n (t) = −2πfcτ

(ij)n (t) = ϕ

(ij)n +

ω(ij)n t, respectively. The carrier frequency is ωc, and the an-

gular Doppler shift is given by ω(ij)n = 2π(υ/λ) cos φ

(ij)n =

2πfD cos φ(ij)n , where υ is the relative velocity between the

transmitter and receiver, λ is the wavelength, φ(ij)n are the

different angles of arrival relative to the direction of motion,and fD is the maximum Doppler shift.

Since the dominant cause of mobile channel variability is themultipath phase variation due to the Doppler effect, the fol-lowing approximations were considered for model simulation:1) fc � fD → τ

(ij)n (t) = τ

(ij)n , α(ij)

n (t) = α(ij)n , and Nij(t) =

Nij , for the small-scale motion of the receiver unit. 2) Eachmultipath component arrives at different receiver antennasby parallel paths, i.e., φ

(ij)n = φ

(1j)n ; moreover, α

(ij)n = α

(1j)n .

3) Nij = N . 4) Multipath components from different transmit-ter antennas are independent. 5) The receiver moves in parallelto the axis of the receiver array, as shown in Fig. 4; thus

ϕ(ij)n = −2πfc

(τ (1j)n − (d/c)(i − 1) cos φ(1j)

n

)(68)

where β = 2π/λ, d is the antenna separation, and c is the speedof light. The antennas will be either equispacedly fit withindmax = 0.5λ or freely scale, i.e., dmax = dnR = 0.5λnR.

The multipath delays follow the exponential model: τ(1j)n =

−στ ln(1 − u(1j)n ), where fτ (τ (1j)

n ) = (1/στ )e−τ(1j)n /στ , and

u(1j)n ∼ U(0, 1). The other delays are given by τ

(ij)n = τ

(1j)n −

Fig. 4. Different multipath phases of a moving array with collinear axis.

(d/c)(i − 1) cos φ(1j)n . Equivalently, the attenuations are ex-

ponentially decaying, i.e., α(1j)n = e−τ

(1j)n /2στ , and the angles

of arrival φ(1j)n ∼ U(0, 2π) simulate the isotropic scattering

model. The RMS delay spread will be the rule of thumbστ = 3 μs, and the number of multipath components will beN = 8. Since it is a natural candidate for fourth-generationLTE licensing, the frequency of operation fc = 2.6 GHz willbe chosen for the simulation, and the maximum Doppler shiftwill be fD ≈ 200 Hz.

The pulse-shaping filter is the one-third rolloff square-root-raised-cosine filter (SRRC) with impulse response

srrc(τ) = (4roll√

fs)[π(1 − (4rollfsτ)2

)]−1

·[cos (π(1 + roll)fsτ) +

sin (π(1 − roll)fsτ)(4rollfsτ)

](69)

delayed an integer Δ of symbols and truncated on both sides

g(τ) ={

srrc(τ − Δ/fs), 0 ≤ τ ≤ 2Δ/fs

0, otherwise(70)

which yields the full pairwise channel response

h ij(τ, t) =N∑

n=1

α(1j)n e

j(ϕ

(ij)n +ω

(1j)n t

)g(τ − τ (ij)

n

). (71)

B. Discrete MIMO Channel Model for Simulation

To avoid aliasing, the noiseless MIMO output u(t) will besampled at twice the signaling rate, i.e.,

u(kTs/2) =∑

l

H(kTs/2, lTs)a lΔ= 2u = 2Ha (72)

where the operator 2H has two different discrete rates on eachof its sides. If the noise is confined to a bandwidth BN/2 = fs,then the global noisy discrete channel model is given by

r(kTs/2) =∑

l

H(kTs/2, lTs)a l + n(kTs/2)

Δ= 2r = 2Ha + 2n. (73)

The entries of H(kTs/2, lTs), hij(kTs/2, lTs) are 1 × 1operators that define the response between the jth input and the


Fig. 5. Average 16-QAM error rates for two receiver array sizes, increasingnumber of transmitter antennas, and receiver antennas equispacedly fit withindmax = 0.5λ (Ts = 100 μs).

ith output of the MIMO channel. They can be related to (71) asfollows:

h ij(kTs/2, lTs)

= h ij(kTs/2 − lTs, kTs/2)

=N∑

n=1

α(1j)n e

j(ϕ

(ij)n +ω

(1j)n kTs/2

)g(kTs/2 − lTs − τ (ij)

n

).

(74)

The discrete ORTHO-TS-MMF 2Φ may be computed from2H using a discretized version of the semiorthonormalizationprocedure described in Section III-A. Unfortunately, imple-mentation showed that it was rather slow to complete andhad poor numerical stability. To address this issue, a novelmethod based on backward block-wise Householder reflectionswas conceived, proving to be remarkably fast and stable. It ispresented in Appendix A.

C. Monte Carlo Simulation and Numerical Results

Simulation starts by randomly generating and modulating amessage of 64 complex symbols from a 16-state quadratureamplitude modulation (16-QAM) constellation. The randommessage is applied to 2H, and noise is added, simulating the

MIMO transmission. The output is then filtered by 2ΦH

, andthe unconstrained linear detector derived in Section V-A. Aslicer makes the final decisions on the transmitted constellationsymbols. To yield reliable error rates, this process is repeated,so that at least 1 × 105 symbol transmissions are simulated.Several plots were obtained for randomly chosen channel re-alizations, and each plot assesses the error rate as a function ofthe SNR (stepped at 2 dB) at the output of the MIMO channel.

The plots in Fig. 5 reveal that increasing the number of trans-mitter antennas has the effect of increasing the error probability,which is legitimate, because the symbols transmitted fromdifferent transmitter antennas will interfere with one another

Fig. 6. Average 16-QAM error rates for one transmitter antenna and severalreceiver array sizes (receiver antennas equispacedly fit within dmax = 0.5λand free scaling considered) for Ts = 100 μs.

Fig. 7. Average 16-QAM error rates for several signaling rates, a MIMOconfiguration of nT = 2 and nR = 4, and receiver antennas equispacedly fitwithin dmax = 0.5λ.

and escalate confusion at the receiver. The penalty for higherdata rates (using the same bandwidth and thus improving thespectral efficiency) is an increase in the probability of symbolerror. Nevertheless, by decreasing the signaling interval (at theexpense of a larger bandwidth), much better error performancesare possible.

A different situation appears in Fig. 6: a single transmitterantenna and an increasing number of receiver antennas. Onereadily gathers that increasing the size of the receiver arrayhas the consequence of decreasing the error probability, whichmeans that the receiver is able to use reception diversity to im-prove the error performance. This improvement is even largerwhen free scaling of the receiver array is permitted, which isnatural, given the reduction in cross correlation.

To assess the variation of the error performance with thesignaling rate, plots for several Ts were obtained as shown inFig. 7. The shortest signaling interval is responsible for the


Fig. 8. Average 16-QAM error rates for several maximum Doppler shifts,a MIMO configuration of nT = 2 and several nR, and receiver antennasequispacedly fit within dmax = 0.5λ (Ts = 1 s).

lowest error rates. As the signaling interval increases belowστ = 3 μs, the error rate also steadily increases, which isexplained by an added difficulty of resolving multipath com-ponents. The error performance severely degrades when Ts

becomes somewhat higher than στ , as demonstrated by theerror rate curve of Ts = 0.1 ms. In this situation, there is littlemultipath diversity available for resolution, and the receivercannot compensate for the intersymbol interference (ISI) intro-duced. However, as the signaling rates get lower and lower, theISI will become negligible, and the error rates will significantlybe reduced (despite the eminent fast-fading situation).

The plots in Fig. 8 confirm that, when the channel is fastfading (which is the case for Ts = 1 s), the receiver is ableto use the extra variability introduced by the Doppler shift toimprove the detection, irrespective of the number of receiverantennas. One realizes that a cardinal advantage of the ORTHO-TS-MMF is its “channel variability awareness,” which exploitsthe available Doppler diversity.

Plots for the error rate variation with the maximum antennaseparation are shown in Fig. 9. They display an error per-formance deterioration as the maximum separation decreases.Bringing the antennas increasingly closer to one another has theeffect of increasing correlation to the extent of not permittingthe compensation of both intrasample ISI and time selectivity.The plots corroborate one’s intuition.

VII. CONCLUSION

This paper has proposed a new framework for the studyof digital communications over delay-Doppler dispersive LTVMIMO systems. Using a baseband setup, it started by defininga continuous input/output model of digital transmission thatdid not ignore the time-varying nature of the wireless channel.Then, it aimed at model discretization using a channel-extractedorthogonal basis. This was accomplished by orchestrating atheory of continuous, discrete, and hybrid matrix operators thatimpressively simplified all MIMO calculations. An orthonor-

Fig. 9. Average 16-QAM error rates for several maximum receiver antennaseparations, a MIMO configuration of nT = 2 and nR = 4, and receiverantennas equispacedly fit within dmax (Ts = 1 μs).

malized MMF, which maintained the white nature of the inputnoise and was optimal from an ML detection perspective, hasbeen derived.

The optimal linear detector for the new model has beenderived and tested with success using Monte Carlo simulations.They have purveyed considerable insight into the error per-formance of the time-varying (i.e., Doppler aware) semiortho-normal MIMO matched filter, confirming that time-frequencyselectivity can be put to one’s favor as it is the result ofdelay-Doppler diversity. They have also shown that the errorperformance is boosted when the number of receiver antennasincreases and that the penalty for achieving higher spectralefficiencies with more transmitter antennas is an increase inerror probability.

APPENDIX

BUILDING OPTIMAL TIME-VARYING SEMIORTHONORMAL

MATRIX MATCHED FILTER WITH BACKWARD

BLOCK-WISE HOUSEHOLDER REFLECTIONS

The goal is to perform the decomposition

2H = 2ΦH ΦΔ= H(kTs/2, lTs) =

∑m

Φ(kTs/2,mTs)H Φm,l

(75)

where H Φ is block lower triangular with block size nT × nT ,

and 2Φ is block orthogonal, with each block consisting of nT

columns.The idea is to sequentially eliminate the entries above each

block in the lower right diagonal of 2H, where the lower rightdiagonal is interpreted as the diagonal that starts with the utmostlower right block of 2H. The entries above the utmost lowerright block can be eliminated by finding the QR factorization ofthe rightmost nT columns of 2H, i.e., H(:, LTs) = Q, where Ris nT × nT lower triangular. Next, the utmost lower right blockof Q is retrieved, and P is computed as follows:

B = Q(end, :) → P = B(BHB)−1/2. (76)


Define v = Q − eP(QHQ)−1/2 = Q − eP, where e is thesame size of Q and is all zeros, except on the lower rightdiagonal, where it is all ones. The block v can be used to buildthe first reflector

H r = I − 2v(vHv)−1vH . (77)

Moreover, from (76), it is known that PHP = I and PHB =BHP = (BHB)1/2. Expanding the products vHv and vHQreveals vHv = 2I − 2(BHB)1/2 = 2vHQ, which means that

H rQ = Q − 2v(vHv)−1vHQ = Q − v = eP. (78)

The entries above the utmost lower right block of Q have beeneliminated.

Now, since the goal is to perform the elimination in 2H,(78) is multiplied by R, which yields the same eliminationH rQR = H rH(:, LTs) = ePR. H r is applied to the entire

operator 2H

2Hset= H r2H = 2H − 2v(vHv)−1vH

2H (79)

and then, the bottom block row is normalized with

E = (RHR)−1/2RHPH → H(end, :) set= EH(end, :) (80)

which ends the first step of the decomposition. Further stepsare required to eliminate the rest of the entries above theblock diagonal, except that they are applied to 2H, excludingthe block columns (and corresponding block rows) that havealready been eliminated. The process ends when the first blockcolumn of 2H undergoes elimination, and at this point, thelower LnT × LnT entries of 2H are, in fact, H Φ in its blocklower triangular form. To avoid the inherent loss of numer-ical information in computing the product BHB, the SVDB = USVH is performed, and P = B(BHB)−1/2 = UVH

is computed, which replaces (76).The preceding description explains how to obtain the opera-

tor H Φ for simulation, but the ORTHO-TS-MMF 2ΦH

has yetto be determined. One obvious option is to apply the procedurejust described to the identity matrix, instead of 2H, which, after

normalization with (80), yields 2ΦH

. What this approach hasin obviousness, it also has in cost. It is slow. Experimentationrevealed that the swiftest approach was to perform Gaussian

elimination on the system HH

Φ2ΦH

= 2HH

.

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Pedro M. Marques received the Licenciatura andPh.D. degrees in electrical and computer engineeringfrom the University of Porto, Porto, Portugal, in 2003and 2009, respectively.

In 2003 and for an entire semester, he was aResearch Collaborator with Instituto de Engenhariade Sistemas e Computadores Porto, working innovel areas of next-generation digital communica-tion networks. In particular, the conception and im-plementation of a state-of-the-art monitoring systemfor Mobile-IPv6-based wireless access networks re-

sulted from that collaboration. He is currently with EFACEC Sistemas deElectrónica, S.A., Maia, Portugal, which is the largest Portuguese company inthe field of electronics. His current research interests include spectrally efficientwireless communications, multiple-input–multiple-output systems, orthogonalfrequency-division multiplexing, cooperative communication networks, andradio-over-free space optics. He is also a teacher at Universidade Lusófona doPorto, Porto, Portugal.

Sílvio A. Abrantes (M’90) received the Licenciaturadegree in electrical engineering and the Ph.D. degreein electrical engineering and computers from theUniversity of Porto, Porto, Portugal, in 1976 and1990, respectively.

He is currently an Assistant Professor with theDepartment of Electrical and Computer Engineering,Faculty of Engineering, University of Porto, and aSenior Researcher with INESC Porto, which is anR&D institute affiliated with the same university.In 2004, he was a Visiting Professor with the In-

formation and Telecommunication Technology Center, University of Kansas,Lawrence. He is the author of three books in Portuguese: Adaptive SignalProcessing (Lisbon: Fundação Gulbenkian, 2000), Source Coding: Two ShortVisits (Porto: FEUP Ediçõoes, 2000), and Error-Correcting Codes for DigitalCommunications (Porto: FEUP Edições, to be published). His past researchinterests include high-speed digital subscriber loops, simulation of communica-tion systems, adaptive signal processing, information theory, and error controlcoding. His current research interests are iterative decoding with turbo and low-density parity check codes.

Prof. Abrantes is a member of the Ordem dos Engenheiros (Portugal).

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