Orthogonal time-frequency signaling over doubly dispersive channels

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 11, NOVEMBER 2004 2583

Orthogonal Time–Frequency SignalingOver Doubly Dispersive Channels

Ke Liu, Student Member, IEEE, Tamer Kadous, Member, IEEE, and Akbar M. Sayeed, Senior Member, IEEE

Abstract—This paper develops a general framework for commu-nication over doubly dispersive fading channels via an orthogonalshort-time Fourier (STF) basis. The STF basis is generated froma prototype pulse via time–frequency shifts. In general, the or-thogonality between basis functions is destroyed at the receiverdue to channel dispersion. The starting point of this work is apulse scale adaptation rule first proposed by Kozek to minimizethe interference between the basis functions. We show that theaverage signal-to-interference-and-noise (SINR) ratio associatedwith different basis functions is identical and is maximized by thescale adaptation rule. The results in this paper highlight the crit-ical impact of the channel spread factor, the product of multipathand Doppler spreads, on system performance. Smaller spreadfactors result in lesser interference such that a scale-adaptedSTF basis serves as an approximate eigenbasis for the channel.A highly effective iterative interference cancellation techniqueis proposed for mitigating the residual interference for largerspread factors. The approximate eigendecomposition leads to anintuitively appealing block-fading interpretation of the channelin terms of time–frequency coherence subspaces: the channelis highly correlated within each coherence subspace whereas itis approximately independent across different subspaces. Theblock-fading model also yields an approximate expression for thecoherent channel capacity in terms of parallel flat-fading channels.The deviation of the capacity of doubly dispersive channels fromthat of flat-fading channels is quantified by studying the momentsof the channel eigenvalue distribution. In particular, the differ-ence between the moments of doubly dispersive and flat-fadingchannels is proportional to channel spread factor. The results inthis paper indicate that the proposed STF signaling framework isapplicable for spread factors as large as 0 01.

Index Terms—Capacity, doubly dispersive channels, Gabor sys-tems, random banded matrices, time-varying multipath channels.

I. INTRODUCTION

WIRELESS channels typically exhibit time-varying mul-tipath fading and can be modeled as linear doubly dis-

persive stochastic channels. The signal experiences dispersionin both time and frequency as it passes through the channel.The multipath effect causes dispersion in time, while the time-varying channel gain associated with each path results in dis-persion in frequency. As compared to our understanding of the

Manuscript received June 8, 2001; revised January 15, 2004. This work wassupported in part by the National Science Foundation under CAREER GrantCCR-9875805. The material in this paper was presented at the IEEE Interna-tional Symposium on Information Theory, Lausanne, Switzerland, June/July2002 and at the IEEE Vehicular Technology Conference (VTC-Fall 2002), Van-couver, BC, Canada, September 2002.

K. Liu and A. M. Sayed are with the Department of Electrical and ComputerEngineering, University of Wisconsin–Madison, Madison, WI 53706 USA(e-mail: [email protected]; [email protected]).

T. Kadous is with Qualcomm Technologies and Ventures, Qualcomm Inc.,San Diego, CA 92121 USA (e-mail: [email protected]).

Communicated by D. N. C. Tse, Associate Editor for Communications.Digital Object Identifier 10.1109/TIT.2004.836931

classical additive white Gaussian noise (AWGN) channel, mod-ulation and coding for doubly dispersive fading channels tendto be quite challenging and different from that for the AWGNchannel.

A general approach to digital communication is orthogonalsignaling in which transmitted symbols are modulated ontoa set of orthonormal basis waveforms. An important class ofsuch schemes is orthogonal frequency-division multiplexing(OFDM) or the discrete multitone modulation (DMT) [1]. Inessence, OFDM divides the channel into many small frequencybands and is well suited for a time-dispersive (frequency-se-lective) channel [2]. For a slowly fading frequency-selectivechannel, longer symbol durations (narrow frequency bands) aredesirable for mitigating the effect of multipath. However, longersymbol durations are more prone to frequency dispersion dueto temporal channel variations. Thus, we see that in doublydispersive channels, the symbol duration is constrained byconflicting requirements dictated by temporal and spectral dis-persion. This suggests signaling over short-time Fourier (STF)basis functions whose time–frequency support is matched tochannel characteristics.

STF signaling over doubly dispersive channels has been ex-plored by several researchers [3]–[7]. We briefly review thiswork to put the results of this paper in proper perspective. AnSTF basis is generated from a given prototype pulse via time andfrequency shifts (see (8)). It is also referred to as a Gabor basisor a Weyl–Heisenberg basis in the literature on time–frequencyanalysis. The time separation and frequency separation

between STF basis functions critically affect the time–fre-quency characteristics of an STF basis. Complete orthogonalSTF bases are only possible for (critical sampling)but are known to suffer from poor time–frequency localization(see, e.g., [8], [9]). If we relax the condition of critical samplingand consider , STF bases with better time–frequencylocalization properties can be generated, but they are necessarilyincomplete. Several researchers have investigated such incom-plete systems for doubly dispersive channels; a particular focushas been on pulse shape optimization to attain good time–fre-quency localization [3]–[6]. In this case, two sets of biorthog-onal bases are used: one at the transmitter for modulating thesymbols, and one at the receiver for recovering the symbols. Thetwo bases are intimately related via a duality relationship (see,e.g., [10], [3]).

The time–frequency dispersion induced by a time-varyingmultipath channel destroys the orthogonality/biorthogonalitycondition in the above systems. As a result, there is interfer-ence between different basis functions at the receiver. A keymotivation of the above works on biorthogonal systems is that

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2584 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 11, NOVEMBER 2004

bases with better time–frequency localization lead to lesserinterference between basis functions. While these works studybasis design quite thoroughly from a time–frequency localiza-tion perspective, the communication and information-theoreticaspects of STF signaling over dispersive channels are notfully explored. The most comprehensive study of incompletebi-orthogonal systems in a communication context is donein [3]. However, the improvements due to pulse optimizationare only quantified in terms of the mean-square error (MSE)in reconstructing the information symbols at the receiver.While biorthogonal systems have less interference comparedto orthogonal systems, they suffer a loss in spectral efficiencycompared to orthogonal systems (which have maximum spec-tral efficiency) by a factor of .1 As we demonstratein this paper, the spectral efficiency has a linear impact onoverall system capacity whereas interference has a logarithmicimpact (due to improvements in effective signal-to-interfer-ence-and-noise ratio (SINR)). Thus, it is attractive to considerorthogonal systems in conjunction with techniques for miti-gating interference. In fact, the numerical results reported in[3] strongly suggest this: with a choice of (halfthe spectral efficiency of orthogonal systems), they report anMSE/interference improvement by a factor of at most.

In this paper, we develop a general framework for orthog-onal STF signaling over time-varying multipath channels andstudy its performance from both communication and informa-tion-theoretic viewpoints. Our analysis is based on the wide-sense stationary uncorrelated scattering (WSSUS) model fordoubly dispersive channels [11], [12]. We consider completeorthogonal STF bases and assume that the pulse prototype isgiven a priori. A key to reliable communication is a funda-mental understanding of the interaction between the signalingbasis and the channel. In this context, the key channel param-eters are the multipath spread and the Doppler spread .In particular, our focus is on underspread wireless channels (forwhich ) since most radio-frequency wireless chan-nels fall within this category [12]. The key corresponding basisparameters are the support of basis functions in time and in fre-quency. The starting point of our work are two attractive prop-erties of orthogonal STF basis functions first reported in [4],[3]: 1) An appropriately chosen STF basis serves as a set of ap-proximate eigenfunctions for underspread linear time-varyingsystems, and 2) the prototype pulse’s duration and bandwidthcan be matched to the delay and Doppler spreads of the channel(pulse scale adaptation) to minimize the interference betweenbasis functions.

The first contribution of this paper is a rigorous investiga-tion of these two properties from a communication-theoreticviewpoint. Specifically, we derive an exact expression for thereceived SINR associated with each basis function and showthat it is identical for all basis functions due to the channelstationarity in time and frequency in the WSSUS model. TheSINR depends on an interference index, governed by the inter-

1The dimension of the space of signals with duration T and bandwidth Wis approximately TW for large TW . The number of basis functions in an STFbasis is given by N = TW=T F . Thus, N = TW for an orthogonal systemwhereas N < TW for a biorthogonal system, yielding N =N=1=T F <1since T F > 1.

action between the channel and the pulse prototype, and max-imizing the SINR is equivalent to minimizing the interferenceindex. In [4], a similar interference minimization criterion wasproposed from the viewpoint of “diagonalization” (eigenprop-erty) of the system operator, and in [3] it was proposed fromthe viewpoint of minimizing the MSE in the reconstruction ofthe transmitted symbols at the receiver. We then cast minimiza-tion of the interference index as an optimization problem overthe pulse scale and analytically derive the optimal pulse scaleadaptation rule for a class of pulses. The pulse scale adaptationrule was proposed in [4], [3] based on heuristic arguments bystudying specific pulse shapes. To our knowledge, our derivationof the optimal pulse scaling rule is the most rigorous and mostappropriate from a communication-theoretic viewpoint (SINRmaximization). Furthermore, the derivation of the pulse scalingrule is the only overlap of our work with existing work on STFsignaling.

We now provide a summary of the remaining results in thispaper that build on optimal pulse scale adaptation. Overall,our results highlight the critical effect of the channel spreadfactor from a communication and information-theoreticviewpoint. For a given delay spread , small spread factorscorrespond to slowly fading channels (small ), while largespread factors correspond to fast fading channels (large ).For sufficiently small spread factors, the residual interferencecan be made very small after pulse scale adaptation and thusthe corresponding STF basis functions serve as approximateeigenfunctions of the channel. We show that the approx-imate eigendecomposition yields an intuitively appealingblock-fading interpretation of the effect of the channel in termsof time–frequency coherence subspaces: the channel remainshighly correlated within each coherence subspace whereas itis approximately uncorrelated across different coherence sub-spaces. The number of independent coherence subspaces equalsthe delay-Doppler diversity afforded by the channel which isproportional to [13]. On the other hand, channels withlarger spread factors exhibit significant residual interferenceeven after pulse-scale adaptation. For such rapidly time-varyingchannels, we propose a highly effective residual interferencecancellation technique, sequential iterative interference cancel-lation (SIIC), which yields an impressive performance gain.Thus, the approximate block-fading model is applicable tolarger values of when pulse scale adaptation is used inconjunction with interference cancellation at the receiver.

The block-fading channel interpretation in terms of time–fre-quency coherence subspaces also yields an approximateexpression for the coherent capacity of doubly dispersivechannels. Essentially, the capacity of the dispersive channelcan be viewed as the capacity of parallel independentflat-fading channels, where is the number of independentcoherence subspaces (and the level of delay-Doppler diversity).Both ergodic and outage capacities can be estimated usingthis block-fading model and also facilitate the comparisonbetween orthogonal and biorthogonal systems from a capacityperspective. As the channel spread factor increases, the channeldiversity increases thereby improving outage capacity per-formance, whereas the ergodic capacity deviates from that ofa flat-fading channel due to increased interference between

LIU et al.: ORTHOGONAL TIME–FREQUENCY SIGNALING OVER DOUBLY DISPERSIVE CHANNELS 2585

basis functions. We quantify this deviation by investigating themoment behavior of the eigenvalue distribution associated withdoubly dispersive channels using perturbation analysis. Ourresult bounds the discrepancy between the moments associatedwith doubly dispersive channels and flat-fading channels, andshows that the discrepancy is proportional to channel spreadfactor.

Our analytical and numerical results indicate that the pro-posed framework for orthogonal STF signaling is effective forvalues of as large as and is most advantageous overconventional methods for channels with relatively large spreadfactors – . We note that this range of spread fac-tors covers all practical wireless communication channels andis also applicable to various other radio channels. For example,[12, Table 14-2-1] lists typical values of spread factor for sev-eral radio channels, whose spread factors are within the appli-cable range of our framework in most cases. For wireless cel-lular applications, the CDMA2000 “Vehicular B” channel givesa spread factor of – at a maximum vehicle spreed of40 km/h.

The paper is organized as follows. Section II briefly reviewsgeneral orthogonal signaling over doubly dispersive channelsand introduces STF signaling. Section III focuses on perfor-mance analysis of STF signaling and derivation of the pulsescale adaptation rule. Section IV discusses the proposed gen-eral time–frequency signaling framework, including the SIICalgorithm for interference cancellation and the block-fading in-terpretation in terms of time–frequency coherence subspaces.Information-theoretic aspects related to capacity of doubly dis-persive channels are studied in Section V. Concluding remarksand pointers for future work are presented in Section VI andmany of the proofs are relegated to the Appendix.

II. SYSTEM MODEL

The (complex) baseband doubly dispersive channel can bemodeled as a random linear operator with kernel [12],[11], [14]

(1)

where is the channel input and the kernel is calledthe delay-Doppler spread function, which is a random process inboth and [11]. The largest delay produced by the channelis called the multipath spread and the largest Doppler shift iscalled the Doppler spread. A wide variety of wireless environ-ments can be fairly accurately described by the WSSUS model,under which different delays and Doppler shifts are uncorrelated

(2)

where denotes the complex conjugation and the nonnegativeis called the scattering function of the channel. Projec-

tions of the scattering function along and are called the delay

power profile and Doppler power profile, respectively. Withoutloss of generality, we assume channel multipath coefficients

to be zero mean with total unit power, that is,

(3)

The Doppler spread is a measure of time variation inthe channel—the larger the value, the more rapidly the channelchanges in time. Its reciprocal, , is called the coher-ence time, within which channel remains strongly correlated.Analogously, channel frequency response within the channelcoherence bandwidth, , is strongly correlated. Theproduct is called the channel spread factor. If ,the channel is said to be underspread; otherwise, it is over-spread. The spread parameters critically control communicationperformance over doubly dispersive channels.

Let be a (complex) orthonormal basis in. Orthogonal signaling modulates transmitted symbols

onto the orthonormal basis by

(4)

Given signaling duration and (two-sided) bandwidth , thebasis functions span a signal subspace with dimension approx-imately being , where is the least integer notless than . After matched filter processing at the receiver, thereceived symbol can be written as

(5)

or, equivalently, in a matrix form

(6)

where transmit symbol has power and the AWGN hasvariance , that is, . The coefficientof represents the coupling produced by the channel betweenthe transmit basis function and the receive basis func-tion

(7)where

denotes the inner product and

is the induced -norm.Ideally, choosing to be eigenfunctions of the

channel will render a diagonal channel matrix , in which casethe channel is said to be flat fading, that is, the channel effectfor each basis function reduces to a multiplicative scalar andthere is no interference among basis functions. However, unliketime-invariant linear channels for which sinusoids are alwayseigenfunctions, there are no fixed eigenfunctions for general


time-varying linear channels. If the transmitter has the knowl-edge of channel realization, it may choose to dynamically adaptits transmit basis to channel eigenfunctions. But this procedurecan be very undesirable, if not totally impractical, from theviewpoint of implementation and system complexity.

Instead, we focus on STF basis for it is matched to thetime–frequency characteristics of doubly dispersive channel:the channel produces time and frequency shifts of the transmitsignal and the STF basis is generated from a prototype functionvia time and frequency shifts. We shall demonstrate that STFbasis functions exhibits good signaling performance, thuspresenting an attractive choice for underspread channels forwhich they serve as approximate eigenfunctions.

Definition 1: The STF basis is defined as

(8)

where is the normalized prototype pulse, isthe time separation, and is the frequency separation betweenbasis functions. The mean of in time and frequency aredefined to be

(9)

where is the Fourier transform of . The variance in timeand frequency are defined correspondingly as

(10)The product is called the time–frequency spread ofthe pulse. Without loss of generality, pulse is assumed tobe centered in time and frequency, that is, and .

Remark 1: The variance in frequency for a rectangular pulsedoes not exist due to slow decay of its spectrum. In this case, wedefine and .

The STF basis falls within the framework of the so-calledGabor systems (see, e.g., [15], [6]). The pulse time–frequencyspread is a measure of its time–frequency localization—thesmaller the spread, the better the localization. The Heisenberg’suncertainty principle (see, e.g., [15]) states that

(11)

with the equality attained by Gaussian pulses. Since STF basisfunctions are generated from the prototype via time–frequencyshifts, they all have the same time–frequency spread parameters.

As we will see later, a simple scaling operation, which shrinksor dilates the pulse in time and frequency, has a profound ef-fect on the performance of STF basis over doubly dispersivechannels.

Definition 2: Scaling by parameter is a mapping de-fined as2

(12)

2Note that the scaling operation preserves pulse energy.

If the original STF basis function has parameters , , ,and , then after scaling those parameters become , ,

, and , respectively. However, note that the products(pulse spread) and are unchanged.

The product is critical to the completeness of STF basisin . For undercritical sampling, , orthonormalSTF bases exist but are not complete. For critical sampling,

, complete orthonormal STF bases exist if the proto-type satisfies certain conditions [9]. Unfortunately, a com-plete STF basis has poor time–frequency localization due to theBalian–Low theorem [8], [9]. For overcritical sampling,

, the basis becomes linearly dependent or redundant. We willprimarily consider critically sampled case since ityields complete orthonormal systems. However, the relativelypoor time–frequency localization of basis functions in this casemakes interference minimization even more critical.

From communications viewpoint, time–frequency localiza-tion of basis functions alone does not determine communica-tion performance. It is the interaction between the basis and thechannel that determines signaling performance over doubly dis-persive channels. It turns out that the pulse ambiguity functionis key to understanding the interaction between the basis and thechannel [3].

Definition 3: The ambiguity function of the pulse ,, is defined as

(13)

One can easily verify the following well-known properties.

Proposition 1:

(14)

Moreover, the following holds for orthonormal basis functions:

and(15)

III. PERFORMANCE ANALYSIS AND INTERFERENCE

MINIMIZATION

In general, interference exists among basis functions indoubly dispersive channels. We rewrite (6) as

(16)

where the second term represents the interference towardsymbol by other symbols. The interference term in (16)encompasses two types of interference. One is the intersymbolinterference (ISI), which exists between adjacent time slots

and is caused by channel delay spread, the other isthe intercarrier interference (ICI), which happens betweenadjacent frequencies and is induced by channelDoppler spread. In principle, better time–frequency localizationof the basis offers better immunity against channel dispersion.Well-localized undercritical Gabor systems have been con-structed in [3], [5], [6]. Since an undercritical basis


often incurs a loss in transmission rate, we are primarilyconcerned with a complete STF basis . We alsoassume that the prototype pulse has been given a priori. (Pulseprototype design in the context of this work has been studied in[4], [3], [6].) We are interested in the impact of interference onsystem performance and methods of reducing interference fora given prototype.

A. Signal-to-Interference-and-Noise Ratio

We first calculate the entries of the channel matrix in (6).

Proposition 2:

(17)

(18)

Proof: See Appendix I.

Because of the time–frequency shift structure of the STFbasis and the WSSUS channel assumption, the variance ofchannel coefficients is invariant under time–frequency shifts,which can be seen from (18) to depend on the difference oftime–frequency indices ( and ). Consequently, theentries on the main diagonal of all have the same variance

(19)Our next result quantifies the amount of interference associ-

ated with STF signaling over doubly dispersive channels andrelates it to channel and basis parameters. We assume that thechannel coefficients are independent of transmitted informationsymbols.

Lemma 1: Define the SINR of receive symbol as

SINR (20)

Then the following holds:

SINR SINR (21)

where is given in (19) and is called the interferenceindex.

Proof: The completeness of the underlining orthonormalbasis implies that

(22)

and hence,

(23)

It follows from the definition of and power normalization of(3) that

So we have

Next we show

from which the lemma follows.

Remark 2: The average distortion for a given basis functionis

(24)

One can show that , that is, the interference indexis equal to the average distortion encountered by STF basis func-tions [16]. Thus, maximizing SINR is equivalent to minimizingthe distortion of basis functions passing through the doubly dis-persive channel.

B. Interference Minimization

Fig. 1 illustrates the basic notion of pulse scaling in thetime–frequency domain to minimize interference. If thechannel is nonselective in time and frequency ( and

), degenerates to a random scalar and hence anychoice of orthonormal basis will completely avoid interference.When the channel is time selective only ( and ),the optimal pulse is peaky in time, which is analogous to


Fig. 1. A schematic illustrating the optimal pulse scaling.

time-division multiplexing (TDM). When the channel is fre-quency selective only ( and ), the optimal pulseis peaky in frequency, which corresponds to frequency-divisionmultiplexing (FDM). For the general case , theoptimal pulse scale varies between the above two extremes.

When a channel is dispersive in both time and frequency, thuscausing both ISI and ICI, the pulse scale can significantly affectsystem performance. A long pulse in time can reduce ISI butis prone to frequency dispersion due to temporal channel varia-tions. On the other hand, a short pulse helps reduce ICI but is atthe cost of higher ISI due to relatively large delay spread withrespect to pulse duration. Therefore, one can expect an optimalpulse scale that minimizes both ICI and ISI jointly. Such a pulseadaptation, given channel statistics and pulse prototype, can becast systematically as the following optimization problem inlight of Lemma 1:

subject to (25)

where is the pulse time–frequency spread, a constantdetermined by the given pulse prototype. We note that the pulseambiguity function is parameterized by pulse scale.

Most of the pulse energy is concentrated within a time dura-tion and a frequency band (such as may be measured fromthe support of the pulse ambiguity function). So, if is largecompared to the channel delay spread while small comparedto coherence time, it will approximately experience flat fading,thus resulting in small distortion. Analogous conditions can alsobe expressed in the frequency domain as well. In summary, thefollowing conditions are necessary for small interference:

(26)

Multiplying together both inequalities in (26) and using, we obtain , which implies that basis design for

low interference is feasible for underspread channels but not forgeneral overspread channels.

Solving the optimization problem in (25) often involves com-plicated numerical methods. The above intuitive discussion oninterference minimization suggests an “equal” footing for timeand frequency. We next give a simple scaling rule whose deriva-tion is relegated to Appendix II. Also shown in Appendix II, theintegral in (25) can be written as a function of where .

Theorem 1 (Optimal Pulse Scale): Assume flat multipathand Doppler power profiles for the channel. Further assumethat the pulse is separable and symmetric in time and frequency,that is,

(27)

where is differentiable. Define

(28)

Then, a stationary point of (25) is 3

(29)

which is a local maximum if . A sufficient conditionfor local maximum is given by

(30)

The corresponding optimal interference index is given by

(31)

Remark 3: When is sufficiently small, the minimalinterference index in (31) can be approximated by

Since is a decreasing function in the neighborhood of , wesee that smaller results in smaller interference. Moreover,the minimal interference depends only on the channel spreadfactor and the pulse spread factor .

Remark 4: The optimal scaling rule matches the duration andbandwidth of prototype pulse relatively to the channel delay andDoppler spread: .

Remark 5: This work assumes no relative time delay be-tween receiver and transmitter. It turns out one can optimizesuch time offset to further improve receiving signal-to-noiseratio (SNR). As studied in [17], offset yields best perfor-mance, in which case the above scaling rule shall be modifiedaccordingly as .

We now apply the above methodology to study the perfor-mance of Gaussian and rectangular pulses. Due to space limi-

3The same relation has been derived in [4], [3] using different arguments.


Fig. 2. Interference index of a Gaussian prototype as a function of time scale for different channel spread factors. Flat multipath and Doppler power profiles areused.

tation, the derivations of our analytical expressions are left outbut can be found in [16].

The Gaussian pulse with variance and is defined as

where the normalization factor is chosen tonormalize pulse energy. Under flat delay and Doppler powerprofiles, one has

(32)where . Here we emphasize that ingeneral is a function of and , but the optimal value ofdepends approximately on the product as in Theorem 1.

In Fig. 2, we evaluate (32) for a Gaussian pulse and plotthe interference index versus the relative scale for dif-ferent channel spread factors. Note that due to the time–fre-quency duality and optimal pulse scaling rule, it is the spreadfactor rather than the individual or that determinesthe optimal performance. The plots, being based on the relativescale , reflect system performance for arbitrary values of

and such that is fixed. We also plot thelocus, denoted as the approximate optimal line, correspondingto the approximate scaling rule in Theorem 1. It is evident fromthe figure that this line intersects with the interference curves attheir global minimum points. Moreover, the figure illustrates the

effect of the channel spread factor: the smaller the spread factor,the smaller the interference. The Gaussian pulse exhibits excel-lent immunity against channel distortion thanks to its excellenttime–frequency localization. For example, for a spread factorof , the minimal interference is less than , whichgives about 30-dB SINR floor in high-SNR regime. However, as

, STF basis generated by a Gaussian pulse becomesunstable [18], thereby limiting its practical utility. Nevertheless,it provides an upper bound on performance.

The rectangular pulse with variance and isdefined as

otherwise.

Under flat-delay and Doppler power profiles

(33)

where . And under the more realistic “bathtub”Doppler profile

(34)

one has

(35)


Fig. 3. Interference index for a rectangular pulse as a function of time scale for different channel spread factors. Flat multipath power profile is used. Both“bathtub” and flat Doppler profiles are used.

We plot the interference index in Fig. 3 by numerically eval-uating (33) and (35). The interference curves are shown to bevery close for the two Doppler profiles. It is seen from the figurethat the approximate optimal line intersects the two interferencecurves near their true minimum points.

Fig. 4 plots the (approximate) minimal interference indexversus the channel spread factor for both rectangular andGaussian pulses ( 10 Hz). The scale adaptation rule inTheorem 1 is used. The figure includes the analytical curvesand the actual simulation data. The simulated channel is gen-erated by the Jakes model [19], which inherently correspondsto a “bathtub” Doppler profile. It is seen that the minimum in-terference index is an increasing function of the spread factor. Itis also evident that the Gaussian pulse outperforms the rectan-gular pulse. However, the performance gap varnishes asdecreases. The analysis and simulation results for the rectan-gular pulse show that it undergoes rather significant channel dis-tortion, even with optimal pulse scale, due to its poor time–fre-quency localization.

IV. GENERAL SIGNALING FRAMEWORK AND INTERFERENCE

CANCELLATION

We have shown that an appropriate pulse scale can be chosento match the channel spread parameters of underspread chan-nels so as to minimize interference among STF basis functions.Thus, the channel matrix will have dominant diagonal entriescorresponding to a strong signal energy component. For suffi-ciently small channel spread factors, interference can be made

so small that is approximately diagonal, in which case theSTF basis functions serve as approximate eigenfunctions. Thechoice of an orthogonal STF basis is motivated by the fact thatit is complete and hence preserves bandwidth efficiency. How-ever, some residual interference remains even after pulse scaleadaptation due to the relatively poor time–frequency localiza-tion of the STF basis. To mitigate the residual interference, wepropose a low-complexity but highly effective interference can-cellation technique, SIIC, which will be described later in thissection.

A. General Signaling Framework

Pulse scale adaptation for interference minimization and theuse of SIIC for residual interference removal suggest a gen-eral orthogonal time–frequency signaling framework for under-spread channels as depicted in Fig. 5. Pulse design can be incor-porated into the framework but that is beyond the scope of thepaper. We focus on a single-user context to illustrate the frame-work. It is a natural extension of conventional OFDM signalingto doubly dispersive channels. The transmitter modulates the(coded) symbols onto the STF basis functions for communica-tion over the channel. At the receiver, a bank of correlators ormatched filters are used to generate the sufficient statistics fordecoding the information symbols. Coherent communication re-quires channel estimation, which can be tackled in a variety ofways using our framework (see, e.g., [20]). Here we assume thatperfect estimates of the matrix are available for simplicity.


Fig. 4. Interference index for Gaussian and rectangular prototypes using the approximately optimal pulse scale. Flat multipath and “bathtub” Doppler profilesare used for analysis and simulation.

Fig. 5. General time–frequency signaling framework.

One important aspect of the STF signaling scheme is that itclearly reveals and facilitates the exploitation of channel diver-sity afforded by the time-varying multipath fading channel. The

auto-term coefficient of the th receive signal is given inProposition 2 as

(36)After pulse scale adaptation, and are much less thanfor interference minimization. Under these conditions,can be approximated by in the integration range, and hence,

(37)

where the time–frequency kernel of the doubly disper-sive channel is defined as

(38)

It is easy to check that the correlation function of isrelated to the channel scattering function as

(39)


Fig. 6. A schematic illustrating the notion of time–frequency coherencesubspaces.

Consequently, the correlation between different auto-term coef-ficients can be approximated by

(40)This demonstrates that the channel memory structure is pre-served by the STF signaling scheme. When two basis functionsare separated beyond coherence time and/or bandwidth

, they encounter independent fading. On the other hand,when basis functions lie within a time–frequency region withsupport , their channel coefficients are strongly cor-related. Therefore, we can partition the entire time–frequencyplane by these coherence subspaces as illustratedin Fig. 6. Since , the number of basis functions in acoherence subspace is

For a total signal duration and bandwidth , the total level ofdiversity afforded by channel equals the number of coherencesubspaces contained within the signal space. More precisely

(41)

where and are, respectively, thelevels of multipath diversity and Doppler diversity [13]. We notethat the is also the number of channel parameters to be esti-mated and that for underspread channels. Codes can bedesigned under the proposed signaling framework to fully uti-lize channel diversity for reliable communication. For example,a simple time–frequency diversity scheme using this frameworkwas proposed in [21].

B. Residual Interference Cancellation

Generally speaking, both ISI and ICI contribute to interfer-ence in STF signaling. We assume interference can be wellapproximated by Gaussian distribution (by the usual centrallimit theorem argument provided the number of interferingbasis functions is large). Therefore, the variance of zero-meaneffective noise (interference plus channel noise) is actually

quantified by Lemma 1, where is the received signalpower, is the interference variance, and is thenoise variance. As in Fig. 4, the interference index can bereduced to be much less than for a wide range of channelparameters, which results in an SNR loss much less than 1 dB.However, the presence of a relatively significant interferencecan severely limit system performance, for as signal powerincreases, that is,

SINR (42)

which is fixed by . For moderate channel spread factors onthe order of (whose interference index is around severalpercent), the limiting SINR is roughly 15–20 dB according to(42).

We consider binary phase-shift keying (BPSK) modulationwith diversity order to demonstrate the effect of interferenceon system performance. Diversity signaling of order can berealized by a simple scheme that repeats the same bit on dif-ferent basis functions over disjoint coherence subspaces. Theprobability of bit error with diversity order is given by[12]

(43)

where and is the effective SNR of each basis

function. In our case, it is SINR of each basis function due tothe Gaussian assumption for interference.

We first show simulated performance of BPSK modulationwithout diversity for different channel spread factors. The trans-mitter and receiver are as in Fig. 5 except that the residual in-terference cancellation is not used. 10 Hz is kept fixedand is changed according to the spread factor . TheSTF basis is generated by a rectangular prototype whose scaleis chosen by the scaling rule in Theorem 1. The system has fi-nite bandwidth such that the number of basis functions in fre-quency is ; that is, . The channel is simulated viathe Jakes model assuming flat multipath profile and “bathtub”Doppler profile. Fig. 7 plots both the simulated and analyticalcurves for as a function of SNR. The figure clearly showsthe existence of floors due to interference—flattening ofas SNR increases. For , the transition point is be-tween 15 and 20 dB. Furthermore, it can be seen that the floordecreases as decreases due to reduced interference.

Next we present in detail our proposed SIIC technique forresidual interference reduction. The SIIC is a decision feedbacktype algorithm that jointly removes both ISI and ICI in time–fre-quency domain. It works sequentially in time. For each time slot,it first cancels the ISI caused by channel delay spread, then it it-eratively cancels the ICI among different frequencies within thesame time slot, analogous to the parallel interference cancella-tion (PIC) algorithms in multiuser detection applications (see,e.g., [22], [23]). Due to its decision feedback nature, the perfor-mance of SIIC is critically affected by the quality of the deci-sions used in interference cancellation.


Fig. 7. Performance of STF signaling with BPSK modulation for different spread factors. Rectangular prototype, flat multipath, and “bathtub” Doppler profiles.The pulse scale is determined by approximate pulse scale rule.

In the following, uncoded systems are considered to illustratethe SIIC algorithm. The algorithm can be readily extended to in-corporate many coding scheme by modifying its decision gen-erating component. Stacking all the frequency variables corre-sponding to the same time index , that is, let

and

the matrix channel (6) can be written as

(44)

where is the multipath spread in symbols. The within (44) represents the ISI from previous symbols, while

represents the ICI among different frequencies at the sametime slot . The receiver is assumed to have perfect knowledgeabout matrices . For rectangular pulses, afterpulse scale adaptation to minimize interference. Thus,and hence only adjacent time slots interfere. The residual ICIsignal can be calculated by

(45)

where is a diagonal matrix whose diagonal elements are thesame as those of .

Algorithm 1 (SIIC): The SIIC algorithm with iterationsproceeds as follows.

1) Initiate iteration index . For time slot , performISI cancellation by calculating the ISI using decoded bitsfrom previous time slots and then subtracting it from

where are previous bit decisions and is the ISI-removed signal.

2) Decode assuming no interference. This generates theth bit decisions .

3) Reconstruct the th residual ICI signal by using (45)and subtracting it from to generate the signalfor the next iteration

4) Iterate Steps 2 and 3 for times.

Remark 6: Many practical slow-fading channels exhibit rel-atively small Doppler dispersion, so the ICI effect is small forthese systems. The optimal pulse scaling for this scenario co-incides with OFDM signaling, where a cyclic prefix is used tocombat ISI due to multipath delay. In this case, the SIIC canbe avoided because of the negligible residual interference in thesystem. In fact, SIIC is intended for fast-fading channels withlarge multipath dispersion.

Remark 7: The exact performance analysis of the SIICalgorithm seems difficult. But approximation can be done bytracking each step of the algorithm and approximating the


Fig. 8. SIIC performance for spread factor 0:01. No diversity signaling is used. The lower bound for the bite-error rate (BER) corresponds to perfect interferencecancellation.

residual interference as Gaussian noise. We refer to [24] forsuch an analysis technique.

Fig. 8 shows the performance of SIIC-based reception of theBPSK modulation with no diversity over a channel with a spreadfactor of . The channel simulation setup is the same as inFig. 7. We also plot the SIIC performance if perfect decisionsare supplied, which serves as a lower bound for . As evidentfrom the figure, the SIIC algorithm demonstrates an impres-sive performance enhancement. A large performance gain canbe achieved by only a small number of iterations. For example,the performance for (ISI removal and one ICI removal)is already very close to the lower bound for SNR ranging from5 to 25 dB. However, the performance gain diminishes as thenumber of iteration increases, which is typical behavior of de-cision feedback techniques.

Shown in Fig. 9 is the SIIC performance for essentially thesame BPSK simulation setup but with two-level of diversity.Comparing it with Fig. 8, we see that SIIC benefits also fromdiversity due to the enhancement in decision feedback quality,which is critical to overall performance. For instance, two it-erations of SIIC are sufficient to attain the performance lowerbound as shown in the figure.

V. SYSTEM CAPACITY

In this section, we study doubly dispersive channels and ourproposed signaling framework from an information-theoretic

viewpoint. We focus on the impact of channel parameters, es-pecially , on system capacity.

A. A Moment Theorem for Doubly Dispersive Channels

We shall work with a discrete description of doubly disper-sive channels for capacity analysis. In this context, the channeloutput and input are related by

(46)

where is the total number of resolvable paths and noiseis white in time. The path coefficient process

is the discrete analog of the continuous processin (1) with the path index corresponding to and the time index

corresponding to . In view of the WSSUS assumption ondoubly dispersive channels, we may assume that different pathprocesses are independent zero-mean complex Gaussian andthat sum of total variance is , that is, whereis the variance of the th wide-sense stationary (WSS) process

.The channel equation in (46) can be concisely rewritten in a

multiple-input multiple-output (MIMO) form as

(47)


Fig. 9. SIIC performance with diversity signaling of order 2.

where the semi-infinite channel matrix is given by

. . .. . .

. . .. . .. . .

. . .. . .

(48)The matrix has only an number of diagonal lines,each corresponding to a path coefficient process. Unlike time-invariant linear channels, doubly dispersive channels cause vari-ations along the diagonals of , whose statistical characteris-tics are strongly connected to channel Doppler spread , orequivalently, to channel coherence time . The larger , thelarger the variation. Since the discrete channel essentially comesfrom discretizing the underlining (continuous) doubly disper-sive channel, one can similarly define the notion of channelcoherence length where the signaling band-width is assumed to be sufficiently large. Therefore, diagonalentries within a range of remain strongly correlated. Since

, one has

(49)

Hence, the channel spread factor can also be seen as ameasure of relative ratio between the number of diagonal linesof and the “constant” length of those lines.

When channel state information is available at the receiverbut not at the transmitter, the ergodic channel capacity per di-mension given power constraint is given by [25], [26]

(50)

where and the subscript means a truncationof the corresponding infinite-dimensional matrix to dimension

. Denote by the limiting empirical distribution of eigen-values of random matrix

(51)

where denotes the cardinality of a set . As shown in[26], the capacity in (50) can also be expressed as

(52)

It is easy to observe that the random matrix also exhibitsa finite banded structure as . The eigenvalue distribution ofsuch banded random matrices is therefore a key to evaluatingchannel capacity. We refer readers to a recent excellent reviewon spectral properties of large random matrices [27], whereseveral types of random matrices have been analyzed. Butthe problem for banded random matrices seems elusive andremains largely unsolved [28]. Although at this stage we arenot able to give the limiting eigenvalue distribution of , wehave characterized its moment behavior.


Fig. 10. A schematic illustrating matrix products using MBG representation: (a) corresponds to HHHHHH and (b) corresponds to BBB where L = 1 and localbranches relative to the i th row are shown.

The th moment of the empirical distributionis

(53)

where denotes the trace of a matrix. As

a.s. (54)

where is the th entry in and representsa typical entry in [26].4 Therefore, the problem boils downto evaluating entries in higher matrix products.

For this purpose, we will briefly touch upon the usage of ma-trix bipartite graph (MBG). The MBG for a matrix is definedto be a bipartite graph consisting of two rows of vertices anddownward arcs between them. Vertices in the top row corre-spond to row indices of and vertices in the bottom row tocolumn indices of . The arc connecting the top vertex tothe bottom vertex has value , the th entry in . Letthe MBG associated with matrix be andthe MBG with be . Then, the MBGcorresponding to can be simply represented by cas-cading and , and the arc connecting to is the sumof all directed paths from of to of , or symbolically,

. Shown in Fig. 10 are the MBG rep-resentations of and whose row is represented bya subgraph fanning out from the common top th vertex. It isstraightforward to observe that the finite bandedness of im-plies the finite bandedness of , which has number ofdiagonals ranging from to . To facilitate exposition, weintroduce a reindexing of matrix entries for banded matrix as

(55)

which denotes the entry at the intersection of the th row and theth diagonal line of .

Fig. 10 clearly reveals the local nature of computation of en-tries in . More specifically, the th row of is only affectedby rows of ranging from to , or equivalently,

4i is sufficiently large for fixed k to avoid the edge effects.

by rows of ranging from to ,denoted by .

Now suppose that the channel coherence length is suffi-ciently large compared to so that each diagonalline is almost constant within the row range . One has

(56)where denotes the correlation operation.Similarly, the th row of is given by iterated correlationof , that is, , times. Let

be the (discrete-time) Fourier transform of the se-quence with respect to the diagonal line index . Sinceare independent complex Gaussian with total variance normal-ized to , it is easy to check that is white infrequency . Since the Fourier transform of is simply re-lated to that of by

(57)

one has

(58)that is, the eigenvalue distribution is exponential. In this case,the capacity of doubly dispersive channels degenerates to thatof flat fading channels.

However, channel variation always exists for general doublydispersive channels where the spread factor is nonzerobut may be small in most practical situations. Suppose that vari-ance of such variation in the local range is uniformlyupper-bounded by , that is,

(59)

Since , one can choose.

Theorem 2: Given an integer , suppose the channelvariance within the range is upper-bounded by . Then,the th moment of limiting eigenvalue distribution satisfies

(60)

where the constant depends on .Proof: See Appendix III.


Fig. 11. Eigenvalue distribution of a random channel matrix corresponding to L+ 1 = 4 paths and N = 100000 symbols.

Theorem 2 characterizes the effect of channel spread parame-ters on capacity. Since the time span in is ,the channel variation is proportional to the relative ratio of thetime span and the channel coherence time and hence,

(61)

by (49). The smaller the channel spread , the smaller theand thus the moment difference up to the th order. Higher ordermoments tend to exhibit more deviation than lower ones. How-ever, more and more moments of the eigenvalue distribution ofdoubly dispersive channels will agree with those of flat-fadingchannels provided that is small enough, which impliesthat capacity of doubly dispersive channels converges to that offlat-fading channels as channel spread decreases.

Fig. 11 plots the empirical eigenvalue distribution of simu-lated doubly dispersive channels for various spread factors. Thedistribution is seen to deviate from the exponential distributionfor large . But the deviation is small as long as channelremains underspread. Shown in Fig. 12 is the channel capacitycomputed from the empirical eigenvalue distribution.As evi-dent from the figure, capacity of underspread channels is upper-bounded by that of flat-fading channels and the difference be-tween them is fairly small for small SNR values.

B. Block Fading in the Time–Frequency Domain

The notion of coherence subspace in our signaling frame-work leads to a time–frequency block-fading view for doubly

disper-sive channels. As illustrated in Fig. 6, channel coeffi-cients within the same coherence subspace are assumed to bethe same while they vary in an independent fashion from onesubspace to another. The operational capacity of STF signalingis affected both by channel dispersion parameters and by the per-formance of interference cancellation technique such as SIIC inour framework. Here we treat interference as Gaussian noise tosimplify the analysis.

The channel ergodic capacity per dimension is given by

SINR (62)

where is exponentially distributed with unit mean. The SINRin the framework depends on many factors such as channelparameters, pulse characteristics, and the interference cancella-tion technique used. It is lower bounded by the minimum SINRwithout interference cancellation and upper-bounded by themaximum SINR with perfect interference cancellation, that is,

SINR (63)

If the interference is left untreated, the system capacity willsuffer from a limiting cap analogous to the error floor in theBER performance. More precisely, as , SINRand hence .

Fig. 13 plots the ergodic capacity of the STF signalingframework for a scale-adapted rectangular pulse over a channelwith spread factor . The lower and upper bounds for SINRwere estimated from the simulation data. The capacity offlat-fading channel is also plotted as a benchmark. As evident


Fig. 12. Ergodic capacity for different spread factors calculated from the estimated eigenvalue distribution.

from the figure, system capacity levels off in the high-SNRregion when no interference cancellation is used. In contrast,the SIIC effectively remedies the capacity bottleneck. In thefigure, the mutual information curves for different iterations ofthe SIIC are plotted using the estimated residual SINR fromsimulation data. More iterations in SIIC narrow the capacitygap to the upper bound. Furthermore, system capacity benefitsfrom small channel spread factors, which is demonstrated byFig. 14, where system capacity corresponding to a spread factor

is plotted. Fig. 14 suggests that interference cancellationtechniques may be avoided without incurring significant ca-pacity loss when the channel spread factor is fairly small. Forexample, the lower bound in Fig. 14 almost coincides with theflat-fading capacity up to an SNR of 10 dB.

It is instructive to point out the difference between under-critical sampling and critical sampling

with respect to system capacity. Undercritcal sampling of-fers better time–frequency localization than the critical sam-pling orthogonal basis in our STF framework. But its spectralefficiency is reduced by a factor of . More specifically,the system capacity per dimension of an undercritical samplingbasis is given by

SINR (64)

where is the SINR improvement over the critical samplingbasis. Comparing it with the capacity of the critical samplingbasis (62), we want to find out how large needs to be in orderto compensate for the loss in spectral efficiency such that it still

would attain the same capacity as that of (62). Using Jensen’sinequality, one has

SINR SINR

SINR (65)

which, together with , implies that

SINR

SINRSINR

as SINR (66)

The above computation illustrates the merit of critical samplingSTF basis: it attains the largest spectral efficiency. The slightimprovement on SINR by using undercritical sampling may notbe warranted from capacity perspective.

Our framework also facilitates the evaluation of the outagecapacity [29]. Given a total signal duration and bandwidth ,the level of multipath-Doppler diversity isand the number of basis function in a coherence subspace is

as discussed in Section IV. So, the maximalmutual information per dimension can be written as

SINR (67)

where is a vector of the fading coefficient foreach coherence subspace. Then, the outage capacity is defined


Fig. 13. Ergodic capacity of STF signaling for spread factor 0:01. The capacity depends on the residual SINR. Capacity when SIIC is used is calculated fromthe estimated residual SINR from the SIIC simulations.

Fig. 14. Ergodic capacity of signaling framework for spread factor 0:001.


Fig. 15. Outage capacity curves for different levels of diversity at SNR = 10 dB.

to be the maximum rate that can be guaranteed with a certainoutage probability . More precisely

(68)

The outage performance curves for different levels of diversityare plotted in Fig. 15. Large channel spread factors give rise tomore diversity in the system. Diversity improves outage perfor-mance although such improvement diminishes at high diversitylevel.

VI. CONCLUSION

We have proposed a general framework for orthogonalsignaling over doubly dispersive fading channels via STF basisfunctions. Our quantitative results reveal the effect of pulsetime–frequency properties and channel characteristics on sig-naling performance. A simple scale adaptation rule is derived,also proposed in [3], [4], to match the pulse characteristicswith the channel. For sufficiently small channel spread factors,a scale-adapted STF basis serves as a set of approximateeigenfunctions of the channel and yields a simple block-fadinginterpretation of the channel in terms of time–frequencycoherence subspaces. We propose an efficient interference can-cellation technique to further reduce the residual interference.System capacity of doubly dispersive channels is studied usingrandom matrix theory, which reveals the important role of chan-nel spread factor on capacity. In particular, the capacity devia-tion of doubly dispersive channels from flat-fading channels isproportional to channel spread factor.

In relation to the work on undercritically sampled biorthog-onal systems [3], [5]–[7], the results in this paperdemonstrate the attractiveness of using orthogonal STF basesfor communication over doubly dispersive channels, despitetheir poor time–frequency localization properties. This is be-cause undercritical systems result in a loss in spectral efficiencythat affects overall system capacity linearly, whereas the im-provement in SINR due to better time–frequency localizationonly yields a logarithmic gain in capacity. Furthermore, thesimplicity and impressive performance of the proposed SIICscheme suggests that orthogonal systems with interferencecancellation could yield SINR performance comparable tobiorthogonal systems. This is also significant from a practicalperspective since the critically sampled case is much easier toimplement in practice. The same set of basis functions are usedat the transmitter and receiver and incorporating our results inexisting systems requires a simple modification: using existingprototype pulses with appropriate scale adaptation that onlyrequires knowledge of channel delay and Doppler spreads. Inbiorthogonal systems, two (dual) basis functions are neededat the transmitter and receiver and the determination of dualprototypes requires numerical optimization, which is alwaysapproximate and changes with channel statistics [3].

Finally, we note that other recent results on the noncoherentcapacity of time-varying multipath channels suggest signalingwaveforms that are peaky in time or frequency [30], [31] as op-posed to noise-like spread-spectrum waveforms. The notion of


time–frequency coherence subspaces introduced in this papersuggests that partial spreading in time–frequency, commensu-rate with the dimension of the coherence subspaces, warrantsfurther investigation in a noncoherent context. Furthermore, theblock-fading interpretation in terms of time–frequency coher-ence subspaces facilitates the application of coding techniquesfor block-fading channels for communication over underspreaddoubly dispersive channels [32], [33].

APPENDIX IPROOF OF PROPOSITION 2

From (7), we have

(69)

For the STF basis, we calculate the inner integral in (69)

Substituting this into (69) yields (17). Next we evaluateto yield (18):

APPENDIX IIPROOF OF THEOREM 1

It is easy to see from the definition of ambiguity function thatand . Then, (19) becomes

(70)

where and . Since is aconstant, we rewrite (70) as

(71)

Taking the derivative with respect to , we have

(72)from which is seen to be a stationary point

, thus proving (29). The second derivative at thispoint is given by

(73)

where is defined in (28). So, will guarantee tobe a local maximum. Since , one has .If condition (30) is satisfied, then

(74)

APPENDIX IIIPROOF OF THEOREM 2

We bound the perturbation along the computation of entriesin . Let be the (random) differencebetween the true value and the desired target . Forconvenience, corresponds to . One has

(75)


where

(76)

Using Jensen’s inequality , onehas

(77)

By the Cauchy–Schwarz inequality ,the first term in the summation can be bounded by

(78)

because the variance of complex Gaussian random variablesand are bounded by and , respectively. Sim-

ilarly

and

Therefore, the th moment of is bounded by

(79)

where is a constant depending on and is upper-boundedby .

Before proceeding to compute , we need to bound themoments of

(80)

where we have used the integral version of the Jensen’s in-equality to push power inside the integral.

Now we compute as

(81)

where

(82)

Similar as the case of , one has, by using (79) and (80)

(83)

where is a constant depending on .In general, is given by

(84)

Hence, repeating the above argument to give

(85)

where is a constant depending on and . Since

the theorem is proved by substituting in (85).

ACKNOWLEDGMENT

The authors gratefully appreciate comments from anonymousreviewers for improving the exposition.

REFERENCES

[1] J. A. C. Bingham, “Multicarrier modulation for data transmission: Anidea whose time has come,” IEEE Commun. Mag., vol. 28, pp. 5–14,May 1990.

[2] E. Biglieri, J. Proakis, and S. Shamai (Shitz), “Fading channels: In-formation-theoretic and communications aspects,” IEEE Trans. Inform.Theory, vol. 44, pp. 2619–2692, Oct. 1998.

[3] W. Kozek and A. F. Molisch, “Nonorthogonal pulse shapes for multi-carrier communications in doubly dispersive channels,” IEEE J. Select.Areas Commun., vol. 16, pp. 1579–1589, Oct. 1998.

[4] W. Kozek, “Adaptation of Weyl–Heisenberg frames to underspreadenvironments,” in Gabor Analysis and Algorithm: Theory and Ap-plications, H. G. Feichtinger and T. Strohmer, Eds. Boston, MA:Birkhäuser, 1997, pp. 323–352.

[5] R. Haas and J. C. Belfiore, “A time-frequency well-localized pulse formultiple carrier transmission,” Wireless Personal Commun., vol. 5, pp.1–18, 1997.

[6] T. Strohmer, “Approximation of dual Gabor frames, window decay, andwireless communications,” Appl. Comp. Harm. Anal., vol. 11, no. 2, pp.243–262, 2001.

[7] H. Bölcskei, R. Koetter, and S. Mallik, “Coding and modulation for un-derspread fading channels,” in Proc. 2002 IEEE Intl. Symp. InformationTheory, Lausanne, Switzerland, June 2002, p. 358.

[8] I. Daubechies, Ten Lectures on Wavelets, ser. cBMS-NSF Reg. Conf.Series in Applied Mathematics. Philadelphia, PA: SIAM, 1992.

[9] H. Jensen, T. Høholdt, and J. Justesen, “Double series representation ofbounded signals,” IEEE Trans. Inform. Theory, vol. 34, pp. 613–623,July 1988.

[10] A. J. E. M. Janssen, “Duality and bioorthogonality for Weyl–Heisenbergframes,” J. Fourier Anal. Appl., vol. 1, no. 4, pp. 403–436, 1995.

[11] P. A. Bello, “Characterization of randomly time-variant linear channels,”IEEE Trans. Commun. Syst., vol. CS-11, pp. 360–393, Dec. 1963.

[12] J. G. Proakis, Digital Communications: McGraw-Hill, 1995.[13] A. M. Sayeed and B. Aazhang, “Joint multipath-Doppler diversity in

mobile wireless communications,” IEEE Trans. Commun., vol. 47, pp.123–132, Jan. 1999.

[14] R. S. Kennedy, Performance Estimation of Fading Dispersive Chan-nels. New York: Wiley, 1968.

[15] S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. San Diego,CA: Academic, 1998.

[16] K. Liu, “Orthogonal time-frequency signaling over doubly dispersivechannels,” M.Sc. thesis, Univ. Wisconsin–Madison, 2001.


[17] P. Schniter, “A new approach to multicarrier pulse design for doublydispersive channels,” in Proc. Allerton Conf. Communication, Control,and Computering, Monticello, IL, Oct. 2003.

[18] I. Daubechies, “The wavelet transform, time–frequency localization, andsignal analysis,” IEEE Trans. Inform. Theory, vol. 36, pp. 961–1005,Sept. 1990.

[19] G. Stüber, Principles of Mobile Communication, 2nd ed. Norwell,MA: Kluwer Academic, 2001.

[20] M. R. Baissas and A. M. Sayeed, “Pilot-based estimation oftime-varying multipath channels for coherent CDMA receivers,”IEEE Trans. Signal Processing, vol. 50, pp. 2037–2049, Aug. 2002.

[21] T. Kadous, K. Liu, and A. Sayeed, “Optimal time-frequency signalingfor rapidly time-varying channels,” in Proc. 2001 IEEE Int. Conf. Acous-tics, Speech and Signal Processing (ICASSP’01), May 2001.

[22] S. Moshavi, “Multi-user detection for DS-CDMA communications,”IEEE Commun. Mag., pp. 124–136, Oct. 1996.

[23] M. K. Varanasi and B. Aazhang, “Multistage detection in asynchronouscode division multiple access channels,” IEEE Trans. Commun., vol. 38,pp. 509–519, Apr. 1990.

[24] K. Liu and A. M. Sayeed, “Improved layered space-time processing inmultiple antenna systems,” in Proc. 39th Annu. Allerton Conf. Communi-cation, Control and Computing, Monticello, IL, Oct. 2001, pp. 757–767.

[25] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Tech. Rep.Bell Labs., 1995.

[26] V. Kafedziski, “Capacity of frequency selective fading channels withside information,” in Proc. 32nd Asilomar Conf. Signals, Systems andComputers, vol. 2, 1998, pp. 1758–1762.

[27] Z. D. Bai, “Methodologies in spectral analysis of large dimensionalrandom matrices, a review,” Stat. Sinica, vol. 9, pp. 611–677, 1999.

[28] S. A. Molchanov, L. A. Pastur, and A. M. Khorunzhii, “Limiting eigen-value distribution for band random matrices,” Theor. Math. Phys., vol.90, pp. 108–118, 1992.

[29] L. H. Ozarow, S. Shamai (Shitz), and A. D. Wyner, “Informationtheoretic considerations for cellular mobile radio,” IEEE Trans. Veh.Technol., vol. 43, pp. 359–378, May 1994.

[30] R. G. Gallager and M. Medard, “Bandwidth scaling for fading chan-nels,” in Proc. 1997 IEEE Intl. Symp. Information Theory, Ulm, Ger-many, 1997, p. 471.

[31] I. E. Telatar and D. N. C. Tse, “Capacity and mutual information ofwideband multipath fading channels,” IEEE Trans. Inform. Theory, vol.46, pp. 1384–1400, July 2000.

[32] R. Knopp and P. A. Humblet, “On coding for block fading channels,”IEEE Trans. Inform. Theory, vol. 46, pp. 189–205, Jan. 2000.

[33] G. Hariharan, Z. Hong, and A. Sayeed, “A full diversity scheme for thedoubly dispersive channel under the orthogonal signaling framework,”in Proc. 41th Allerton Conf. Communication, control and Computing,Monticello, IL, Oct. 2003, pp. 1486–1495.

Orthogonal time-frequency signaling over doubly dispersive channels

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