Non-Coherent Capacity and Reliability of Sparse Multipath Channels in the Wideband Regime

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1 Non-Coherent Capacity an Reliability of Sparse Multipath Channels in the Wieban Regime Gautham Hariharan an Abar M Sayee Department of Electrical an Computer Engineering University of Wisconsin-Maison gauthamh@caewisceu, abar@engrwisceu Abstract In contrast to the prevalent assumption of rich multipath in information theoretic analysis of wireless channels, physical channels exhibit sparse multipath, especially at large banwiths We propose a moel for sparse multipath faing channels an present results on the impact of sparsity on non-coherent capacity an reliability in the wieban regime A ey implication of sparsity is that the statistically inepenent egrees of freeom in the channel, that represent the elay- Doppler iversity affore by multipath, scale at a sub-linear rate with the signal space imension (time-banwith prouct) Our analysis is base on a training-base communication scheme that uses short-time Fourier (STF) signaling waveforms Sparsity in elay-doppler manifests itself as time-frequency coherence in the STF omain From a capacity perspective, sparse channels are asymptotically coherent: the gap between coherent an non-coherent extremes vanishes in the limit of large signal space imension without the nee for peay signaling From a reliability viewpoint, there is a funamental traeoff between channel iversity an learnability that can be optimize to maximize the error exponent at any rate by appropriately choosing the signaling uration as a function of banwith I INTRODUCTION Recent avances in the emerging areas of ultra-wieban communication systems an wireless sensor networs have renewe the search for a complete unerstaning of the funamental performance limits in the wieban/low SNR regime The impact of multipath signal propagation, which leas to faing, on the capacity an reliability of wieban channels, epens critically on nowlege of the channel state information (CSI) at the receiver The seminal wor in [] best illustrates this: with perfect CSI at the receiver, pea-power limite QPSK achieves secon-orer optimality, whereas with no receiver CSI, peay signals are necessary to even achieve first-orer optimality, although they fail to be secon-orer optimal Motivate by the fact that channel learning can brige this gap an the sharp cut-off, the authors in [2] stuy wieban capacity by assuming a coherence time scaling with SNR of the form T coh = SNR µ, µ > 0 () However there is no explanation for why such scaling laws shoul hol in practice Accurate moeling of the channel characteristics in time an frequency, as a function of physical multipath characteristics, is critical in analyzing the performance of channel learning schemes an the impact of CSI on the performance limits While most existing results assume rich multipath, there is growing experimental evience (eg [3], [4]) that physical channels exhibit a sparse structure at wie banwiths an when we coe over long signaling urations In this paper, we use a virtual representation [5] for physical multipath channels to present a framewor for moeling sparsity The virtual representation uniformly samples multipath in elay an Doppler at a resolution commensurate with the signaling banwith an signaling uration, respectively Uner this representation, the virtual channel This wor was partly supporte by the NSF uner grant #CCF coefficients represent the DoF in elay an Doppler Sparse channels correspon to a sparse set of virtual coefficients an a ey implication is the sub-linear scaling of the number of inepenent egrees of freeom (DoF) with signal space imensions This is contrast to rich multipath, where the DoF scale linearly We consier signaling over orthogonal short-time Fourier (STF) basis functions that serve as approximate eigenfunctions for unersprea channels an provie a natural mechanism to relate sparsity in elay-doppler to coherence in time-frequency With no receiver CSI a priori, we consier training-base communication schemes an investigate the wieban ergoic capacity uner the assumption that the time-frequency coherence imension N c scales with SNR accoring to N c = SNR µ, µ > 0 (2) It is observe that the coherence requirements for achieving capacity are share between both time an frequency: the coherence banwith, W coh, increases with banwith, W (ue to sparsity in elay), an the coherence time, T coh, increases with signaling uration T (ue to sparsity in Doppler) As a result, the scaling requirements on T coh with W neee in [2] for first- an secon-orer optimality are replace by scaling requirements on N c = T coh W coh This leas to ramatically relaxe requirements on T coh scaling with banwith/snr compare to those assume in [2] In particular, sparse multipath channels are asymptotically coherent; that is, for a sufficiently large but fixe banwith, the conitions for firstan secon-orer optimality can be achieve by simply maing the signaling uration sufficiently large accoring to T T δ 2 m W δ W µ +δ 2 (3) P µ Equation (3) relates the signaling parameters (T,W,P ), as a function of the channel parameters (T m,w,δ,δ 2) in orer for the relationship (2) to hol between N c an SNR at any esire value of µ (in particular, µ > for first-orer optimality an µ > 3 for seconorer optimality) The asymptotic coherence of sparse channels also eliminates the nee for peay signaling that has been emphasize in existing results [6], [] for increasing the spectral efficiency of non-coherent communication schemes Our investigation of the reliability of sparse channels is through ranom coing error exponents [7] For training-base communication schemes, our results reveal a funamental learnability versus iversity traeoff in sparse channels At any transmission rate less than the coherent capacity, there is an optimal choice of signal parameters (as a function of channel parameters) that optimizes the traeoff an yiels the largest error exponent

2 2 II SYSTEM SETUP A Sparse Multipath Channel Moeling A physical iscrete multipath channel can be moele as h(τ, ν) = X n r(t) = X n β nδ(τ τ n)δ(ν ν n) β nx(t τ n)e j2πνnt + w(t) (4) where h(τ, ν) is the elay-doppler spreaing function of the channel, β n, τ n [0, T m] an ν n [ W /2, W /2] enote the complex path gain, elay an Doppler shift associate with the n-th path T m an W are the elay an Doppler spreas respectively an w(t) is aitive white Gaussian noise (AWGN) We assume a sufficiently unersprea channel, T mw In this paper we use a virtual representation [5], [8] for time- an frequency-selective multipath channels that captures the channel characteristics in terms of resolvable paths an greatly facilitates system analysis from a communication-theoretic perspective The virtual representation uniformly samples the multipath in elay an Doppler at a resolution commensurate with signaling banwith W an signaling uration T, respectively [5], [8] DOPPLER DOPPLER W /2 0 - W /2 /W (a) /T T m MULTIPATH MULTIPATH W f Coherence Subspace T (b) T coh W coh Basis Functions t y(t) = h l,m T mw X l=0 X TW /2 X m= TW /2 h l,m x(t l/w)e j2πmt/t (5) n S τ,l S ν,m β n (6) The sample representation (5) is linear an is characterize by the virtual elay-doppler channel coefficients {h l,m } Each h l,m consists of the sum of gains of all paths whose elays an Doppler shifts lie within the (l, m)-th elay-doppler resolution bin as shown in Fig (a) Distinct h l,m s correspon to approximately isjoint subsets of paths an are hence approximately statistically inepenent (ue to inepenent path gains an phases) In this wor, we assume that the channel coefficients {h l,m } are perfectly inepenent We also assume Rayleigh faing in which {h l,m } are zero-mean Gaussian ranom variables an the channel statistics are thus characterize by the power in the virtual channel coefficients Ψ(l, m) = E[ h l,m 2 ] (7) which is a measure of the (sample) elay-doppler power spectrum Let D enote the number of ominant non-zero channel coefficients The parameter D reflects the statistically inepenent egrees of freeom (DoF) in the channel an also signifies the elay-doppler iversity affore by the channel It can be boune as D = D TD W D max = D T,maxD W,max D T,max = TW, D W,max = T mw (8) where D T,max enotes the maximum number of resolvable paths in Doppler (maximum Doppler or time iversity) an D W,max enotes maximum number of resolvable paths in elay (maximum elay or frequency iversity) In rich multipath, D T = D T,max an D W = D W,max an each elay-doppler resolution bin in Fig (a) is populate by a path In this case D scales linearly with the signal space imensions, N = TW However, recent measurement campaigns [3], [4] for UWB channels show that ominant channel coefficients get sparser in the elay omain as the banwith increases As we consier large banwiths an/or long signaling urations, the resolution of paths in both elay For which Ψ(l, m) > γ for some prescribe threshol γ > 0 (c) Fig (a) Delay-oppler sampling commensurate with signaling uration an banwith (b) Time-frequency coherence subspaces in STF signaling (c) Illustration of the training-base communication scheme in the STF omain One imension in each coherence subspace (ar squares) represent the training imension an the remaining imensions are use for communication an Doppler omains gets finer, leaing to the scenario in Fig (a) where the elay-doppler resolution bins are sparsely populate with paths, ie D < D max Thus physical multipath channels get sparser with increasing W ue to fewer than D W,max resolvable elays an with increasing T ue to fewer than D T,max resolvable Doppler shifts We moel such sparse behavior with a sub-linear scaling in D T an D W with T an W : D T (TW ) δ, D W (T mw) δ 2, δ, δ 2 [0, ] (9) where {δ i} represent channel sparsity; smaller the value of {δ i}, the slower (sparser) the growth in the resolvable paths in the corresponing omain This implies that the elay-doppler DoF, D = D TD W, scale sub-linearly with the number of signal space imensions N Note that with perfect CSI at the receiver, D reflects the elay- Doppler iversity affore by the channel, whereas with no CSI, it reflects channel uncertainty B Orthogonal Short-Time Fourier Signaling We consier signaling using an orthonormal short-time Fourier (STF) basis [9], [0] that is a natural generalization of orthogonal frequency-ivision multiplexing (OFDM) for time-varying channels An orthogonal STF basis for the signal space is generate from a fixe prototype waveform g(t) via time an frequency shifts: φ l,m (t) = g(t lt o)e j2πwot where T ow o = l = 0,, N T, m = 0,, N W N T = T T o, N W = W W o an N = N TN W (0) The N transmitte symbols x l,m are moulate onto the STF basis x(t) = X l,m x l,m φ l,m (t)

3 3 For a signaling uration T an banwith W, the basis functions span the signal space with imension equal to N = TW The receive signal in (4) is projecte onto the STF basis functions to yiel the receive symbols r l,m = r, φ l,m = X, m x l, m + w l,m () l,m h l,m;l Equivalently, we can represent the system in STF-omain using an N-imensional matrix system equation r = SNR Hx + w (2) where w represents the aitive noise vector whose entries are ii CN(0, ) The N N matrix consists of the channel coefficients {h l,m;l, m } in () The parameter SNR represents the transmit energy per moulate symbol an for a given transmit power P equals SNR = P W (E[ x 2 ] = ) In this wor, our focus is on the wieban regime, where SNR 0 For sufficiently unersprea channels, the parameters T o an W o can be matche to T m an W so that the STF basis waveforms serve as approximate eigenfunctions of the channel [0], [9] Thus the N N channel matrix H is approximately iagonal In this wor, we will assume that H is exactly iagonal, that is, h H = iag h, h,nc, h 2, h 2,Nc {z } {z } Subspace Subspace 2 h D, h D,Nc {z } Subspace D i (3) Furthermore, the iagonal entries of H in (3) amit an intuitive bloc faing interpretation in terms of time-frequency coherence subspaces [9] illustrate in Fig (b) The signal space is partitione as N = TW = N cd (4) where D represents the number of statistically inepenent timefrequency coherence subspaces (elay-doppler iversity), reflecting the DoF in the channel (see (8)), an N c represents the imension of each coherence subspace, which we will refer to as the timefrequency coherence imension In the bloc faing moel in (3), the channel coefficients over the i-th coherence subspace h i, h i,nc are assume to be ientical, h i, whereas the coefficients across ifferent coherence subspaces are inepenent an ue to the stationarity of the channel statistics across time an frequency, ientically istribute Thus, the D istinct STF channel coefficients, {h i}, are ii zero-mean Gaussian ranom variables (Rayleigh faing) with variance E[ h i 2 ] = Using the DoF scaling for sparse channels in (9), the coherence imension of each coherence subspace can be compute as T coh = T D T = T N c = T coh W coh = T W δ W δ W δ 2 T δ 2 m, W coh = W D W = W δ 2 T δ 2 m (5) ı = N c,min (6) T mw where T coh is the coherence time an W coh is the coherence banwith of the channel, as illustrate in Fig (b) Note that δ = δ 2 = correspons to a rich multipath channel in which N c = N c,min = /(T mw ) is constant an D = D max increases linearly with N = TW This is the assumption prevalent in existing wors In contrast, for sparse channels, (δ, δ 2) (0, ), an both N c an D increase sub-linearly with N In terms of channel parameters, N c increases with ecreasing T mw as well as with smaller δ i In terms of signaling parameters, N c can be increase by increasing T an/or W On the other han, when the channel is rich, N c epens only on T mw an oes not scale with T or W In this paper, our focus is on computing the sparse channel capacity an reliability an, as we will see later, both metrics turn out to be functions only of the parameters N c an SNR Furthermore, in the wieban limit they critically epen on the following relation between N c an SNR N c = where > 0 is a constant SNR µ, µ > 0 (7) C Training-Base Communication Using STF Signaling We use the bloc faing moel inuce by STF signaling to stuy the impact of time-frequency coherence on channel capacity an reliability in sparse multipath channels Within the non-coherent regime, we focus our attention on a communication scheme in which the transmitte signals inclue training symbols to enable coherent etection Although it is argue in [2] that training-base schemes are sub-optimal from a capacity point of view, the restriction to training schemes is motivate by practical consierations We provie an outline of the training-base communication scheme, aapte from [2], suitable to STF signaling (see [] for etails) The total energy available for training an communication is PT, of which a fraction η is use for training an the remaining fraction ( η) is use for communication Since the quality of the channel estimate over one coherence subspace epens only on the training energy an not on the number of training symbols [2], our scheme uses one signal space imension in each coherence subspace for training an the remaining (N c ) for communication, as illustrate in Fig (c) We consier minimum mean square error (MMSE) estimation uner which the channel estimation performance is measure in terms of the resulting mean square error (MSE) III ERGODIC CAPACITY OF THE TRAINING-BASED COMMUNICATION SCHEME We first characterize the coherent capacity of the wieban channel with perfect CSI at the receiver The coherent capacity per imension (in bps/hz) is efine as E ˆlog C coh (SNR) = sup 2 et `I N + HQH H Q: Tr(Q) TP N where P enotes transmit power an H is the iagonal channel matrix in (3) with the iagonal elements following the bloc-faing structure Due to the iagonal nature of H, the optimal Q is also iagonal In particular, uniform power allocation Q = TP N IN = SNR I N achieves capacity an we have P D i= C coh (SNR) = E ˆlog 2 ` hi 2 + TP N D (a) = E ˆlog 2 ` 2 + SNR h (8) where (a) follows since {h i} are ii with h representing a generic ranom variable, N = TW an SNR = P = TP W N The next proposition provies a lower boun to the coherent capacity in the low SNR regime [] Proposition : The coherent capacity, C coh (SNR), satisfies C coh log 2 (e) `SNR SNR 2 (9) Moreover the capacity converges to the lower boun in the limit of SNR 0 The lower boun in Proposition shows that the minimum energy per bit necessary for reliable communication is given by E b N 0 = min log e (2) an the wieban slope S 0 =, the two funamental metrics of spectral efficiency in the wieban regime efine in []

4 4 In terms of the scaling law, N c =, µ > 0, as efine SNR µ in (7), we are intereste in computing the value of µ such that the training-base communication scheme achieves first- an seconorer optimality The result is summarize in the following theorem Theorem : The average mutual information of the training-base scheme, with the scaling law N c =, satisfies SNR h µ i I tr log 2 (e) SNR O SNR +µ 2 (20) In particular, the first- an secon-orer optimality conitions are met if an only if µ > an µ > 3, respectively Proof: Omitte for this version See [] for etails It can be seen that the coherence imension N c plays a critical role in etermining the capacity of the training an communication scheme Both channel an signaling parameters impact N c in sparse channels an for a given T mw an {δ i}, the signal space parameters T an W can be suitably chosen to obtain any esire value for N c Recalling the expression for W coh in (5), we note that W coh = W δ 2 (T m) δ 2 = P δ2 (T m) δ 2 SNR δ 2 (2) an thus W coh naturally scales with SNR Using (2) the expression for N c in (6) becomes () T = N c = T P δ 2 (W ) δ (T m) δ 2 SNR δ 2 (22) Equating (7) with (22) leas to the following canonical relationship W µ +δ 2 T δ 2 m W δ P µ (23) that relates the signaling parameters (T,W,P ), as a function of the channel parameters (T m,w,δ,δ 2) in orer for the relationship (7) to hol between N c an SNR = P/W Equations (7) an (23) are the two ey equations that capture the essence of the results in this paper A Discussion of Results on Ergoic Capacity In the context of existing results in [2] that assume rich multipath (δ = δ 2 = ), Theorem shows that the requirement on T coh is now the requirement on the coherence imension N c = T coh W coh Thus, the coherence cost is share in both time an frequency resulting in significantly weaene scaling requirements for T coh If we have W coh = O `W δ 2, then the T coh scaling requirement reuces to T coh = N c/w coh = O W 2+δ 2 (24) to achieve secon-orer optimality This is significantly less stringent than the T coh = O `W 3 require in the framewor of [2] Combining Theorem with (23) lea to scaling rules for the locus of points (T,W,P ) in orer to achieve a esire value of µ (Recall µ > for first-orer optimality an µ > 3 for secon-orer optimality) Specifically, log (T) = log δ µ δ W δ T δ 2 m + log (P) µ + δ2 δ log (W) (25) It is observe that smaller δ i s imply a slower scaling of T with W Conversely, for any system operating at a particular T an W, (25) can be use to etermine the effective value of µ as µ eff = ( δ) log(t/c) + ( δ2) log(p) log(w/p) + ( δ 2) (26) where c = T δ 2 m W δ Note that µ eff as T for sparse channels, which implies that first- an secon-orer optimality can be achieve by simply increasing T This is ue to the impact of sparsity in Doppler an in irect contrast to the case of rich multipath where the coherence requirement is inepenent of signaling uration We provie numerical illustration of the results by consiering the low SNR asymptote of the coherent capacity in (9) The coefficients of the first- an seconorer terms are λ = log 2 (e) an λ 2 = log 2 (e), respectively In Fig 2, we plot the numerically estimate values c an c 2 of λ an λ 2, respectively, for the training-base communication scheme, which are estimate using Monte-Carlo simulations We observe that for a large enough T such that µ eff > 3, the secon-orer constant c 2 λ 2 = log 2 (e) Also shown in the figure is the behavior of the first-orer constant an it is seen that c λ for a much smaller value of T since all we nee is µ > Contrary to the traitional emphasis on peay signaling to improve the spectral efficiency of non-coherent communication, our results imply that elay-doppler sparsity, along with a suitable choice of T, W an P as in (25) is sufficient to achieve a esire level of coherence with non-peay signaling schemes It is shown in [] that the T coh requirements for non-peay signals uner our framewor are still better than those for peay training-base communication schemes propose in [2] base on a rich multipath assumption c log 2 (e) W = GHz, P/N 0 = 40 B, T m W = 0 6, δ 2 = 04 c 2 log 2 (e) δ = 02 δ = 02 δ = 03 δ = 03 δ = 04 δ = Signaling uration T (secs) Fig 2 Convergence of the coefficients of the SNR an SNR 2 terms in capacity as a function of T IV RELIABILITY OF SPARSE MULTIPATH CHANNELS The reliability function of the channel is efine as [7] log Pe (N, R) E (R) = lim sup N N where P e (N, R) is the average probability of error over an ensemble of coes (ranom coing) in which each coewor spans the signal space imensions N = T W an communication taes place at transmission rate R For any finite N, while the ranom coing exponent E r(n, R) provies a lower boun to E(R), the spherepacing exponent E sp(n, R) is an upper boun to E(R) [7] We recall the ranom coing upper boun on P e [7] given by P e e N[Er(N,R)] (33) E r (N, R) = max max 0 ρ Q [Eo (N, ρ, Q) ρr] (34)

5 5 E tr r (N, R) = R cr = R max = K = 8 N >< c log + (Nc )K ( (K ) ǫ ) R o() 0 R R 2 cr N c log + (Nc )K ( (K ) ǫ )ρ ρ R o() R +ρ >: cr R R max 0 R > R max N c 2N c K ( (K ) ǫ ) [2+(N c )K ( (K ) ǫ )] N c N c K ` (K ) ǫ η ( η )(N csnr) 2 (N c )(+η N csnr)+( η )N csnr (30) η = NcSNR+Nc (N c 2)N csnr ρ = (2 + ) +»q + (Nc 2)NcSNR N csnr+n c q (2 + ) 2 + 4( + )( N cr ) 2( + ) = (N c )K ` (K ) ǫ (27) (28) (29) (3) (32) E o (N, ρ, Q) = N log E H» R y h R i q(x)p(y x,h) +ρ +ρ x x y (35) We compute the ranom coing error exponent in (34) for the training an communication scheme escribe in Sec II-C The result is summarize in the following theorem Theorem 2: The average probability of error for the training-base communication scheme is upper-boune by, P e e N[Etr r (N,R)] where Er tr (N, R) is given in (27) on the next page R max in (29) efines the maximum rate until which we have a non-zero error exponent (ecaying P e) The critical rate, R cr in (28) elineates the regime of the optimal parameter ρ that maximizes the exponent ρ = for 0 < R < R cr an ρ is given in (32) for R cr < R < R max The constant ǫ > 0 an is chosen very small (ǫ 0) so that the o() terms are negligible See [3] for more etails Note that the error exponent of the training an communication scheme in (27) epens only on SNR an N c = A Discussion of Results on Reliability SNR µ We investigate the behavior of the ranom coing exponent for ifferent values of µ as illustrate in Fig 3 for the given channel parameter set It is observe that for any transmission rate R, there exists an optimum value of µ = µ opt(r) for which the error exponent in (27) is maximum For any N, we formally efine µ opt(n, R) = arg max [Er (R, N, µ)] (36) µ ˆE tr r (N, R) in (27) explic- where we have written E r (R, N, µ) = itly as a function of µ to emphasize its epenance As we traverse from R = 0 to R = C coh (as in (8)), the optimal operating point at each rate is ictate by the value of µ opt in (36) an can be achieve by choosing T, W an P as in (23) Furthermore, the optimizing µ opt increases monotonically as we consier larger transmission rates In fact, using the results on capacity from Theorem, it follows that with µ =, we only obtain first-orer optimality an therefore the error exponent in Fig 3 is non-zero only for a fraction of the coherent capacity, C coh On the other han, at R = C coh, we woul require µ opt > 3 (secon-orer optimal) in orer to achieve a positive error exponent In Fig 4, we plot the error exponent in (27) as a function of the parameter µ for two ifferent transmission rates For each scenario, we observe that the error exponent is concave as a function of µ an is maximize at µ = µ opt Also illustrate in the figure is the error exponent with perfect CSI at the receiver that is an upper boun to Er tr (N, R) in (27) an ecreases monotonically with µ These plots reveal a funamental learnability versus iversity traeoff in sparse channels For any rate R, when µ < µ opt(r) (too little coherence), the system is in the learnability-limite regime an the error exponent of the training-base communication scheme is smaller ue to poor channel estimation performance On the other han when µ > µ opt(r) (too much coherence), we are in the iversity-limite regime an the hit taen by the error exponent here is ue to the inherent reuction in the egrees of freeom (DoF) (or elay-doppler iversity, D) The best exponent is obtaine at µ = µ opt(r), which emarcates the two regimes an escribes the optimal traeoff E r (N,R) / C coh µ = 09 µ = 08 W = 0 5 Hz, P/N 0 = 40 B, T m W = 0 6, δ = δ 2 = 05 µ = 0 N c = / SNR µ µ = µ = 2 µ = 3 µ = R / C coh Fig 3 Ranom coing error exponent versus rate for a sparse channel an for the training-base communication scheme Different curves correspon to ifferent values of µ in the ey relationship N c = SNR µ REFERENCES [] S Verú, Spectral Efficiency in the Wieban Regime, IEEE Trans Inform Theory, vol 48, no 6, pp , June 2002 [2] L Zheng, M Mear, an D N C Tse, Channel Coherence in the Low SNR Regime, Submitte, IEEE Trans Inform Theory, 2005 [3] A F Molisch, Ultrawieban Propagation Channels - Theory, Measurement an Moeling, IEEE Trans Veh Tech, vol 54, no 5, pp , Sept 2005

6 6 W = 0 5 Hz, P/N 0 = 40 B, T m W = 0 6, δ = δ 2 = R = 0, Perfect CSI R = 095 C coh, Perfect CSI R = 0, Training scheme R = 095 C coh, Training scheme E r (R, µ) / C coh µ Fig 4 Illustration of the learnability versus iversity traeoff for sparse channels The value of µ at which the maximum is attaine in each case efines µ opt(r) at the corresponing transmission rate R [4] R Saaane, D Aboutajine, A M Hayar, an R Knopp, On the Estimation of the Degrees of Freeom of Inoor UWB Channel, IEEE Veh Tech Conf (Spring), 2005 [5] A M Sayee an B Aazhang, Joint Multipath-Doppler Diversity in Mobile Wireless Communications, IEEE Trans Commun, pp 23 32, Jan 999 [6] R S Kenney, Faing Dispersive Communication Channels, Wiley InterScience, NY, 969 [7] R G Gallager, Information Theory an Reliable Communication, John Wiley an Sons Inc, 968 [8] A M Sayee an V V Veeravalli, Essential Degrees of Freeom in Space-Time Faing Channels, Proc 3th IEEE Intern Symp Pers Inoor, Mobile Raio Commun, vol 4, pp 52 56, Sept 2002 [9] K Liu, T Kaous, an A M Sayee, Orthogonal Time-Frequency Signaling over Doubly Dispersive Channels, IEEE Trans Inform Theory, vol 50, no, pp , Nov 2004 [0] W Koze, Aaptation of Weyl Heisenberg Frames to Unersprea Environments, in Gabor Analysis an Algorithm: Theory an Applications, H G Feichtinger an T Strohmer, Es Boston, MA, Birhäuser, pp , 997 [] V Raghavan, G Hariharan, an A M Sayee, Capacity of Sparse Multipath Channels in the Ultra-Wieban Regime, Submitte, IEEE Journal of Selecte Topics in Signal Proc, Dec 2006 [2] B Hassibi an B Hochwal, How Much Training is Neee in a Multiple Antenna Wireless Lin?, IEEE Trans Inform Theory, vol 49, no 4, pp , Apr 2003 [3] G Hariharan an A M Sayee, Minimum Probability of Error in Sparse Wieban Channels, 44th Annual Allerton Conference on Communication, Control an Computing, Sep 2006