Minimum-Energy Bandlimited Time-Variant Channel Prediction with Dynamic Subspace Selection Zemen, Thomas; Mecklenbräuker, Christoph; Fleury, Bernard Henri Published in: The Proceedings of the 14th European Signal Processing Conference (EUSIPCO 2006)

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Citation for published version (APA): Zemen, T., Mecklenbräuker, C., & Fleury, B. H. (2006). Minimum-Energy Bandlimited Time-Variant Channel Prediction with Dynamic Subspace Selection. In The Proceedings of the 14th European Signal Processing Conference (EUSIPCO 2006)

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MINIMUM-ENERGY BANDLIMITED TIME-VARIANT CHANNEL PREDICTION WITH DYNAMIC SUBSPACE SELECTION Thomas Zemen1, Christoph F. Mecklenbr¨ auker1 and Bernard H. Fleury2 1

Forschungszentrum Telekommunikation Wien Donau-City Str. 1/3, 1220 Vienna, Austria E-mail: {thomas.zemen, cfm}@ftw.at

ABSTRACT In current cellular communication systems the time-selective fading process is highly oversampled. We exploit this fact for time-variant flat-fading channel prediction by using dynamically selected predefined low dimensional subspaces spanned by discrete prolate spheroidal (DPS) sequences. The DPS sequences in each subspace exhibit a subspace-specific bandwidth matched to a certain Doppler frequency range. Additionally, DPS sequences are most energy concentrated in a time interval matched to the channel observation interval. Both properties enable the application of DPS sequences for minimum-energy (ME) bandlimited prediction. The dimensions of the predefined subspaces are in the range from one to five for practical communication systems. The subspace used for ME bandlimited prediction is selected based on a probabilistic bound on the reconstruction error. By contrast, time-variant channel prediction based on non-orthogonal complex exponential basis functions needs Doppler frequency estimates for each propagation path which requires high computational complexity. We compare the performance of this technique under the assumption of perfectly known complex exponentials with that of ME bandlimited prediction augmented with dynamic subspace selection. In particular we analyze the mean square prediction error of the two schemes versus the number of discrete propagation paths. 1. INTRODUCTION In mobile communication systems channel state information at the transmitter proves to be beneficial for increasing the system capacity. In a time-division duplex (TDD) system channel state information can be obtained while a data block is received and used for the subsequent transmission period by exploiting channel reciprocity. However, for moving users at vehicular speed the channel state information gets outdated rapidly and thus appropriate prediction is necessary. In mobile communication systems the Doppler bandwidth is much smaller than the actual channel bandwidth. Thus the time-selective fading process is highly oversampled. Time-limited snapshots of the sampled fading process span a subspace with small dimension. The same subspace is also spanned by time-limited discrete prolate spheroidal (DPS) sequences [1]. The energy concentration of the DPS sequences in time is matched to the length of the observation interval, i.e. the length of the data block. The bandlimitation is chosen according to the Doppler bandwidth of The work of Thomas Zemen and Christoph F. Mecklenbr¨ auker was funded by the Wiener Wissenschafts- Forschungs- und Technologiefonds (WWTF) in the ftw. project “Future Mobile Communications Systems (Math+MIMO)”. The cooperation with Bernard Fleury was sponsored by the EU Network of Excellence in Wireless Communications (NEWCOM).

2 Aalborg University Department of Communication Technology Fredrik Bajers Vej 7, A3 9220 Aalborg, Denmark Email: [email protected]

the time-selective fading process. Slepian [2] showed that time-limited snapshots of a bandlimited sequence can be continued uniquely in the minimum-energy (ME) bandlimited sense. We use this property for ME bandlimited prediction of time-variant flat-fading channels [3]. In [4, Sec. 12.7] a solution to the prediction problem is shown using a linear MMSE (Wiener) filter. This filter can be closely approximated by a reduced-rank maximum likelihood (ML) filter according to [5]. For both predictors full knowledge of the Doppler power spectral density is required. By contrast the ME bandlimited predictor [3] needs knowledge of the Doppler bandwidth only. We show in [6] that ME bandlimited prediction is equivalent to reduced-rank maximum likelihood prediction for a process with flat Doppler spectrum. Using analytic performance results we prove that the prediction error is strongly dependent on the Doppler bandwidth, while the actual shape of the Doppler spectrum is of minor importance [6]. The ME bandlimited predictor in [3] utilizes the Dopplerbandwidth estimator proposed in [7]. The estimator in [7] assumes a Doppler spectrum according to Clarke’s model [8]. Thus specular Rice components lead to biased estimates as well as channels with a small number of specular propagation paths. The estimator proposed in [9] is less sensitive to deviations from Clarke’s model, however requires large observation intervals and an SNR larger than 30 dB. In this paper we develop a subspace selection procedure for ME bandlimited prediction that avoids the need for an explicit Doppler bandwidth estimate. Contributions of the Paper • We use a predefined set of subspaces. The DPS sequences in each subspace exhibit a subspace-specific bandwidth matched to a certain Doppler frequency range. The energy concentration of the DPS sequences is matched to the channel observation time interval. We obtain probabilistic bounds on the reconstruction error of each subspace using the method from [10]. The subspace with the smallest reconstruction error is used for channel prediction. • We present numerical performance results in terms of mean square error versus Doppler-bandwidth for ME bandlimited prediction with dynamic subspace selection. The results are compared with those of a Wiener predictor [4, Sec. 12.7] and a predictor that is derived based on the assumption that the channel is composed by a finite number of specular paths [11]. Notation We denote a column vector by a and its i-th element with a[i]. Similarly, we denote a matrix by A and its (i, ℓ)-th element by [A]i,ℓ . The transpose of A is given by AT , its con-

jugate transpose by AH . A diagonal matrix with elements a[i] is written as diag(a) and the Q × Q identity matrix as I Q . The absolute value of a is denoted by |a| and its complex conjugate by a∗ . The largest (smallest) integer that is lower (greater) or equal than a ∈ R is denoted by ⌊a⌋ (⌈a⌉). Organization of the Paper We introduce the signal model in Section 2. ME bandlimited prediction is reviewed in Section 3. In Section 4 we describe the dynamic subspace selection method and MonteCarlo simulation results are presented in Section 5. Finally, concluding remarks are provided in Section 6.

3.1 Channel Estimation The channel coefficients for a single block of length M are collected in the vector h = [h[0], h[1], . . . , h[M − 1]]T and the covariance matrix is defined as Σh = E{hhH } with elements [Σh ]ℓ,m = Rh [ℓ − m]. The noisy observation vector y = [y[0], y[1], . . . , y[M − 1]]T is used for channel prediction. Its covariance matrix reads Σy = Σh + (1/SNR)I M . We consider a subspace-based approximation which expands the vector h in terms of D orthonormal basis vectors ui = [ui [0], ui [1], . . . , ui [M − 1]]T , i ∈ {0, . . . , D − 1}:

2. SIGNAL MODEL FOR TIME-VARIANT FLAT-FADING CHANNELS We consider a time division duplex (TDD) communication system transmitting data in blocks of length M over a timevariant channel. The symbol duration is much longer than the delay spread of the channel, i.e. TS ≫ TD . Hence we assume the channel as frequency-flat. Discrete time at rate RS = 1/TS is denoted by m. The channel incorporates the transmit filter, the transmit antenna, the physical channel, the receive antenna, and the receive matched filter. The data symbols b[m] are randomly and evenly drawn from a symbol alphabet with constant modulus, i.e. |b[m]| = 1. The discrete-time signal at the matched filter output y ′ [m] = h[m]b[m] + n′ [m]

(1)

is the superposition of the data symbol multiplied by the sampled time-variant channel weight h[m], and complex white Gaussian noise n′ [m] with variance σ 2 . We assume an error-free decision feedback structure [12, 13]. Thus we are able to obtain noisy channel observations using the error-free data symbol estimates ˆb[m] = b[m]: y[m] = y ′ [m]ˆb[m]∗ = h[m] + n′ [m]ˆb[m]∗ = h[m] + n[m] .

(2)

(3)

1 ˆ 2 = 1 kU H nk2 + 1 kV H hk2 kh − hk M M M

z=

(4)

ˆ = Uγ where h ˆ , V = [uD , . . . , uM −1 ] contains the basis vectors spanning the subspace orthogonal to the signal subspace U and the noise vector is defined as n = [n[0], n[1], . . . , n[M − 1]]T . The mean square reconstruction error writes MSE = E{z}. We seek basis vectors ui minimizing the reconstruction error per data block. Discrete prolate spheroidal (DPS) sequences {ui [m]} time limited to IM form such basis vectors for a fading process with flat Doppler spectrum [1]. The properties of DPS sequences are analyzed by Slepian in [2]. DPS sequences are bandlimited to the frequency range [−νD , νD ] and simultaneously most concentrated in IM . They are defined as

X

M −1

ℓ=0

sin(2πνD (ℓ − m)) ui [ℓ] = λi ui [m] , π(ℓ − m)

(5)

for m ∈ Z where Z denotes the set of integers. The sequences {ui [m]} and the eigenvalues λi depend on the Doppler bandwidth νD and the observation interval length M . We omit this dependence to keep the notation simple. Both, the DPS sequences {ui [m]} and their restriction on IM form orthogonal sets. The eigenvalues λi decay exponentially for i ≥ D′ . The essential subspace dimension is defined as D′ = ⌈2νD M ⌉ + 1 . (6) The vectors ui satisfy the eigenvalue decomposition Cui = λi ui , where [C]l,m =

sin(2πνD (ℓ − m)) , π(ℓ − m)

ℓ, m ∈ IM .

(7)

Knowing ui we can continue the sequence {ui [m]} over Z in the ME bandlimited sense [2] by evaluating (5). The subspace dimension minimizing the MSE for a given SNR is found to be [14]

3. MINIMUM-ENERGY BANDLIMITED PREDICTION In this section we review the concept of ME bandlimited prediction of a time-variant flat-fading channel [3, 6].

γi u i .

i=0

In this expression U = [u0 , . . . , uD−1 ] and γ = [γ0 , . . . , γD−1 ]T contains the basis expansion coefficient. The least square estimate of γ simplifies to γ ˆ = U H y due to the orthogonality of the basis functions. The reconstruction error per data block is defined as

′

Notice that n[m] has the same statistical properties as n [m]. The signal-to-noise ratio (SNR) is SNR = 1/σ 2 . The transmission is block oriented. A data block corresponds to the time interval IM = {0, . . . , M − 1}. The noisy channel observations y[m], m ∈ IM , obtained during a single date block are used to predict the channel weight up to N symbols into the future. The electromagnetic field at the receiver is the superposition of the contribution of the individual fields of P impinging wave fronts. Each wave front is conceived as originating from a specific scatterer. For a user moving with velocity v the time-variant fading process {h[m]} will be bandlimited by the one-sided normalized Doppler bandwidth νD = vfC TS /c0 where fC is the carrier frequency and c0 stands for the speed of light. Without loss of generality {h[m]} is a zero-mean, circularly symmetric, unit-variance (due to power control) process. We assume a time-variant block-fading channel model. Hence the fading process {h[m]} with correlation function Rh [k] = E{h∗ [m]h[m + k]} is wide-sense stationary over the limited time interval IM +N (c.f. Section 5.1).

X

D−1

h ≈ Uγ =

D=

argmin ˜ D∈{1,...,M }

0

1 2νD M

X

M −1

˜ i=D

1

˜ 1 D A. λi + M SNR

(8)

(1)

4.2 Subspace Selection

(2)

In [10] an information theoretic subspace selection scheme is proposed. This method uses the observable data error

νD

...

νD

xq =

(Q)

νD

ν −νDmax

− νDmax 0 Q

νDmax Q

where

3.2 Channel Prediction So far we treated the channel estimation problem for a channel observed over a time interval IM . We used orthogonal basis vectors that result from time limiting an infinite sequence to the interval IM . The DPS sequences are most energy concentrated in this interval. However, the main interest of this paper lies on channel prediction. Slepian points out [2, Sec. 3.1.4] that there are infinitely many ways to choose the channel samples h[m], m ∈ Z\IM such that the infinite sequence {h[m]} is bandlimited. However, there exists a unique way to extend a bandlimited sequence in the sense of a ME continuation. This is achieved by using the DPS sequences {ui [m]}. We can express the ME bandlimited prediction of the fading process for any m ∈ Z as

X

D−1

ˆ h[m] = f [m]T γ ˆ=

γˆi ui [m] ,

(9)

i=0

where f [m] = [u0 [m], . . . , uD−1 [m]]T . For a processes with flat spectrum the mean square prediction error per symbol MSE[m] can be expressed by PD−1 λ |u [m]|2 . For a 2 ˆ MSE[m] = E{|h[m] − h[m]| } = 1 − i=0 i i general Doppler power spectral density (mismatched case) this expression provides a lower bound (for more details see [6]).

4.1 Subspace Definition We define the maximum Doppler bandwidth νDmax = vmax fC TS /c0 as system parameter given by the maximum user velocity vmax . We define a set of Q subspaces with (one (q) sided) Doppler bandwidth νD = q/Q νDmax , q ∈ {1, . . . , Q}, (q) see Fig. 1. The DPS sequences {ui [m, νD ]} correspond(q) ing to a Doppler bandwidth νD are calculated according to (q) (q) (5). We define the subspace U q = [u0 (νD ), . . . , uDq (νD )], (q) (q) where the vectors ui (νD ) are the sequences {ui [m, νD ]} time-limited to IM . The dimension of the subspace Dq spanned by U q grows with increasing q due to the increasing (q) spectral support 2νD (c.f. (6) and (8)).

(11)

to obtain a bound on the reconstruction error zq =

1 ˆ q k2 kh − h M

(12)

which cannot be observed directly. For the subspace selection h is considered deterministic. We adapt the results from [10] to complex valued variables. The reconstruction error zq is a sample of a random variable Zq which is distributed as [10, Lemma 1] 2M σ2

Zq −

1 2 kV H q hk M

∼ χ22Dq

(13)

where χ22Dq is a Chi-square random variable of order 2Dq . 2 Assuming kV H is known, the reconstruction error is q hk bounded with probability p1 according to zq′ ≤ zq ≤ zq′ , where

2 zq′ = Dq σ 2 /M + 1/M kV H q hk − Gq (p1 , σ, 2Dq )

2 and zq′ = Dq σ 2 /M + 1/M kV H q hk + Gq (p1 , σ, 2Dq ). The term Gq (p1 , σ, 2Dq ) is calculated by solving p1 = F (2Dq + 2Gq M/σ 2 , 2Dq ) − F (2Dq − 2Gq M/σ 2 , 2Dq ) numerically for Gq . The Chi-square cumulative distribution function with n degrees of freedom is denoted by F (x, n). The data error xq is a sample of a random variable Xq which is distributed as [10, Lemma 2]

2M Xq ∼ χ22(M −Dq ) . σ2

(14)

Because 2(M − Dq ) is large we can invoke the Central Limit Theorem to approximate Xq with a Gaussian random vari2 able. The term 1/M kV H q hk is bounded with probability p2 , H 2 i.e. Bq ≤ 1/M kV q hk ≤ Bq . The bounds Bq and Bq are given in [10, Theorem 1]. Finally, we use the upper bound

4. DYNAMIC SUBSPACE SELECTION We are interested in a low-complexity channel predictor. To this end we develop a dynamic subspace selection scheme for ME bandlimited prediction that does not need an explicit Doppler bandwidth estimate. Firstly, we define a set of subspaces. The orthogonal basis vectors spanning each subspace are calculated once and then kept fixed. Secondly, we propose a subspace selection method based on the observation of a single data block.

(10)

ˆq = U qU H h qy,

νDmax

Figure 1: Symmetric bandlimiting regions with (one sided) (q) bandwidth νD = q/Q νDmax , q ∈ {1, . . . , Q} used to define a set of Q subspaces. The q-th subspace is spanned by the columns of U q .

1 ˆ q k2 ky − h M

zq =

Dq 2 σ + Bq + Gq (p1 , σ, 2Dq ) ≥ zq′ M

(15)

on the reconstruction error to select the appropriate subspace q spanned by the columns of U q , q = argminq zq . The chosen subspace U q is used for ME bandlimited prediction. 5. MONTE-CARLO SIMULATIONS 5.1 Physical Wave Propagation Channel Model We simulate the fading process {h[m]} using physical wave propagation principles [8, 15]. The electromagnetic field at the receiver is the superposition of the contribution of the individual fields of P impinging plane waves. Each plane wave is conceived as originating from a specific scatterer. Under these assumption the channel weight is of the form

X

P −1

h[m] =

p=0

X

P −1

ap ej2πfp TS m =

ap ej2πνp m .

(16)

p=0

Here fp is the Doppler shift of wave p. For easier notation we define the normalized Doppler frequency as νp = fp TS . Notice that |νp | ≤ νD < 1/2. The gain and phase shift of

5.2 Simulation Parameters The symbol duration TS = 20.57 µs is chosen according to the system parameters considered in [1]. The speed of the receiver varies in the range 0 ≤ v ≤ vmax = 100 km/h = 27.8 m/s. The carrier frequency is fC = 2 GHz. This results in a Doppler bandwidth range 0 ≤ BD ≤ 180 Hz. Thus, the normalized Doppler bandwidth ranges 0 ≤ νD ≤ νDmax = 3.8 · 10−3 . The channel is observed over M = 256 symbols. We are interested in the prediction error at a prediction horizon of m − M + 1 ∈ {32, 128} symbols. At speed vmax the prediction horizon {32, 128} relates to a distance of {λ/8, λ/2} where λ = c0 /fC denotes the wavelength. For all simulations the SNR is 10 dB. All simulation results are averaged over 1000 independent channel realizations. The probabilities p1 and p2 for the subspace selection are Rα √ 2 chosen as p1 = p2 = −α (1/ 2π)e−x /2 d x with α = 8, i.e. −15 p1 = p2 ≈ 1 − 10 . 5.3 Choice of the Number of Subspaces Q We use the Cramer Rao lower bound (CRLB) for frequency estimation of a single complex exponential in white Gaussian noise [4, Sec. 15.10] to obtain a suitable choice on the number of subspaces Q. In the case of multiple paths P the energy per paths will be reduced by 1/P and we can utilize θ=

1 νDmax

s

p

var(ˆ ν) 6P σ 2 ≤ (2π)2 M (M 2 − 1) νDmax

(17)

as p lower bound on the relative frequency resolution var(ˆ ν )/νDmax . In Fig. 2 we plot θ versus the number of paths P for an SNR ∈ {0, 10, 20} dB. Fig. 2 documents that the lower bound on the frequency resolution is in the order of 0.01νDmax to 0.1νDmax for P = 20 propagation paths. We choose to partition the range of the Doppler bandwidth 0 ≤ νD ≤ νDmax into Q = 10 intervals, see Fig. 1. 5.4 Complexity The complexity of the proposed ME bandlimited predictor with dynamic subspace selection is mainly determined by the

frequency resolution l.b. θ

path p are embodied in the complex weight ap ∈ C. We model the random parameter sets ap and νp , p ∈ {0, . . . , P − 1} as independent. The random variables in each set are independent and identically distributed. The path angles αp are uniformly distributed over [−π, π) and the Doppler shift per path √ νp = cos(αp )νD . The path weights are defined as ap = 1/ P [cos(ψp ) + j sin(ψp )] where ψp are uniformly distributed over [−π, π). Under the above assumptions, the covariance function of h[m] converges to Rh [k] = J0 (2πνD k), for P → ∞ where J0 is the zeroth order Bessel function of the first kind [8]. We assume a time-variant block-fading channel model comprised of P paths. Hence the random path parameters ap and νp are assumed to be constant over a block of M + N symbols. However, the path parameters ap and νp change independently from block to block therefore the short-time Doppler spectrum changes as well. We note that the over-idealized simulation models from Jakes [16] or Zheng [17] are not suitable for the evaluation of channel prediction algorithms. This is because a symmetric distribution of the scatterers with equidistant spacing is assumed in [16, 17]. However real-world channels will not show equidistantly spaced scatterers. Prediction algorithms assuming a finite number of specular paths [18] show optimistically biased performance due to this over-idealized scatterer distribution.

0.12

SNR=0dB SNR=10dB SNR=20dB

0.1 0.08 0.06 0.04 0.02

5

10

15

20

number of paths P

25

30

Figure 2: Lower bound on the frequency resolution (17) versus number of paths P . The SNR ∈ {0, 10, 20} dB. complexity of projecting the observation vectorPy on all Q subspaces U q in (11). This operation requires 2( Q q=1 Dq )M complex multiply accumulate instructions. The calculation of the lower bound on the reconstruction error involves simple arithmetics or look-up tables of which the complexity and can be neglected. A predictor based on complex exponentials needs Doppler shift estimates for all P paths. Most methods for Doppler shift estimation rely on an eigenvalue decomposition of the channel’s sample covariance matrix [19]. The complexity of the eigenvalue decomposition grows with P M 2 . Hence the overall complexity of complex-exponential-based predictors is much higher than the one of the ME bandlimited predictor with dynamic subspace selection. 5.5 Simulation Results Monte Carlo simulations have been performed to contrast the performance of the three following predictors: The Wiener predictor [4, Sec. 12.7], the ME bandlimited predictor and a predictor derived based on the specular path model (16) [11]. The later predictor knows the number of paths and their Doppler shift. The MSE of all three predictors are reported in Fig. 3 and Fig. 4 for two different prediction horizons m − M + 1 ∈ {32, 128}. The ME bandlimited predictor with dynamic subspace selection (denoted ‘ME bandlimited’) performs better than a Wiener predictor (denoted ‘Wiener predictor’) which is designed using the long-term Clarke [8] Doppler power spectral density. Comparing the ME bandlimited predictor to a predictor based on complex exponentials (denoted ‘compl. exponential’) we can see the following results: For a short prediction horizons m − M + 1 = 32 (Fig. 3) the predictor based on complex exponentials perfroms better for P = 2 and νD M > 0.8 only. For a longer prediction horizon m − M + 1 = 128 (Fig. 4) the crossover point for P = 2 is at νD M = 0.4 and for P = 4 at νD M = 0.8. 6. CONCLUSION We presented a minimum-energy (ME) bandlimited predictor with dynamic subspace selection. Each subspace is matched to a certain Doppler bandwidth. The dimensions of the predefined subspaces are in the range from one to five for practical communication systems. The subspace applied for ME bandlimited prediction is selected based on a probabilistic bound on the reconstruction error. The ME bandlimited predictor with dynamic subspace selection performs better

0

10

[5]

−1

10

MSE

[6]

−2

10

ME bandlimited, P=2 ME bandlimited, P=30 compl. exponential, P=2 compl. exponential, P=4 compl. exponential, P=30 Wiener predictor

−3

10

0

0.2

0.4

νD M

0.6

0.8

[7]

1

[8]

Figure 3: Mean square prediction error MSE[m] at prediction horizon m − M + 1 = 32. We compare the minimumenergy (ME) bandlimited predictor with dynamically selected subspace (‘ME bandlimited’), the Wiener predictor (‘Wiener predictor’) and a predictor using perfectly known complex exponential basis functions (‘compl. exponential’).

[9]

[10]

0

10

[11]

−1

10

MSE

[12]

−2

10

−3

10

0

[13]

ME bandlimited, P=2 ME bandlimited, P=30 compl. exponential, P=2 compl. exponential, P=4 compl. exponential, P=30 Wiener predictor 0.2

0.4

νD M

0.6

0.8

1

Figure 4: Mean square prediction error MSE[m] at prediction horizon m − M + 1 = 128.

[14]

[15]

than a predictor based on complex exponentials for channels with more than ten propagation paths.

[16]

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[17]

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