optimal power loading for ofdm transmissions over

OPTIMAL POWER LOADING FOR OFDM TRANSMISSIONS OVER UNDERSPREAD RAYLEIGH TIME-VARYING CHANNELS Anna Scaglione1, Sergio Barbarossa2 1

Dept. of ECE, Univ. of Minnesota Minneapolis, MN 55455 [email protected]

2 Infocom

Dept., Univ. of Rome \La Sapienza", via Eudossiana 18, 00184 Roma, ITALY [email protected], Fax: (+39) 06-4873300

ABSTRACT

OFDM renders multipath channels with nite memory equivalent to parallel at fading subchannels, over which equalization amounts to simple phase compensation. If the channel status information is known at the transmitter side, optimal power loading across subchannels is possible. Users mobility, time and carrier asynchronism introduce variations in the equivalent channel impulse response setting the channel coherence time as a boundary for the OFDM symbol duration. Optimal design should incorporate such variations, possibly avoiding the overhead of frequent training. Most adaptive modulation schemes assume knowledge of the channel response and at fading. Modeling the channel taps as correlated Gaussian random processes, in this paper we develop the optimal power/bit loading strategies for frequency selective fading, based on the knowledge of the channel past estimates and correlation. In practice the channel sample correlation are obtained as a by-product of channel estimation which, however, does not need to be performed on a block by block basis.

1. INTRODUCTION Many researchers dug the eld of communications over timevarying channels from various perspectives. For narrowband, time-selective, at fading channels, maximum information rate is achieved by optimally distributing power/bits as a function of time. Adaptive modulation schemes were thus proposed to achieve optimality using dierent criteria (see e.g. [3],[9],[7]), under a transmit power constraint and assuming the channel perfectly known at both receiver and transmitter sides. Knowledge of the channel is possible in the presence of a feedback channel or in time division duplex (TDD) systems, where transmitter and receiver use the same carrier frequency, because of the reciprocity law. In practice, however, the channel is known only within a certain estimation/prediction error, which inevitably causes performance degradation [1]. Partial knowledge of the channel was considered in [6], where the proposed adaptive modulation assumed that the channel was a Gaussian process (Rayleigh fading), so that the pdf of the channel at a certain instant, conditioned to its past, followed a Gaussian law with mean value equal to the MMSE channel linear prediction and variance equal to the prediction variance. While in [6] the channel correlation was assumed to be known, in

[4] a similar approach was used for equalization but based on the channel correlation estimates, obtained by modeling the channel taps as AR processes and using spectral estimation techniques to retrieve the AR parameters. Both [6] and [4] assumed the channel to be stationary and at fading within blocks of data. Very recently, the adaptive modulation idea was extended to time and frequency selective channels using OFDM (see e.g. [10]). Although OFDM decomposes a time-invariant channel into a set of parallel

at fading subchannels, the extension of the aforementioned adaptive modulation techniques to OFDM is not immediate because the average power and the bits should be loaded in a jointly optimal way across dierent frequency bins and time blocks. Moreover, unlike adaptive modulation on at fading channels [9], OFDM cannot achieve capacity, not even asymptotically. In fact, for deterministic underspread frequency selective time-varying channels, optimal precoding corresponds to convey the information through particular channel-dependent FM modulated signals, that operate water- lling in the time-frequency domain [2]. Ideally, to achieve capacity one should code in nite size blocks of data and predict the channel exactly. In practice, this entails in nite delay and complexity. To keep complexity to a minimal, in this paper we use an OFDM strategy, where power loading is optimized in order to maximize average SNR or minimize average error probability. We assume that the channel is approximately constant during one OFDM symbol, but unlike [10] and similarly to [6],[4],[1] we incorporate the uncertainty in channel knowledge in our design. Specifically, we assume that the channel is an FIR lter of order L and time varying impulse response h(n; l). For an input u(n), the channel output samples are

y(n) =

L X l=0

h(n; l)u(n ; l) + v(n);

(1)

where v(n) is AWGN. We adopt the random model and assume that h(n; l) are correlated zero mean jointly Gaussian random variables. Thus, the ensemble mean value and correlation describe the pdf od any order completely (sample averages will replace the expected values in our design). We also assume that h(n; l) is stationary both in n and l, i.e.

Rhh (n; l) = E fh (p; k)h(p + n; k + l)g:

(2)

2. SYSTEM MODEL The basic assumption of OFDM is that the channel, including the eects of time asynchronism, multipath and pulse shaping lters, can bePmodeled as an FIR lter of nite order L, i.e. y(n) = Ll=0 h(l)u(n ; l) + v(n). In OFDM, information symbols are parsed into consecutive M -long blocks s(n) := (s(nM ); : : : ; s(nM + M ; 1))T ; and mapped through a precoding matrix F onto the vector u(n) = F s(n) whose entries are modulated and transmitted through the channel. The columns of F are complex exponentials, fF gn;m := m (n)ej 2M m(n;L) m 2 [0; M ;1], with M > L, n 2 [0; P ; 1]. The coecients m (n) in fF gn;m are used to load dierent powers across subchannels and blocks. Matrix F is tall (P  M where P := M + L), for the rows of F are increased by the addition of a pre x or sux of length L, equal or greater than the channel order. Thus u(n) := (u(nP ); : : : ; u(nP + P ; 1))T . Cyclic pre x or trailing zeros (TZ) can be adopted. For the sake of clarity, we assume the use of a cyclic pre x, but all the conclusions drawn in the following apply to the TZ case as well. After discarding the guard interval, inter-block interference (IBI) free data blocks y(n) := (y(nP + L); : : : ; y(nP + P ; 1))T are obtained. Assuming that h(nP + k; l)  h(nP; l), for k = 0; : : : ; P ; 1, the kth entry of y(n) (omitting additive noise v(nP + k) and using a block dependent coecient m (n)) is

y(nP + k)  =

L X

h(nP; l)

l=0 M ;1 X m=0

M ;1 X m=0

fF gk;l;m fs(n)gm

H (nP; fm )ej

2

(3)

m M (k;L) m (n)sm (nM );

for k = L; : : : ; P ; 1 and where fm = m=M , m = 0; : : : ; M ; 1, indicates a speci c normalized frequency and sm (nM ) := s(nM + m). Subsequently, an M points FFT is performed over y (n) and the th FFT output sample is

Y (nP; f ) :=

PX ;1 k=L

e;j M k y(nP + k) 2

(4)

 H (nP; f ) (n)s (nM ) + V (nP; f ); (5) P

where V (nP; f ) := Pk=;L1 exp(;j 2M k)v(nP + k) are also AWGN samples for  2 [0; M ; 1]. In (5) we neglected the eect of inter-symbol interference (ISI), arising because of the time selectivity. To eliminate ISI, the LTV channel singular functions should be used instead of the complex exponentials (c.f. [2]). As anticipated in the introduction, and assuming h(nP + k; l)  h(nP; l), for k = 0; : : : ; P ; 1, the channel is h(n) := (h(nP; 0); : : : ; h(nP; L))T  N (0; Rhh (0))

where N (; C ) indicates a multivariate complex Gaussian distribution with mean  and covariance C , while Rhh (n) is de ned as

fRhh (n)gk;l := E fh (qP; k)h(qP + nP; l)g

(6)

Assuming that the channel impulse response h(n) is exactly known at the 0th block, the conditional pdf of h(n) given h(0) is N (h^ LP (n); C LP (n)) where: h^ LP (n) := Rhh (n)R;hh1 (0)h(0); (7) ; 1 H C LP (n) := Rhh (0) ; Rhh (n)Rhh (0)Rhh (n); (8) with h^ LP (n) indicating the linearly predicted channel impulse response and C LP (n) its covariance. To incorporate the initial error on the knowledge of h(0), i.e.
h(0) := h(0) ; h^ (0), we can also assume
h(0)  N (0; C E (0)) and independent of h(n). This assumption is true if pilot tones are used to estimate h^ (0), since linear estimation is ecient in AWGN and in such a case C E (0) is the CRLB of the estimate. From the pdf of h(n), because H (nP; fm ) = eHm h(n) where em = (1; exp(j 2fm ); : : : ; exp(j 2fm L))T , we can easily derive the pdf of H (nP; fm ). In fact, H (nP; fm )  2 (nP; fm )) where H ^ LP (nP; fm ) := eHm N (H^ LP (nP; fm ); LP 2 H h^ LP (n) and LP (nP; fm ) := em C LP (n)em .

2.1. Max. average SNR under constraint power

In this section we derive a closed form solution for the optimal power loading maximizing the average SNR, subject to an average power constraint. In formulas, using the Lagrange multipliers, we seek the m (n) that maximize the functional Nb M ;1 X X

J () := N1 SNR(nP; fm ) b n=1 m=0

;1 b M X X jm (n)j2 ; P0 ); ; ( N1b n=1 m=0 N

(9)

where  := (0 (1); : : : ; M ;1 (Nb ))T is the subcarrier's amplitudes matrix and P0 is the average power over Nb consecutive blocks; SNR(nP; fm ) is the signal to noise ratio in the mth subcarrier of the nth block, where the noise incorporates both thermal noise and prediction error: ^ (nP; fm )j2 jm (n)j2 s2 SNR(nP; fm ) = j2HLP fm )jm (n)j2 s2 + v2 (10) LP (nP;   = m (n) 1 ;  (n)j 1 (n)j2 + 1 ;(11) m m where 2 (nP; fm ) 2 ^ (nP; fm )j2 LP s; m (n) := jHLP 2 (nP; fm ) and m (n) := v2 LP (12) and s2 is the symbol variance. Taking Nb = 1 in (9), we have a short term power constraint that yields an instantaneous optimal distribution, whereas increasing Nb we have a long term constraint. Of course, Nb cannot be too large for not incurring in excessive decoding delays. Setting to zeros the derivatives of J () with respect to m (n) in (9), we get m (n)m (n)   (13) (m (n)jm (n)j2 + 1)2 m (n) ; m (n) = 0;

which implies either m (n) = 0 or:

jm (n)j2 =

r

!

m (n)m (n) ; 1 1  m (n) :

(14)

Substituting m (n) as a function of  in the constraint we nd: P b PM ;1 ;1 Nb P0 + Nn=1 1 p = PNb PM ;1 p m=0 m (n) ; (15)  n=1 m=0 m (n)=m (n) which plugged back into (14) provides the closed form expression for jm (n)j2 . If the solution is negative, the smallest jm (n)j2 are progressively nulled and the power redistributed across the subchannels, until all remaining jm (n)j2 are positive. Substituting jm (n)j2 in (14) inside (10) we have:  r   SNR(nP; fm ) = m (n) 1 ;  (n) (n) : (16) m m Practical systems usually target a prescribed quality of service (QoS), expressed in terms of maximum tolerable BER Pe . Assuming a QAM constellation of order Q, we can use the bound [7]: SNR(nP;fm )

Pe (nP; fm )  0:2 e; M ; ; (17) valid at high SNR and assuming Gray encoding, to determine if the QoS can be attained as a function of n, without updating the channel estimate, and if the SNR over the worse subchannels is too low to render power investment on them worthwhile. 3

2(

1)

2.2. Min. average BER under constrained power

We assume here DPSK modulation so that the probability of error is simply 0:5 exp(;SNR). Nevertheless this expression, through appropriate scaling, can be used also as a bound for the BER in BPSK coherent detection or for QAM constellations according to (17). Minimizing the probability of error Pe (n) in the nth block is equivalent Q to maximizing its complement Pc (n) := (1 ; Pe (n)) = i [1 ; 0:5 exp(;SNR(nP; fi ))]. Since P usually Pe (n)  1, it is reasonable to assume Pc (n)  1 ; i 0:5 exp(;SNR(nP; fi )), which implies that the average BER Pe (n)  Pe (n). Therefore, minimizing the average BER is almost equivalent to minimize the block error probability, for all the values of Pe (n) that guarantee a reasonable QoS. In this case, the Lagrangian is

J () := N1 b

Nb M ;1  X X 1 e;SNR(nP;fm ) + j (n)j2 ;P : m 0 n=1 m=0

2

(18) Setting to zero the partial derivative of J () with respect to m (n), we obtain   ;0:5 e;SNR(nP;fm) (m (nm)j(nm)(nm)(j2n)+ 1)2 +  m (n) = 0; (19) so that m (n) is either equal to 0 or is the value that nulls the term between parentheses in (19). Unfortunately, the

resulting equation is non linear in m (n), for SNR(nP; fm ) is a function of m (n) (c.f. (10)) and the exact solution requires non linear programming techniques. Here we propose a suboptimal, yet practical, approach to solve the problem, based on the following observations: i) if, on the m-th channel, L2 P (nP; fm )jm (n)j2 s2  v2 , we deduce from (10) that the error probability on that channel is lower bounded by 0:5e;m (n) ; in such a case, we may adopt the following conservative strategy to enforce a target constant Pe on all subchannels: We order the m (n) in decreasing sense and progressivley channels with the smallest m (n) Q  discard;the m (n) ) > 1 ; Pe and determine the until M (1 ; : 5 e m=1 values m (n) on the remaining M channels as solutions of 0:5 e;SNR(fm ) = 1 ; (1 ; Pe )1=M ; (20)

setting jm (n)j2 = 0 in case of negative solutions; ii) con2 (nP; fm )   2 we can approximate the value versely, if LP v of SNR(nP; fm ) in (10) as SNR(nP; fm )  m (n) := jH^ LP (fm )j2 jm (n)j2 =v2 ; (21) which allows us to obtain a closed form solution. In fact, in this case (19) becomes: ; m (n) e; m (n)jm (n)j2  (n) +  (n) = 0 : (22) m m 2 Solving (22) with respect to m (n), we get   ( n ) 1 m 2 (23) jm (n)j = max m (n) ln 2 ; 0 ; ! where Nb M ;1 X X 1 1 1 ln(2) = M N b (n) ln m (n) ; P0 : (24) n=1 m=0 m jm (n)j2 given in (23),

The values of with  given by (24) can then be used to initialize a nonlinear iterative search algorithm to solve (19).

3. PERFORMANCE To analyze the performance of our transmission strategy we consider the case of uncorrelated scattering (US), i.e. fRhh (n)gk;l = Rhh (nP; k ; l) ' l2 (k ; l)Rt (nP ) (25) where l2 is the power delay pro le. Because in the US model Rhh (n) := Rt (nP )diag(02; : : : ; L2 ;1 ) is diagonal, using (7) and (8) we obtain H^ LP (nP; fm ) = eHm h^ LP (n) = Rt (nP )H (0; fm ); (26) 2 (nP; f ) = eH C (n)e LP m m m LP  L 2 n)  X = Rt (0) 1 ; RRt2((0) l2 : (27) t l=0 From (27) we observe that the decay of Rt (n) dictates 2 how fast we converge from PLLP 2(0; fm ) = 0 to the limit 2 of LP (nP; fm ) = Rt (0) l=0 l , corresponding to total uncertainty on the channel taps. Moreover, the US model

0

H(t,f)

10

6

−1

10

Relative power

5 4 3 −2

2

10

1 0 0 10

0

0 20

10

−3

50

100

150

200 Delay (nsec)

250

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30

Figure 1: Channel power-delay pro le. 2 (nP; fm ) which is constant across subchanleads to a LP nels. Simple choices for Rt (nP ) are: i) exponential: Rt (nP ) = e;nP=Nc (28) where Nc := Tc =Ts is the coherence time normalized to the sampling rate Ts  1=B , and B is the bandwidth; or ii), the isotropic scattering model in [8, p.31], Rt (nP ) = J02 (2nP=Nc ) (29) where J0 (x) is the Bessel function of rst kind. Example In this example we consider a wireless LAN indoor scenario. The standard High Performance Radio LAN (HIPERLAN) provides short distance, high speed radio links between computers systems using the 5.2 GHz and the 17.1 GHz frequency bands. We generated random impulse responses according to the US model, using a power delay pro le named \Channel A" (Fig.1), chosen as a typical indoor multipath scenario for HIPERLAN/2 in [5], operating at 5.2 GHz, with B = 200 MHz (Ts = 10 nsec). The channel order is L  19, for the impulse response samples beyond the 19th are strongly attenuated. We used OFDM blocks of size M = 64 (P = M + L) and modeled Rt (nP ) as in (28), with coherence time of 10sec, which corresponds to Nc  104 and approximately to 100 blocks. Fig. 2 shows a realization of the channel time-varying transfer function H (nP; fm ). In Fig. 3 we report the SNR(nP; fm ) resulting from the optimal short term power loading in (14), obtained with P0 = M , for an average SNR = 10 dBs. Subchannels with SNR(n; fm ) below the value necessary to achieve Pe (n; fm )  10;3 for BPSK, are discarded. This strategy is appropriate for data transmissions that must guarantee BER values below a certain level rather than maximizing the data rates. From Fig. 3 we notice that, as n and thus the prediction error increases, i) the power tends to be equally distributed across channels and ii) the number of channels able to guarantee the required BER decreases and thus the data rate decreases. Imposing a bound on the minimal data rate, from curves like Fig. 3 we are able to assess after how many blocks it is necessary to update the channel estimate for not incurring into severe data rate losses.

REFERENCES [1] M.S. Alouini A.J. Goldsmith, \Adaptive modulation for Nagakami fading channels", Proc. of IEEE

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Figure 2: Time varying transfer function. SNR(t,f)

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Figure 3: SNR(n; fm ) resulting from optimal loading [2] [3] [4] [5] [6] [7] [8] [9] [10]

GLOBECOM' 97, Phoenix, November 1997. S. Barbarossa, A. Scaglione, \Optimal precoding for transmissions over linear time-varying channels", Proc. of IEEE GLOBECOM '99, Rio de Janeiro, Dec. 1999. J.K. Cavers, \Variable-rate transmission for Rayleigh fading channels", IEEE Trans. on Comm., Vol.20, pp.15-22, February 1972. T. Eyceoz, A. Duel-Hallen, H. Allen, \Deterministic channel modeling and long range prediction of fast fading mobile radio channels", IEEE Comm. Letters, Vol.2, No.9, pp.254-256, September 1998. ETSI Normalization Commitee, Norme ETSI, doc. 3ERI085B, Sophia-Antipolis, Valbonne, France 1998 D.L. Goekel, \Adaptive coding for time-varying channels using outdated fading estimates", IEEE Trans. on Comm., Vol.47, pp.844-855, June 1999. A.J. Goldsmith, S.C. Chua, \Variable-rate variable power M-QAM for fading channels", IEEE Trans. on Comm., Vol.45, pp.1218-1230, October 1997. W.C. Jakes, Microwave Mobile Communication. Pitaskaway, NJ; IEEE Press, second ed., 1994. W.T. Webb, R. Steele, \Variable rate QAM for mobile radio", IEEE Trans. on Comm., Vol.43, pp.2223-2230, July 1995. C. Y. Wong, R. S. Cheng, K. B. Letaief, R. D. Ross,\Multiuser OFDM with adaptive subcarrier, bit, and power allocation", IEEE Trans. on Sel. Areas in Comm., Vol.17, pp.1747-1757, October 1999.