The Effect of Feedback Delay to the Closed-Loop Transmit Diversity in FDD WCDMA

Transcription

1 The Effect of Feedback Delay to the Closed-Loop Transmit Diversity in FDD WCDA Jyri Hämäläinen Risto Wichman Nokia Networks, P.O. Box 319 Nokia Research Center, P.O. Box 407 FIN Oulu, Finland FIN NOKIA GROUP, Finland Abstract Transmit diversity techniques provide attractive solutions for increasing downlink capacity in 3G systems within low-mobility environments. Open-loop and closed-loop transmit diversity modes with two transmit antennas have already been included into 3GPP WCDA Release 4, and at the moment schemes exploiting a larger number of transmit antennas are being developed. This paper analyzes the performance of a simple closed loop transmit diversity scheme suitable for 3GPP WCDA. The goal is to study the effect of different feedback delays to the performance of the given closed loop scheme. Although the analysis applies only in the case of the speci c algorithm (phase only feedback) it is expected that the performance of the more general magnitude and phase feedback shows similar properties. We use signal-to-noise ratio (SNR) gain as a performance measure, which corresponds to the gain in expected received SNR in mobile terminal when feedback information from the receiver is applied in the transmitter. I. INTRODUCTION The uplink capacity of the proposed third generation CDA systems can be enhanced by various techniques including multi-antenna reception and multi-user detection. However, the expected breakthrough of wireless internet services will probably load more heavily the downlink than the uplink, and therefore it is important to nd techniques that improve the capacity of the downlink channel. Bearing in mind the strict complexity requirements of terminals, cost ef ciency, and the characteristics of the downlink channel, a brute-force application of advanced detectors with multiple receive antennas is not seen as the desired solution to the downlink capacity problem. Alternative solutions have been proposed suggesting that multiple transmit antennas at the base station will increase downlink capacity with only a minor increase in terminal implementation. Open-loop transmit diversity techniques apply various linear or non-linear preprocessing techniques to combat the fading channel. Recently, the linear ltering approach has been generalized in [1]. A coding perspective to transmit diversity was contributed in [2], [3] with the introduction of space-time codes that can be optimized to fading channels. The simplest space-time code [2] has been adopted into 3GPP WCDA standard as an open-loop transmit diversity method. With low mobile speeds the additional side information of the channel state in the transmitter provided by closed loop techniques further improves the performance by increasing the received SNR in the mobile station. This far few analytical results concerning the performance of various transmit diversity feedback algorithms have been presented. Narula et al. [4] evaluate the expected received SNR in multi input single output (ISO) at Rayleigh fading channel containing 1 bits of side information, where refers to the number of transmit antennas. Sandell [5] studied the performance of the feedback schemes with two transmit antennas assuming unquantized channel state information in the transmitter. ISO at Rayleigh fading channels were analyzed assuming quantized phase and gain information in [6], and two transmit one receive system together with multipath Rayleigh fading and quantized side information was studied in [7]. The results above have not addressed the effect of feedback delay which, however, is unavoidable in practical systems. The feedback delay was considered in [8] assuming two transmit one receive system and unquantized feedback. In 3GPP FDD WCDA Release 4, the capacity of the feedback channel is very limited and therefore the feedback delay severely limits the applicability of the closed loop transmit diversity with high mobile speeds. This paper analyzes the performance of quantized phase only feedback using expected received SNR as the performance measure. Furthermore, the effect of feedback errors is discussed. The paper is structured as follows: Section II summarizes the current closed loop transmit diversity modes in 3GPP WCDA Release 4, and Section III introduces the ISO system model. Section IV provides the analysis of the effect of feedback delay to the performance of the example algorithm in the case of at Rayleigh fading. The paper is concluded in Section V. II. 3GPP WCDA FEEDBACK ODES System capacity can be increased from that of open-loop modes if the transmitter is equipped with additional side information of the downlink channel. In an FDD system this means that the receiver has to provide the information through some feedback mechanism. The prevailing 3GPP FDD WCDA transmit diversity concept assumes two transmit antennas in the base station which are suf ciently close to each other so that the propagation delays between each transmitting element and the mobile station are effectively identical. On the other hand it is desirable that transmit antennas are spaced suf ciently far apart so that mobile station experiences independent fading channels from the antennas. In principle, the antennas need not be calibrated unlike in the case of beamforming applications. There are currently two different feedback modes with slightly different tradeoffs in effective constellation resolution and signaling robustness. In mode 1 the length of the feed-

2 back word is one bit, and the base station interpolates between two consecutive feedback words making the transmit weight to follow a time-varying QPSK constellation. In mode 2 the feedback word consists of four bits where three bits are assigned to phase and one bit to gain. Thus, mode 1 maintains equal power transmission from both antennas while with mode 2 the antennas transmit with different powers. A detailed description of frame and slot structures can be found in [9]. It is well know that the optimum transmit weight is the eigenvector corresponding to the maximum eigenvalue of the channel covariance matrix at the mobile station R(t) H(t) H H(t), where H(t) (h 1 (t),...,h (t)) C L and h m (t) refers to the impulse response between the m th transmit antenna and the terminal at time t. It is assumed that channels are constant during each slot, and they vary from slot to slot in a rate depending on the mobile speed. In FDD systems R(t) can be estimated by the terminal only and therefore some related side information must be signaled to the base station. Naturally, this information becomes available in the base station only after some round trip and processing delay τ. In the present WCDA Release 4 system, terminal can obtain ˆR(t) by measuring ĥ 1 (t) and ĥ 2 (t) from the common pilot channel. III. SYSTE ODEL Consider a system with transmit antennas in the base station and a single receive antenna in the mobile station. For the analysis we adopt a single path Rayleigh fading channel model. Since h m (t) is now a complex scalar rather than vector, we denote h m (t) instead of h m (t) and h(t) (h 1 (t),...,h (t)) instead of H(t) (h 1 (t),...,h (t)). obile station receives the signal from the dedicated channel as r(t) (h(t)w)s(t)+n(t), where s(t) is the transmitted symbol, n(t) is zero-mean Gaussian noise, w (w 1,w 2,...,w ) T consists of transmit weights selected based on the feedback from mobile station, and components of the channel vector h(t) (h 1 (t),...,h (t)) are samples of a zero-mean Gaussian process with the common variance σ We assume that channel coef cients h m (t) are samples from independent processes. In later sections we denote α m (t) h m (t). The mathematical formulation for the problem of nding the best possible transmit weight w at each time instant t becomes Find w 0 W : h(t)w 0 max h(t)w (1) w W where W {w (w 1,w 2,...,w ) T : w m C} and it has been assumed that w 1. The performance of the analyzed algorithm is illustrated using the expected SNR gain factor γ E{ h(t)w 2 }, which provides the upper bound for the system capacity improvement in a fading channel because C E{log(1 + SNR)} log(1 + E{SNR}). We remark that the feedback delay τ is taken into account in the analysis. IV. ANALYSIS OF EXAPLE ALGORITH Let us rst give an example algorithm that has been addressed previously in [4], where it has been assumed that a feedback word of length 1 bits is provided to the base station having antenna elements. Each bit contains information about the phase difference of the corresponding antenna and the common reference antenna. A generalization utilizing N feedback bits/antenna is given by the following algorithm [6]. Example Algorithm. Assume that ( 1)N bits of side information is available. Then the transmit weights {ŵ m } m2 at time t + τ are chosen by using the condition h 1 (t)+ŵ m h m (t) max w m W N h 1 (t)+w m h m (t), (2) where 2 m and W N {e j2π(n 1)/2N / : n 1,2,...,2 N }. That is, we adjust each phase independently against the phase of the rst channel. It should be noticed that the complexity of the example algorithm increases linearly with additional antennas, i.e., complexity is proportional to ( 1)2 N. When 2, N 2 the example algorithm resembles FDD WCDA transmit diversity ode 1. The only difference between ode 1 and the example algorithm is that in ode 1 the feedback word results from the interpolation between two consecutive one bit feedback words. For the analysis purposes we approximate the effect of interpolation as an additional half a slot delay. In the following proposition we study the effect of feedback delay to the performance of the given example algorithm. In order to simplify the analysis we rst assume that all feedback bits are chosen based on the channel state at time t τ and they are applied at time t, where τ is the feedback delay. Proposition 1: The SNR gain factor γ for the Example Algorithm is given by γ πc ( 2 ) N 1+ c 2 N J0 ( 2πvτ )2, (3) where c N 2N π sin π 2 N and J 0 ( ) is the Bessel function of order zero, and v, and τ refer to the mobile speed, carrier wavelength and feedback delay respectively. Proo f. Let us write rst h(t)w 2 m1 k1 w m h m (t)w k h k (t) : A 1 + A 2 + A 3,

3 where we have denoted A 1 A 2 A 3 m1 k2 w m 2 α m (t) 2 2Re ( w 1 w k h 1 (t)h k (t) ) m 1 m3 k2 2Re ( w m w k h m (t)h k (t) ). Since each sample α m (t) follows Rayleigh distribution and weight vector w is normalized we nd that E{A 1 } 1. oreover, we have chosen w 1 1/ and amplitudes α m (t) have no effect to the phase adjustments given by (2). Therefore we get E{w 1 w k h 1 (t)h k (t)} π 4 E{ej(φ 1 (t) φ k (t) ψ k ) }. Here φ k (t) is the phase of h k (t) and w k 1/ e jψ k. Now we write φ 1 (t) φ k (t) ψ k φ 1 (t) φ k (t)+ϕ k : Ψ k (t), where φ k (t) φ k (t) φ k (t τ). Since ψ k is chosen by using (2) at time instant t τ we see that ϕ k φ 1 (t τ) φ k (t τ) ψ k is uniformly distributed on ( π 2 N, π 2 N ) and there holds E{sinϕ k } 0, E{cosϕ k } 2N π sin π 2 N c N. Thus we have Re ( E{e j(φ 1 (t) φ k (t) ψ k ) } ) E{cos(Ψ k (t))} c N E{cos φ 1 (t)cos φ k (t)+sin φ 1 (t)sin φ k (t))}. Random variables φ 1 (t) and φ k (t) are identically distributed and independent since channels corresponding to separate antennas are assumed to be uncorrelated. Therefore we obtain Re ( E{e j(φ 1 (t) φ k (t) ψ k ) } ) c N ( E{cos( φ(t))} 2 + E{sin( φ(t))} 2), where we have simpli ed notations by dropping out the indices that are not needed. Let us now study the phase difference φ(t) φ(t) φ(t τ). If the velocity of the mobile is v it moves the distance d vτ during the time delay τ and we see that our problem is equivalent to the one where spatially (distance d) separated antennas receive the same sinusoidal wave. Let φ 0 : φ(t τ), so that φ(t) φ 0 + 2π d cosψ. where ψ is uniformly distributed on ( π,π) (we do not know the direction where the mobile moves). We obtain E{cos( φ(t))} 1 2π π π cos( 2π d cosψ)dψ J 0 (2π d) J 0 (2π vτ) Furthermore, since sin-function is odd we nd that the expectation of sin( φ(t)) vanish. By using the above formulae we see that E{A 2 } 1 2 πc N J (2π 0 vτ) 2. It remains to nd E{A 3 }. This case is studied in a similar manner as the case E{A 2 }. Now where arg{w m w k h m (t)h k (t)} φ m (t) φ k (t) ψ m + ψ k φ m (t) φ k (t)+ϕ m ϕ k, ϕ m φ m (t τ) φ 1 (t τ) ψ m, ϕ k φ k (t τ) φ 1 (t τ) ψ k. Variables ϕ m and ϕ k are independent and uniformly distributed on ( π 2 N, π 2 N ). Hence we obtain E{cos(ϕ m ϕ k )} E{cosϕ m }E{cosϕ k } c 2 N. Finally, we get E{A 3 } which completes the proof. ( 1)( 2) πc 2 4 NJ 0 ( 2πvτ )2 In 3GPP FDD WCDA Release 4 system the feedback rate is 1500 bits/s. Thus, there is a single feedback bit available in each slot. oreover, in control channel there are also two feedback bits (in each slot for fast power control. If these bits are also available in our example algorithm the feedback rate becomes 4500 bits/s. When the capacity of the feedback channel is less than N( 1) bits/slot the transmitter may choose to update the transmit weight vector whenever a new bit is received from the feedback channel. In this case the elements of w experience different delays, and it is straightforward to show that Proposition 1 becomes γ 1+ πc ( N 2 J 0 ( 2πvτ k k2 ) 2 + m3 m 1 k2 J 0 ( 2πvτ m )J 0 ( 2πvτ ) k ) (4) Figure 1 compares the SNR gains of the current WCDA transmit diversity ode 1 and Example Algorithm with feedback rates 1500 and 4500 bits/s, one slot (2/3 ms) signaling delay, and 4, N 2 assuming singe path Rayleigh fading channels. Furthermore, N 2 in Example Algorithm is achieved by interpolating two one bit feedback messages in base station in the same vein as with ode 1. We approximate the interpolation by the delay of a FIR lter with a symmetric impulse response so that the corresponding feedback delays become with ode 1 and 4500 bits/s feedback rate with 4. In the case of 1500 bits/s and 4 the delays in (4)

4 SNR gain obile speed [km/h] Fig. 1. Analytical SNR gains (linear scale) of the present ode 1 (- -), and the given Example Algorithm with 4 transmit antennas and 1500 (- -) and 4500 ( ) feedback bits per second, and simulated SNR gains of ode 1 ( ), 4, 1500 bps ( ), 4, 4500 bps ( ). SNR gain obile speed [km/h] Fig. 2. Analytical SNR gains (linear scale) of the present ode 1 (- -), and the given Example Algorithm with 4 transmit antennas and 1500 (- -) and 4500 ( ) feedback bits per second, and simulated SNR gains of ode 1 ( ), 4, 1500 bps ( ), 4, 4500 bps ( ). Probability of feedback errors p are 2.5, 3.5 and 4.5 slots. It is seen that the analysis and simulations match perfectly in the case of ode 1, but with 4 the approximation of the interpolation by the additional delay causes some bias error. The effect becomes more pronounced with 1500 bits/s when the phase of the channel changes rapidly with respect to the feedback rate. When I and Q components of a transmit weight experience different delays, the feedback is able to track the channel better than in the case of approximating the effect by a delay caused by linear ltering. In FDD WCDA fast power control is typically applied to uplink control channel carrying the feedback information. Power control attempts to shorten the correlation time of the channel and maintain a xed received power, and therefore we make a simplifying assumption that feedback bit errors are uniformly distributed in time. A typical (uncoded) feedback error probability p is, for example, 0.04 which roughly corresponds to the order of 10 3 coded BER in a voice system. In the following proposition we study the effect of feedback errors to the SNR gain γ when also time delay is considered. Proposition 2: Let the feedback error probability be p and let N 1. Then the SNR gain factor γ for the Example Algorithm is given by γ 1+ 1 (1 2p)( 1+ 2 (1 2p) ) J π 0 ( 2πvτ )2, where J 0 ( ) is the Bessel function of order zero. Symbols v, and τ refer to the mobile speed, carrier wavelength and feedback delay respectively. Proo f. The feedback errors affect to the distribution of the variables ϕ m φ 1 (t τ) φ m (t τ)+ψ m of the proof of Proposition 1. ore precisely, now the probability that ϕ m is uniformly distributed on ( 2 π, 2 π ) is p while the probability that ϕ m is uniformly distributed on ( 2 π, 2 π )c (here ( ) c means the complement of the set ( )) is 1 p. Thus we nd that c 1 E{cosϕ m } 2 (1 2p). π Employing this equation we obtain the result of this proposition directly from Proposition 1. In the same vein it is straightforward to show that when N 2 substituting c 2 as c 2 (1 2p)c 2 (3) gives the expected SNR gain in the case of feedback errors and QPSK feedback constellation. The same principle applies to N 3 and Gray coded feedback message as well, and it can be shown that N 3 is enough to get SNR gain which is close to the ideal one when the transmitter has complete knowledge of the channel [6]. Figure 2 shows the analytical and simulated performance of the three feedback schemes with 4% feedback error probability. Again, analysis and simulations match well in the case of 2. Furthermore, it is noticed that ode 1 is more robust to feedback errors than the four transmit antenna schemes, and with high mobile speeds the feedback delay is more critical to the performance than the nominal 4% feedback error probability. V. CONCLUSIONS We studied closed-loop transmit diversity techniques suitable to FDD WCDA system. The effect of feedback delay and bit errors to an example phase only feedback algorithm was analyzed in at Rayleigh fading channels using expected

5 received SNR gain as the performance measure. Results show that feedback delay puts strict limits to the applicability of closed loop algorithms within high mobile speeds while the effect of feedback bit errors is not crucial if a constant bit error rate is assumed. However, it is known that the fast power control is not able to compensate fast fading with high mobile speeds are high, and therefore, the feedback bit error rate also depends on the mobile speed and it is expected that the curves of Figures 1 and 2 provide only upper bound for the achievable SNR gain. Finally it is emphasized that if the capacity of the feedback channel in future FDD WCDA releases is not increased then the feedback delay may cause serious problems when introducing closed loop algorithms utilizing more than two transmit antennas. REFERENCES [1] A. Narula,. D. Trott, G. W. Wornell: Performance Limits of Coded Diversity ethods for Transmitter Antenna Arrays, IEEE Trans. Information Theory, V. 45, No. 7, November 1999, pp [2] S.. Alamouti: A Simple Transmitter Diversity Scheme for Wireless Communications, IEEE Selected Areas of Comm., V. 16, No. 8, Oct. 1998, pp [3] V. Tarokh, N. Seshadri, A. R. Calderbank: Space-Time Codes for High Data Rate Wireless Communications, IEEE Trans. Information Theory, V. 44, No. 2, arch 1998, pp [4] A. Narula,. J. Lopez,. D. Trott, G. W. Wornell: Ef cient use of side information in multiple-antenna data transmission over fading channels, IEEE Journal of Selected Areas of Comm., V. 17, No. 8, Oct. 1998, pp [5]. Sandell, Analytical analysis of transmit diversity in WCDA on fading multipath channels, IEEE International Symposium on Personal, Indoor and obile Radio Communications, September [6] J. Hämäläinen, R. Wichman: Closed-loop transmit diversity for FDD WCDA systems, 34th Asilomar Conference on Signals, Systems and Computers, October [7] J. Hämäläinen, R. Wichman: Feedback schemes for FDD WCDA systems in multipath environments, IEEE Vehicular Technology Conference, ay [8] B. Raghothaman, G. andyam, R. T. Derryberry, Performance of closed loop transmit diversity with feedback delay, 34th Asilomar Conference on Signals, Systems and Computers, October 2000, pp [9] 3GPP RAN WG1: Physical layer General description, 3GPP TS , Ver , June 2000.