Cramer-Rao Lower Bound for Channel Estimation in a MUROS/VAMOS Downlink Transmission

Transcription

1 nd IEEE Personal Indoor Mobile Radio Communications Cramer-Rao Lower Bound for Channel Estimation in a MUROS/VAMOS Downlink Transmission Michael A. Ruder, Robert Schober, Wolfgang H. Gerstacker Chair of Mobile Communications, Universität Erlangen-Nürnberg, Cauerstr. 7, D-9058 Erlangen, Germany, {ruder, gersta}@lnt.de Department of Electrical & Computer Engineering, University of British Columbia, 356 Main Mall, Vancouver, BC V6T Z4, Canada, rschober@ece.ubc.ca Abstract Voice over Adaptive Multi-user Channels on One Slot VAMOS is an extension of the Global System for Mobile Communications GSM stard, where two overlaid Gaussian minimum-shift keying GMSK signals are transmitted in the same time slot on the same frequency. The overlaid signals are usually assigned different powers to combat slow fading propagation loss. For channel estimation, specific training sequences are used for both signals. In the downlink, at each mobile station the channel coefficients the sub-channel power imbalance ratio SCPIR have to be estimated. In this paper, the Cramer-Rao lower bound for the training sequence based joint estimation of the SCPIR the channel coefficients at the mobile station is derived compared with the for a conventional GSM transmission. The results are compared with the performance of channel estimation algorithms designed for VAMOS. It turns out that these algorithms perform very close to the. I. INTRODUCTION Due to still growing dems for voice services in Global System for Mobile Communications GSM networks, a new transmission mode called Voice over Adaptive Multi-user channels on One Slot VAMOS has been specified in 3GPP Release 9 []. With VAMOS, the capacity of existing GSM networks can be doubled up to four half rate voice users can share one time slot. This extension of the GSM stard, which has been discussed by 3GPP in the work item Multiple Users Reusing One Slot MUROS, requires efficient receivers in the mobile station to separate the users []. Prior to equalization, the channel coefficients have to be estimated. For VAMOS, where the signal of each user is transmitted on an orthogonal sub-channel OSC, the sub-channel power imbalance ratio SCPIR also has to be estimated. A new set of training sequence codes TSCs [3], [4] has been introduced for the paired sub-channel OSC-. The new TSCs were designed for optimum autocorrelation properties low cross-correlation with the set of legacy training sequence codes that are used for the first sub-channel OSC- []. In the downlink, at each mobile station MS a superposition of both OSCs is received. Due to this fact it is possible to exploit the TSCs of both users to enhance the channel estimation performance []. The Cramer-Rao bound is a lower bound for the variance of unbiased estimators [5]. The for training sequence based channel estimation for GSM without VAMOS extension is well known, cf. [6]. As a performance bound the for the joint estimation of the channel impulse response the SCPIR indicates whether existing estimators perform already close to the optimum or there is still room for improvement of the algorithms. Furthermore, by comparing the for training sequence based channel estimation in conventional non VAMOS GSM with the for channel estimation in VAMOS, the degradation in channel estimation performance of VAMOS compared to the legacy system can be analyzed. This paper is organized as follows. Section II introduces the system model of a VAMOS downlink transmission, the state-of-the-art channel estimation algorithms for VAMOS are revisited. In Section III, the for the joint estimation of the channel coefficients the SCPIR is derived. Sections IV V provide simulation results conclusions, respectively. II. SYSTEM MODEL AND CHANNEL ESTIMATION A. System Model In the considered scenario of a VAMOS downlink transmission [4], the base station transmits two user signals in the same time slot on the same frequency resource. Both signals are Gaussian minimum-shift keying GMSK modulated, where the signal of the user on OSC- is rotated by 90 scaled by a real valued factor b > 0 that is related to the SCPIR. After GMSK derotation at the receiver, the discrete-time received signal at time index k in equivalent complex baseb notation at one of the two involved MSs can be written as r[k] = q h κ=0 h[κ]a [k κ] + j b q h κ=0 h[κ]a [k κ] + n[k]. Here, the discrete-time channel impulse response h[κ] of order q h comprises the effects of GMSK modulation, the mobile channel from the base station BS to the considered user, receiver input filtering, GMSK derotation at the receiver. The channel impulse response h[κ] is always assumed to be constant within a transmission burst, but varies romly between bursts block fading. a i [k] denotes the kth binary phase-shift keying BPSK training sequence symbol of user i {, } with variance σ a. Both TSCs are time-aligned. Knowledge of q h is assumed. In GSM, q h 4 is valid in most cases. Thus, assuming q h = 4 in system design is sufficient. GMSK modulation can be well approximated by filtered BPSK //$ IEEE 438

2 The discrete-time additive white Gaussian noise AWGN of variance is denoted by n[k]. In the following, it is assumed that a [k] is the signal corresponding to the MS of interest. B. Channel Estimation For channel estimation one can exploit the fact that both user signals propagate through the same channel []. Therefore, the overall channel impulse response of OSC- is that of OSC- scaled by a factor b rotated by 90 factor j. If the received symbols corresponding to the TSCs of both users are collected in a vector r, can be rewritten as r = A h + ba h + n, where A i represents the K q h q h + Toeplitz convolution matrix corresponding to the TSC symbols of user i {, }, with training sequence length K, h = [h[0] h[]... h[q h ]] T T : transposition. Please note that the first q h received symbols have to be discarded for channel estimation because they contain unknown data symbols. n is a vector with statistically independent AWGN entries. For an easier notation, the factor j in has been absorbed in A. In the following, a short review of the joint maximumlikelihood ML estimation of h b that has been already outlined in [] will be given. The joint ML estimates for h b result from minimizing the L -norm of the error vector e = r A ĥ ˆbA ĥ, where ĥ ˆb denote the estimated quantities. Differentiating e H e H : Hermitian transposition with respect to ĥ : complex conjugation ˆb, the following two conditions for the ML estimates of h b are obtained after setting the derivatives to zero: ĥ = V V H V H r 3 ˆb = ĥh A H A ĥ ĥh A H r A ĥ + r H ĥh A H A ĥ with V = A + ˆbA. Eqs. 3 4 can also be interpreted as the ML estimate of the channel for given b the ML estimate of b for given channel vector, respectively []. Both equations are coupled it does not seem to be possible to obtain a closed-form solution for ĥ ˆb. An iterative solution was proposed in [], where ˆb is blindly estimated this result is used as an initial choice for ˆb in 3. The resulting channel vector is then used for refining ˆb via 4, etc., until convergence is reached. Due to the iterative refinement of the estimates, an unbiased estimation cannot be always ensured. III. CRAMER-RAO LOWER BOUND For a real-valued parameter vector λ, containing unknown parameters to be estimated from observations, the Fisher information matrix FIM is defined as [5] Jλ = E r λ { ln pr λ λ 4 } ln T pr λ, 5 λ where p r λ is the conditional probability density function of the vector with observations r also called likelihood function E{ } sts for the expectation operation. The Cramer-Rao bound is defined as λ = Jλ. 6 For an unbiased estimate ˆλ the estimation error λ = λ ˆλ, the error covariance matrix C λ = E { λ λt } satisfies C λ λ 0, 7 where A 0 means that matrix A is positive semidefinite [5]. A. Conventional GSM Transmission The for TSC based channel estimation for a conventional GSM transmission is given by [6] h = A H A, 8 only depends on the values of the symbols of the TSC the noise variance. The lower bound for the mean-squared error MSE of an estimate of h is therefore { MSE h trace A H A }, 9 where trace{ } sts for the trace of a matrix. For optimum TSCs minimizing MSE h, A H A is a scaled version of the identity matrix [6]. B. VAMOS Transmission For the derivation of the for a downlink VAMOS transmission, we rewrite as r[k] = a [k] + j b a [k] h + n[k], 0 with k {q h +, q h +,..., K}, the row vector a i [k] = [a i [k] a i [k ] a i [k q h ]] of user i {, } containing a section of the training sequence. This can be condensed to r[k] = s[k] + n[k], with the information bearing part of the received signal s[k] = a [k] + j b a [k] h. The vector with parameters that need to be estimated is λ = [ b Re {h[0]} Re {h[]}... Re {h[q h ]} Im {h[0]} Im {h[]}... Im {h[q h ]} ], where the channel impulse response has been separated into its real imaginary parts, whereas b is per definition purely real-valued. The likelihood function of the received signal given the parameter vector λ [5] is obtained as p r λ r λ = π K q exp h r[k] s[k λ]. k=q h + 439

3 The log-likelihood function can be expressed as ln p r λ r λ = r[k] s[k λ] r [k] s [k λ] k=q h + K q h lnπ. The derivative of the log-likelihood function with respect to λ m, the mth element of vector λ, can be calculated to ln p r λ r λ = k=q h + n [k] + n[k] s [k λ], where r[k] s[k λ] was substituted by n[k]. The element in the mth row nth column of the FIM Jλ is defined by { ln pr λ r λ [J] mn = J mn = E ln p } r λr λ. Inserting into yields J mn = E 4 k=q h + l=q h + n [k]n[l] s [l λ] n[k]n[l] s [k λ] s [l λ] n [k]n [l] s[l λ] + + n[k]n [l] s [k λ] s[l λ] + }. 3 For k l the expected value is zero, therefore we only sum over values with k = l. Furthermore, for k = l, E{n [k]} = E{n [k]} = 0 E{n [k]n[k]} = E{n[k]n [k]} =. Therefore, 3 can be simplified to J mn = k=q h + + s [k λ] s [k λ]. 4 It is obvious from this expression, that J mn = J nm always holds therefore the FIM is a symmetric matrix. The partial derivatives are = j a [k] h, b = j a [k] h, b Re {h[x]} = [a [k] + j b a [k]] x+, Re {h[x]} = [a [k] j b a [k]] x+, Im {h[x]} = j [a [k] + j b a [k]] x+, Im {h[x]} = j [a [k] j b a [k]] x+, where [ ] x+ denotes the x + th element of a vector, with x {0,,..., q h }. The q h + 3 q h + 3 FIM J can be separated into smaller blocks, J = J j j 3 j 4 J 5 J 6 j 7 J 8 J 9. 5 With the partial derivatives the blocks of the FIM can be calculated. The scalar J can be determined as J = a [k]h a [k]h. The x+th elements x {0,,..., q h } of the q h + vectors j j 3 are [j ] x+ = [j 3 ] x+ = a [k] b [a [k]] x+ Re {h} [a [k]] x+ Im {h}, a [k] [a [k]] x+ Re {h} +b [a [k]] x+ Im {h}, respectively. Due to the symmetry properties of the FIM the q h + vectors j 4 j 7 are given by j 4 = j T j 7 = j T 3. The element in the x + th row the y + th column x, y {0,,..., q h } of the q h + q h + matrix J 5 is given by [J 5 ] x+y+ = the elements of J 6 are [J 6 ] x+y+ = [a [k]] x+ [a [k]] y+ +b [a [k]] x+ [a [k]] y+, b [a [k]] x+ [a [k]] y+ +b [a [k]] x+ [a [k]] y+. Again due to symmetry J 8 = J T 6 holds. The elements of the q h + q h + matrix J 9 are [J 9 ] x+y+ = i.e., J 5 = J 9. [a [k]] x+ [a [k]] y+ + b [a [k]] x+ [a [k]] y+, 440

4 0 MSEb Fig.. b =. MSE versus SNR for the estimation of the channel coefficients Fig.. MSE versus SNR for the estimation of the SCPIR b =. The lower bound for the MSE of the SCPIR estimation of a VAMOS transmission is now obtained as MSE b [J ], 6 the lower bound for the MSE for the estimation of the channel impulse response coefficients is MSE h [J ] mm. 7 q h +3 m= In contrast to the for the channel impulse response estimation in conventional GSM, now the performance does not only depend on the TSCs, but also on the actual channel impulse response the SCPIR value. Due to the inversion of the FIM in 5, a closed-form solution for the MSE cannot be given. IV. SIMULATION RESULTS In the following, numerical simulation results are presented in order to analyze the for VAMOS in more detail. The MSEs of the joint channel SCPIR estimation algorithm according to Section II-B are compared with the for the joint estimation of the parameters. Thereby, the is evaluated for deterministic parameters b h. For the calculation of the FIM the actual values of the parameters are used. For each simulation the SCPIR value is kept constant, while 5000 different channel impulse responses of order q h = 5 are generated with a rom complex normal distribution with variance σh =. The resulting s for the estimation of h b, respectively, are averaged over the realizations of the channel impulse response. They are plotted for different values of the signal-to-noise ratio SNR of the composite signal of both users SNR = + b σ a Fig. 3. MSE versus SNR for the estimation of the channel coefficients b =. Estimation algorithm according to [7] has been used for the simulation. Due to the fact that the channel estimation algorithm also exploits the signal of the orthogonal user, the factor + b is incorporated in the SNR definition. Additionally, the for channel estimation for a conventional GSM transmission SNR = σ a σ is given as a reference. This can be used to n analyze the loss in channel estimation accuracy due to the necessity of estimating the SCPIR factor. TSC 0 of the TSC set for the conventional GSM system the VAMOS TSC 0 [4] have been used for the simulations. Figure depicts the MSE for channel estimation for b =. For a broad range of SNR values the proposed estimator matches the tightly only for low SNRs a minor degradation in channel estimation accuracy is visible. A loss compared to a conventional GSM channel estimation is barely visible for this SCPIR value. For the estimation of b the 44

5 results are depicted in Fig.. For moderate-to-high SNR, the bound the simulation results match well, while for low SNR values the simulation results are slightly better than the. This is because for low SNR, the estimation of b is not unbiased anymore. Also a computationally less complex estimator [7] has been analyzed the corresponding results for the channel coefficient estimation are shown in Fig. 3. This estimator also matches the lower bound very tightly in a broad range of SNRs. The results for the estimation of b are similar to the ones depicted in Fig. therefore not shown. For b {/, }, Fig. 4 depicts the MSE for the channel estimation. For b = / the s for VAMOS GSM are even closer than for b =, while for b = the bounds are further apart. For both values of b the channel estimation technique proposed in Section II-B matches the VAMOS quite well. Hence, the proposed channel estimation technique is well suited for the use in a VAMOS system. For increasing b the gap between the s for VAMOS conventional GSM grows. Although a small b value is beneficial for the channel estimation performance of the user on OSC-, this also results in a worse channel estimation performance for the orthogonal user on OSC-. Fig. 5 shows the corresponding results for the SCPIR estimation. For an easier comparison the normalized mean-squared error NMSE, NMSE b = MSE b /b, has been used. Due to the normalization the for both values of b is identical. Again, for moderate-to-high SNR the bound the simulation results match well. For low SNR values the simulation results are slightly better than the for b = 0.5, while for b = the performance of the estimation algorithm is noticeably better than the bound for low SNR because the estimation is not unbiased anymore in this regime 3. V. CONCLUSIONS The Cramer-Rao lower bound for channel SCPIR estimation in a VAMOS/MUROS GSM system has been derived in this paper. For channel estimation the for VAMOS has been compared to the for a conventional GSM system. It has been shown that only a minor loss in channel estimation accuracy occurs due to the additional estimation of the sub-channel power imbalance ratio. The mean-squared error of the channel estimation technique from [] has been compared with the bound, showing that this channel estimation algorithm performs very close to the optimum. Also a second channel estimation technique [7] has been tested found to perform equally well. REFERENCES [] M. Säily, G. Sébire, E. Riddington, Eds., GSM/EDGE: Evolution Performance. Wiley, 00. [] R. Meyer, W. H. Gerstacker, F. Obernosterer, M. A. Ruder, R. Schober, Efficient receivers for GSM MUROS downlink transmission, in Proc. IEEE 0th Int Personal, Indoor Mobile Radio Communications Symp, 009, pp b = 0.5 b = 0.5 b = b = Fig. 4. MSE versus SNR for the estimation of the channel coefficients b {/, }. NMSEb b = 0.5 b = 0.5 b = b = Fig. 5. NMSE versus SNR for the estimation of the SCPIR b {/, }. [3] X. Chen, Z. Fei, J. Kuang, L. Liu, G. Yang, A scheme of multiuser reusing one slot on enhancing capacity of GSM/EDGE networks, in Proc. th IEEE Singapore Int. Conf. Communication Systems ICCS 008, 008, pp [4] TR V Circuit switched voice capacity evolution for GSM/EDGE Radio Access Network GERAN, 3rd Generation Partnership Project 3GPP Std. [5] S. Kay, Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory v.. Prentice Hall, 993. [6] E. De Carvalho D. T. M. Slock, Cramer-Rao bounds for semiblind, blind training sequence based channel estimation, in Proc. First IEEE Signal Processing Workshop Signal Processing Advances in Wireless Communications, 997, pp [7] ST-NXP Wireless France Com-Research, MUROS downlink receiver performance for interference sensitivity, 3GPP TSG GERAN, GERAN Telco 9 on MUROS, Jan., This is because also for low SNRs, the estimate of b tends to converge to a limited value. 44