... GSM/GPRS networks. In the second part, a lower bound on the bit error probability ... bit error probability (BEP) for the desired user is introduced. It is conceptually ..... The simulation result for cancelled residual ISI does not show this effect.
Single Antenna Interference Cancellation: Iterative Semi-Blind Algorithm and Performance Bound for Joint Maximum-Likelihood Interference Cancellation Hendrik Schoeneich and Peter A. Hoeher Information and Coding Theory Lab, University of Kiel Kaiserstr. 2, D-24143 Kiel, Germany Phone: +49 431 880 6133 (or 6127), Fax: +49 431 880 6128 Email: {hs,ph}@tf.uni-kiel.de Abstract— In the first part, an iterative receiver for joint channel estimation and cochannel interference cancellation suitable for time-division multiple-access (TDMA) cellular radio systems is proposed. The receiver is based on joint maximum-likelihood sequence estimation (JMLSE). It is blind with respect to the data of the interfering users but training-based with respect to the data of the desired user and is therefore referred to as semi-blind. As opposed to concurrent receiver structures where the number of receiving antennas must exceed the number of cochannel interferers, the proposed receiver is suitable for just one receive antenna. No knowledge on the burst structure of the interferer is used. Hence, the proposed receiver is suitable for (but not restricted to) the downlink in asynchronous networks. The proposed receiver is able to operate at a signal/interference ratio of 0 dB in asynchronous GSM/GPRS networks given a single dominating interferer. Furthermore, the results indicate that with proper interference cancellation it is not necessary to synchronize GSM/GPRS networks. In the second part, a lower bound on the bit error probability for joint maximum-likelihood sequence estimation in the presence of cochannel interference (CCI) and intersymbol interference (ISI) is presented. This lower bound is valid for an arbitrary number of interferers.
I. I NTRODUCTION Typically, the network capacity of 2nd generation TDMA cellular radio networks is limited by cochannel interference rather than noise, unless cochannel interference cancellation is done. Single antenna (cochannel) interference cancellation (SAIC) is currently a hot research topic, since mobile stations are commonly (due to cost, volume, power consumption, and design aspects) equipped with just one antenna and since especially with the introduction of multimedia services the downlink became the bottleneck in 2nd generation TDMA networks. From a research point of view, SAIC is more challenging than making use of multiple antennas. In fact, many algorithms fail if the number of receive antennas does not exceed the number of interferers. Motivated by recent GSM/GPRS field measurements reporting a significant improvement of voice capacity and data throughput due to SAIC, it is likely that the requirement of the next GERAN (GSM/EDGE radio access network) release will be tightened. A corresponding working group has been established recently. Nevertheless,
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the proposed algorithm may also be applied in receivers with multiple antennas, although this is not the focus here. Assuming perfect knowledge of the channel coefficients of the desired user and J interfering users, the set of optimal sequences in terms of the maximum-likelihood criterion can be obtained by joint maximum-likelihood sequence estimation (JMLSE) [5]. The semi-blind receiver proposed in the first part of this paper is based on JMLSE. For related work see [6]. In the second part, for JMLSE a tight lower bound on the bit error probability (BEP) for the desired user is introduced. It is conceptually based on a lower bound originally derived for CDMA multiuser detection [9, Chapter 4.3], which is adapted here to JMLSE and which is moreover extended to frequencyselective fading channels and partial-response modulation. The organization of the remainder of this paper is as follows. In Section II, the channel model under investigation is introduced. In Section III, JMLSE and joint least squares channel estimation (JLSCE) are briefly outlined, since a JMLSE/JLSCE receiver serves as a reference. The novel semiblind receiver is described in Section IV. The lower bound on the bit error probability in the presence of CCI and ISI is derived in Section V. In Section VI, analytical as well as simulation results are presented. Finally, conclusions are drawn in Section VII. II. C HANNEL M ODEL Throughout this paper, the complex baseband notation is used and a linear transmission scheme is assumed. The equivalent, baud-rate sampled, discrete-time channel model can be written as y(k) =
L l=0
hl (k) a(k − l) +
J L
gl,j (k) bj (k − l) + n(k),
j=1 l=0
(1) where y(k) is the k-th sample at the output of the receive filter, h(k) := [h0 (k), . . . , hL (k)]T is the channel coefficient vector of the desired user (E{||h(k)||2 } = 1), gj (k) := [g0,j (k), . . . , gL,j (k)]T is the channel coefficient vector of the j-th interferer (1 ≤ j ≤ J), L is the effective memory length, a(k) is the k-th data symbol of the desired user (E{a(k)} = 0,
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E{|a(k)|2 } = 1), bj (k) is the k-th data symbol of the jth interferer (E{bj (k)} = 0, E{|bj (k)|2 } = 1), n(k) is the k-th Gaussian distributed noise sample (E{n(k)} = 0, E{|n(k)|2 } = N0 /Es ), k is the time index (0 ≤ k ≤ K − 1), and K is the burst length. The channel coefficients comprise the pulse shaping filter, the physical time-varying multipath fading channel, the receiving filter, and the sampling. In vector/matrix form, the channel model can be written as y = A h(k) +
J
Bj gj (k) + n := C f + n,
(2)
j=1
where C := [A, B1 , . . . , BJ ] comprises the data matrices of the desired user A and all J interfering users Bj , 1 ≤ j ≤ J and f := [hT (k), g1T (k), . . . , gJT (k)]T all channel coefficient vectors, respectively. III. JMLSE AND JLSCE Given perfect knowledge of all channel coefficients, the optimal detector in terms of maximum-likelihood sequence estimation solves ˆ J = arg max p y|˜ ˜ 1, . . . , b ˜ J , (3) ˆ 1, . . . , b ˆ, b a, b a ˜J ˜ 1 ,...,b ˜ ,b a
where a := [a(0), . . . , a(K − 1)]T and bj := [bj (0), . . . , bj (K − 1)]T . For white Gaussian noise, the solution simplifies as ˆ 1, . . . , b ˆ J = arg min y − C ˜ f 2 , ˆ, b a (4) ˜J ˜ 1 ,...,b ˜ ,b a
which can efficiently be solved by means of the Viterbi algorithm (VA) [5], [1]. The computational complexity of a joint Viterbi detector (VD) is O(K M (J+1)L ), where M is the cardinality of the symbol alphabet. The complexity can be further reduced by applying the principles of decision feedback [4], [3] and/or set partitioning [4]. Given perfect knowledge of all data sequences and assuming block fading, the optimal channel estimator in terms of joint least-squares channel estimation can be written as [7] ˆf = (CH C)−1 CH y.
ˆ 0 ) = (AH At )−1 AH yt , h(k t t
(6)
where At is a sub-matrix of A representing only training symbols and yt contains the corresponding received samples. The initial estimate is corrupted by CCI. The initial estimate of g(k) is chosen to be of the form ˆ (k0 ) = [0, . . . , 0, c, 0, · · · , 0]T , where c = 0. Experience g indicates that the channel coefficients are centered around the non-zero coefficient. This aids to mitigate shift ambiguity [2]. B. Joint Data and Channel Estimation In order to perform joint data and channel estimation, we propose to integrate an adaptive channel estimator into a joint Viterbi detector (JVD). Adaptive channel estimation is done (i) to improve the noisy estimates of the channel coefficients of the desired user, (ii) to acquire the channel coefficients of the interfering user(s), and (iii) to track slowly time-varying channels. In the following, we use the LMS algorithm for adaptive channel estimation due to its simplicity compared to more sophisticated adaptive channel estimators. The update is done in a decision-directed fashion. Good results have been obtained by using the most likely survivor as a tentative decision of the data sequences {a(k)} and {b(k)} in conjunction with a small delay δ. Alternatively, one may use per-survivor processing or the list Viterbi algorithm to provide tentative decisions of the data sequences. Let us consider a full-state trellis. (A simplification to reduced-state sequence estimation [4], [3] is straightforward.) Let T ˜ν,µ (k) := a ˜ν,µ (k), a ˜ν,µ (k − 1), · · · , a ˜ν,µ (k − L) (7) a and
˜ ν,µ (k) := ˜bν,µ (k), ˜bν,µ (k − 1), · · · , ˜bν,µ (k − L) T b
(5)
Joint least-squares channel estimation is useful in systems with overlapping training sequences, i.e., particularly in synchronous networks. A JMLSE/JLSCE receiver serves as a reference in the following. IV. I TERATIVE S EMI -B LIND D ETECTION AND C HANNEL E STIMATION In this section, an iterative semi-blind algorithm based on a combination of the VA and the least-mean-squares (LMS) algorithm is presented. We introduce the proposed receiver for J = 1 dominating interferer and one receive antenna. The receiver structure can easily be extended to take more interferers into account or to make use of antenna diversity. It can be used in any TDMA mobile radio system. However, the specific implementation below is based on the popular GSM/GPRS system.
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A. Initial Channel Estimation The first step of the algorithm provides initial estimates of the channel coefficients h(k) and g(k). (We drop index j = 1 in g1 (k) for convenience.) An initial estimate of h(k) can be obtained, for example, by correlative channel estimation exploiting the training sequence of the desired user, or by leastsquares channel estimation (LSCE) according to
(8)
be hypotheses of the transmitted data symbol sequences at time index k corresponding to the respective state transition ˆ ˆ (k) be estimates of from state ν to state µ. Let h(k) and g the channel coefficients at time index k. Then, joint data and channel estimation is done as follows: 1) Calculate all branch metrics ˆ T (k) a ˜ν,µ (k + 1) γν,µ (k + 1) = y(k + 1) − h ˜ ν,µ (k + 1)2 . (9) −ˆ gT (k) b
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2) Calculate the path metrics for all transitions from k to k + 1 and determine the best path for each state µ at time index k + 1: Γµ (k + 1) = min {Γν (k) + γν,µ (k + 1)} , ν
(10)
i.e., let only the best path per state survive.
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3) The most likely survivor is traced back δ symbols ˆ − δ), ˆ(k − δ) and b(k to obtain tentative decisions a respectively. The decision delay δ is a design parameter of the algorithm. The choice of δ is based on a trade-off between the quality of the tentative decisions and the tracking capability. 4) The channel estimates are updated using the LMS algorithm: ˆ + 1) h(k ˆ (k + 1) g
ˆ ˆ∗ (k − δ) = h(k) + ∆(k) e(k) a ˆ ∗ (k − δ), (11) ˆ (k) + ∆(k) e(k) b = g
where T ˆ (k) a ˆ − δ) ˆ(k − δ) + g ˆ T (k) b(k e(k) := y(k − δ) − h (12) is an error signal common for both recursions and ∆(k) is the step size of the algorithm, which may be timevarying. 5) Increment the time index k and go to Step 1) until the end of the burst is reached. C. Iterative Update Process As mentioned before, the proposed algorithm can be used in conjunction with an arbitrary burst structure. We use the GSM normal burst shown in Fig. 1 as an example. An application to other structures can be implemented in a similar way.
000000 111111 0 1 0 1 00000 11111 00000 11111 0 1 0 1 k
0
3
11 00 00 11 tail
61
data
3
2
87
midamble
58
26
training
60
86
1 2
data 58
1 0 0 1 0 1 0 1 000 111 000 111 0 1 0 0 1 1 0 1 145
148
tail
guard
3
144
8
147
1 0 0 1
155
n
Fig. 1.
GSM normal burst and sketch of the update process.
ˆ 0 ) and g ˆ (k0 ) are determined Initial channel estimates h(k according to Section IV-A. Then, channel and data estimation for both the desired and the interfering user are performed based on the algorithm described in Section IV-B. The update process begins at the left end of the midamble (k0 = 61 + L) and proceeds to the right end of the burst (forward recursion). This is illustrated in Fig. 1. When the right end of the burst is reached, the last channel estimates are stored and used as initial estimates for a subsequent update process proceeding from the right end of the midamble to the left end of the burst (backward recursion). The last channel estimates are again stored and can be used as initial estimates for further iterations. One iteration consists of one forward recursion and one backward recursion. The total number of iterations is denoted by n. The final decision on the user’s data sequence corresponds to the most likely survivor after the last iteration.
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V. L OWER B OUND ON B IT E RROR P ROBABILITY In this section, we introduce a lower bound on the BEP of the desired user using JMLSE. The bound is conceptually based on a lower bound originally derived for CDMA multiuser detection [9, Chapter 4.3], which is adapted here to JMLSE and which is moreover extended to frequency-selective fading channels and partial response modulation. The bound is valid for a arbitrary number of interferers, J. Binary antipodal mapping and perfect channel knowledge is assumed. T Let x(k) := [a(k), b1 (k), · · · , bJ (k)] , a, bj ∈ {−1, +1}, j = 1, . . . , J, be the joint data vector at time index k, ˆ (k) be an estimate of this vector. 0 ≤ k ≤ K − 1, and x The transmitted data can be represented by the matrix X := [x(0), · · · , x(K − 1)]. Correspondingly, an error matrix E is defined as ˆ X−X . (13) E := 2 T
Each column vector ej of E = [e0 , e1 , · · · , eJ ] ∈ (J+1)×K {−1, 0, +1} represents the bit errors of a certain user j, 0 ≤ j ≤ J, where index 0 represents the desired user. The set of error matrices that affect the desired user is denoted as
(J+1)×K e0 = 0 . (14) ΦK := E ∈ {−1, 0, +1} A similar set can also be defined for every interfering user by considering other rows of E, but in this paper focus is merely on the errors affecting the desired user. Given an error matrix E ∈ ΦK , the set of data matrices X being compatible with E is defined as
(J+1)×K E ∈ ΦK . (15) ΨK (E) := X ∈ {±1} Using JMLSE, the BEP for the desired user, Pb,JMLSE , can be lower-bounded as follows: Assume there exists a genie who knows the sent data sequences of the desired user and all J interfering users. Given an error matrix E ∈ ΦK that affects the desired user, the genie verifies whether the sent matrix X is compatible with E or not. If X ∈ ΨK (E), the genie tells the receiver that the correct X is in the set {X, X − 2E}. The receiver determines the most likely of the two hypotheses employing the ML criterion, which is equivalent to a simple threshold decision with two decision regions. If X ∈ / ΨK (E), the genie reveals X to the receiver, so that no bit error is made. Because for X ∈ ΨK (E) the number of hypotheses is two and no errors occur for X ∈ / ΨK (E), the BEP of this genieaided receiver is a lower bound with respect to Pb,JMLSE . For uniformly distributed data, the BEP of this genie-aided receiver is d2 (E) Es −w(E)−1 · · erfc Pb,genie1 (E) = w (e0 ) · 2 4 N0 ≤ Pb,JMLSE , (16) where
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w (ej ) :=
K−1
|ej (k)| ,
0 ≤ j ≤ J,
(17)
k=0
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denotes the weight of error vector ej (which is equal to the number of bit errors of user j), w (E) :=
J
w (ej ) =
j=0
J K−1
|ej (k)|
PLB,1
(18)
j=0 k=0
denotes the error weight of matrix X ∈ ΦK , and 2 L L+K−1 J 2 e0 (k − l) hl + d (E) = 4 ej (k − l) gl,j j=1 k=0 l=0 (19) ˆ correspondis the squared Euclidean distance between y and y ing to error matrix E. The channel coefficients are assumed to be (quasi) time-invariant during a burst, but may change from burst to burst. The error matrices E and −E are compatible to disjoint sets of data matrices, i.e., ΨK (E) ∩ ΨK (−E) = ∅, but lead to the same distance and weight profile and therefore to the same lower bound, Pb,genie1 . By making use of this insight, the genie-aided receiver can be improved as follows. Before revealing X to the receiver in case of X ∈ / ΨK (E), the genie verifies if X ∈ ΨK (−E). If it is, the genie tells the receiver that the correct X is in the set {X, X + 2E}. The BEP of this second genie-aided receiver is d2 (E) Es · Pb,genie2 (E) = w (e0 ) · 2−w(E) · erfc 4 N0 ≤ Pb,JMLSE . (20) The inequality in (20) holds for any E ∈ ΦK . Nevertheless, it is desirable to tighten the bound by picking an error matrix Emax that maximizes Pb,genie2 : (21) Emax = arg max Pb,genie2 (E) . E∈ΦK
The corresponding bit error probability PLB = Pb,genie2 (Emax ) ≤ Pb,JMLSE
(22)
is the tightest lower bound that can be attained by the preceding considerations. For a given data sequence length K, the lower bound in (22) requires an exhaustive search over the set ΦK . As the cardinality of ΦK grows exponentially with K, the computational complexity of such an exhaustive search is prohibitive for large K. Alternatively, the maximization can be done on a subset of ΦK . The resulting lower bound is termed simplified lower bound. A straightforward way to choose a suitable subset is to restrict the length of the considered error sequences. Let K be an error sequence length such that 1 ≤ K ≤ K. Then a simplified lower bound of the minimum error probability is obtained as (23) PLB,K = max Pb,genie2 (E) . E∈ΦK
A simple example of a suitable subset of ΦK is Φ1 . The corresponding simplified lower bound can be displayed in the
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following form:
max Pb,genie2 (E) L 1 2 Es erfc |hl | , = max 2 N0 l=0 L 1 E s 2 erfc |hl ± gl | 4 N0 =
E∈Φ1
(24)
l=0
It can be clearly seen that the optimum lower bound given a subset of error matrices is not necessarily related to the minimum distance of the subset. This is due to the different factors 2−w(E) which take into account that different error matrices have different numbers of compatible data matrices X. VI. A NALYTICAL AND S IMULATION R ESULTS The performance of the proposed semi-blind receiver has been investigated by means of Monte Carlo simulations, which are supported by the analytical lower bound. The GSM/GPRS system was chosen to exemplify the performance of the proposed receiver. According to the GSM 05.05 test specifications, one dominating interferer (J = 1) modeled by a continuous, GMSK-modulated random sequence is assumed. (Note that according to these severe test conditions the dominating interferer has no burst structure which may be exploited.) In the numerical examples, emphasis is on the GSM 05.05 typical urban (TU) channel model, because urban environments are most likely to be affected by cochannel interference. Frequency hopping is done from burst to burst, which is the worst case with respect to channel estimation. Only one transmit and one receive antenna are used. An analog root-raised cosine receive filter in conjunction with baud-rate sampling is used. Since the effective channel memory length is about L = 3, 2L = 8 states are considered in the conventional receiver and 22L = 64 states are considered in the joint trellis. A signal/noise ratio (SNR) of Es /N0 = 20 dB is assumed. In Fig. 2, the average raw bit error rate (BER) of the desired user is plotted versus the signal/interference ratio (SIR) for the case of perfect channel knowledge. The top curve represents the conventional GSM receiver ignoring CCI. The next curve represents the performance for JVD. A comparison between these curves demonstrates that for low SIRs the raw BER significantly improves if CCI is compensated at the receiver. The increased BER for low SIR values is due to residual ISI from the interfering user. The simulation result for cancelled residual ISI does not show this effect. Note that the simplified lower bound virtually delivers the same result than JVD without residual ISI for K = 5. For JVD/JMLSE it is interesting to note that a local BER maximum exists, which is confirmed by the lower bound. In a first-order approximation, the worst-case SIR is about the same as the average SNR, because in this case noise and interferer are difficult to distinguish. In Fig. 3, corresponding results are plotted for the case of real channel estimation given the TU50 channel model. The top curve represents the conventional GSM receiver (LSCE/VD) ignoring CCI. The next two curves illustrate the
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performance of the proposed semi-blind receiver with n = 2 and n = 6 iterations, respectively. The step size of the LMS algorithm was changed once per iteration and was chosen to be in the range from ∆ = 0.04 to ∆ = 0.01; the first iteration started with the highest value. The tentative decision delay was chosen to be δ = 1. The average BER improves with the number of iterations. Additional simulations indicate that more than six iterations do not lead to further improvements. Finally, the bottom line features the performance for JLSCE/JVD in a synchronous environment. The simulation results show that the proposed receiver significantly outperforms the conventional GSM receiver. For SIRs as low as 5 dB, the raw BER performance is comparable to JLSCE/JVD if six iterations are applied, even though JLSCE/JVD requires synchronized training sequences. This result indicates that synchronous GSM/GPRS networks are not necessary if proper interference cancellation is done. For SIRs as low as 0 dB (where the average signal power of the interferer is equal to the average signal power of the desired user), the raw BER is below 10−2 . The corresponding gain with respect to the conventional receiver is about 16 dB. 0
Bit Error Rate of Desired User
10
Conventional receiver (VD) Opt. receiver (JVD), sync. network Opt. receiver (JVD), sync. network, no residual ISI Simplified lower bound (K’=5) Simplified lower bound (K’=1)
-1
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VII. C ONCLUSIONS An iterative receiver combating CCI and ISI is presented. Data and channel estimation is done jointly. The proposed receiver is semi-blind in the sense that the training sequence of the desired user is assumed to be known, but the training sequences of the J interferer are assumed to be unknown. Besides burst acquisition, tracking of slowly time-varying channels is performed. Processing is done burst-by-burst rather than exploiting long-term statistics. The receiver can be implemented in any – even existing – TDMA radio system. It is downward compatible to conventional TDMA receivers. The branch metric can easily be extended to take antenna diversity into account. However, unlike concurrent receivers, antenna diversity is not mandatory. Therefore, the proposed receiver is suitable both for uplink and downlink processing. Furthermore, a universal approach to lower-bound the BEP of single-antenna JMLSE in the presence of CCI and ISI is introduced. Simulation results are presented for the popular GSM/GPRS system. It is shown that in spite of its simplicity, the proposed receiver is capable to achieve an excellent bit error performance. For the example of a typical urban channel and an SNR of Es /N0 = 20 dB, an average raw BER of 10−2 can be maintained even at a SIR of C/I = 0 dB (!) if one dominant interferer is present. The simulation results were interpreted and confirmed by the analytical lower bound. A modification to accept a priori information and to deliver soft outputs [8] is straightforward. The proposed receiver is also suitable for the case that training symbols of the interfering users are available. In this case, the performance will improve further in the sense that less (or even no) iterations are required. Unlike JLSCE, the proposed receiver does not need overlapping training sequences and can therefore be used with arbitrary large shifts of the training sequences in asynchronous networks. Finally, it is argued that synchronous networks are not necessary if proper processing is done at the receiver.
-4
10 -10.0
-5.0
0.0
5.0 20.0 10.0 15.0 Signal-to-Interference Ratio (dB)
25.0
30.0
Fig. 2. BER versus SIR, TU0 channel model, perfect channel knowledge, Es /N0 = 20 dB.
0
Bit Error Rate of Desired User
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10
16 dB (!)
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Conventional receiver (LSCE/VD) Semi-blind CE/JVD (n=2) Semi-blind CE/JVD (n=6) Opt. receiver (JLSCE/JVD), sync. network
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R EFERENCES [1] J.-T. Chen, J.-W. Liang, H.-S. Tsai, and Y.-K. Chen, “Low-complexity joint MLSE receiver in the presence of CCI,” IEEE Commun. Letters, vol. 2, pp. 125-127, May 1998. [2] X.-M.Chen and P.A.Hoeher, “Blind equalization with iterative joint channel and data estimation for wireless DPSK systems,” in Proc. IEEE GLOBECOM ’01, San Antonio, Texas, USA, pp. 274-279, Nov. 2001. [3] A. Duel-Hallen and C. Heegard, “Delayed decision-feedback sequence estimation,” IEEE Trans. Commun., vol. 37, no. 5, pp. 428-436, May 1989. [4] M.V. Eyuboglu and S.U. Qureshi, “Reduced-state sequence estimation with set partitioning and decision feedback,” IEEE Trans. Commun., vol. 36, no. 1, pp. 13-20, Jan. 1988. [5] K. Giridhar, J.J. Shynk, A. Mathur, S. Chari, and R.P. Gooch, “Nonlinear techniques for the joint estimation of cochannel signals,” IEEE Trans. Commun., vol. 45, no. 4, pp. 473-484, Apr. 1997. [6] B. Lo and K. Letaief, “Adaptive equalization and interference cancellation for wireless communication systems,” IEEE Trans. Commun., vol. 47, no. 4, pp. 538-545, Apr. 1999. [7] P.A. Ranta, A. Hottinen, and Z.-C. Honkasalo, “Co-channel interference cancellation receiver for TDMA mobile systems,” in Proc. IEEE ICC ’95, Seattle, pp. 17-21, June 1995. [8] P. Robertson, P. Hoeher, and E. Villebrun, “Optimal and sub-optimal maximum a posteriori algorithms suitable for turbo decoding,” Europ. Trans. on Telecommun., pp. 119-125, Mar./Apr. 1997. [9] S. Verdu, Multiuser Detection. Cambridge University Press, 1998.
Fig. 3. BER versus SIR, TU50 channel model, real channel estimation, Es /N0 = 20 dB.
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