AbstractâThis contribution deals with symbol rate detection algorithms for multi-rate receivers. From the maximum likeli- hood criterion, we derive two blind rate ...
ML-based blind symbol rate detection for multi-rate receivers Henk Wymeersch and Marc Moeneclaey Digital Communications Research Group Dept. of Telecommunications and Information Processing Ghent University, Sint-Pietersnieuwstraat 41, 9000 GENT, BELGIUM E-mail: {hwymeers,mm}@telin.UGent.be Abstract— This contribution deals with symbol rate detection algorithms for multi-rate receivers. From the maximum likelihood criterion, we derive two blind rate detection algorithms: the first follows from a low SNR approximation of the likelihood function, while the second makes use of the expectation-maximization (EM) algorithm. We compare the proposed detectors with a cyclic correlation-based scheme from literature in terms of their rate detection error probability. The proposed ML-based algorithms turn out to substantially outperform the correlationbased algorithm.
I. I NTRODUCTION In multi-rate systems with burst transmission, the symbol rate is constant within a frame, but can vary from one frame to the next. The transmitter can decide to change the symbol rate depending on various criteria, such as the channel conditions, dynamic bandwidth constraints, content type, etc [1]. The receiver can obtain the value of the symbol rate in two ways: either this value is sent from the transmitter to the receiver through a separate channel, or the receiver (which knows the possible values of the symbol rate), estimates the symbol rate, based on the incoming packet. The latter strategy is known as rate detection (RD). Clearly, the former strategy leads to some overhead in terms of bandwidth, and requires the presence of a separate channel. Hence, in some scenarios, rate detection is preferred. The problem of blind rate detection has already been treated extensively for DS-CDMA systems for 2nd and 3rd generation wireless devices (see [2]–[4] and references therein). In DS-CDMA the symbol rate is changed by fixing the chip rate and varying the length of the spreading codes (i.e., the number of chips per symbol). However, the considered rate detection algorithms are very application-specific, are often developed specifically for BPSK, and, more importantly, are derived under the assumption that all channel parameters (such as channel gains, propagation delays etc.) are known. Apart from the abovementioned work for DS-CDMA, most blind rate detectors exploit the cyclostationarity of the received signal [5], [6]. Although such detectors have some very attractive properties, they fail to operate properly when the excess bandwidth decreases or when the SNR is not sufficiently high. Since the data will generally be protected by an errorcorrecting code, a low SNR operating point can be assumed, so that these algorithms are no longer suitable. Another type of
symbol rate detector was proposed in [7]: the received signal was filtered using an analog filter bank. Through an ad-hoc criterion, the most likely signal bandwidth was determined. The authors reported an estimation accuracy of 99.5% (i.e., a rate detection error probability of 0.005). In this paper, we will investigate the problem of symbol rate detection for linear modulation in the presence of unknown channel parameters. We propose two algorithms: one based on a low SNR approximation of the maximum likelihood (ML) criterion and one based on the Expectation-Maximization (EM) algorithm. These algorithms will be compared to a cyclic correlation-based approach from literature in terms of their rate error probability performance. Additionally, the impact of the proposed algorithms on the overall BER performance will be discussed. Through computer simulations, we will evaluate the BER performance of the considered RD algorithms, applied to a turbo code [8]. II. S YSTEM M ODEL The transmitter forms packets as follows: a (possibly coded) bit sequence is mapped onto a sequence of K data symbols1 taken from some constellation Ω. The transmitted (baseband) signal is given by sT (t) =
p
Es
K−1 X k=0
ak pT (t − kT )
(1)
where Es denotes the energy per transmitted symbol, a = [a0 , . . . , aK−1 ] is the vector of data symbols and pT (t) is the transmit pulse corresponding to symbol rate 1/T . We assume that pT (t) is a Rsquare-root Nyquist pulse, i.e. gT (kT ) = δk where gT (t) = p∗T (−u) pT (t − u) du. The symbol interval T belongs to a finite set of equiprobable values: T ∈ S = {Tmin , . . . , Tmax }. The receiver is assumed to know the set S. For a flat, quasi-static channel, the complex envelope of the received signal can be expressed as r (t) = hsT (t − τ ) + n (t)
(2)
1 K is assumed to be independent of the symbol rate 1/T . The proposed algorithms are easily adapted if this is not the case.
with
EST
p∗T1 (−t)
COR
p∗T2 (−t)
r(t) filter
COR
ADC EST
p∗T|S| (−t) (a)
selection
EST
other DSP
(c)
COR
where h = A exp (jθ) denotes the complex channel gain, τ the propagation delay and n (t) a complex AWGN process with spectral density N0 . We model the phase θ as uniformly distributed in (0, 2π); the probability density function of the magnitude is arbitrary (e.g., in case of fading, a Rayleigh distribution is appropriate), while τ is uniformly distributed in [−∆, +∆]. We consider a fully digital receiver whereby the signal r (t) is band-limited through analog filtering and sampled at a fixed rate 1/Ts : (3)
with E [n (kTs ) n (lTs )] = N0 Ts δk−l . The main goal of the receiver is to recover the data symbols. In order to do this, the receiver requires reliable estimates of h, τ , and T . A generic multi-rate receiver is shown in Fig. 1. Its main blocks are an Analog-to-Digital Convertor (ADC), a matched filter bank (i.e., one matched filter per symbol rate), parameter estimators and correctors. The parameter estimators (denoted by EST) estimate τ and h, while the parameter correctors (denoted by COR), perform compensation for carrier phase, amplitude and delay. The latter operation, which includes timing correction and sample rate conversion, is performed by a digital interpolator. The cyclic correlationbased algorithm from [5] performs rate detection in front of the matched filter bank and then the branch corresponding to the estimated rate is selected. Although such an approach has the advantage of very low complexity, we will show that it leads to large degradations in low SNR regimes. ∗
where C and do not depend on T 2 , and √ P C ∗ yT (t) = Ts k r (kTs ) pT (t − kTs ) / Es . The quantities yT (kT + τ ) are obtained by applying the received signal to a (digital) filter, matched to the transmit pulse, corresponding to symbol rate 1/T . Unfortunately, averaging in (5) w.r.t. the unknown data symbols is generally intractable. We propose two rate detection algorithms that avoid this averaging. But first, we briefly describe the rate detection algorithm from [5]. A. Cyclic correlation based algorithm In order to eliminate excess noise, we first apply the signal samples r (kTs ) to a low-pass filter. This results in a sequence of N samples, denoted by y. We then define, for some Υ ∈ N: y2 (n) rN (T )
. = [y (n − Υ) y ∗ (n) , . . . , y (n + Υ) y ∗ (n)] N −1 1 X . y2 (n) e−j2πnTs /T = N n=0
while the final rate detection algorithm is given by 2 Tˆ = arg max |rN (T )| . T
This corresponds to performing rate detection at point (a) in Fig. 1. B. Low-SNR method Here we expand p (r |a, τ, h, T ) in a Taylor series. Neglecting terms beyond second order and assuming the data symbols to be pairwise uncorrelated3 , we obtain (see Appendix): p (r |τ, T ) ∝
K−1 X k=0
Averaging over τ yields: K−1 XZ p (r |T ) ∝
2
We denote by r the expansion of r (t) onto a suitable basis. The ML estimate of the symbol interval is obtained by maximizing the likelihood function [9]: (4)
=
Z
−∆
(6)
|yT (kT + τ )| .
+∆
2
|yT (kT + τ )| dτ
k=0 −∆ (K−1)T +∆
III. S YMBOL RATE DETECTION
T ∈S
where Ea,τ,h [.] denotes the expectation w.r.t. all possible data sequences, τ and h. Taking into account the AWGN noise, we can write p (r |a, τ, h, T ) ! 1 X 2 = C exp − |r (kTs ) − hsT (kTs − τ )| N0 k ( K−1 )! X 2Es 0 ∗ ∗ = C exp < h ak yT (kT + τ ) N0 k=0
Fig. 1. Multi-rate receiver consisting of a bank of matched filters parameter estimation (EST) and correction (COR)
TˆM L = arg max p (r |T )
(5)
0
(b)
r (kTs ) = hsT (kTs − τ ) + n (kTs )
p (r |T ) = Ea,τ,h [p (r |a, τ, h, T )]
2
wT (u) |yT (u)| du
(7)
2 Note that dependence of C and C 0 on h, τ or a is irrelevant for the maximization in (4). 3 This holds not only for uncoded transmission, but also for many types of coded transmission.
where wT (u) is a window function that depends on K, ∆ and T . Assuming KT ≥ 2∆, wT (u) is constant and proportional to 1/T in the interval ∆ ≤ u ≤ KT − ∆. For ∆/T K, we can approximate (7) with Z 1 (K−1)T +∆ 2 p (r |T ) ∝ |yT (u)| du T −∆ ≈
Ts T
d((K−1)T +∆)/Ts e
X
k=0
2
|yT (kTs )| .
(8)
Substituting (8) into (4) yields the final symbol rate detection algorithm. This algorithm can be interpreted as selecting the branch in the matched filter bank that yields the largest output power. It is important to note that this algorithm requires no knowledge of τ or h. However, it is computationally more complex than the cyclic correlation approach from [5], since now the incoming signal has to be filtered by each of the matched filters. Only after considering the outputs of the matched filter bank, a decision is made w.r.t. the symbol rate. This corresponds to performing rate detection at point (b) in Fig. 1. C. EM approach The Expectation-Maximization (EM) algorithm [10] is a method that iteratively solves the problem of ML estimation of a parameter b. The algorithm makes use of the so-called complete data z. The complete data is related to the observation r through some many-to-one mapping r = g (z). The EM algorithm at iteration k consists of two steps: the E-step Q b| ˆb(k) =
and the M-step
h i Ez log p ( z| b) r; ˆb(k)
ˆb(k+1) = arg max Q b| ˆb(k) . b
(9)
(10)
Note that the EM algorithm requires an initial estimate (ˆb(0) ) of b. When b can take on values in a discrete set S only, the EM algorithms suffers from convergence problems [11]. To avoid this, we propose the following solution [12] ˆb = arg max Q ( b| b) b∈S
(11)
which has been shown to have very good performance in other detection problems [13] and does not require an initial estimate of b. Note that (11) can be applied only when S is finite. Application of the EM algorithm to the rate detection problem requires the following steps: 1) The received signal is applied to the matched filter bank, and to each filter output, an algorithm for estimating τ and h is applied. This estimation can be performed by means of a classical algorithm (see [9], [14]) or by means of a more sophisticated EM algorithm (see [15]– [17]). Hence, for each branch in the matched filter bank there are corresponding estimates of τ and h
2) We then apply the EM algorithm to determine b = T . Selecting as complete data z = [r, a], it can be verified that [15]: ( K−1 ) h i X ∗ ∗ ˆ ˆ Q (T| T) = < h Ea a | T, r, τˆ, h yT (kT + τˆ) k
k=0
(12) ˆ are estimates of the propagation delay where τˆ and h and the channel gain, respectively. h i ˆ can The a posteriori symbol expectations Ea a∗k | T, r, τˆ, h be interpreted as soft symbol decisions; they are weighted averages of all possible constellation points: h i X ˆ . (13) ˆ = a∗ p ak = a| T, r, τˆ, h Ea a∗k | T, r, τˆ, h a∈Ω
If we can obtain the marginal a posteriori symbol ˆ , the symbol rate probabilities p ak = a| T, r, τˆ, h estimate can be easily computed. In the case of uncoded transmission, computation of the a posteriori ˆ symbol probabilities is trivial: p ak = a| T, r, τˆ, h = 2 √ ˆ Es a γ exp − N10 yT (kT + τˆ) − h with γ denoting a
normalizing constant. In the case of coded transmission, the posterior symbol probabilities can be computed using a MAP detector [18], [19]. As the latter approach will drastically increase the overall computational complexity, it is preferred to treat the data symbols as uncoded during the rate detection process. Clearly, the EM algorithm is more complex than the lowSNR method proposed in the previous section. As an additional disadvantage, it requires estimates of both τ and h. However, we will show that the EM-based algorithm has superior performance. This EM-based approach corresponds to performing rate detection at point (c) in Fig. 1. D. Effect of rate detection errors on BER performance
The performance measure we consider is the RD error probability (RDEP), i.e., the fraction of frames for which the symbol rate is not correctly detected. As a detection error results in the loss of an entire frame, the RDEP should be sufficiently low. More specifically, if we denote the BER of a (coded or uncoded) system under perfect symbol rate detection by BER0 , then the BER in the presence of occasional RD errors is upper-bounded by BERRD
< ≈
BER0 (1 − RDEP ) + 1 × RDEP RDEP BER0 1 + . BER0
Consequently, in order to obtain a low BER degradation due to RD, the ratio RDEP/BER0 should be below 1, in which case we obtain BERRD < 2BER0 . A similar reasoning can be applied to frame error rate performance.
K=128
0
10
0
K=512
0
10
10
−1
10 −1
−1
10
10
−2
10 −2
−2
10
10
−3
−3
10
−4
BER
RDEP
RDEP
10 −3
10
−4
10
10 cyclostationary low SNR EM−NDA−est.(τ,θ) EM−NDA−perfect(τ,θ) EM−DA−est.(τ,θ) EM−DA−perfect(τ,θ)
−5
10
−5
10
−6
10 −10
−6 −4 E /N [dB] s
Fig. 2.
−2
0
−5
cyclostationary low SNR EM−NDA−est.(τ,θ) EM−NDA−perfect(τ,θ) EM−DA−est.(τ,θ)
−6
10
10 −10
−8
0
−6 −4 E /N [dB]
−7
10 −10
s
−2
−9
0
−8
−7
−6
−5 E /N [dB] s
−4
−3
−2
−1
0
0
0
Rate detection error rate vs. SNR for K = 128 and K = 512
IV. N UMERICAL RESULTS We have carried out computer simulations for BPSK 4 transmission, with pT (t) a square-root cosine roll-off pulse with roll-off α = 0.5. We set S = {4Ts , 8Ts , 16Ts } and consider frame lengths K=128 and 512. The propagation delay τ and the carrier phase θ will be estimated using conventional algorithms from [20] and [21], respectively. We assume A = 1 and A is known to the receiver. In Fig. 2 we show the RDEP performance of three RD algorithms: the cyclic correlation-based approach from [5] (outlined in section III-A), the low-SNR method from section III-B and the EM-based approach from section III-C. In the EM-based approach we discern four subcases. The propagation delay τ and the carrier phase θ are assumed to be known at the receiver or are estimated using conventional algorithms from [20] and [21], respectively. These cases are denoted by perf ect(τ, θ) and est.(τ, θ) in Fig. 2, respectively. On the other hand, the data symbols a can assumed to be known at the receiver (to be denoted EM-DA, for Data-Aided) or unknown (to be denoted EM-NDA, for Non-DA). Simulations for a given RD algorithm were halted after at least 100 rate detection errors. We observe that the cyclic correlation-based approach exhibits very poor performance in the considered SNR range. This is due to two factors: for low rates, excess noise will degrade the performance, while for high rates, the number of samples per symbol interval is too low to accurately detect T . Both proposed ML-based algorithms achieve fairly good performance, with the EM-NDA algorithm outperforming the low-SNR algorithm with about one order of magnitude. The performance of the NDA EM-based algorithm can be further improved by around 1 dB by applying more advanced estimation algorithms for τ and θ (see EM-NDA-perfect(τ, θ)). On the other hand, an improvement of around 3.5 dB is 4 Of
0
cyclic correlation low SNR EM−NDA−est.(τ,θ) EM−NDA−perfect(τ,θ)
10
−6
−8
BER −4
10
course, the algorithms can be applied to any signaling constellation.
Fig. 3.
BER performance for turbo code for K = 128
visible when the data symbols are assumed to be known at the receiver. This gives us some idea of the performance improvements attainable by exploiting code properties during RD. Combining these two (i.e., exploiting the code during RD and using superior algorithms for estimation), can give us a total improvement of around 7 to 8 dB. However, we remind that exploitingcode properties implies that in (13), we should ˆ using a MAP decoder [18]. compute the p ak = a| T, r, τˆ, h The resulting computational complexity may not justify this approach. In Fig. 3, the impact of rate detection on the overall BER can be observed. We consider a rate 1/3 turbo code whereby the constituent convolutional codes are recursive, systematic and separated by a pseudo-random interleaver. The codes have generator polynomials (21, 37)8 and constraint length 5. For K = 512 (results not shown), we have verified that both the low-SNR and EM-based algorithm yield excellent performance. For K = 128, the situation is somewhat different: the low-SNR method gives rise to a BER degradation of around 2 dB. Application of the EM-based algorithm reduces this degradation to around 1 dB. For the sake of illustration, we have included the BER performance of the EM-based method when τ and θ are perfectly known at the receiver. The resulting degradation is around 0.5 dB. This means that by applying superior propagation delay and carrier phase estimation algorithms, a small reduction in BER degradation is possible. In order to reduce the degradation even further, one must consider code-aided rate detection algorithms. V. C ONCLUSION We have proposed two symbol rate detection algorithms. One is derived from a low-SNR approximation of the likelihood function, while the other is based on the EM algorithm. The latter has the higher complexity as it requires a posteriori probabilities of the data symbols and estimates of all channel parameters. On the other hand, the EM-based detector achieves
superior performance. We have shown that both proposed algorithms perform fairly well at the low-SNR operating point of powerful codes, whereas a cyclic correlation-based detector from literature is totally unreliable at low SNR. The proposed algorithms can be straightforwardly modified to other setups (such as MIMO systems, DS-CDMA systems etc.) and extended to exploit code properties during rate detection (as in [2], for CDMA). A PPENDIX Low-SNR method derivation We start from p (r |T ) = Ea,τ,h [p (r |a, τ, h, T )] with (up to an irrelevant multiplicative constant): (K−1 )! X 2Es p (r |a, τ, h, T ) = exp < zk N0 k=0
. where zk = h∗ a∗k yT (kT + τ ). Expansion in a Taylor series yields (K−1 ) X 2Es p (r |a, τ, h, T ) ≈ 1 + < (14) zk N0 k=0 (K−1 )!2 2 X 1 2Es + zk < 2 N0 k=0
The first term in (14) is independent of T and can be dropped. The second term is linear in ak , so that, since E [ak ] = 0, Ea [zk ] = 0. Within irrelevant constants, this results in (K−1 )!2 X zk p (r |τ, h, T ) = Ea < = =
Ea
k=0
1X
2
1X 2
k
k,k0
(zk + zk∗ ) (zk0 + zk∗0 )
h i 2 Ea (zk + zk∗ )
(15)
since Ea [zk zk∗0 ] h= 0 and Eai [zk zk0 ] = 0 when k 6= k 0 . We 2 now evaluate Ea (zk + zk∗ ) , abbreviating yT (kT + τ ) with yk : h i 2 Ea (zk + zk∗ ) = h i h i 2 2 2 2 2 2 h2 (yk∗ ) Ea a2k + (h∗ ) yk2 Ea (a∗k ) + 2 |h| |yk | Ea |ak | i h 2 = 1, Ea a2k = It can easily been seen that Ea |ak | i h 2 Ea (a∗k ) = C where C = 0 for complex constellations and C = 1 for real constellations. Additionally, since 2 h = Aejθ with θ uniformly distributed in (0, 2π), E h = θ h i ∗ 2 = 0, so that averaging over θ of (15) yields Eθ (h ) (irrespective of the distribution of A): p (r |τ, T ) ∝
X k
2
|yT (kT + τ )| .
ACKNOWLEDGEMENT This work has been supported by the Interuniversity Attraction Poles Program P5/11- Belgian Science Policy. R EFERENCES [1] A.J. Goldsmith and S. Chua. ”Adaptive coded modulation for fading channels”. IEEE Trans. on Comm., 46(5):595–602, May 1998. [2] S. Buzzi and A. De Maio. ”Code-aided blind rate detection for multirate DS/CDMA”. In Proc. IEEE 7th International Symposium on Spread Spectrum Techniques and Applications, pages 19–23, vol. 1, 2002. [3] A. Sharma and U. Mitra. ”Blind rate detection for multirate UMTS DS-CDMA signals”. In Proc. IEEE International Conference on Communications (ICC’01), pages 2504–2509, vol. 8, Helsinki, Finland, June 2001. [4] A. Gutierrez, Y.K. Lee and G. Mandyam. ”A ML rate detection algorithm for IS-95 CDMA”. In IEEE Vehicular Technology Conference (VTC Spring’99), pages 417–421, vol. 1, May 1999. [5] P. Ciblat, P. Loubaton, E. Serpedin and G.B. Giannakis. ”Asymptotic Analysis of blind cyclic correlation-based symbol-rate estimators”. IEEE Trans. on Information Theory, 48(7):1922–1934, July 2002. [6] A.V. Dandawat´e and G.B. Giannakis. ”Statistical tests for presence of cyclostationarity”. IEEE Trans. on Signal Processing, 42:2355–2369, September 1994. [7] Jong Youl Lee, Young Mo Chung, and Sang Uk Lee. ”On a timing recovery technique for a variable symbol rate signal”. In Proc. IEEE Vehicular Technology Conference, pages 1724–1728, Phoenix, AZ, USA, May 1997. [8] C. Berrou, A. Glavieux and P. Thitimajsima. ”Near Shannon limit errorcorrecting coding and decoding”. In Proc. IEEE International Conference on Communications (ICC), pages 1064–1070, Geneva, Switzerland, May 1993. [9] U. Mengali and A.N. D’Andrea. ”Synchronization Techniques for Digital Receivers”. Plenum Press, 1997. [10] A.P. Dempster, N.M. Laird and D.B. Rubin. ”Maximum likelihood from incomplete data via the EM algorithm”. Journal of the Royal Statistical Society, 39(1):pp. 1–38, 1977. Series B. [11] P. Spasojevic and C.N. Georghiades. ”On the (non) convergence of the EM algorithm for discrete parameter estimation”. In Proc. Allerton Conference, Monticello, Illinois, Oct. 2000. [12] H. Wymeersch and M. Moeneclaey. ”Code-aided frame synchronizers for AWGN channels”. In Proc. International Symposium on Turbo Codes & related topics, Brest, France, September 2003. [13] H. Wymeersch and M. Moeneclaey. ”Code-aided phase and timing ambiguity resolution for AWGN channels”. In Proceedings IASTED Intl. Conf. on Acoustics, Signal and Image Processing (SIP03), Honolulu, Hawaii, August 2003. [14] H. Meyr, M. Moeneclaey, S.A. Fechtel. ”Synchronization, Channel Estimation, and Signal Processing”, volume 2 of ”Digital Communication Receivers”. John Wiley & Sons, 1997. [15] N. Noels, C. Herzet, A. Dejonghe, V. Lottici, H. Steendam, M. Moeneclaey, M. Luise and L. Vandendorpe. ”Turbo-synchronization: an EM algorithm approach”. In Proc. IEEE International Conference on Communications (ICC), Anchorage, May 2003. [16] C. Herzet, V. Ramon, L. Vandendorpe and M. Moeneclaey. ”EM algorithm-based timing synchronization in turbo receivers”. In Proc. International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Hong Kong, April 2003. [17] M. Guenach, H. Wymeersch and M. Moeneclaey. ”Joint estimation of path delay and complex gain for coded systems using the EM algorithm”. In Proc. International Z¨urich Seminar (IZS’04), Z¨urich, Switserland, February 2004. [18] L.R. Bahl, J. Cocke, F. Jelinek and J. Raviv. ”Optimal decoding of linear codes for minimising symbol error rate”. IEEE Trans. Inform. Theory, 20:pp. 284–287, March 1974. [19] S. ten Brink, J. Speidel and J. C. Yan. ”Iterative demapping and decoding for multilevel modulation”. In IEEE GLOBECOM’98, 1998. [20] M. Oerder and H. Meyr. ”Digital filter and square timing recovery”. IEEE Trans. on Comm., 36:605–611, May 1988. [21] A.J. Viterbi and A.M. Viterbi. ”Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission”. IEEE Trans. Inform. Theory, IT-29:pp. 543–551, July 1983.
