- Email: [email protected]
Frequency offset estimation for GSM and EDGE
Digital Signal Processing 17 (2007) 311–318 www.elsevier.com/locate/dsp
Frequency offset estimation for GSM and EDGE Jan C. Olivier ∗ Department of Electrical, Electronic and Computer Engineering, University of Pretoria, Lynnwood Road, Pretoria, South Africa Available online 7 August 2006
Abstract A low complexity method for estimating the frequency offset in global system mobile (GSM) and enhanced data rates for GSM evolution (EDGE) is proposed. The method is based on minimising a suitable least squares (LS) metric utilising both the pilot (training) and tail symbols available in GSM and EDGE. It is shown that the tail symbols are important in estimating frequency offset even though they are few relative to the pilot (training) symbols. Via computer simulation it is shown that overall performance compares favourably to standard methods for estimating the frequency offset in GSM such as the method by Luise and Regiannini (L&R). © 2006 Elsevier Inc. All rights reserved. Keywords: Frequency offset; Estimation; Least squares approximation; Maximum likelihood
1. Introduction Due to hardware impairments and relative motion between the basestation and the mobile station (MS), a frequency offset exists between the transmitter and receiver. This offset needs to be estimated in the receiver and then compensated for if a serious degradation in performance is to be avoided. The estimation of frequency offset in general is a mature topic that has been studied extensively over a long period of time [1]. For discrete time observations the optimal maximum likelihood (ML) estimator [2] has been considered but it requires prohibitively large computational resources, which is impractical especially in the MS where a low complexity implementation suitable for a digital signal processor (DSP) is desired. Consequently many researchers considered instead suboptimal procedures with low complexity. Among these, linear regression [3], linear prediction [4], and autoregressive models [5,6] have been studied. Burst mode transmission is common in wireless communication systems such as global system mobile (GSM) [7]. Each GSM burst contains a short known pilot (training) sequence of 26 symbols as a midamble, flanked by 58 data symbols on either side, while a small number of tail symbols terminate the burst on both sides. Effective and low complexity methods for frequency offset estimation have been developed for flat fading [8–10], but since the radio channel in GSM is a frequency selective fading one, those methods are not directly applicable to GSM. A frequency offset estimator based on the pilot sequence in GSM was proposed by Luise and Regiannini (L&R) [11]. In [11] numerical results are provided for the frequency offset estimation in GSM for slow mobiles and promising results were so obtained. These are used as a benchmark to compare the results of this paper to. * Fax: +27 12 362 5000.
E-mail address: [email protected]. 1051-2004/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.dsp.2006.07.002
312
J.C. Olivier / Digital Signal Processing 17 (2007) 311–318
Enhanced data rates for GSM evolution (EDGE) was recently introduced [12] and uses the same radio burst format as GSM where the training/pilot symbols are placed in a midamble. The modulation used in EDGE is 8 phase shift keying (8 PSK), while Gaussian minimum shift keying (GMSK) is used in GSM [12,13]. The difference in modulation however does not affect the estimation of the frequency offset based on known training symbols, and hence the work reported in [11] are therefore applicable to EDGE as well. This paper presents and applies a new and low complexity frequency offset estimator to GSM/EDGE which is based on a least squares (LS) minimisation of the maximum likelihood formulation. It does not require any assumption of the correlation of the fading process, nor does it require phase unwrapping. The LS minimisation enables the use of both the training sequence in the midamble and the tail symbols on both ends of the GSM/EDGE burst. The paper will show that the use of the tail symbols in addition to the training symbols is important in estimating the GSM/EDGE frequency offset, even though the tail symbols are few relative to the pilot symbols. Numerical results obtained from computer simulation will be provided to indicate that the new method outperforms that of Luise and Regiannini [11]. The paper is organised as follows. The baseband model and an analytical expression for the new LS frequency offset estimator is derived in Section 2, while results obtained via computer simulations are presented in Section 3. Conclusions are presented in Section 4. 2. The baseband model and the LS frequency offset estimator for GSM/EDGE The frequency offset estimator is located after the prefilter in the receiver chain as indicated in Fig. 1. The prefilter transforms the impulse response to a minimum phase form, with the effect that the leading taps of the impulse response are dominant [14]. A model for a received sample at discrete time n for the burst under consideration (see Fig. 2) and denoted as r[n] is given by r[n] = Aej ωo nT +θ
L
h[k]d[n − k] + ns [n].
(1)
k=0
r[n] is the received symbol at time n, T the sampling interval, ωo and θ denote the frequency offset in terms of an angular frequency and a phase, and the pair ωo , θ is the objective of the frequency offset estimation. A is the sinusoid amplitude (a nuisance parameter), and ns is white Gaussian noise since the prefilter is a whitening filter [14]. d[n] denotes data, pilot (training) or tail symbols depending on the value of n. h denotes the estimated impulse response
Fig. 1. The GSM receiver chain with the frequency offset estimator indicated.
Fig. 2. The radio burst in GSM and EDGE. The evaluation of B[n] for the typical urban channel in GSM/EDGE where the 2 leading taps of the impulse response after the minimum phase prefilter are dominant, is indicated for S = {2, 3, 63, . . . , 87, 148, . . . , 150}.
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with L + 1 taps after the prefilter for a multipath (frequency selective) Raleigh fading channel and is valid for the burst under consideration. 2.1. Resolving inter symbol interference Consider now the GSM/EDGE burst depicted in Fig. 2 where two sets of tail symbols and the pilot/training symbols in the midamble are known, while the rest of the symbols are data symbols that are unknown during frequency offset estimation. Using the known tail and pilot symbols, intersymbol interference of L + 1 taps can be resolved at certain times n where enough pilot and tail symbols are available. This set of values for n is denoted as S and contain N entries. At such time instants, i.e. when n ∈ S, define B[n] as B[n] =
L
h[k]d[n − k]
∀n ∈ S.
(2)
k=0
d[n] in Eq. (2) now denotes either known tail or known pilot symbols since the values of n are restricted to be in S. As an example, Fig. 2 indicates the positions where B[n] can be computed with S = {2, 3, 63, . . . , 87, 148, . . . , 150} for the typical urban channel [12] where the impulse response contains 2 dominant taps after the prefilter. Note that the prefilter requires an estimated impulse response, and hence the impulse response provided to any frequency offset estimator following the prefilter is an estimate of the actual impulse response [11,12]. 2.2. The frequency offset estimator Assuming normally distributed white noise, where N values of B[n] in the set S have been computed, the loglikelihood is given by r[n] − Aej ωo nT +θ B[n]2 . (3) Λ(ωo , θ ) = − n∈S
A ML estimate for the frequency offset pair ωo , θ can be found as [15] r[n] − Aej ωo nT +θ B[n]2 . ωˆ o , θˆ = arg min ωo ,θ
(4)
n∈S
The ML estimate (4) can be found by searching over a grid of values for ωo , θ and efficient search algorithms can be devised for DSP implementation. However, even with such search algorithms the computational complexity required is still prohibitive, especially in the MS where computational resources need to be conserved. Thus a new and low complexity solution for the above minimisation is derived here. First of all it is straightforward to show that a complex quadratic series such as (4) can be minimised without differentiation by equating each term to zero, thereby setting up an over determined matrix equation for ωˆ o , θˆ . As an example, for S = {2, 3, 63, . . . , 87, 148, . . . , 150} depicted in Fig. 2 it can be shown that Eq. (4) leads to r[2] ˆ = ej ωˆ o 2T +θ , AB[2] r[3] ˆ = ej ωˆ o 3T +θ , AB[3] r[63] ˆ = ej ωˆ o 63T +θ , AB[63] .. . r[87] ˆ = ej ωˆ o 87T +θ , AB[87] r[147] ˆ = ej ωˆ o 147T +θ , AB[147] .. . r[150] ˆ = ej ωˆ o 150T +θ . AB[150]
(5)
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After eliminating the unknown sinusoid amplitude A by equating phases in (5) a matrix equation for ωˆ o , θˆ can be found as ⎡ r[2] ⎤ ⎤ ⎡ arg( B[2] ) 2T 1 ⎢ ⎥ r[3] ) ⎥ ⎢ ⎢ arg( B[3] 3T 1⎥ ⎥ ωˆ o ⎢ ⎥=⎢ , (6) .. ⎥ ⎢ ⎥ ⎢ .. ⎦ θˆ ⎣ ⎦ ⎣ . . r[150] 150T 1 ) arg( B[150] where arg(z) denotes the phase of a complex number z. By solving for ωˆ o , θˆ using a LS solution an expression for the offset estimate is given by ⎛⎡ ⎤ ⎡ ⎤⎞−1 ⎡ ⎤ ⎡ arg( r[2] ) ⎤ B[2] 2T 1 T 2T 1 2T 1 T ⎟ ⎢ ⎢ ⎜⎢ r[3] ⎥ ⎥ ⎥ ⎥ ⎢ ⎜⎢ 3T 1 ⎥ ⎢ 3T 1 ⎥⎟ ⎢ 3T 1 ⎥ ⎢ arg( B[3] ) ⎥ ωˆ o ⎟ ⎜ ⎢ ⎥, (7) = ⎜⎢ .. ⎥ ⎢ .. ⎥⎟ ⎢ .. ⎥ ⎢ ⎥ .. ⎦ ⎣ ⎦ ⎦ ⎣ ⎣ θˆ . . . ⎠ ⎝ ⎣ ⎦ . 150T 1 150T 1 150T 1 arg( r[150] ) B[150]
where ( denotes the transpose. Equation (7) represents the estimate ωˆ o , θˆ given S. No phase unwrapping is performed in Eq. (7) in simulation results to follow. Equation (7) yields the following final expression for the frequency offset pair ωˆ o , θˆ : ⎡ ⎤T 0.2423 −727.4998 ⎢ 0.2396 −717.8726 ⎥ ⎢ ⎥ ⎢ 0.0736 −140.2371 ⎥ ⎢ ⎥ ⎢ 0.0709 −130.6098 ⎥ ⎢ ⎥ ⎢ 0.0681 −120.9826 ⎥ ⎢ ⎥ ⎢ 0.0653 −111.3553 ⎥ ⎢ ⎥ ⎢ 0.0626 −101.7280 ⎥ ⎢ ⎥ ⎢ 0.0598 −92.1008 ⎥ ⎢ ⎥ ⎢ 0.0570 −82.4735 ⎥ ⎢ ⎥ ⎢ 0.0543 ⎡ arg( r[2] ) ⎤ −72.8463 ⎥ ⎢ ⎥ B[2] ⎢ 0.0515 ⎥ −63.2190 ⎢ ⎥ ⎢ r[3] ⎥ arg( ⎢ 0.0487 ⎢ −53.5917 ⎥ B[3] ) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0.0460 ⎢ r[63] ⎥ −43.9645 ⎥ )⎥ ⎢ ⎥ ⎢ arg( B[63] ⎢ ⎢ ⎥ −34.3372 ⎥ ⎥ ⎢ ⎥ ⎢ 0.0432 .. ⎢ ⎥ ⎢ ⎥ ωˆ o 0.0404 −34.3372 ⎥ ⎢ . ⎢ ⎥. (8) =⎢ ⎥ ⎢ −34.3372 ⎥ ⎢ arg( r[87] ) ⎥ θˆ ⎢ 0.0377 B[87] ⎥ ⎢ 0.0349 ⎥ ⎢ ⎥ −5.4554 ⎥ ⎢ ⎢ r[148] ⎥ )⎥ ⎢ 0.0321 ⎢ arg( B[148] 4.1718 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0.0294 ⎥ ⎢ ⎥ .. 13.7991 ⎥ ⎣ ⎢ ⎦ . ⎢ 0.0266 ⎥ 13.7991 ⎥ ⎢ r[150] arg( B[150] ) ⎢ 0.0238 33.0536 ⎥ ⎢ ⎥ ⎢ 0.0211 ⎥ 42.6808 ⎥ ⎢ ⎢ 0.0183 42.6808 ⎥ ⎢ ⎥ ⎢ 0.0155 61.9354 ⎥ ⎢ ⎥ ⎢ 0.0128 71.5626 ⎥ ⎢ ⎥ ⎢ 0.0100 81.1899 ⎥ ⎢ ⎥ ⎢ 0.0072 90.8171 ⎥ ⎢ ⎥ ⎢−0.1615 678.0799 ⎥ ⎢ ⎥ ⎣−0.1642 687.7072 ⎦ 697.3344 −0.1670 )T
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Fig. 3. The angular frequency ωˆ o and phase θˆ weights for S = {2, 3, 63, . . . , 87, 148, . . . , 150}.
2.3. The role of the training and the tail symbols for computing ωˆ o and θˆ It is evident that the role of the matrix in Eq. (8) is that of a weight matrix, containing the optimal weights for r[n] combining all the available observations arg( B[n] ) ∀n ∈ S to produce the frequency offset pair estimate ωˆ o and θˆ . It is interesting to inspect the relative absolute values of these weights shown in Fig. 3. These indicate that the observations r[n] ) for n ∈ S close to the burst symmetry point contribute least to the estimate of ωˆ o and θˆ . The tail observations arg( B[n] on the other hand are far away from the burst symmetry point and have the largest weights. Hence even though the tails are few compared to the training symbols, the LS frequency offset estimator indicate they do play an important role in the estimate ωˆ o and θˆ . This seems to suggest that the approach taken in this paper of using both tails and training symbols is well justified. 2.4. Computational complexity of the LS frequency offset estimator The arg( ) operation can be performed efficiently via a look-up table, so that N lookup table operations are required. 2N multiplication and addition operations are needed to multiply the weight matrix and the vector in Eq. (8). However many DSP processors perform a multiply and accumulate (MAC) operation as a single instruction, and under such conditions a DSP implementation of Eq. (8) thus requires N devision and lookup table operations to compute the arg( ) values, and then 2N MAC operations. For the numerical results presented in the next section, N = 30. 3. Numerical results Results obtained via simulation of the LS frequency offset estimator in GSM1 are compared to the results given in [11] (Fig. 10 in that reference) also for GSM. The simulation assumptions used in [11] were applied here as well. First 1 The simulator used to generate the results for the LS frequency offset estimator is open source at http://opensource.ee.up.ac.za.
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Fig. 4. Comparison of performance for the L&R frequency offset estimator and the LS frequency offset estimator in GSM.
of all, frequency hopping is enabled so that the frequency offset in each burst is different. This eliminates the use of averaging techniques over multiple bursts, i.e. the frequency offset for each burst is estimated using information from the burst under consideration only. Secondly perfect symbol timing recovery, a small mobile velocity (3 km/h) and frequency selective Raleigh fading are assumed here. The small mobile velocity makes the assumption of a constant impulse response h over the burst valid, as was the case also in [11]. An estimation of the channel impulse response is performed in the GSM simulator [11,12]. For the LS frequency offset estimator the choice of S is depicted in Fig. 2 and the simulation results are shown in Fig. 4 for the typical urban (TU) channel (frequency selective Raleigh fading). There it is compared to the results from [11]. The LS frequency offset estimator performs better than the L&R algorithm under both noisy and virtually noiseless conditions (Eb /No = 10 dB, and at Eb /No = 200 dB). A second observation is that the LS frequency offset estimator shows a smaller bias than does the L&R frequency offset estimator at Eb /No = 10 dB. Under the very low noise conditions found at Eb /No = 200 dB both estimators shows virtually no bias. The small offsets used in Fig. 4 were intended to study bias and compare the two algorithms. To study the performance of the LS frequency offset estimator at large offsets, offsets up to 900 Hz (corresponding to an oscillator with 1 part per million stability at 900 MHz carrier frequency) were considered and the simulation results are shown in Fig. 5. It will be unusual for a GSM system to experience larger offsets than 900 Hz, as the GSM standard recommends levels of acceptable uncompensated frequency offset of a few hundred Hz [7,11]. Clearly, the LS frequency offset estimator performs well also for large offset. Finally, the performance of the frequency offset estimator is studied for the Rural area (RA) model at 250 km/h. Under these conditions, fading is fast and the impulse response is not stationary over the burst. In the computer simulation no frequency offset due to hardware impairments is added, but the Doppler shift manifests itself as a frequency offset, which is estimated by the offset estimator and then corrected for in the receiver. Figure 6 shows the raw (uncoded) bit error rate (BER) for EDGE with 8 PSK, where it is indicated that the offset estimator is able to recover some of the losses incurred due to the Doppler shift. Since the impulse response is varying over the burst, some unrecoverable losses are incurred in the equaliser, hence there is still loss relative to the RA 3 km/h test case.
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Fig. 5. The performance of the LS frequency offset estimator for large frequency offset in GSM/EDGE.
Fig. 6. The performance of the LS frequency offset estimator for a fast mobile in the RA 250 km/h model, for EDGE with 8 PSK modulation.
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4. Conclusions A low complexity LS estimator for estimating the frequency offset in GSM and EDGE were presented. The formulation makes possible the use of both pilot/training symbols as well as tail symbols available in GSM and EDGE. It was shown that the tail symbols play an important role in estimating the offset, in spite of the fact that they are few relative to the training symbols in GSM/EDGE. Numerical results comparing the performance of the LS estimator to the L&R method indicate the LS frequency offset estimator outperforms the L&R algorithm at realistic SNR, and also in the limit when SNR is large. In computer simulations small offsets were first considered to study the bias of both methods, and it was concluded that the LS frequency offset estimator shows less bias. Subsequently large frequency offsets were also simulated, and results indicated that the LS estimator offers good performance also in that case. References [1] C. Heegard, J.A. Heller, A.J. Viterbi, A microprocessor-based PSK modem for packet transmission over satellite channels, IEEE Trans. Commun. COM-26 (5) (1978) 552–564. [2] S. Kay, Modern Spectral Estimation: Theory and Applications, Prentice Hall, Englewood Cliffs, NY, 1988. [3] S.A. Tretter, Estimating the frequency of a noisy sinusoid by linear regression, IEEE Trans. Inform. Theory IT-31 (6) (1985). [4] S. Kay, A fast and accurate single frequency estimator, IEEE Trans. Acoust. Speech Signal Process. 37 (12) (1989). [5] D.W. Tufts, R. Kumaresan, Estimation of frequencies of multiple sinusoids: Making linear prediction perform like maximum likelihood, Proc. IEEE 70 (9) (1982) 975–989. [6] G.W. Lank, I.S. Reed, G.E. Pollon, A semi-coherent detection and Doppler estimation statistic, IEEE Trans. Aerospace Electron. Syst. AES9 (2) (1973) 151–165. [7] ETSI: GSM 05.05, Radio transmission and reception, ETSI EN 300 910 v8.5.1, November 2000. [8] O. Besson, P. Stoica, On frequency offset estimation for flat-fading channels, IEEE Commun. Lett. 10 (5) (2001) 402–404. [9] W. Kuo, M.P. Fitz, Frequency offset compensation of pilot symbol assisted modulation in frequency flat fading, IEEE Trans. Commun. 45 (1997) 1412–1416. [10] M. Morelli, U. Mengali, G.M. Vitetta, Further results in carrier frequency estimation for transmission over flat fading channels, IEEE Commun. Lett. 2 (1998) 327–330. [11] M. Luise, R. Regiannini, Carrier frequency recovery in all-digital modems for burst-mode transmission, IEEE Trans. Commun. 43 (2–4) (1995) 1169–1178. [12] W. Gerstacker, R. Schober, Equalisation concepts for EDGE, IEEE Trans. Wireless Commun. 1 (2002) 190–199. [13] J.C. Olivier, S.-Y. Leong, C. Xiao, K. Mann, Efficient equalisation and symbol detection for 8-PSK EDGE cellular system, IEEE Trans. Vehicular Tech. 52 (3) (2003) 525–529. [14] N. Al-Dhahir, J.M. Cioffi, MMSE decision-feedback equalisers: Finite-length results, IEEE Trans. Inform. Theory 41 (4) (1995) 961–975. [15] L. Sharf, Statistical Signal Processing, Detection, Estimation and Time Series Analysis, Addison–Wesley, Reading, MA, 1991.
Jan C. Olivier is a Professor of electronics engineering at the University of Pretoria, Pretoria, South Africa. He received the B.Sc. Eng. (electronics), M.Sc. Eng. (electronics), and Ph.D. degrees summa cum laude from the University of Pretoria, South Africa in 1985, 1986, and 1990, respectively. In 1988 he joined the National Institute for Defence Research in Pretoria as a Research Scientist, then joined Northwestern University in 1991, and then the University of Pretoria in 1994 as an Associate Professor. He joined Bell Northern Research in Ottawa, Canada, in 1997, then joined Nokia Research Centre in Dallas, Texas, USA, in 1999, where he worked as a Principle Scientist until 2003 when he rejoined the University of Pretoria as a Full Professor. His research interests are in estimation and detection theory, communications, and numerical analysis.
Frequency offset estimation for GSM and EDGE Jan C. Olivier ∗ Department of Electrical, Electronic and Computer Engineering, University of Pretoria, Lynnwood Road, Pretoria, South Africa Available online 7 August 2006
Abstract A low complexity method for estimating the frequency offset in global system mobile (GSM) and enhanced data rates for GSM evolution (EDGE) is proposed. The method is based on minimising a suitable least squares (LS) metric utilising both the pilot (training) and tail symbols available in GSM and EDGE. It is shown that the tail symbols are important in estimating frequency offset even though they are few relative to the pilot (training) symbols. Via computer simulation it is shown that overall performance compares favourably to standard methods for estimating the frequency offset in GSM such as the method by Luise and Regiannini (L&R). © 2006 Elsevier Inc. All rights reserved. Keywords: Frequency offset; Estimation; Least squares approximation; Maximum likelihood
1. Introduction Due to hardware impairments and relative motion between the basestation and the mobile station (MS), a frequency offset exists between the transmitter and receiver. This offset needs to be estimated in the receiver and then compensated for if a serious degradation in performance is to be avoided. The estimation of frequency offset in general is a mature topic that has been studied extensively over a long period of time [1]. For discrete time observations the optimal maximum likelihood (ML) estimator [2] has been considered but it requires prohibitively large computational resources, which is impractical especially in the MS where a low complexity implementation suitable for a digital signal processor (DSP) is desired. Consequently many researchers considered instead suboptimal procedures with low complexity. Among these, linear regression [3], linear prediction [4], and autoregressive models [5,6] have been studied. Burst mode transmission is common in wireless communication systems such as global system mobile (GSM) [7]. Each GSM burst contains a short known pilot (training) sequence of 26 symbols as a midamble, flanked by 58 data symbols on either side, while a small number of tail symbols terminate the burst on both sides. Effective and low complexity methods for frequency offset estimation have been developed for flat fading [8–10], but since the radio channel in GSM is a frequency selective fading one, those methods are not directly applicable to GSM. A frequency offset estimator based on the pilot sequence in GSM was proposed by Luise and Regiannini (L&R) [11]. In [11] numerical results are provided for the frequency offset estimation in GSM for slow mobiles and promising results were so obtained. These are used as a benchmark to compare the results of this paper to. * Fax: +27 12 362 5000.
E-mail address: [email protected]. 1051-2004/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.dsp.2006.07.002
312
J.C. Olivier / Digital Signal Processing 17 (2007) 311–318
Enhanced data rates for GSM evolution (EDGE) was recently introduced [12] and uses the same radio burst format as GSM where the training/pilot symbols are placed in a midamble. The modulation used in EDGE is 8 phase shift keying (8 PSK), while Gaussian minimum shift keying (GMSK) is used in GSM [12,13]. The difference in modulation however does not affect the estimation of the frequency offset based on known training symbols, and hence the work reported in [11] are therefore applicable to EDGE as well. This paper presents and applies a new and low complexity frequency offset estimator to GSM/EDGE which is based on a least squares (LS) minimisation of the maximum likelihood formulation. It does not require any assumption of the correlation of the fading process, nor does it require phase unwrapping. The LS minimisation enables the use of both the training sequence in the midamble and the tail symbols on both ends of the GSM/EDGE burst. The paper will show that the use of the tail symbols in addition to the training symbols is important in estimating the GSM/EDGE frequency offset, even though the tail symbols are few relative to the pilot symbols. Numerical results obtained from computer simulation will be provided to indicate that the new method outperforms that of Luise and Regiannini [11]. The paper is organised as follows. The baseband model and an analytical expression for the new LS frequency offset estimator is derived in Section 2, while results obtained via computer simulations are presented in Section 3. Conclusions are presented in Section 4. 2. The baseband model and the LS frequency offset estimator for GSM/EDGE The frequency offset estimator is located after the prefilter in the receiver chain as indicated in Fig. 1. The prefilter transforms the impulse response to a minimum phase form, with the effect that the leading taps of the impulse response are dominant [14]. A model for a received sample at discrete time n for the burst under consideration (see Fig. 2) and denoted as r[n] is given by r[n] = Aej ωo nT +θ
L
h[k]d[n − k] + ns [n].
(1)
k=0
r[n] is the received symbol at time n, T the sampling interval, ωo and θ denote the frequency offset in terms of an angular frequency and a phase, and the pair ωo , θ is the objective of the frequency offset estimation. A is the sinusoid amplitude (a nuisance parameter), and ns is white Gaussian noise since the prefilter is a whitening filter [14]. d[n] denotes data, pilot (training) or tail symbols depending on the value of n. h denotes the estimated impulse response
Fig. 1. The GSM receiver chain with the frequency offset estimator indicated.
Fig. 2. The radio burst in GSM and EDGE. The evaluation of B[n] for the typical urban channel in GSM/EDGE where the 2 leading taps of the impulse response after the minimum phase prefilter are dominant, is indicated for S = {2, 3, 63, . . . , 87, 148, . . . , 150}.
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with L + 1 taps after the prefilter for a multipath (frequency selective) Raleigh fading channel and is valid for the burst under consideration. 2.1. Resolving inter symbol interference Consider now the GSM/EDGE burst depicted in Fig. 2 where two sets of tail symbols and the pilot/training symbols in the midamble are known, while the rest of the symbols are data symbols that are unknown during frequency offset estimation. Using the known tail and pilot symbols, intersymbol interference of L + 1 taps can be resolved at certain times n where enough pilot and tail symbols are available. This set of values for n is denoted as S and contain N entries. At such time instants, i.e. when n ∈ S, define B[n] as B[n] =
L
h[k]d[n − k]
∀n ∈ S.
(2)
k=0
d[n] in Eq. (2) now denotes either known tail or known pilot symbols since the values of n are restricted to be in S. As an example, Fig. 2 indicates the positions where B[n] can be computed with S = {2, 3, 63, . . . , 87, 148, . . . , 150} for the typical urban channel [12] where the impulse response contains 2 dominant taps after the prefilter. Note that the prefilter requires an estimated impulse response, and hence the impulse response provided to any frequency offset estimator following the prefilter is an estimate of the actual impulse response [11,12]. 2.2. The frequency offset estimator Assuming normally distributed white noise, where N values of B[n] in the set S have been computed, the loglikelihood is given by r[n] − Aej ωo nT +θ B[n]2 . (3) Λ(ωo , θ ) = − n∈S
A ML estimate for the frequency offset pair ωo , θ can be found as [15] r[n] − Aej ωo nT +θ B[n]2 . ωˆ o , θˆ = arg min ωo ,θ
(4)
n∈S
The ML estimate (4) can be found by searching over a grid of values for ωo , θ and efficient search algorithms can be devised for DSP implementation. However, even with such search algorithms the computational complexity required is still prohibitive, especially in the MS where computational resources need to be conserved. Thus a new and low complexity solution for the above minimisation is derived here. First of all it is straightforward to show that a complex quadratic series such as (4) can be minimised without differentiation by equating each term to zero, thereby setting up an over determined matrix equation for ωˆ o , θˆ . As an example, for S = {2, 3, 63, . . . , 87, 148, . . . , 150} depicted in Fig. 2 it can be shown that Eq. (4) leads to r[2] ˆ = ej ωˆ o 2T +θ , AB[2] r[3] ˆ = ej ωˆ o 3T +θ , AB[3] r[63] ˆ = ej ωˆ o 63T +θ , AB[63] .. . r[87] ˆ = ej ωˆ o 87T +θ , AB[87] r[147] ˆ = ej ωˆ o 147T +θ , AB[147] .. . r[150] ˆ = ej ωˆ o 150T +θ . AB[150]
(5)
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After eliminating the unknown sinusoid amplitude A by equating phases in (5) a matrix equation for ωˆ o , θˆ can be found as ⎡ r[2] ⎤ ⎤ ⎡ arg( B[2] ) 2T 1 ⎢ ⎥ r[3] ) ⎥ ⎢ ⎢ arg( B[3] 3T 1⎥ ⎥ ωˆ o ⎢ ⎥=⎢ , (6) .. ⎥ ⎢ ⎥ ⎢ .. ⎦ θˆ ⎣ ⎦ ⎣ . . r[150] 150T 1 ) arg( B[150] where arg(z) denotes the phase of a complex number z. By solving for ωˆ o , θˆ using a LS solution an expression for the offset estimate is given by ⎛⎡ ⎤ ⎡ ⎤⎞−1 ⎡ ⎤ ⎡ arg( r[2] ) ⎤ B[2] 2T 1 T 2T 1 2T 1 T ⎟ ⎢ ⎢ ⎜⎢ r[3] ⎥ ⎥ ⎥ ⎥ ⎢ ⎜⎢ 3T 1 ⎥ ⎢ 3T 1 ⎥⎟ ⎢ 3T 1 ⎥ ⎢ arg( B[3] ) ⎥ ωˆ o ⎟ ⎜ ⎢ ⎥, (7) = ⎜⎢ .. ⎥ ⎢ .. ⎥⎟ ⎢ .. ⎥ ⎢ ⎥ .. ⎦ ⎣ ⎦ ⎦ ⎣ ⎣ θˆ . . . ⎠ ⎝ ⎣ ⎦ . 150T 1 150T 1 150T 1 arg( r[150] ) B[150]
where ( denotes the transpose. Equation (7) represents the estimate ωˆ o , θˆ given S. No phase unwrapping is performed in Eq. (7) in simulation results to follow. Equation (7) yields the following final expression for the frequency offset pair ωˆ o , θˆ : ⎡ ⎤T 0.2423 −727.4998 ⎢ 0.2396 −717.8726 ⎥ ⎢ ⎥ ⎢ 0.0736 −140.2371 ⎥ ⎢ ⎥ ⎢ 0.0709 −130.6098 ⎥ ⎢ ⎥ ⎢ 0.0681 −120.9826 ⎥ ⎢ ⎥ ⎢ 0.0653 −111.3553 ⎥ ⎢ ⎥ ⎢ 0.0626 −101.7280 ⎥ ⎢ ⎥ ⎢ 0.0598 −92.1008 ⎥ ⎢ ⎥ ⎢ 0.0570 −82.4735 ⎥ ⎢ ⎥ ⎢ 0.0543 ⎡ arg( r[2] ) ⎤ −72.8463 ⎥ ⎢ ⎥ B[2] ⎢ 0.0515 ⎥ −63.2190 ⎢ ⎥ ⎢ r[3] ⎥ arg( ⎢ 0.0487 ⎢ −53.5917 ⎥ B[3] ) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0.0460 ⎢ r[63] ⎥ −43.9645 ⎥ )⎥ ⎢ ⎥ ⎢ arg( B[63] ⎢ ⎢ ⎥ −34.3372 ⎥ ⎥ ⎢ ⎥ ⎢ 0.0432 .. ⎢ ⎥ ⎢ ⎥ ωˆ o 0.0404 −34.3372 ⎥ ⎢ . ⎢ ⎥. (8) =⎢ ⎥ ⎢ −34.3372 ⎥ ⎢ arg( r[87] ) ⎥ θˆ ⎢ 0.0377 B[87] ⎥ ⎢ 0.0349 ⎥ ⎢ ⎥ −5.4554 ⎥ ⎢ ⎢ r[148] ⎥ )⎥ ⎢ 0.0321 ⎢ arg( B[148] 4.1718 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0.0294 ⎥ ⎢ ⎥ .. 13.7991 ⎥ ⎣ ⎢ ⎦ . ⎢ 0.0266 ⎥ 13.7991 ⎥ ⎢ r[150] arg( B[150] ) ⎢ 0.0238 33.0536 ⎥ ⎢ ⎥ ⎢ 0.0211 ⎥ 42.6808 ⎥ ⎢ ⎢ 0.0183 42.6808 ⎥ ⎢ ⎥ ⎢ 0.0155 61.9354 ⎥ ⎢ ⎥ ⎢ 0.0128 71.5626 ⎥ ⎢ ⎥ ⎢ 0.0100 81.1899 ⎥ ⎢ ⎥ ⎢ 0.0072 90.8171 ⎥ ⎢ ⎥ ⎢−0.1615 678.0799 ⎥ ⎢ ⎥ ⎣−0.1642 687.7072 ⎦ 697.3344 −0.1670 )T
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Fig. 3. The angular frequency ωˆ o and phase θˆ weights for S = {2, 3, 63, . . . , 87, 148, . . . , 150}.
2.3. The role of the training and the tail symbols for computing ωˆ o and θˆ It is evident that the role of the matrix in Eq. (8) is that of a weight matrix, containing the optimal weights for r[n] combining all the available observations arg( B[n] ) ∀n ∈ S to produce the frequency offset pair estimate ωˆ o and θˆ . It is interesting to inspect the relative absolute values of these weights shown in Fig. 3. These indicate that the observations r[n] ) for n ∈ S close to the burst symmetry point contribute least to the estimate of ωˆ o and θˆ . The tail observations arg( B[n] on the other hand are far away from the burst symmetry point and have the largest weights. Hence even though the tails are few compared to the training symbols, the LS frequency offset estimator indicate they do play an important role in the estimate ωˆ o and θˆ . This seems to suggest that the approach taken in this paper of using both tails and training symbols is well justified. 2.4. Computational complexity of the LS frequency offset estimator The arg( ) operation can be performed efficiently via a look-up table, so that N lookup table operations are required. 2N multiplication and addition operations are needed to multiply the weight matrix and the vector in Eq. (8). However many DSP processors perform a multiply and accumulate (MAC) operation as a single instruction, and under such conditions a DSP implementation of Eq. (8) thus requires N devision and lookup table operations to compute the arg( ) values, and then 2N MAC operations. For the numerical results presented in the next section, N = 30. 3. Numerical results Results obtained via simulation of the LS frequency offset estimator in GSM1 are compared to the results given in [11] (Fig. 10 in that reference) also for GSM. The simulation assumptions used in [11] were applied here as well. First 1 The simulator used to generate the results for the LS frequency offset estimator is open source at http://opensource.ee.up.ac.za.
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Fig. 4. Comparison of performance for the L&R frequency offset estimator and the LS frequency offset estimator in GSM.
of all, frequency hopping is enabled so that the frequency offset in each burst is different. This eliminates the use of averaging techniques over multiple bursts, i.e. the frequency offset for each burst is estimated using information from the burst under consideration only. Secondly perfect symbol timing recovery, a small mobile velocity (3 km/h) and frequency selective Raleigh fading are assumed here. The small mobile velocity makes the assumption of a constant impulse response h over the burst valid, as was the case also in [11]. An estimation of the channel impulse response is performed in the GSM simulator [11,12]. For the LS frequency offset estimator the choice of S is depicted in Fig. 2 and the simulation results are shown in Fig. 4 for the typical urban (TU) channel (frequency selective Raleigh fading). There it is compared to the results from [11]. The LS frequency offset estimator performs better than the L&R algorithm under both noisy and virtually noiseless conditions (Eb /No = 10 dB, and at Eb /No = 200 dB). A second observation is that the LS frequency offset estimator shows a smaller bias than does the L&R frequency offset estimator at Eb /No = 10 dB. Under the very low noise conditions found at Eb /No = 200 dB both estimators shows virtually no bias. The small offsets used in Fig. 4 were intended to study bias and compare the two algorithms. To study the performance of the LS frequency offset estimator at large offsets, offsets up to 900 Hz (corresponding to an oscillator with 1 part per million stability at 900 MHz carrier frequency) were considered and the simulation results are shown in Fig. 5. It will be unusual for a GSM system to experience larger offsets than 900 Hz, as the GSM standard recommends levels of acceptable uncompensated frequency offset of a few hundred Hz [7,11]. Clearly, the LS frequency offset estimator performs well also for large offset. Finally, the performance of the frequency offset estimator is studied for the Rural area (RA) model at 250 km/h. Under these conditions, fading is fast and the impulse response is not stationary over the burst. In the computer simulation no frequency offset due to hardware impairments is added, but the Doppler shift manifests itself as a frequency offset, which is estimated by the offset estimator and then corrected for in the receiver. Figure 6 shows the raw (uncoded) bit error rate (BER) for EDGE with 8 PSK, where it is indicated that the offset estimator is able to recover some of the losses incurred due to the Doppler shift. Since the impulse response is varying over the burst, some unrecoverable losses are incurred in the equaliser, hence there is still loss relative to the RA 3 km/h test case.
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Fig. 5. The performance of the LS frequency offset estimator for large frequency offset in GSM/EDGE.
Fig. 6. The performance of the LS frequency offset estimator for a fast mobile in the RA 250 km/h model, for EDGE with 8 PSK modulation.
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4. Conclusions A low complexity LS estimator for estimating the frequency offset in GSM and EDGE were presented. The formulation makes possible the use of both pilot/training symbols as well as tail symbols available in GSM and EDGE. It was shown that the tail symbols play an important role in estimating the offset, in spite of the fact that they are few relative to the training symbols in GSM/EDGE. Numerical results comparing the performance of the LS estimator to the L&R method indicate the LS frequency offset estimator outperforms the L&R algorithm at realistic SNR, and also in the limit when SNR is large. In computer simulations small offsets were first considered to study the bias of both methods, and it was concluded that the LS frequency offset estimator shows less bias. Subsequently large frequency offsets were also simulated, and results indicated that the LS estimator offers good performance also in that case. References [1] C. Heegard, J.A. Heller, A.J. Viterbi, A microprocessor-based PSK modem for packet transmission over satellite channels, IEEE Trans. Commun. COM-26 (5) (1978) 552–564. [2] S. Kay, Modern Spectral Estimation: Theory and Applications, Prentice Hall, Englewood Cliffs, NY, 1988. [3] S.A. Tretter, Estimating the frequency of a noisy sinusoid by linear regression, IEEE Trans. Inform. Theory IT-31 (6) (1985). [4] S. Kay, A fast and accurate single frequency estimator, IEEE Trans. Acoust. Speech Signal Process. 37 (12) (1989). [5] D.W. Tufts, R. Kumaresan, Estimation of frequencies of multiple sinusoids: Making linear prediction perform like maximum likelihood, Proc. IEEE 70 (9) (1982) 975–989. [6] G.W. Lank, I.S. Reed, G.E. Pollon, A semi-coherent detection and Doppler estimation statistic, IEEE Trans. Aerospace Electron. Syst. AES9 (2) (1973) 151–165. [7] ETSI: GSM 05.05, Radio transmission and reception, ETSI EN 300 910 v8.5.1, November 2000. [8] O. Besson, P. Stoica, On frequency offset estimation for flat-fading channels, IEEE Commun. Lett. 10 (5) (2001) 402–404. [9] W. Kuo, M.P. Fitz, Frequency offset compensation of pilot symbol assisted modulation in frequency flat fading, IEEE Trans. Commun. 45 (1997) 1412–1416. [10] M. Morelli, U. Mengali, G.M. Vitetta, Further results in carrier frequency estimation for transmission over flat fading channels, IEEE Commun. Lett. 2 (1998) 327–330. [11] M. Luise, R. Regiannini, Carrier frequency recovery in all-digital modems for burst-mode transmission, IEEE Trans. Commun. 43 (2–4) (1995) 1169–1178. [12] W. Gerstacker, R. Schober, Equalisation concepts for EDGE, IEEE Trans. Wireless Commun. 1 (2002) 190–199. [13] J.C. Olivier, S.-Y. Leong, C. Xiao, K. Mann, Efficient equalisation and symbol detection for 8-PSK EDGE cellular system, IEEE Trans. Vehicular Tech. 52 (3) (2003) 525–529. [14] N. Al-Dhahir, J.M. Cioffi, MMSE decision-feedback equalisers: Finite-length results, IEEE Trans. Inform. Theory 41 (4) (1995) 961–975. [15] L. Sharf, Statistical Signal Processing, Detection, Estimation and Time Series Analysis, Addison–Wesley, Reading, MA, 1991.
Jan C. Olivier is a Professor of electronics engineering at the University of Pretoria, Pretoria, South Africa. He received the B.Sc. Eng. (electronics), M.Sc. Eng. (electronics), and Ph.D. degrees summa cum laude from the University of Pretoria, South Africa in 1985, 1986, and 1990, respectively. In 1988 he joined the National Institute for Defence Research in Pretoria as a Research Scientist, then joined Northwestern University in 1991, and then the University of Pretoria in 1994 as an Associate Professor. He joined Bell Northern Research in Ottawa, Canada, in 1997, then joined Nokia Research Centre in Dallas, Texas, USA, in 1999, where he worked as a Principle Scientist until 2003 when he rejoined the University of Pretoria as a Full Professor. His research interests are in estimation and detection theory, communications, and numerical analysis.