Feedforward carrier frequency estimation with MSK-type signals

IEEE COMMUNICATIONS LETTERS, VOL. 2, NO. 8, AUGUST 1998

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Feedforward Carrier Frequency Estimation with MSK-Type Signals M. Morelli and U. Mengali, Fellow, IEEE Abstract— We propose a non-data-aided technique for the estimation of the carrier frequency offset in minimum-shift keying (MSK)-type modulations. The proposed algorithm has a feedforward structure and is suited for burst-mode transmissions. Computer simulations are used to assess its performance and make comparisons with other existing methods in terms of estimation accuracy and minimum operating signal-to-noise ratio (threshold). Numerical results are provided for modulation schemes of practical interest such as MSK and Gaussian MSK.

I. INTRODUCTION

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ONTINUOUS PHASE modulation (CPM) is a signaling method that allows bandwidth and signal energy savings [1]. Furthermore, it generates constant envelope waveforms which are attractive in radio channels with nonlinear amplifiers. Unfortunately, maximum-likelihood (ML) detection of CPM signals is generally complex since it requires a Viterbi algorithm operating on a large trellis. Simpler receivers have been devised for the subclass of minimum-shift keying (MSK)type formats [2], which are characterized by a binary alphabet and a modulation index of 0.5. The Gaussian MSK (GMSK) modulation adopted in the GSM European cellular mobile digital system belongs to this sub-class. The demodulation of MSK-type signals requires knowledge of the carrier frequency offset caused by oscillator instabilities and Doppler effects. Several methods for frequency estimation have been recently investigated. In [3] a data-aided feedforward algorithm is proposed for GMSK. Its performance is remarkably close to the modified Cramer–Rao bound (MCRB) [4, Ch. 2], but its acquisition range is limited to a few percents of the bit rate. The method illustrated in [5] is suitable for MSK and has a non-data-aided (NDA) feedforward structure. Although its performance does not attain the MCRB, it can manage frequency offsets on the order of 25% of the bit rate. In [6] frequency recovery is approached by means of an NDA algorithm having a delay-and-multiply structure. Its acquisition range is generally larger than with the method in [5] but the estimates are plagued by a large amount of self-noise. In this letter we propose an NDA frequency estimation algorithm for MSK-type modulation, which is suited for fully digital implementation. The algorithm exploits the knowledge of the symbol timing. This is justified by the fact that, with moderate frequency offsets (on the order of 10%–15% of the bit rate), timing information can be derived by one of the

methods proposed in [5], [7], and [8] and then exploited for frequency estimation. The remainder of the letter is organized as follows. Section II introduces the signal model for MSK-type formats and overviews the Laurent decomposition for binary CPM waveforms [9]. Section III describes the frequency estimation algorithm. Section IV illustrates some simulation results and makes comparisons with other existing methods. Conclusions are drawn in Section V. II. SIGNAL MODEL The complex envelope of an MSK-type signal is expressed by [1] (1) are the transmitted symbols taking values where is the symbol period and is the phase-shaping pulse of the modulator, which is normalized in such a way that (2) The parameter in (2) is referred to as the correlation length. In [9] it is shown that the exponential in (1) can be expressed PAM waveforms, i.e., as a superposition of (3) are related to the information where the coefficients and the pulses depend on the shape of data With full response systems is unity and there is only one PAM component in (3). With partial response , vice versa, may be quite large. Most of the formats signal power, however, is concentrated in the first component , which is referred to as the in (3) (the one with index fundamental component. In [9] it is also demonstrated that the (in the minimum mean square error best approximation to sense) by means of a single PAM component is obtained by truncating the expansion (3) to the first term, i.e., (4) where

Manuscript received October 8, 1997. The associate editor coordinating the review of this letter and approving it for publication was Prof. N. C. Beaulieu. M. Morelli is with CSMDR, Italian Centro Nazionale delle Ricerche (CNR), 56100 Pisa, Italy. U. Mengali is with the Department of Information Engineering, Universit`a degli Studi di Pisa, 56100 Pisa, Italy. Publisher Item Identifier S 1089-7798(98)06958-0. 1089–7798/98$10.00  1998 IEEE

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IEEE COMMUNICATIONS LETTERS, VOL. 2, NO. 8, AUGUST 1998

and (7) elsewhere. is time limited to the interval Note that the pulse and achieves a maximum at III. CARRIER RECOVERY ALGORITHM The complex envelope of the received waveform is composed of signal plus noise (8) In this equation, is the residual carrier frequency offset and is the phase shift introduced by the channel. The noise is Gaussian and has independent real and imaginary components, each with a two-sided power spectral density where is the signal energy per symbol. The incoming waveform is first fed to a low-pass filter to eliminate out-ofThe filter band noise and then is sampled at symbol rate bandwidth is large enough not to distort the signal component even in the presence of frequency offsets on the order of be the filter 10–15% of the symbol rate. Thus letting output, we can write (9) is the filtered noise. Denoting by the samples where taken at and using the of approximation (4), yields (10) and The term accounts where for the intersymbol interference (ISI) and has the form (11) with (for example, It is worth noting that is time limited to the interval . with MSK) since Our goal is to estimate in the absence of any information on the transmitted symbols (NDA estimation). To this end the are fed to a nonlinear device with the purpose samples of eliminating the modulation. Here we consider the quadratic nonlinearity (QNL) (12) To see the physical meaning of (12) let us assume negligible in (10)). Then, noise and ISI levels (i.e., set [see (5)] yields bearing in mind that (13) This equation indicates that the QNL generates a discretetime sinewave at frequency . Thus the parameter can be estimated by noncoherently measuring the frequency of and then dividing by two. is To see how this can be done, let us imagine what and are not negligible. Intuitively, like when becomes some noisy version of (13). It follows that can be estimated by means of one of the methods available in the literature [10]–[14]. In the sequel we concentrate on the Rife & Boorstyn (R&B) method [10], which provides the best performance in terms of minimum operating signal-to-noise

Fig. 1. Normalized frequency variance versus Es =N0 for MSK.

ratio (SNR) (threshold). Denoting by R&B method produces

a trial value for

the

(14) This equation is referred to as the QNL estimator in the sequel. The following remarks are of interest. , is 1) It is clear that the absolute value in (14), say . Thus, the maximum a periodic function of of period of lies in the interval and, in consequence, the This explains estimation range of the QNL is the previous statement that the QNL can manage frequency offsets on the order of 25% of the bit rate. can have many local maxima, the largest 2) As maximum must be sought in two steps. The first step (coarse over a discrete set of -values covsearch) calculates ering the uncertainty range of and determines that which maximizes . The second step (fine search) interpolates and computes the local maximum between samples of nearest to the -value chosen earlier. 3) Occasionally may be so distorted by noise that its peak lies far from . When this happens the estimator makes large errors (outliers). The SNR value at which outliers start occurring is referred to as the threshold of the estimator. Since large errors lead to intolerable degradation in the receiver performance, operation above threshold is mandatory. 4) In practice the coarse search can be efficiently performed using pruned fast Fourier transform (FFT) techniques [15].

MORELLI AND MENGALI: FEEDFORWARD CARRIER FREQUENCY ESTIMATION

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We see that the estimation variance of the QNL comes close to the MCRB at intermediate/high SNR values. As SNR decreases, the variance increases first slowly and then very rapidly. The rapid change in the slope of the curves represents the threshold of the estimator. The figures indicate that the QNL has a much better performance than the MCM scheme. It should be stressed, however, that this superiority does not come for free. In fact the QNL is more complex. For example, for and a pruning factor it turns out that the QNL requires about 4736 (real) multiplications and 6528 (real) additions whereas the MCM needs only 640 multiplications and 640 additions. Further simulations (not shown) say that and the threshold value of the QNL is strongly related to increases. This feature is of great importance decreases as with coded modulations for it makes the QNL suitable for by adequately increasing operation at very low V. CONCLUSIONS We have proposed a new non-data-aided frequency estimation scheme for MSK-type modulation. It operates on signal samples taken at bit rate and has a feedforward structure that is useful in applications where short acquisitions are required. Its performance is much better than that of other existing methods, even though its computational complexity is higher. Simulations indicate that its accuracy is close to the MCRB at values. Its threshold decreases as the sufficiently high length of the estimation interval increases. Fig. 2. Normalized frequency variance versus Es =N0 for GMSK.

REFERENCES

The choice of the pruning factor considerably affects the estimator performance. Reference [10] recommends -values in the range 2–8 to keep the threshold low. IV. SIMULATION RESULTS AND COMPARISONS In this section we discuss some simulation results for MSK and GMSK signals. The following system parameters have been chosen: 1) the pruning factor equals two; 2) the ; 3) the true estimation interval corresponds to ; 4) the front-end low-pass filter frequency offset is set to is an eight-pole Butterworth FIR filter with a 3-dB bandwidth for MSK and for GMSK; and 5) the fine search consists of a parabolic interpolation. Simulations indicate that the QNL is unbiased (meaning that its bias is much less than the error standard deviation). Figs. 1 and 2 illustrate the normalized frequency variance versus for MSK and GMSK, respectively. The 3-dB bandwidth of the premodulation filter is set to for GMSK. The modified Cramer–Rao bound [4] (15) is also shown as a benchmark. The curve labeled MCM pertains to the estimator investigated by Mehlan, Chen, and Meyr in [5], which has the form (16)

[1] J. B. Anderson, T. Aulin, and C.-E. Sundberg, Digital Phase Modulation. New York: Plenum, 1986. [2] P. Galko and S. Pasupathy, “On a class of generalized MSK,” in Proc. Int. Conf. Commun., Denver, CO, June 1981, pp. 2.4.1–2.4.5. [3] M. Luise and R. Reggiannini, “An efficient carrier frequency recovery scheme for GSM receivers,” in GLOBECOM ’92, Conf. Rec. Commun. Theory Mini-Conf., Orlando, FL, Dec. 6–9, pp. 36–40. [4] U. Mengali and A. N. D’Andrea, Synchronization Techniques for Digital Receivers. New York: Plenum, 1997. [5] R. Mehlan, Y-E. Chen, and H. Meyr, “A fully digital feedforward MSK demodulator with joint frequency offset and symbol timing estimation for burst mode mobile radio,” IEEE Trans. Veh. Technol., vol. 42, pp. 434–443, Nov. 1993. [6] A. N. D’Andrea, A. Ginesi, and U. Mengali, “Frequency detectors for CPM signals,” IEEE Trans. Commun., vol. 43, pp. 1828–1837, Apr. 1995. [7] U. Lambrette and H. Meyr, “Two timing recovery algorithms for MSK,” in Proc. ICC ’94, New Orleans, LA, May 1994, pp. 1155–1159. [8] A. N. D’Andrea, U. Mengali, and M. Morelli, “Symbol timing estimation with CPM modulation,” IEEE Trans. Commun., pp. 1362–1372, Oct. 1996. [9] P. A. Laurent, “Exact and approximate construction of digital phase modulations by superposition of amplitude modulated pulses,” IEEE Trans. Commun., vol. COM-34, pp. 150–160, Feb. 1986. [10] D. C. Rife and R. R. Boorstyn, “Single tone parameter estimation from discrete-time observations,” IEEE Trans. Inform. Theory, vol. IT-20, pp. 591–598, Sept. 1974. [11] S. Kay, “A fast and accurate single frequency estimator,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 37, pp. 1987–1990, Dec. 1989. [12] M. Luise and R. Reggiannini, “Carrier-frequency recovery in all-digital modems for burst-mode transmissions,” IEEE Trans. Commun., vol. 43, pp. 1169–1178, Mar. 1995. [13] M. P. Fitz, “Planar filtered techniques for burst mode carrier synchronization,” in IEEE Globecom’91, Phoenix, AZ, Dec. 1991, paper 12.1. [14] M. Morelli and U. Mengali, “Data-aided frequency estimation for burst digital transmission,” IEEE Trans. Commun., vol. COM-45, pp. 23–25, Jan. 1997. [15] J. D. Markel, “FFT pruning,” IEEE Trans. Audio Electroacoust., vol. AU-19, pp. 305–311, Dec. 1971.