Improved Pilot Symbol Aided Estimation of Rayleigh Fading Channels with Unknown Autocorrelation Statistics

Transcription

1 Improved Pilot Symbol Aided Estimation of Rayleigh Fading Channels with Unknown Autocorrelation Statistics Kareem E. Baddour, Student Member, IEEE Department of Electrical and Computer Engineering Queen s University, Kingston, ON, Canada K7L 3N6 Norman C. Beaulieu, Fellow, IEEE Department of Electrical and Computer Engineering University of Alberta, Edmonton, AB, Canada T6G 2V4 Abstract New pragmatic pilot symbol assisted modulation (PSAM) techniques are proposed for the accurate estimation of time-selective Rayleigh fading channels. Unlike the optimal approach, the suggested techniques do not rely on knowledge of the fading autocorrelation function, and yet are shown to provide near-optimum channel estimation performance. While simple PSAM estimators published in the literature may appear to match this claim, we demonstrate that the gap in terms of estimation mean-squared error between existing techniques which do not rely on the channel statistics and the optimum estimator is about an entire order of magnitude for small Doppler rates. This gap is significantly reduced by employing estimators which operate with partial knowledge of the channel statistics. Moreover, we develop two novel channel interpolation methods which can exploit estimates of the Doppler frequency, and optionally, the channel SNR. The accuracies of the developed estimators are characterized and compared to those of existing suboptimal PSAM approaches. We demonstrate that the proposed estimators provide significant performance gains for slow fading scenarios. For example, a gain of approximately 2.5 db in SNR is found for BPSK error rate calculations and simulations in channels with a normalized Doppler rate of I. INTRODUCTION Mobile communication systems are characterized by channels with severe amplitude and phase fluctuations. The channel fading must be estimated accurately and then compensated to achieve high-quality coherent digital communications. This process is especially critical in emerging high-rate wireless systems, for which the use of adaptive transmission techniques combined with higher-order modulations are being envisioned. Pilot symbol-assisted modulation (PSAM) is a popular method with well-known merits as a fading countermeasure for time-selective Rayleigh channels. For such channels, the optimum PSAM approach in the minimum mean-squared error (M) sense is the interpolator [1]. This solution relies on accurate knowledge of the autocorrelation function (ACF) of the channel process and knowledge of the SNR. However, the ACF is unknown in wireless systems and varies over different channels [2, pp. 88]. Since a long history of channel observations is required for its accurate estimation, the optimum solution is generally impractical to implement but useful for benchmarking the performance of PSAM systems. Many suboptimum PSAM approaches have been considered in the literature. The authors in [3] proposed the use of smoothing of the channel estimates at the pilot locations followed by a simple linear interpolator. In this case, only a single filter needs to be designed which reduces the complexity at the receiver. However, the ACF and noise statistics are again assumed to be known. Simple interpolation techniques which do not require any statistical knowledge of the channel have also been proposed, including a low order Gaussian interpolator [4], a raised-cosine filter [5], low-pass windowed sinc interpolation [6], [7], and interpolation in the frequency domain based on the fast Fourier transform (FFT) [8]. A popular approach has proven to be the interpolator with windowed sinc coefficients, which was shown by the authors in [6] to significantly outperform the Gaussian filter and to provide nearoptimum BER performance for the particular PSAM parameters and channel characteristics used in their computations. While performance comparisons between the raised-cosine, windowed sinc and FFT PSAM interpolators have not appeared in the literature, such a comparison is performed in the first part of this paper and all three approaches are found to provide a similar order of estimation performance. More significantly, the gap in terms of mean-squared error () for small Doppler rates between these simple interpolators, which do not use the channel statistics, and the optimum filter is shown to be about an entire order of magnitude for typical scenarios. Thus, the observed -like performance achieved by the windowed sinc PSAM estimator does not hold in general and depends on the Doppler rate of the channel. These observations are critical, for example, to adaptive modulation implementations, which typically work more effectively in slow fading environments and require high quality fading estimates. Based on these observations, we are motivated to search for pragmatic PSAM approaches which do not require knowledge of the channel ACF and which provide near-optimum channel estimation performance over a broad range of Doppler rates. In this paper, we investigate PSAM estimators which operate with only partial knowledge of the channel statistics. Moreover, we develop two novel interpolation methods which can exploit knowledge of the Doppler frequency and optionally, the channel SNR. The accuracy of the proposed estimators is then characterized and compared to alternative PSAM approaches. We also consider the important practical case where mismatched parameters or imperfect estimates of the Doppler rate and SNR are available. Finally, examples are provided to demonstrate the performance of the PSAM estimators when used in conjunction with a practical Doppler estimator. Our results confirm

2 that to achieve near-optimum channel estimates, knowledge of the channel autocorrelation is not required and only a reliable Doppler estimate is needed. II. BACKGROUND A. System Model Detailed descriptions of the PSAM technique are provided in [1],[7]. Essentially, known pilot symbols are periodically multiplexed into the transmitted symbol stream to sound the channel. The data is formatted into frames of Å symbols, with the first symbol in each frame reserved for the pilot. In this paper, ideal receiver matched filtering, symbol synchronization and carrier recovery are assumed. The symbol-spaced matched filter output samples can thus be represented by the baseband equivalent ÜÒ Ü ÒÌ µ ÒÌ µ ÒÌ µ Û ÒÌ µ, where Ì is the symbol duration, ÒÌ µ represents the complex fading distortion, ÒÌ µ is the complex baseband signal and pilot symbols, and Û ÒÌ µ is additive white Gaussian noise with variance ¾ Û. For simplicity, we assume that all the pilot symbols take on the same value and that the pilot and information symbols have a constant modulus ¾. In our work, ÒÌ µ are complex samples of a zero mean bandlimited WSS Rayleigh flat fading process with maximum Doppler frequency, variance ¾ and with ACF Ö µ. To test the estimator performances in a general scattering environment, we use the plausible fading channel correlation model proposed in [9, eqn. (2)] and justified in that paper. The classic Rayleigh isotropic scenario is a special case of this model, which leads to the well-known ACF Ö µ ¾ ¼ ¾ µ. The PSAM receiver obtains a fading estimate at the pilot locations by dividing the received signal by the pilots ÝÅ ÜÅ Å ÛÅ where ÝÅ is the fading at the pilot symbol in the th frame. As long as the signaling rate ½Ì and maximum Doppler frequency satisfies the Nyquist condition Å ¾, the fading at the data symbols can then be recovered by interpolation. The fading estimate at the Ñth symbol time in the th frame is Ñ Å À Ñ Ý Ä¾ Ä ½µ¾ Ñ Ý µå (1) where Ñ ½ Å ½ is the symbol index within each frame, Ü is the largest integer smaller than or equal to Ü, Ý is a length Ä column vector representing the set of observed fading samples Ý µå for Ä ½µ¾ ľ, is a length Ä vector containing the interpolation filter coefficients, and Ñ is the th interpolator coefficient. The coefficients are in general time-variant, which can be interpreted as each symbol location having an interpolator of its own. B. PSAM Techniques Various interpolation filters have been proposed for PSAM. The optimum M coefficients, which minimize ¾ Ñ ÒÑ Å Ñ Å ¾Ó, are obtained by solving [1] Ê ÓÔØ Ñ ÚÑ (2) where ÚÑ is defined as the length Ä covariance vector ÚÑ Ý Ñ Å (3) and where Ê is the Ä Ä autocorrelation matrix given by In (4), Ê has µth element Ê ÝÝ À Ê Á (4) Ê Ö µåì µ ½ ¾ Ä (5) ¾ Û ¾ and Á is the Ä Ä identity matrix. The th element of ÚÑ can be expressed in terms of the channel correlation as Ú Ñ Ö Ä ½µ¾ ½ µå Ñ Ì µ Ä ½µ¾ ½ Ö Ä ½µ¾ ½ µå Ñ Ì µ Ä ½µ¾ ½ for ½ ¾ Ä For any particular choice of the PSAM coefficients Ñ in (1), the can be expressed as [1] ¾ Ñ ¾ ¾ Ê Ú À Ñ Ñ À Ñ ÊÑ (7) which is independent of. For the solution, the minimum is achieved and (7) simplifies to ¾ ÑÒ Ñ ¾ Ú À Ñ Ê ½ ÚÑ. Simple interpolation filters which do not require any statistical knowledge of the channel have been proposed [4]-[8]. Among these, the raised-cosine [5], windowed sinc [6] and FFT [8] interpolators are the most promising. A popular approach has proven to be the windowed sinc interpolator, which was reported by the authors in [6], [7] to provide near-optimum performance for the particular parameters used in their computations. Therefore, sinc interpolation will be used as a benchmark in this paper for its simplicity and performance. For this estimator, the interpolation coefficients are chosen as Ñ Ñ sinc Å (8) where Ä ½µ¾ ľ. As suggested in [6], a Hamming window is then applied to the sinc coefficients to prevent the sharp truncation of a rectangular window. The raisedcosine interpolator has coefficients given by [5] Ñ sinc (6) Ñ Å Ó Ñ Å µå µ ½ ¾ Ñ Å µå µ ¾ ¼ ½ where is the roll-off factor. Optimization of the roll-off has been investigated by Lo et. al. in [5] in which it was concluded based on the M criterion that the optimum roll-off varies with the channel sampling rate and Doppler bandwidth. When the pilot symbol rate is close to Nyquist or when ½Å ¾ Ì, the optimum roll-off should approach zero to avoid aliasing, and approach one at higher sampling rates. (9)

3 C. Optimum Versus Suboptimum PSAM While performance comparisons between the raised-cosine, windowed sinc and FFT PSAM interpolators have not appeared in the literature, such a comparison was performed by the authors using the model in [9, eqn. (2)] and is summarized here. For the, raised-cosine and Hamming windowed sinc interpolators, the estimation was calculated using (7). The FFT PSAM estimator was implemented using the zero insertion method proposed in [8], and as in [8], its performance was evaluated by simulations. For accurate and rapid simulation of the correlation model in [9, eqn. (2)], the filtering technique detailed in [10] was adopted in this paper using 50 filter taps. Monte Carlo results are averaged over 1000 independent channel realizations. For the numerical and simulation results in this paper, we make the convenient normalization ¾ ½ and define the channel estimation SNR ¾ ¾ Û. Based on evaluations over a wide range of channel parameters, our results indicate that the three suboptimum interpolation approaches generally provide a similar order of channel estimation performance [13]. For example, a typical result is illustrated in Fig. 1., which plots the average over the data symbol locations as a function of the normalized Doppler rate when the various estimators operate in Rayleigh isotropic channels with a frame size of Å ½¼, interpolation order of Ä ¾ and SNR ¾¼ db. For the FFT estimator, a 32 point FFT was used to provide a fair comparison. The M curve, which is achieved by the filter, is also plotted. For the curve, the interpolator is optimized at every point on the graph. In Fig. 1 we observe the expected result that the increases rapidly when the Doppler frequency becomes large enough to cause the sampling rate to fall below the Nyquist value. Fig. 1 also illustrates that the ½ raised-cosine estimator performs very poorly at high Doppler rates. This occurs because the Doppler bandwidth exceeds that of the PSAM filter in these cases and useful channel energy is discarded from the interpolation process. If the filter is to be implemented nonadaptively, it must therefore be designed for worst case Doppler conditions as advocated in [1] for the filter. For small Doppler rates, the raised-cosine interpolator performs at its best with ½ and marginally outperforms both the windowed sinc and FFT estimators. More significantly and unexpectedly, however, the gap in for small Doppler rates between the interpolators which do not use the channel statistics and the M of the filter can be seen to be about an entire order of magnitude. Similar results were obtained for a broad range of channel parameters [13]. BER results are considered later in the paper. Thus, the observation of -like performance for the windowed sinc PSAM results of [6] and [7] does not hold in general and clearly depends on the Doppler rate. In the next section, we investigate PSAM estimators which can exploit partial knowledge of the channel statistics to close the performance gap between the simple estimators and the solution. III. ESTIMATORS WHICH EXPLOIT PARTIAL KNOWLEDGE OF THE CHANNEL STATISTICS In this section, we propose two novel fading channel interpolation methods which can exploit side information regarding the Doppler frequency and the channel SNR. Initially, we assume that these parameters are known perfectly, but we also consider the effects of mismatches later in the paper. A. Estimator Based on Yen Interpolation The first novel estimator is motivated by some results in the signal processing literature on the interpolation of bandlimited Gaussian stochastic processes that are corrupted by additive white noise. When the power spectral density of the random process is unknown, it is natural to suppose that the spectrum is flat over the frequency region inside the bandlimit, or equivalently, that the autocorrelation is given by Ö Ò Ö ÒÌ µ sinc ¾ ÒÌ µ (10) In the following, we consider solving for the channel interpolation filter coefficients using the equations (2)-(5) from the solution but with the interpolation kernel set to the sinc function in (10). That is, the PSAM coefficients in (1) for the Ñth symbol time are computed as Ñ Ê ½ sinc Ú sincñ (11) where Ê sinc is chosen to be the Ä Ä autocorrelation matrix as defined in (4)-(5) but with the unknown correlations Ö ÒÌ µ set to (10). Correspondingly, the elements of the length Ä covariance vector Ú sinc Ñ are set to (6) with correlation values selected from (10). Such a solution is called the Yen interpolator in the signal processing literature and is known to be optimal in the M sense when there is no prior knowledge of the bandlimited channel PSD or autocorrelation function and when only the bandlimit and additive white noise variance ¾ Û are known [11]. In the special case when the channel spectrum is flat, the Yen interpolator matches the solution and is the globally optimum interpolator in the M sense. In Section IV, we benchmark the performance of Yen interpolators of Rayleigh fading channels when both perfect and mismatched parameters and ¾ Û are available to the receiver. B. Estimator Based on Smoothing and Interpolation The structure of this PSAM receiver is similar to the one in [3], but here the channel ACF is not assumed to be known. First, a time-invariant finite-length smoothing filter is used to provide noise-filtered channel estimates at the pilot symbol locations. The fading estimate at the pilot location in the th frame is ¾ Å À Ý Â ½µ¾ Ý µå (12)

4 where  is the length of the smoothing filter, is a length  vector containing the smoothing coefficients, and Ý is a length  column vector representing the set of observed fading samples Ý µå for  ½µ¾ ¾. The smoothed estimates are then interpolated to obtain the fading at the data symbol times. Moreover, the channel estimate at the Ñth symbol time in the th frame is obtained as ľ Ñ Å Ä ½µ¾ Ñ µå (13) where Ä is the length of the interpolator and Ñ is the th interpolator coefficient for the Ñth symbol time. The optimal smoother coefficients in a M sense are time-invariant and given by a solution [3]. For the unknown ACF case, we consider a novel smoother designed in a similar fashion to the Yen filter by selecting a bandlimited sinc kernel function. Moreover, the smoother coefficients in (12) are computed using Ê ½ sinc Û sinc, where Ê sinc is now a   autocorrelation matrix as previously defined in (4)-(5) and with the unknown correlations Ö ÒÌ µ set to (10). Correspondingly, the elements of the length  covariance vector Û sinc are set to the values in (6) with Ñ ¼ and Ö ÒÌ µ sinc ¾ ÒÌ µ For convenience, we set the smoother length to Â Ä in the rest of the paper. Since the smoother has reduced the noise in the channel samples at the pilot locations, the interpolation step is now less demanding. To estimate the fading at the symbol times, a simple interpolator can be used. In this work, we adopt the windowed sinc filter [6] for its simplicity and performance. The filter is easily computed by multiplying (8) with a Hamming window. IV. PERFORMANCE EVALUATION Based on performance evaluations over a wide range of channel parameters in [9, eqn. (2)], the proposed estimators were benchmarked against the optimized and windowed sinc estimators. The complete results are found in [13] and are summarized in this section. For the and the proposed estimators, the results show that ¾ Ñ is almost identical over all symbol positions for practical conditions [13]. For simplicity, therefore, all results are averaged over the data symbol locations. As well, we have observed that for a given Doppler rate, SNR and fixed PSAM parameters Ä and Å, the of a particular estimator is almost identical regardless of the underlying bandlimited channel Doppler spectrum for typical in [9, eqn. (2)] with ¼ [13]. Therefore, unless otherwise indicated, the results presented in this section are for isotropic scattering scenarios and are representative of the trends for general scattering environments. A. Performance with Known and SNR Fig. 2 illustrates the effect of the frame size Å on the average of the various estimators for Ä ¾ and SNR ¾¼ db when Ì ¼¼¼ ¼¼¾ ¼¼. As expected, all interpolators suffer the effects of aliasing and their accuracy degrades severely when the frame size becomes large enough to cause the sampling rate to fall below the Nyquist value. We find that both of the proposed PSAM methods provide performance very close to the bound in the region of interest and their improves as the pilot spacing is reduced. However, the windowed sinc interpolator cannot take advantage of an increase in the pilot rate and maintains the same level of performance regardless of Å when the sampling rate is greater than the Nyquist value. A frame size of Å is adopted for benchmarking in the rest of this paper, which allows for Doppler rates up to about Ì ¼¼ to be accomodated [1]. Next, we consider the effect of the filter orders Ä on the performance of the proposed PSAM estimators. Fig. 3 shows the dependence of the on Ä for Å, Ì ¼¼½ and with SNR ¾¼ db and SNR ¼ db. From Fig. 3, we observe that the improvement with increasing Ä becomes marginal beyond 5 to 10 coefficients for the windowed sinc interpolator and beyond 16 to 20 coefficients for the two proposed estimators as well as for the filter. We also find that the proposed Yen interpolator performs very close to the optimum solution. The proposed estimator based on smoothing and interpolation performs similar to the sinc interpolator for small filter orders, but its performance rapidly approaches that of the Yen and interpolators as Ä increases. The effect of noise is considered next. Fig. 4 plots the average versus the SNR for scenarios with Å, Ì ¼¼¼ and when Ä ½¾. These results illustrate the strong influence of the noise on the interpolator accuracy. We observe that the Yen and interpolators provide almost identical s and their accuracy generally improves rapidly with increasing SNR. For low SNR levels, the proposed estimator based on both smoothing and windowed sinc interpolation performs as well as the Yen and interpolators, but as the SNR increases, the estimation gains due to the smoother become negligible and its performance converges with that of the windowed sinc filter. Both of the estimators based on sinc interpolation suffer from an irreducible at higher SNR levels. In contrast, the Yen and interpolators do not exhibit an floor for practical SNR levels [13]. B. Performance with Mismatched Parameters The performance results described in the previous section for the and proposed estimators are unrealistic in that the Doppler frequency and SNR are assumed to be known. In practice, either the PSAM coefficients would be optimized for one particular operating point or an estimate of the Doppler or SNR would be used to compute the interpolator or smoother coefficients. Thus, a highly relevant question is how sensitive the estimation performance is to errors in these parameters. In the following, we calculate the effect of parameter mismatch, i.e., when the Doppler frequency and white noise variance ¾ Û are over-estimated or under-estimated rather than known perfectly.

5 In Fig. 5, we plot the of the proposed estimators for isotropic scenarios with Å, Ä ½¾, Ì ¼¼¼ and SNR ¾¼ db versus the Doppler mismatch ratio, defined as, where and are the estimated and exact Doppler frequency, respectively. The of the simple windowed sinc estimator, which is obviously unaffected by the mismatch ratio, is also included for reference. The curves quantify the degree to which Doppler misestimation affects the Ä ½¾ PSAM estimates. We find that under-estimating significantly degrades the performance of the and of the proposed estimators by discarding useful channel energy, while over-estimating by several times only results in a moderate degradation due to additional noise. Note that the best performance for the proposed PSAM estimators does not always coincide precisely for, since the assumed sinc ACF does not generally match with the actual channel ACF [13]. Next, we consider the effect of a mismatch in the noise variance parameter. Fig. 6 illustrates the as a function of the mismatch ratio ¾ Û ¾ Û for the same scenario as Fig. 5, where ¾ Û and ¾ Û represent the estimated and exact noise variances, respectively. The results reveal that the accuracy of the estimators is relatively insensitive to this parameter. A significant performance degradation does not occur unless ¾ Û is overestimated by at least several times. Under-estimating ¾ Û or, equivalently, over-estimating the SNR does not produce a noticeable degradation of the. Similar results were observed for general scenarios [13]. For simplicity, therefore, SNR estimation is not considered in this paper. Instead, a nominal SNR estimate of 30 db is adopted for the remaining experiments, as this value is expected to exceed actual SNR values. C. Performance with Practical Estimation In the previous section, we observed that the availability of accurate Doppler rate information is critical to obtaining high quality fading channel estimates. This strongly motivates us to characterize the performance of the proposed estimators in conjunction with a practical Doppler spread estimator. In our work, we adopt the FFT based Doppler spread estimator proposed in [12], which was shown to be a strong competitor to existing Doppler estimators by virtue of its accurate performance in a wide range of Doppler rates, propagation scenarios and noise levels. Here, we stress the need to consider nonisotropic scenarios because of the sensitivity of many estimators to the underlying Doppler spectrum (e.g., see [12]). The FFT Doppler rate estimator was implemented as described in [12, eqn. (14)] with exponent ½. The unknown maximum Doppler frequency was estimated for each independent trial based on the observation of Æ ½¼¼ consecutive pilot symbols. This sequence was zero-padded and a 256 point FFT was used to evaluate a coarse spectrum estimate. The Doppler bandwidth estimate was then computed as the minimum frequency range within which the power spectrum exceeds a particular ratio of the total power in the spectrum estimate. Full details and characterization of the FFT estimator are available in [12]. The Doppler estimator of [12] is biased for general scenarios, with the bias depending largely on as well as on the actual channel SNR and. A nominal threshold of ¼ was chosen for all experiments in this paper, as this value was found to result in a small bias for a broad range of channels [13]. Of importance, and not demonstrated here due to space limitations, when the FFT Doppler estimator degrades due to low SNR, the Doppler bandwidth becomes overestimated [13]. Based on the results in the previous section, this characteristic is clearly preferable to under-estimating the Doppler rate, and it permits the PSAM filters that are adjusted using the FFT Doppler estimates to outperform the windowed sinc interpolator at low SNR [13]. In Fig. 7 the average of the proposed PSAM techniques with estimated Doppler and mismatched SNR is plotted as a function of the normalized Doppler rate for nonisotropic channels with ¾ and «in [9, eqn. 2], and with Ä ¾¼ and actual SNR ¾¼ db. The results of the ideal matched filter and the windowed sinc solution are also included for reference. The results confirm the -like estimation accuracy of the proposed PSAM solutions when combined with a reliable Doppler estimator. For Ì ¼¼¼½, almost an entire order of magnitude improvement to the is obtained when compared to the sinc interpolator. Note that Ì ¼¼¼½ represents, for example, a typical cellular system operating at 2 GHz with 50 kilosymbols per second signaling and a mobile velocity of 27 km/h. For this case and with Å, the 100 pilots used for Doppler estimation corresponds to an observation window of seconds. Fig. 8 illustrates the corresponding BER averaged over the data symbols for the scenario with Ì ¼¼¼ when uncoded BPSK modulation is used. To generate the simulation results, 5000 errors were processed for each point. The theoretical BER results were obtained by averaging the exact conditional BER in [1, eqn. (24)] over the symbol locations. The BER with perfect channel knowledge, given by [14, eqn. (5.58)], is also plotted for reference. The proposed estimators are found to significantly outperform windowed sinc PSAM, by approximately 2.5 db in SNR at a BER of ½¼. V. CONCLUSION In this paper, we have proposed novel pragmatic PSAM techniques for the accurate estimation of time-selective Rayleigh channels. Significant performance gains were observed for slow time-selective fading channels when compared to simple PSAM methods. Our theoretical and simulation results demonstrate that to achieve near optimum pilot symbol based channel estimates, knowledge of the channel ACF and SNR are not required and only a reliable Doppler estimate is needed. REFERENCES [1] J. K. Cavers, An analysis of pilot symbol assisted modulation for Rayleigh fading channels, IEEE Trans. Veh. Technol., vol. 40, pp , Nov [2] K. Pahlavan and A. Levesque, Wireless Information Networks, New York, NY: Wiley, 1995.

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7 with mismatched f d Yen with mismatched f d sinc smooth + interp with mismatched f d (sim) Ideal (matched) Yen (matched) Yen with Doppler est. & mismatched SNR (sim) Smooth + interp with Doppler est. and mismatched SNR (sim) Fig Doppler Mismatch Ratio The average of the various PSAM estimators as a function of the Doppler mismatch ratio for Å, Ä ½¾, Ì ¼¼¼ and SNR ¾¼ db. Fig Normalized Doppler Rate Average of the proposed PSAM estimators as a function of Ì for Ä ¾¼, Å when the SNR is unknown and when is estimated in a nonisotropic Rayleigh channel with SNR ¾¼ db and ¾, «in [9, eqn. 2]. The Doppler rate is estimated using Æ ½¼¼ pilots for each trial based on the method in [12] with a 256 point FFT. The theoretical s of ideal PSAM and windowed sinc PSAM are also shown for reference. (mismatched) Yen (mismatched) sinc smooth + interp mismatched (sim) Coherent BPSK (perfect estimation) (matched) Windowed Sinc Yen (matched) Yen with Doppler Est. & mismatched SNR (sim) Smooth + Interp with Doppler Est. & mismatched SNR (sim) BER Fig Noise Variance Mismatch Ratio The average of the various PSAM estimators as a function of the noise variance mismatch ratio for Å, Ä ½¾, Ì ¼¼¼ and SNR ¾¼ db SNR per bit (db) Fig. 8. The average BER when using uncoded BPSK for the proposed PSAM estimators when the SNR is unknown and when is estimated. The results are for the same simulation scenario used for Fig. 7. The theoretical BER curves of ideal PSAM, Yen PSAM with matched Doppler and SNR, and windowed sinc PSAM are also included for reference.