Abstract:
Adaptive low complexity minimum mean square error (MMSE) channel estimator for OFDM systems operating over mobile channel. Complexity of the estimator is reduced by partitioning sub-carriers into windows where, window size is optimized by considering channel model mismatch error (MME). Three types of adaptive windowed MMSE (W-MMSE) estimators include: A first type, a simplified delay profile applied as channel reference model, and optimum window size adaptive to the estimated signal-to-noise ratio. A second type, a group of candidate channel reference models are considered. The receiver roughly estimates and selects current reference model from candidate group, then adapts optimum window size based on the estimated SNR and selected channel model. A third type, the current channel statistics are finely estimated and window size is iteratively optimized at receiver. The first two adaptive W-MMSE estimators are tolerant to channel model mismatch error and the third captures channel variations to realize real-time estimation.

Description:
TECHNICAL FIELD OF THE INVENTION 
     The proposed estimator can apply to receivers for any communications system using OFDM (orthogonal frequency-division multiplexing) or COFDM (coded orthogonal frequency-division multiplexing), including digital TV broadcast receivers, cell phones, or data systems such as Wi-Fi. 
     BACKGROUND 
     Mobile reception in OFDM (orthogonal frequency-division multiplexing) systems has received increasing attention with the growing demands for applications such as for communication in high speed railways. In mobile OFDM systems, the accurate estimation and tracking of time-varying and frequency selective fading is critical to the design of the frequency domain equalization and detection. Pilots can be inserted in transmitted OFDM symbols to assist channel estimation in various standards, e.g., scattered and edge pilots are inserted in time and frequency directions to assist channel estimation in the second generation digital terrestrial television broadcasting system (DVB-T2) and advanced television systems committee (ATSC) 3.0 systems. Channel estimations are used to operate the frequency domain equalizer to improve signal reception. 
     Generally, the receiver first obtains estimates of the channel frequency response using pilot sub-carriers, then interpolates this response to provide a channel estimate for the data sub-carriers. The channel estimate is applied to the equalizer in the receiver and is subsequently updated to track changing channel conditions. In the frequency domain the data sub-carriers can be estimated using a time-frequency two-dimensional (2D) filter to get a good estimation of the data, but this technique results in high computation complexity. Thus two one-dimensional (1D) filters, the first one for time direction interpolation and the second for frequency direction interpolation, may be used alternatively although with a slight performance loss. The channel estimator disclosed herein can be applied to the time-frequency 2D estimation, 1D time direction estimation and 1D frequency direction estimation, respectively. 
     Frequency domain interpolation may be performed using many different interpolation techniques such as linear, cubic and spline interpolations or a combination thereof. Linear interpolation is popular in practical implementations due to its low complexity, but provides relatively poor performance. Due to the nature of broadcast RF (radio frequency) signals, there are some similarities between different channel coefficients even when the channel experiences fast changes. These similarities may be quantified by a function referred to as channel correlation. Exploiting the channel correlation and noise variance characterizing a given channel, a MMSE (minimum-mean-square error) estimator can significantly improve the estimation accuracy of the data sub-carriers. However, two constraints impede the implementation of the MMSE estimator in a practical system. The first is that the channel statistics and noise variance is usually unknown, especially in a mobile scenario. The second is the high computing complexity caused by the matrix inversion operation in the MMSE. Many prior art works focus on the design of a practical MMSE estimator to overcome one or both of the above two constraints. 
     Corresponding to the first constraint, channel estimators have been proposed by either constructing a reference channel model or by real-time estimation of the current channel statistics. An ideal band-limited time domain correlation and a rectangular delay profile have been suggested for estimators incorporating the reference channel model. Similarly, a uniform power delay profile has been used in the cases where the worst correlation is expected. The foregoing reference channel models are insensitive to variations in channel statistics and are relatively simple to implement. Another method is to roughly estimate the actual current channel correlation and then design the MMSE estimator to be adaptive to the current channel. An extra computing effort is required to obtain the accurate channel model, thus implementation complexity is increased. 
     In case of the reference model based MMSE channel estimator, due to fast fading variations, channel model mismatch errors (MME), i.e., the errors caused by the difference between the reference channel model and the real channel, are usually inevitable in practical mobile communication systems. Designing a MMSE estimator that is naturally tolerant of this mismatch provides significant improvement over the prior art estimators. 
     Turning to the second constraint, various techniques have been proposed to reduce the complexity of the MMSE estimator, such as singular value decomposition (SVD), which works for the block-type pilots with the assumption of perfect channel correlation. U.S. Pat. No. 7,801,230, in order to reduce complexity, discloses the use of a split MMSE (S-MMSE) that separates the sub-carriers into groups, and a separate filter is applied into each sub-group independently. However, some significant limitations hinder implementation of this application. First, the proposed method only works for block type pilots. It does not work for comb-type pilots such as those in DVB-T2 or ATSC 3.0. Second, it is assumed the channel statistics are perfectly known, while the perfect prior information is not available in mobile OFDM systems. Finally, the window size obtained under the ideal channel correlation assumption does not work well for systems with channel model mismatch. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a functional block diagram of an OFDM transmitter. 
         FIG. 2  is a functional block diagram of an OFDM receiver. 
         FIG. 3A  is a first exemplary comb-type pilot pattern useful in certain embodiments of the invention in which D x =4 and D y =1. 
         FIG. 3B  is a second exemplary comb-type pilot pattern useful in certain embodiments of the invention where D x =2 and D y =2. 
         FIG. 4  illustrates a 2D interpolation of the data and pilots of  FIG. 3B . 
         FIG. 5A  illustrates a 1D time interpolation of the pilots of  FIG. 3B . 
         FIG. 5B  illustrates a 1D frequency interpolation of the data and pilots of  FIG. 5A . 
         FIG. 6  illustrates the procedure of windowing according to certain embodiments of the invention. 
         FIG. 7  is a functional block diagram of an exemplary 1D frequency direction estimation with the W-MMSE estimator of a certain embodiment of the invention. 
         FIG. 8A  is a graph that illustrates a first exemplary channel delay function used as a reference channel model according to certain embodiments of the invention. 
         FIG. 8B  is a graph that illustrates a second exemplary channel delay function used as a reference channel model according to certain embodiments of the invention. 
         FIG. 9  is a flowchart of a first type W-MMSE estimator according to certain embodiments of the invention which is adaptive to the SNR and a predefined channel model. 
         FIG. 10  is a flowchart of a second type W-MMSE estimator according to certain embodiments of the invention which is adaptive to both an estimated SNR and an estimated reference channel model. 
         FIG. 11  is a flowchart of the pre-calculation of the optimum window size under a different SNR and a reference channel model. 
         FIG. 12  is an exemplary table of the optimum window size obtained from the pre-calculation in  FIG. 11 . 
         FIG. 13  is a flowchart of the third type iterative W-MMSE estimator according to certain embodiments of the invention which iteratively optimizes the window size in real-time. 
         FIG. 14  is a graph that illustrates an exemplary performance of the W-MMSE estimator of certain embodiments of the invention as a function of the window size under ideal channel statistics. 
         FIG. 15  is a graph that illustrates an exemplary performance of the W-MMSE estimator of certain embodiments of the invention as a function of the window size under a channel model mismatch. 
         FIG. 16  is a graph that illustrates an exemplary symbol error rate comparison when the window sizes are obtained under an ideal channel model and under a reference channel model in the W-MMSE channel estimation. 
         FIG. 17  is a graph that illustrates an exemplary performance comparison between the proposed W-MMSE estimator and a conventional linear interpolator under D x =16. 
         FIG. 18  is a graph that illustrates an exemplary performance comparison between the proposed W-MMSE estimator and a conventional linear interpolator under D x =8. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  is a block diagram of a baseband OFDM (orthogonal frequency-division multiplexing) transmitter. Data bits are encoded, interleaved and mapped into constellations (e.g., QPSK (quadrature phase shift keying) or M-QAM (multiple quadrature amplitude modulation)) by a processor  100  (labeled as coding, interleaving, mapping in the Figure), then provided to a Serial-to-Parallel (S/P) converter  110 . Pilots, such as pilots having a comb-type scattered pattern are then inserted in the time-frequency hyper plane by a pilot inserter  120  to assist channel estimation. For example, comb-type pilot patterns used in standards like DVB-T2 (digital video broadcasting—second generation terrestrial) and ATSC (Advanced Television Systems Committee) 3.0 may be employed. In  FIG. 1 , comb-type scattered pilots are used as an example to illustrate the channel estimation algorithms described herein. Note that the channel estimation algorithms proposed herein are applicable not only to the comb-type pilots in DVB-T2 and ATSC 3.0, but also to the block-type training sequence such as used in IEEE (Institute of Electrical and Electronics Engineers) 802.11. 
       FIGS. 3A and 3B  illustrate exemplary comb-type pilot patterns. The separation of pilot bearing carriers is denoted as D x  and the number of symbols forming one scattered super-pilot sequence is D y . Thus the pilot is scattered by D x  sub-carriers over D y  OFDM symbols. For further reference, the total number of sub-carriers in one symbol may be denoted as N, the number of pilots as N p  and the number of data as N d , respectively. Referring back to  FIG. 1 , after pilot insertion, an inverse fast Fourier transform (IFFT)  130  is performed to transform frequency domain X(k) into time domain x(n). Both of the data X d  and pilot X p  are included in X. The output of the IFFT is supplied to a Parallel-to-Serial (P/S) converter  140  to provide a serial time domain OFDM signal for transmission. Before transmission, the OFDM signal is supplied to a guard interval inserter  150  which inserts a cyclic prefix (CP) in the signal large enough to prevent inter-symbol-interference (ISI). The OFDM signal may experience time-varying frequency-selective fading with additive white Gaussian noise (AWGN) as it is transmitted through a transmission channel  160 . 
       FIG. 2  is a block diagram of a baseband OFDM receiver. At the receiver, the CP (cyclic prefix) is first removed by a guard interval remover  200  from the received signal stream from the transmission channel  160 . The received signal stream is then supplied to a S/P converter  210 . A fast Fourier transform (FFT)  220  follows to transform the time domain signal y(n), comprising a version of x(n) as distorted by the channel, into the frequency domain Y. The frequency domain signal Y comprises the received data Y d  and received pilot Y p . The received data Y d  is applied from FFT  220  to a frequency domain equalizer  230  (to reduce errors in Y d ) and therefrom to a processor  240  (labeled in the Figure as Demod., Deinterleay., Decod.) for the demodulating, deinterleaving, and decoding the equalized signal. The received pilot Y p  is extracted from Y by a channel estimator  250 . Channel estimator  250  outputs the channel coefficients Ĥ d  of data sub-carriers. The estimated channel coefficients Ĥ d  are taken as the initial setting of the frequency domain equalizer  230  for channel equalization. The channel estimator  250  preferably implements a W-MMSE (windowed minimum mean square error) channel estimation according to the invention to derive coefficients Ĥ d . 
     The channel estimator  250  first obtains an estimate of the channel frequency response at the pilot sub-carriers. The estimation of pilots may be performed based on the Least Square (LS) method for simplification. It also can be estimated using the W-MMSE method disclosed herein. After getting the estimated channel coefficients Ĥ d  of the pilot sub-carriers, the channel estimates of data sub-carriers are obtained through interpolating the estimated coefficients of pilot sub-carriers. A time-frequency 2D estimation can be performed as shown in  FIG. 4 . The 2D estimator results in a high computation complexity. In a more practical system, two 1D filters may be used for implementation, as shown in  FIGS. 5A and 5B . The time direction interpolation is first performed as shown in  FIG. 5A , and then interpolation in the frequency direction follows as shown in  FIG. 5B . The channel estimator  250  can be implemented using the time-frequency 2D estimation method shown in  FIG. 4 , the 1D time direction estimation in  FIG. 5A  and 1D frequency direction estimation in  FIG. 5B . In the following description, the 1D frequency direction interpolation is used as an example to explain the invention. 
     Since only frequency direction interpolation is considered in this example, the symbol index is omitted for brevity. With the help of CP, the orthogonality between sub-carriers can be fully preserved, thus the observations of N sub-carriers in one symbol can be written as
 
 Y=XH+Z,   (1)
 
where XεS N×N  is a diagonal matrix with the transmitted pilot and data signal on its diagonal, HεC N×1  is the channel frequency response vector, and, ZεC N×1  is the AWGN sample with variance σ e   2 . S is the modulation alphabet set and C is the set of complex number.
 
     With the comb-type scattered pilot pattern in the  FIG. 3A , the pilots are extracted from the N sub-carriers by the channel estimator  250  and represented in matrix form as:
 
 X   p   H   p   +Z   p ,  (2)
 
where Y p εC N     p     ×1 , X p εS N     p     ×N     p   , H p εC N     p     ×1 , Z p εC N     p     ×1  are the samples of the extracted pilots of Y, X, H, Z in (1), respectively.
 
     The extracted pilots are then used by the channel estimator  250  to estimate the data through interpolation.  FIG. 7  is a block diagram illustrating a 1D frequency direction estimation including a novel W-MMSE (window minimum mean squared error) estimator that may be used to derive the channel coefficients. 
     In this exemplary embodiment, the LS estimation performed by block  700  is used to obtain the estimate of the channel coefficients at the pilot locations. The estimated results of these channel coefficients are represented as
 
 Ĥ   p,ls   =X   p   −1   Y   p ,  (3)
 
A time direction interpolator  710  follows to derive Ĥ p,d  by interpolating Ĥ p,ls  using an interpolation algorithm such as a simple linear interpolation. The following portion of  FIG. 7  depicts the output  715  of the novel W-MMSE channel coefficient estimator. Conventionally, in the frequency direction, the MMSE estimate of channel coefficients at the data sub-carriers is expressed as,
 
                       H   ^     d     =           R   dp     ⁡     (       R   pp     +     1     γ   p         )         -   1       ⁢       H   ^       p   ,   d                 (   4   )               
where R pp  is the channel auto-correlation matrix of pilot sub-carriers, R dp  is the cross-correlation matrix between pilot and data sub-carriers, and
 
               γ   p     =       E   p       σ   e   2             
is the SNR with E p  being the power of pilots.
 
     The matrices R pp  and R dp  depend on the channel conditions. Due to the communication environment variations such as the relative movement between the transmitter and receiver in the mobile case, the channel statistics change with respect to time. Therefore, the ideal assumption of channel statistics and noise variance cannot be matched to the real values any more. In other words, the prior knowledge of the matrices R pp , R dp , and γ p  in equation (4) are not available. Therefore, in  FIG. 7 , a typical SNR is estimated by a noise estimator  720 . Note that the SNR estimation is usually required by the other parts of the receiver such as for synchronization, thus the need for an estimation of the SNR does not increase complexity. Meanwhile, a well predefined reference channel model is established by block  730 . Two exemplary delay profiles that may be used as channel reference models are shown in  FIG. 8A  and  FIG. 8B , respectively. 
     The large dimension of the correlation matrix R pp  causes high computing complexity. For example, if an FFT size of 8192 (8 k) is used and D x =16, then the size of the matrix R pp  is as high as 512×512. To reduce the complexity, in certain embodiments of this invention, the entire span of sub-carriers of an OFDM symbol is partitioned into equal-sized windows or non-overlapping groups of sub-carriers. The groups of sub-carriers may be analyzed along one dimension (time or frequency) or two dimensions (time and frequency). The receiver performs an estimation in each window independently while neglecting the correlation between the windows. The concept of windowing is shown in  FIG. 6  where the span of sub-carriers is partitioned into multiple windows. 
     The optimization of the window size or equivalently the number of windows is critical to the design of the W-MMSE estimator and determined by a processor  740  (labeled in the Figure as Optimum W-MMSE window size). With a hypothetical ideal SNR (i.e. a value perfectly known at the receiver) and an ideal channel model from blocks  720  and  730 , a larger window size will provide better channel estimation results. This can be explained by the fact that the larger window size means more pilots are used for the estimation, thus leading to a better estimation. This may be confirmed by simulation as shown in the  FIG. 14 . In  FIG. 14 , the MSE (minimum square error) is shown as a function of the window size under different SNRs. The FFT size is 8 k, QPSK modulation is used, an elementary period is 0.146 μs, and D x =16. The channel model in the simulation is TU6 (six tap typical urban). The ideal SNR and channel model are explored in the simulation. It is observed that the channel estimation MSE decreases as the window size increases under various SNR. Thus a larger window size can benefit the channel estimation. 
     In contrast, the trend differs when the practical estimated SNR and reference channel model are applied. With the channel model mismatch, on one hand, the larger window size can positively contribute to the estimation due to the availability of more pilots. On the other hand, more pilots result in a larger channel model mismatch error, and this negatively affects the estimation result. This tradeoff relationship is further revealed in  FIG. 15 . The MSE is shown in  FIG. 15  as a function of the window size under different SNR. The curves in  FIG. 15  differ from the simulation in  FIG. 14  because a uniform power delay profile is adopted as the predefined reference channel model. It is observed that the MSE is implicitly convex in the window size. Thus, the statement that the larger window sizes provide the better performance does not hold for such a system. Instead, the window size which leads to the minimum MSE needs to be determined. 
     In  FIG. 15 , it is also shown that the optimum window size varies with the SNR. Thus, the optimum window sizes are 32, 32 and 16 for SNR at 10 dB, 20 dB, 30 dB, respectively. Based on the above observation, in operation block  740  is designed so that the optimum window size adapts to the estimated SNR and reference channel model. 
     Referring back to  FIG. 7 , after the optimized window size is determined in block  740 , the W-MMSE is applied within each window by a plurality of estimators as shown at block  750  as: 
                         H   ^     d     (   i   )       =             R   ^     dp     (   i   )       ⁡     (         R   ^     pp     (   i   )       +     1       γ   ^     p         )         -   1       ⁢       H   ^       p   ,   d       (   i   )           ,           (   5   )               
where i=1, . . . N w  and N w  is the optimized window number from operation  740 .
 
     In a collecting step, the estimations of the channel coefficients are collected in block  760  from all the windows in sequence, which provides the channel coefficients for use. 
       FIG. 9  is a flowchart illustrating the operation of the W-MMSE estimator of  FIG. 7  which is adaptive to the SNR and a predefined channel model according to certain embodiments of the invention. The LS is applied to estimate the channel coefficients of the received pilots at  900 . The practical prior information at step  930 , including the SNR estimated at step  910  and the robust reference channel model at step  920 , is used to determine the optimum window size at step  940 . This can be implemented using appropriate look-up tables which will be explained hereinafter with reference to  FIG. 11  and  FIG. 12 . Finally, the channel coefficients are independently calculated for each window at step  950  according to expression (5) and the results are collected to get the entire estimates of the channel coefficients at  960 . 
     To further improve the estimation performance, a second type of W-MMSE estimator, which is adaptive to both the SNR and the selected channel model is shown in the flowchart of  FIG. 10 . Steps  1000  and  1010  conform to steps  900  and  910  of  FIG. 9 . In step  1030 , there is provided a group of candidate channel reference models. In step  1020 , the best matched channel reference model is selected. The channel model selection can be realized by analyzing the received scatter pilot or CP. For example, the channel impulse response (CIR) may be obtained by taking the IFFT of the frequency domain channel estimate which is calculated at pilot locations. The power delay profile is generated from the estimated CIR or from the average of estimated CIR over several OFDM symbols. Then the reference channel is selected by comparing the parameters such as root-mean-squared (rms) delay spread of the estimated and candidate power delay profiles. Both of the estimated SNR and selected reference channel model are used to determine the optimum window size at  1040 . The channel coefficients are then estimated within each window at step  1050  and collected at step  1060 . This type of adaptive W-MMSE estimator is better to cope with channel variations in complicated communication environments, while the extra computing is required to obtain the channel model in  1030 . 
     The calculation of optimum window size in step  940  of  FIG. 9  and in step  1040  of  FIG. 10  can be realized by using a pre-calculated look-up table comprising part of the processor  740  of  FIG. 7 . The pre-calculation to populate the look-up table is shown in  FIG. 11 . Assume a transmitted reference channel A comprising an approximate simple TU6 type channel and a receiver having an exponential delay profile. Referring to  FIG. 11 , the pre-calculations to populate the look-up table comprise the steps of: 
     1) Start an initial trial at step  1101  which will ultimately result in the creation of a single curve of the type shown in  FIG. 15 . 
     2) First, at step  1102  simulate the transmission (Tx) of the approximate TU6 signal which includes both data and pilots at  1102 . 
     3) At the receiver, in frequency domain, calculate the channel coefficients Hd at the data sub-carriers using an estimation algorithm such as MMSE at step  1103 . The estimated coefficients at the data sub-carriers are approximately taken as the perfect channel information for the optimization of window size in W-MMSE. The noise variance is also estimated at step  1103 . 
     4) Set the initial window size such as n i =4 for the W-MMSE at step  1104 . 
     5) At step  1105  perform LS to get the channel estimation at the pilot sub-carriers, and perform W-MMSE interpolation to obtain the channel estimate at data sub-carriers. 
     6) At step  1106 , calculate the MSE between the real channel coefficients from step  1103  and the W-MMSE estimated coefficients from step  1105 . The calculated MSE at window size=4 establish, for example, a first point x on a given curve of  FIG. 15 . 
     7) Next, the window size is increased by an increment (such as 4) at step  1107 , i.e., n i+1 =n i +4. 
     8) If the incremented window size is determined to be less than or equal to a predetermined limit N p  at step  1108  the steps  1105 - 1108  are repeated to establish a second point (such as y) on the curve of  FIG. 15 . Steps  1105 - 1108  are successively repeated with the updated window sizes to derive additional points on the given curve of  FIG. 15  until limit N p  is achieved and the curve has been completed. 
     9) When the limit N p  is achieved the trial index j is increased at step  1109  to start a new trial. Each new trial will result in the development of a new curve as shown in  FIG. 15  until the total number of trials N loop  is achieved  1110 . 
     10) At this point, at step  1111  the average MSE and noise variance is calculated at each window size for each trial curve to produce a single averaged curve. 
     11) At step  1112  the optimal window size is established by finding the minimum average MSE 1112 characteristic of the curve produced according to the foregoing calculations. 
     The calculations set forth in  FIG. 11  result in an optimum window size for a given range of SNR. The optimum window size for different SNRs can be obtained by repeating the calculations of  FIG. 11  with the updated transmission power. 
     With the procedure illustrated in  FIG. 11 , an exemplary pre-calculated optimum window size look-up table is shown in  FIG. 12  under different SNR and channel mode. At the receiver, such as in step  940  of  FIG. 9 , with the estimated SNR in step  910  and the reference channel model in step  920  (such as if 20 dB SNR and Ch. A channel are used), the receiver finds the optimum window size is 32 in the table of  FIG. 12 , and uses it as the optimum window size to facilitate the W-MMSE estimation. 
     The above two types of reference channel model based W-MMSE (those illustrated in  FIGS. 9 and 10 , respectively) may be implemented using look-up tables as herein described. W-MMSE can also be performed iteratively to track changes and/or variations of the channel in real-time with a higher computation complexity. This is shown in the flow chat in  FIG. 13 . 
       FIG. 13  illustrates the use of W-MMSE with a real time (current) channel used in place of a reference channel as in  FIGS. 9 and 10 . As in  FIG. 9  and  FIG. 10 , LS is performed to estimate channel coefficients at the pilot sub-carriers at step  1300  and SNR is estimated at step  1301 . At  1302  the current channel statistics are estimated. Different from the channel selection in the second type W-MMSE shown in  FIG. 10 , this channel statistics estimation is more accurate with higher computation efforts. It can be realized by using more resources such as the preamble, side information, CP, or pilots of the OFDM signal at the receiver and taken as the prior information for the W-MMSE optimization. An initial MSE of W-MMSE is performed at step  1303  with the window size n=n 0 . The MSE of W-MMSE under window size n j  at j-th iteration (see steps  1303  and  1305 ) is expressed as 
                     mse   ⁡     (     n   j     )       =       ∑     i   =   1       N   w       ⁢     (       R   pp     (   i   )       -           R   dp     (   i   )       ⁡     (       R   pp     (   i   )       +     1     γ   p         )         -   1       ⁢     R   dp     H   ,     (   i   )             )               (   6   )               
where R pp , R dp  are obtained in step  1302  and the total number of windows
 
               N   w     =         N   p       n   j       .           
The window size is increased, by an increment such as 4, in step  1304  and the MSE of W-MMSE under the updated window size is calculated at step  1305 . If the condition of 1306 is satisfied, where mse(n j ) is the MSE under current window size is calculated at step  1305 . If the stop condition of 1306 is not satisfied, where mse(n j ) is the MSE under current window size n j  at j-th iteration, mse(n j-1 ) is the MSE of the previous window size at (j−1)-th iteration, and c is a threshold decided in the implementation, the window size is increased at  1304  and a next iteration of steps  1305  and  1306  is performed. If the stop condition is satisfied, which means the current MSE is larger than the previous one, then the optimum window size is established at step  1307  as n j-1 . With the optimum window size obtained in  1307 , channel coefficients are estimated within each window at step  1308  and collected at step  1309 .
 
     It should be noted that, for some applications, the optimum window size in steps  940 ,  1040 , or  1307  may yield to a predefined hardware threshold if desired or necessary. When the optimum window size is smaller than the threshold, the optimum window size is used to calculate the channel coefficients. Otherwise, the predefined threshold is used as the optimum window size for calculating the channel coefficients. 
       FIG. 16  shows the simulated SER (symbol error rate) with the W-MMSE and MMSE channel estimations. The performances of W-MMSE with two window sizes are compared in the figure: one obtained under ideal channel model and another obtained under reference channel model. Both of these two are not iterative. The D x =32, and TU6 are used in the simulation. Uniform delay profile is taken as the reference channel model. Other parameters are the same as those in  FIG. 14 . It is shown that the MMSE with the window size obtained under the ideal channel assumption does not perform well in certain cases of channel model mismatch. 
       FIG. 17  and  FIG. 18  illustrate SER comparison with the optimized W-MMSE estimator and the conventional linear interpolator. D x =16 for  FIG. 17  and D x =8 for  FIG. 18 . The other simulation parameters are the same as those in  FIG. 14 . Comparing the widely used linear interpolator,  FIG. 17  shows a 0.4 dB performance gain with the adaptive W-MMSE estimator. For D x =8, the performance gain can be as high as 0.7 dB in  FIG. 18 . 
     As previously indicated, the above description takes the frequency direction interpolation in  FIG. 5B  as an example. Some variations for the applications to estimation of pilots, 2D interpolation in  FIG. 4 , and 1D time direction interpolation in  FIG. 5A  are briefly explained here. To the application of estimating pilots, the cross-correlation R dp  in equation (4) will become the auto-correlation R pp . The reference channel model differs and the correlation between OFDM symbols is utilized. For 2D interpolation, the dimension and matrix structure of R dp  and R pp  in equation (4) will be different. The channel reference model becomes 2D, which considers both of the time correlation and the correlation among the multiple paths. 
     Although preferred embodiments of the method and apparatus of the present invention have been illustrated in the accompanying Drawings and described in the Detailed Description, it is understood that the invention is not limited to the embodiments disclosed, but is capable of numerous modifications and substitutions.