Orthogonal frequency division multiplexing (OFDM) is a predominant transmission technology for ubiquitous wireless broadband networks because it can mitigate severe effects of frequency selective fading and provide good spectrum efficiency [1-4]. There have been many attempts to apply the multiple-input multiple-output (MIMO) technology to OFDM transmission systems, such as [5,6].

For coherent detection of the received data symbols in OFDM transmission, channel frequency responses (CFRs) must be estimated and equalized. One of the OFDM channel estimation methods is pilot-aided channel estimation (PACE) where pilot symbols are assigned to pilot subcarriers, which are multiplexed with data subcarriers, and channel estimation for data symbols is performed by interpolation techniques. There are three basic factors affecting the performance of the PACE method. These are pilot patterns, estimation methods, and signal detection. The choice of these factors depends on OFDM system specifications and wireless channel conditions.

In PACE, CFRs at pilot subcarriers are estimated using least squares (LS) or minimum mean square error (MMSE) estimators, and then, interpolation techniques are performed in order to estimate CFRs at data subcarriers by using the estimated CFRs at pilot subcarriers, as shown in Fig. 1. As an interpolation technique of PACE methods, discrete Fourier transform (DFT)-based interpolation is often used. However, it may have the leakage problem caused by virtual carriers in an OFDM symbol. In order to deal with the leakage problem, some suppression methods were proposed in [7-11]. In particular, optimal and suboptimal linear estimators were proposed to estimate the equally spaced virtual carriers in [11], but whose performances are very sensitive to the coherence bandwidth of wireless channels. Therefore, in this study, optimal and suboptimal linear estimators are reviewed and theoretically analyzed in terms of coherence bandwidth.

The rest of this paper is organized as follows: in Section II, DFT-based channel estimation and its leakage problem are described. Section III reviews optimal and suboptimal linear estimators and analyzes the effect of coherence bandwidth on them. In Section IV, numerical results are presented to show the performance degradation due to the effect of coherence bandwidth. Conclusions are presented in Section V.

DFT-based interpolation is an efficient interpolation technique because of its good performance and low complexity [8,9]. The PACE method using DFT-based interpolation is called DFT-based channel estimation, as shown in Fig. 2, where an inverse DFT (IDFT) operation is executed first to obtain the estimated channel impulse responses (CIRs) by using the LS-estimated CFRs at pilot subcarriers, and then, the estimated CIRs are transformed back into the frequency domain by the DFT operation to obtain the final CFRs at data subcarriers.

In most commercialized OFDM systems, virtual carriers are exploited to ease the implementation of spectral masking filters and ensure guard bands to avoid interferences between adjacent systems [10].

However, these virtual carriers have a bad influence on the performance of DFT-based channel estimation. In other words, they may cause leakage effects. Virtual carriers correspond to rectangular windowing in the frequency domain, which results in the convolution of CIRs with the sinc function in the time domain. Hence, the channel taps of CIRs are leaked to one another. Further, time-domain windowing is performed to reduce the noise and interference, which causes spectral leakage or Gibbs phenomenon, as shown in Fig. 3.

In order to review the performance of DFT-based channel estimation and the leakage problem from virtual carriers, the normalized mean square error (NMSE) is represented as follows: Let the number of total subcarriers be N = N_U + N_V + 1, where N_U + 1 and N_V are the number of useful subcarriers and the number of virtual carriers, respectively. The number of cyclic prefix samples is defined as N_G. The received symbol at the k-th subcarrier is represented by

where X[k] is a transmitted symbol, W[k] is a circularly symmetric complex Gaussian noise with zero mean and variance σ², and H[k] is a CFR represented by

where h[l] is the l-th channel gain of an L × 1 circularly symmetric complex Gaussian CIR vector h with zero mean and covariance matrix C_h = E[hh^H]. The pilot symbols are assigned to estimate the CFR at pilot subcarriers

where i_m = −N_U/2 + mD_f denotes a subcarrier location, D_f denotes the minimum pilot spacing, P[m] denotes a pilot symbol with a binary phase shift keying constellation, and N_P denotes the number of pilot subcarriers.

The received pilot vector with entries Y[i_m] is given by

where X is a diagonal matrix with P[m] on its diagonal, and H_P is a CFR vector, W is a noise vector with zero mean and covariance matrix σ²I_P ( I_P is an N_P × N_P identity matrix), and F is a DFT matrix with entries

where 0 ≤ m ≤ N_P − 1 and 0 ≤ n ≤ L − 1. After the LS channel estimation at pilot subcarriers, the observation vector is represented by

where the noise vector is statistically equivalent to W since E[] = 0_P×1 and E[] = σ²I_P . Here, 0_P×1 denotes an N_P × 1 zero vector. The observation vector Z can be rewritten as to address the leakage effects

where is an M × 1 observation vector, and M = N_b + N_P + N_f is the number of pilot subcarriers including leftside virtual carriers (N_b) and right-side virtual carriers (N_f) with minimum pilot spacing. Here, 0_b×1 and 0_f×1 denote an N_b × 1 zero vector and an N_f × 1 zero vector, respectively. H_b×1 and H_f×1 denote an N_b × 1 ideal CFR vector and an N_f × 1 ideal CFR vector, respectively. In addition, is a DFT matrix with entries

where = −N_U/2 + (m − N_b)D_f, 0 ≤ m ≤ M − 1, and 0 ≤ n ≤ L − 1. The IDFT operation transforms into the estimated CIR vector in (9)

Then, the DFT operation transforms into the estimated CFR vector in (10)

where = MI_L, and H = Gh. Here, G is a DFT matrix with entries

where −N_U/2 ≤ m ≤ N_U/2 and 0 ≤ n ≤ L − 1 . Note that the third term in (10) denotes the leakage. Now, the error covariance matrix of H can be given by

where U = F^HF, P = G^H, and V = E[]. To reveal the channel covariance matrix C_h, V is rewritten as

where _v denotes a DFT matrix with entries related to H_v. The NMSE performance can be presented as

where the first term denotes the noise effects and the second term denotes the leakage effects.

In order to minimize the NMSE in (14), optimal and suboptimal linear estimators were proposed in [11]. They can be expressed as

Then, optimal or suboptimal linear estimators are used to estimate = K_optZ or = K_subZ, respectively. Their error covariance matrices can be given by

where V_opt and V_sub are represented as

Finally, their NMSEs are given by

From (21) and (22), we find that the NMSE performance improvement depends on the second terms V_opt and V_sub, respectively.

Although optimal and suboptimal linear estimators provide good performance with moderate complexity, their performance may decrease according to wireless channel conditions such as coherence bandwidth.

Coherence bandwidth is a statistical measurement of the range of frequencies over which the channel can be considered flat [12]. The coherence bandwidth can be approximately defined as

where τ_max is the maximum delay spread of a wireless channel and T_s is the sampling time of an OFDM system. If the coherence bandwidth is divided by the subcarrier spacing Δf = 1/NT_s, the number of subcarriers in the range of the coherence bandwidth can be obtained. For instance,

where we assume N = 512 and L = 20. If the minimum pilot spacing is D_f = 8, three D_f-spaced virtual carriers will be in the range of the coherence bandwidth, as shown in Fig. 4.

Therefore, if optimal and suboptimal linear estimators try to estimate CFRs outside the range of the coherence bandwidth, they may experience a decrease in their performance.

Computer simulations have been run to analyze leakage suppression methods such as optimal and suboptimal linear estimators in terms of the coherence bandwidth of wireless channels. We consider the OFDM system using QPSK modulation in the 1.25-MHz bandwidth at 2.3 GHz with N_G = 32 , N = 512 , N_U + 1 = 481 , N_V = 31 , D_f = 8 , N_P = 61, and M = 64. The multipath fading channel is assumed to have an exponential power delay profile with L = 20.

Fig. 5 shows the NMSE performance of DFT-based channel estimation in the case of no virtual carriers. It has constant values determined by the noise variance or the signal-to-noise power ratio (SNR). In contrast, Fig. 6 shows the NMSE performance in the case of virtual carriers (namely, conventional DFT-based channel estimators), which is theoretically plotted according to the total subcarriers index by using (14). The first term in (14) strongly depends on the noise variance, whereas the second term leads to the leakage. In addition, the second term is the dominant factor in performance degradation, particularly at edge subcarriers.

Fig. 7 shows the performance improvement resulting from the suboptimal linear estimator by using σ² = 40 in (16). The first term is the same as that in Fig. 6, but the second term is highly improved when compared with that in Fig. 6.

Fig. 8 illustrates the NMSE performance of the suboptimal linear estimator with σ² = 40 in (16) as the number of pilot subcarriers (N_P) decreases, which means that the number of D_f-spaced virtual carriers to be estimated increases. Further, Fig. 9 illustrates the NMSE performance of the suboptimal linear estimator with σ² = 10. By comparing Fig. 8 with Fig. 9 when N_P = 61, 59, 57, we find that the suboptimal linear estimator with σ² = 40 is superior to that with σ² = 10. Due to the high SNR assumption in [13], these results are reasonable. However, when N_P = 55, 53, the NMSE performances in Fig. 8 are more severely degraded.

These results denote the cases in which the D_f-spaced virtual carriers to be estimated are out of the range of the coherence bandwidth. In other words, the D_f-spaced virtual carriers outside the coherence bandwidth are sensitive to the SNR mismatch. In addition, the SNR mismatch of the suboptimal linear estimator with σ² = 40 is more critical than that of the suboptimal linear estimator with σ² = 10, when the actual SNR is 20 dB. Hence, the channel estimation errors outside the coherence bandwidth resulting from the SNR mismatch cause the entire performance degradation.

In this study, the effect of the coherence bandwidth of wireless channels on leakage suppression methods such as the optimal and suboptimal linear estimators for OFDM channel estimation was analyzed. The NMSE performances of these methods were very sensitive to the coherence bandwidth of wireless channels. If some of the estimated CFRs were placed out of the range of the coherence bandwidth, a severe channel estimation error occurred at edge subcarriers and the entire NMSE performance decreased. Further, the SNR mismatch of the suboptimal linear estimators was more critical in these cases.