Analysis of the Effect of Coherence Bandwidth on Leakage Suppression Methods for OFDM Channel Estimation
 Author: Zhao Junhui, Rong Ran, Oh ChangHeon, Seo Jeongwook
 Publish: Journal of information and communication convergence engineering Volume 12, Issue4, p221~227, 31 Dec 2014

ABSTRACT
In this paper, we analyze the effect of the coherence bandwidth of wireless channels on leakage suppression methods for discrete Fourier transform (DFT)based channel estimation in orthogonal frequency division multiplexing (OFDM) systems. Virtual carriers in an OFDM symbol cause orthogonality loss in DFTbased channel estimation, which is referred to as the leakage problem. In order to solve the leakage problem, optimal and suboptimal methods have already been proposed. However, according to our analysis, the performance of these methods highly depends on the coherence bandwidth of wireless channels. If some of the estimated channel frequency responses are placed outside the coherence bandwidth, a channel estimation error occurs and the entire performance worsens in spite of a high signaltonoise ratio.

KEYWORD
Coherence bandwidth , DFTbased channel estimation , Leakage , OFDM , Wireless channel

I. INTRODUCTION
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 [14]. There have been many attempts to apply the multipleinput multipleoutput (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 pilotaided 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 [711]. 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, DFTbased 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.
II. OFDM CHANNEL ESTIMATION
> A. DFTBased Channel Estimation
DFTbased interpolation is an efficient interpolation technique because of its good performance and low complexity [8,9]. The PACE method using DFTbased interpolation is called DFTbased 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 LSestimated 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.
> B. Virtual Carriers and Leakage Problem
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 DFTbased 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, timedomain 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 DFTbased 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, whereN_{U} + 1 andN_{V} are the number of useful subcarriers and the number of virtual carriers, respectively. The number of cyclic prefix samples is defined asN_{G} . The received symbol at thek th subcarrier is represented bywhere
X [k ] is a transmitted symbol,W [k ] is a circularly symmetric complex Gaussian noise with zero mean and varianceσ ^{2}, andH [k ] is a CFR represented bywhere
h [l ] is thel th channel gain of anL × 1 circularly symmetric complex Gaussian CIR vectorh with zero mean and covariance matrixC _{h} =E [hh ^{H}]. The pilot symbols are assigned to estimate the CFR at pilot subcarrierswhere
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, andN_{P} denotes the number of pilot subcarriers.The received pilot vector with entries
Y [i_{m} ] is given bywhere
X is a diagonal matrix withP [m ] on its diagonal, andH _{P} is a CFR vector,W is a noise vector with zero mean and covariance matrixσ ^{2}I _{P} (I _{P} is anN_{P} ×N_{P} identity matrix), andF is a DFT matrix with entrieswhere 0 ≤
m ≤N_{P} − 1 and 0 ≤n ≤L − 1. After the LS channel estimation at pilot subcarriers, the observation vector is represented bywhere the noise vector is statistically equivalent to
W sinceE [] =0 _{P×1} and E[] =σ ^{2}I _{P} . Here,0 _{P×1} denotes anN_{P} × 1 zero vector. The observation vectorZ can be rewritten as to address the leakage effectswhere is an
M × 1 observation vector, andM =N_{b} +N_{P} +N_{f} is the number of pilot subcarriers including leftside virtual carriers (N_{b} ) and rightside virtual carriers (N_{f} ) with minimum pilot spacing. Here,0 _{b×1} and0 _{f×1} denote anN_{b} × 1 zero vector and anN_{f} × 1 zero vector, respectively.H _{b×1} andH _{f×1} denote anN_{b} × 1 ideal CFR vector and anN_{f} × 1 ideal CFR vector, respectively. In addition, is a DFT matrix with entrieswhere = −
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 =
M I _{L}, andH =Gh . Here,G is a DFT matrix with entrieswhere −
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 ofH can be given bywhere
U =F ^{H}F ,P =G ^{H}, andV =E []. To reveal the channel covariance matrixC _{h},V is rewritten aswhere _{v} denotes a DFT matrix with entries related to
H _{v}. The NMSE performance can be presented aswhere the first term denotes the noise effects and the second term denotes the leakage effects.
III. EFFECT OF COHERENCE BANDWIDTH ON LEAKAGE SUPPRESSION METHODS
> A. Review of Leakage Suppression Methods
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 _{opt}Z or =K _{sub}Z , respectively. Their error covariance matrices can be given bywhere
V _{opt} andV _{sub} are represented asFinally, their NMSEs are given by
From (21) and (22), we find that the NMSE performance improvement depends on the second terms
V _{opt} andV _{sub}, respectively.> B. Effect of Coherence Bandwidth
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 andT_{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 andL = 20. If the minimum pilot spacing isD_{f} = 8, threeD_{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.
IV. NUMERICAL RESULTS
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 1.25MHz bandwidth at 2.3 GHz with
N_{G} = 32 ,N = 512 ,N_{U} + 1 = 481 ,N_{V} = 31 ,D_{f} = 8 ,N_{P} = 61, andM = 64. The multipath fading channel is assumed to have an exponential power delay profile withL = 20.Fig. 5 shows the NMSE performance of DFTbased channel estimation in the case of no virtual carriers. It has constant values determined by the noise variance or the signaltonoise power ratio (SNR). In contrast, Fig. 6 shows the NMSE performance in the case of virtual carriers (namely, conventional DFTbased 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
σ ^{2} = 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
σ ^{2} = 40 in (16) as the number of pilot subcarriers (N_{P} ) decreases, which means that the number ofD_{f} spaced virtual carriers to be estimated increases. Further, Fig. 9 illustrates the NMSE performance of the suboptimal linear estimator withσ ^{2} = 10. By comparing Fig. 8 with Fig. 9 whenN_{P} = 61, 59, 57, we find that the suboptimal linear estimator withσ ^{2} = 40 is superior to that withσ ^{2} = 10. Due to the high SNR assumption in [13], these results are reasonable. However, whenN_{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, theD_{f} spaced virtual carriers outside the coherence bandwidth are sensitive to the SNR mismatch. In addition, the SNR mismatch of the suboptimal linear estimator withσ ^{2} = 40 is more critical than that of the suboptimal linear estimator withσ ^{2} = 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.V. CONCLUSIONS
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.

[Fig. 1.] Pilotaided channel estimation for orthogonal frequency division multiplexing (OFDM) systems.

[Fig. 2.] Discrete Fourier transform (DFT)based channel estimation. LSCE: leastsquares channel estimation.

[Fig. 3.] Description of the reason for leakage effects.

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[Fig. 4.] Coherence bandwidth and minimum pilot spacing.

[Fig. 5.] Normalized mean square error (NMSE) performance in case of no virtual carriers. SNR: signaltonoise ratio.

[Fig. 6.] Normalized mean square error (NMSE) performance of the conventional method (signaltonoise ratio [SNR] = 20 dB).

[Fig. 7.] Normalized mean square error (NMSE) performance of the suboptimal method (signaltonoise ratio [SNR] = 20 dB).

[Fig. 8.] Normalized mean square error (NMSE) performance of the suboptimal method (σ2 = 40).

[Fig. 9.] Normalized mean square error (NMSE) performance of the suboptimal method (σ2 = 10).