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RADIOENGINEERING, VOL. 18, NO. 4, DECEMBER 2009 497
A Novel Comb-Pilot Transform Domain Frequency
Diversity Channel Estimation for OFDM System
Liu LIU, Cheng TAO, Jiahui QIU, Xiaoyu QI
School of Electronics and Information Engineering, Beijing Jiaotong University, Beijing, P. R. China
bill0715@163.com, chtao@bjtu.edu.cn, 08120165@bjtu.edu.cn, 07120166@bjtu.edu.cn
Abstract. Due to implementation complexity, the transform domain channel estimation based on training symbols or comb-type pilots has been paid more attention because of its efficient algorithm FFT/IFFT. However, in a comb-type OFDM system, the length of the channel impulse response is much smaller than the pilot number. In this case, the comb-pilot transform domain channel estimation only works as interpolation like the Least Squares (LS) algorithm, but loses the noise suppression function. In this paper, we pro- pose a novel frequency diversity channel estimation method via grouped pilots combining. With this estimator, not only the channel frequency response on non-pilot subcarriers can be interpolated, but also the noise can be better suppressed. Moreover, it does not need prior statistical characteristics of the wireless channel.
Keywords
Channel estimation, transform domain, orthogonal fre- quency division multiplexing (OFDM), noise suppres- sion.
1. Introduction
Orthogonal frequency division multiplexing (OFDM) has received a lot of interests recently since it can transmit data at a very high speed. It has been the key technique in wireless LAN (WLAN), 3rd^ Generation Mobile Group Long Term Evolution (3GPP LTE) and WiMAX for its advan- tages in transmitting over frequency selective fading chan- nels, anti-interference of narrow band and combining MIMO easily [1], [2]. However, perfect channel state information (CSI) is required for signal detection and its accuracy de- termines the performance of OFDM communication system directly, especially for those with high order modulation or MIMO techniques.
The comb-pilot OFDM is widely used because it can estimate the time-varying wireless channel without fre- quency efficiency reduction [3], [4]. Then channel frequency response (CFR) of other non-pilot subcarriers used for trans- mitting data information can be obtained with linear inter-
polation [5] or spline interpolation [6] with the comb-pilots. These methods are performed in frequency domain. In re- cent years, more and more attention has been paid to trans- form domain channel estimation because of its efficient and fast algorithm via FFT/IFFT [7]-[10]. With this method, af- ter the initial transformation of the training symbols or pilots, if the length of multipath wireless channel can be obtained, we can zero the terms out in transform domain (time do- main) corresponding to noise while retaining the significant taps corresponding to the true delay taps. However, being lack of the knowledge on the order of wireless channel, when transform domain channel estimator is employed, it is diffi- cult to zero-force the noise in transform domain in a comb- type pilot or training symbol OFDM systems. The traditional method is zero-padding at first, and then directly transform back to frequency domain, thus the noise remains on each subcarrier [8]. In this way, the comb-pilot transform domain channel estimation only works as interpolation on non-pilot subcarriers but causes large loss in noise suppression. If the distance of the pilots is less than the coherence bandwidth of the multipath wireless channel, the performance of this tradi- tional method is the same as LS algorithm [11] for all pilots when the channel CIR taps are sample spaced. If the CIR taps are not sample spaced, error floor will occur. This will be analyzed further in the following sections of this paper. In this paper, we propose a novel comb-pilot trans- form domain based frequency diversity channel estimation for OFDM systems. First of all, all the pilots in one OFDM symbol are divided into several groups, in which the dis- tance between two adjacent pilots in a sub-group should be less than the coherence bandwidth of the multipath wireless channel. Then the pilots in the sub-group estimate the CRF of the wireless channel respectively so that several CFRs of the same wireless channel can be obtained. Finally, we can combine these CFRs with some rules, such as equiprobabil- ity rule. As a result, the combined CFR shows better perfor- mance than LS algorithm due to stronger noise suppression. Furthermore, the prior statistical characteristics of the mul- tipath wireless channel is unnecessary while employing this method. The rest of this paper is organized as follows. Section 2 describes the comb-pilot OFDM system and multipath chan- nel model. In Section 3, the relationship between the comb-
498 L. LIU, C. TAO, J. QIU, X. QI, A NOVEL COMB-PILOT TRANSFORM DOMAIN FREQUENCY DIVERSITY CHANNEL...
pilot transform domain channel estimation and the LS algo- rithm for all pilots is analyzed. And also the equivalence of these two methods is shown if the channel CIR taps are sample spaced. Then a novel comb-pilot transform domain frequency diversity channel estimation for OFDM system is presented. In Section 4 we compare the proposed algorithm results with the traditional methods using computer simula- tion, and the conclusions are drawn in Section 5.
- Comb-Pilot OFDM System Model
In an OFDM transmitter, comb-pilots are inserted in frequency domain after binary bit stream is mapped. Here, we consider Np pilots locate in each OFDM symbol and equidistance between two pilots is ∆P = N/Np without con- sidering the virtual subcarriers and DC tone. The data sub- carrier and the inserted pilot signal in frequency domain can be expressed as:
X(k) =
Xp(m · ∆P) m is an integer, 0 ≤ m ≤ Np − 1 XD otherwise (1)
where X(k) is the signal carried by a subcarrier in frequency domain, XD denotes the data and Xp represents the pilot. Then OFDM transmitter uses an inverse IDFT (IFFT) with size of N to modulate each subcarrier. Thus the time domain signal x(n) can be expressed as:
x(n) =
N
N− 1 ∑ k= 0
X(k)exp( j 2 π kn N
) 0 ≤ n ≤ N − 1 (2)
where n is the signal index of an OFDM symbol in time do- main. Then cyclic prefix (CP) whose length is much longer than the maximum delay tap of multipath wireless channel is added to each OFDM symbol to avoid inter-symbol inter- ference. After that the OFDM symbol passes through the multipath channel. For simplicity, the channel tap is sam- ple spaced. Otherwise the interpolation can be applied to obtain the spaced tap gain. Here, we assume the channel is a time-varying multipth channel. The multipath channel can be modeled as a tapped delay line structure as [8]:
h(n) =
r− 1 ∑ i= 0
hi exp
j 2 π N
fDiT n
δ (λ − τi) (3)
where r is the total number of resolvable delay paths in chan- nel. hi is the complex gain of i-th propagation path. fDi is the i-th path Doppler frequency shift caused by the relative motion of the transceivers. T is the duration of each OFDM symbol. λ is the delay spread index, and τi is the i-th path de- lay time normalized by sampling interval. Then the received signal is as follows:
y(n) = x(n) ∗ h(n) + w(n) (4)
where ′∗′^ denotes the convolution operation, w(n) is additive white Gaussion noise (AWGN).
Because of CP, linear convolution of the transmitted signal x(n) and the channel impulse response h(n) changes into circular convolution. At the receiver side, after perfect synchronization and CP removal, the received signal is de- modulated by DFT (FFT) with size of N. And then the signal on each subcarrier in frequency domain can be written as:
Y (k) = X(k)H(k) +W (k) (5)
where H(k) is the CFR of the subcarrier, W (k) is the AWGN with zero mean and variance σ^2 for the k-th subcarrier.
- Comb-Pilot Transform Domain Frequency Diversity Channel Esti- mation
3.1 Unification of Comb-Pilot Traditional
Transform Domain Channel Estimation
with Frequency Domain LS Approach
According to (5), the CFR of the pilot subcarrier based LS algorithm is given as [11]:
H˜p(k) = X∗ p (k)Yp(k) = |Xp(k)|^2 Hp(k) + X∗ p (k)Wp(k) = Hp(k) +W
′ p(k)^ (6)
where the subscript p indicates that this subcarrier is used for pilot and W ′ p(k)^ also follows AWGN distribution. Here, BPSK is employed for pilot, then |Xp(k)|^2 = 1. Actually, if the channel CIR taps are sample spaced, and the distance between two adjacent pilots is less than the coherence bandwidth, comb-pilot traditional transform do- main channel estimation and frequency domain LS approach perform equivalently. With the assumption that channel taps are sample spaced, the mean square error (MSE) of LS ap- proach is written as [7]:
MSE =
β SNR
where SNR = E{|xk|^2 }/σ^2 is the average signal to noise ra- tio. Here, xk is transmitted signal on each subcarrier. We assume that all the tones are distributed independently and use the same constellation. β is a constellation factor and expressed as β = E{|xk|^2 }E{|xk|−^2 }. For BPSK and QPSK, β = 1 , for 16QAM, β = 17 /. Traditional comb-pilot transform domain channel esti- mation algorithm is expressed as [12]:
HDFT =
NP/NFDFHp H˜LS (8)
where F is the unitary FFT matrix with elements e−^ j^2 πik/N^. (·)H^ denotes conjugate-transpose operation. FHp is the uni- tary IFFT matrix with elements e j^2 πik/Np^ /Np and H˜LS is the
500 L. LIU, C. TAO, J. QIU, X. QI, A NOVEL COMB-PILOT TRANSFORM DOMAIN FREQUENCY DIVERSITY CHANNEL...
where 1 ≤ i ≤ Np/2. It can be found that Xp 0 and Xp 1 correspond to the pilots in the odd location and even loca- tion respectively. According to these two grouped pilots, we can apply them for two channel estimators respectively with comb-pilot traditional transform domain channel estimation. When both the CFRs are obtained, we can combine them with some combining rules, such as equiprobability rule. This result is as follows:
HNew = αHT F 0 + ( 1 − α)HT F 1 (19)
where HNew is the combined channel estimation proposed by us, HT F 0 and HT F 1 are the odd and even group channel estimations, α is the weight of the odd group channel estima- tion. Because of unification of comb-pilot traditional trans- form domain channel estimation with frequency domain LS approach, formula (19) becomes:
HNew = H + e + αW 0 + ( 1 − α)W 1 (20)
where W 0 and W 1 are independent AWGN with zero mean and variance σ^2 respectively. When α = 1 /2 which means equiprobability, (20) can be expressed as:
HNew = H + e +
(W 0 + W 1 ). (21)
It can be found that (W 0 + W 1 )/2 is also AWGN, but with zero mean and variance σ^2 /2. Because of unification which has been analyzed above, MSE of our proposed fre- quency diversity channel estimation can be obtained as:
MSE =
β SNR
This is an exciting result. It shows that when all the pilots are divided into two groups, we can say that the pilots can use two frequency diversity to estimate the channel. If the distance between two adjacent pilots in sub-group is less than the coherence bandwidth of the channel, the diversity gain of MSE will be 3 dB. When four frequency diversity is applied, the diversity gain of MSE will be 6 dB. When eight frequency diversity is used, the diversity gain of MSE will be 9 dB.
Theoretically, regardless of complexity, when the dis- tance between adjacent two pilots in sub-group is less than the coherence bandwidth of channel, L frequency diversity will bring 10log( 1 /L) dB diversity gain of :
MSE =
L
β SNR
This diversity gain of MSE will improve bit error rate (BER) performance greatly. Furthermore, another merit of (23) is that it employs no prior statistical characteristics of the multipath wireless channel.
- Simulation Results Our OFDM system parameters in simulation are listed in Tab. 1. Here, the virtual subcarriers and DC tone are ig- nored. The multipath channel is modeled as a tapped de- lay line structure. The delay time and average gain follow ITU-R vehicular A channel model listed in Tab. 2 [13]. The central carrier frequency is set to 2.3 GHz. Fig. 2 and Fig. 3 present the MSE and BER perfor- mance of our proposed method when L = 2 and L = 4 and the traditional transform domain estimation vs SNR. The aster- isk and the triangle line show the simulation results on con- dition corresponding to Case 1 and Case 2 in Tab. 1. In both cases, the amount of the pilots is not larger than the length of CP. The result of circle line shows the performance of our proposed algorithm corresponding to frequency diversity L = 2 (Np = 64 , ∆P = 16 , Np 2 = 32 , ∆P 2 = 32 ). The result of square line shows the performance of our proposed algo- rithm corresponding to frequency diversity L = 4 (Np = 128 , ∆P = 8 , Np 2 = 32 , ∆P 2 = 32 ). In Fig. 2 and Fig. 3, it can be found that according to the traditional transform do- main estimation, when pilot amount is not larger than the length of CP, MSEs without frequency diversity perform the same as LS algorithm MSE = 1 /SNR in (7). We can see that no improvement is achieved even if pilot amount is increas- ing. When frequency diversity L = 2 , it can be found that diversity gain of MSE is 3 dB compared to the traditional method. Diversity gain of MSE is 6 dB corresponding to frequency diversity L = 4. These gains improve the perfor- mance of BER, when L = 2 about 1.5 dB and when L = 4 more than 2 dB in Fig. 3. Fig. 4 and Fig. 5 present the MSE and BER perfor- mance of our proposed method when L = 8 and the tra- ditional transform domain estimation vs SNR. Here, the amount of pilots is larger than the length of CP. The square line corresponding to the result shows the performance of
Bandwidth 10 MHz FFT/IFFT 1024 Length of CP 128 OFDM symbol duration 115.2 μs Pilot/Data Modulation BPSK( β = 1) Np Case 1: 64 Case 2: 128 Case 3: 256 ∆P Case 1: 1024/64 = 16 Case 2: 1024/128 = 8 Case 3: 1024/256 = 4 Tab. 1. OFDM system parameters. Tap Relative Delay (ns) Average Power (dB) 1 0 0. 2 310 -1. 3 710 -9. 4 1090 -10. 5 1730 -15. 6 2510 -20. Tab. 2. ITU-R vehicular A channel model.
RADIOENGINEERING, VOL. 18, NO. 4, DECEMBER 2009 501
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10 -
10 -
10 -
10 0
SNR dB
MSE
Traditional DFT Estimator with 'P = 8 Traditional DFT Estimator withL=2 Liu proposed Estimator 'P = 16 L=4 Liu proposed Estimator
Fig. 2. MSE of the proposed channel estimation with different diversity and traditional transform domain estimation vs SNR.
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SNR dB
BER
Traditional DFT Estimator with 'P = 8 Traditional DFT Estimator with 'P = 16 L=2 Liu Proposed Estimator L=4 Liu Proposed Estimator Perfect Channel Estimation
Fig. 3. BER of the proposed channel estimation with different diversity and traditional transform domain estimation vs SNR.
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SNR dB
MSE
Traditional DFT Estimator with 'P = 4 L=8 Liu Proposed Estimator
Fig. 4. MSE of the proposed channel estimation with diversity L = 8 and traditional transform domain estimation vs SNR.
our proposed algorithm corresponding to frequency diversity of L = 8 (Np = 256 , ∆P = 4 , Np 8 = 32 , ∆P 8 = 32 ). The di- amond line shows the traditional method. Here, when the amount of total pilots exceeds the length of CP, traditional transform domain estimation can employ noise suppression in time domain as zeroing the elements corresponding to those whose index exceeds CP length. It can be found that our proposed algorithm can also work better than this in noise suppression. However, when L = 8 , the complexity of our proposed algorithm will be increased. In such case, a trade-off should be considered to obtain satisfying perfor- mance while maintaining the system complexity.
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BER
Traditional DFT Estimator with 'P = 4 L=8 Liu Proposed Estimator Perfect Channel Estimation
Fig. 5. BER of the proposed channel estimation with diversity L = 8 and traditional transform domain estimation vs SNR.
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BER
L=2, D=1/2, Liu proposed Estimator L=4, D=1/4, Liu proposed Estimator L=8, D=1/8, Liu proposed Estimator perfect channel estimation
Fig. 6. MSE of the proposed channel estimation with diversity L = 2 , L = 4 , L = 8vs SNR.
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MSE
L=2, D=1/2, Liu proposed Estimator L=4, D=1/4, Liu proposed Estimator L=8, D=1/8, Liu proposed Estimator
Fig. 7. BER of the proposed channel estimation with diversity L = 2 , L = 4 , L = 8 vs SNR.
Fig. 6 and Fig. 7 give the MSE and BER performance with diversity L = 2 , L = 4 , L = 8 vs SNR. We can conclude that our proposed estimator is better with increasing diver- sity because of better noise suppression. When L = 8, the BER gap between the perfect channel estimation is less than 0.8 dB. Additionally, it should be pointed out that in Fig. 4 it can be found that error floor occurs in MSE and BER perfor- mance of our proposed algorithm when SNR exceeds 22 dB. This is caused by not sample spaced channel taps as men- tioned above. When L = 2 and L = 4 this problem also exits, but does not stand out apparently when SNR is not high. This
