Parametric channel modeling based OFDM channel estimation

The Journal of China Universities of Posts and Telecommunications October 2014, 21(5): 1–8 www.sciencedirect.com/science/journal/10058885

http://jcupt.xsw.bupt.cn

Parametric channel modeling based OFDM channel estimation QING Hao-bo1 (

), LIU Yuan-an1, XIE Gang2, LIU Kai-ming1, LIU Fang1

1. Beijing Key Laboratory of Work Safety Intelligent Monitoring, Beijing University of Posts and Telecommunications, Beijing 100876, China 2. Beijing Key Laboratory of Network System Architecture and Convergence, Beijing University of Posts and Telecommunications, Beijing 100876, China

Abstract A simplified parametric channel estimation approach was proposed for orthogonal frequency division multiplexing (OFDM) systems. Based on parametric channel model, this algorithm is composed of two parts: the estimation of channel parameters and channel interpolation. The exponentially embedded family (EEF) criterion is exploited to determine the number of channel paths as well as the multipath time delays. Consequently, the channel frequency responses is acquired via the estimated parameters. Additionally, the authors’ scheme is computationally efficient owing to the needless of the eigenvalue decomposition or the estimation of signal parameters by the rotational invariance technique (ESPRIT). Simulations are provided to validate the performance of this algorithm from perspectives of the probability of correct estimation and the mean square error (MSE). It is demonstrated that this approach exhibits a superior performance over the existing algorithms. Keywords channel estimation, OFDM, mean square error, EEF criterion

1 Introduction The OFDM [1] is a promising multicarrier modulation technique for broadband wireless access. In OFDM, the available bandwidth is partitioned into a series of orthogonal subcarriers. The high-rate data is transformed into parallel low-rate streams, thereby increasing the symbol duration and eliminating the inter symbol interference. OFDM possesses the properties of high spectral efficiency, robustness to multipath delay spread, feasibility to combat frequency selective fading, adaptive modulation as well as power allocation. These features motivate OFDM as a standard for digital audio/video broadcasting, wireless local area network and the fourth generation mobile communication system. If non-coherent OFDM system is adopted, the system complexity will be reduced since differential demodulation does not require estimating the channel at the receiver. However, it suffers from 3 dB~4 dB performance loss Received date: 13-01-2014 Corresponding author: QING Hao-bo, E-mail: [email protected] DOI: 10.1016/S1005-8885(14)60323-X

compared to the coherent OFDM system [2]. On the contrary, the channel estimation [2–5] becomes a necessity for the coherent OFDM system to track the channel parameters. Moreover, the coherent OFDM system is capable of supporting higher data throughput by employing multilevel modulation schemes with non-constant amplitude, e.g. 16 quadrature amplitude modulation (QAM). This article deals with channel estimation for the coherent OFDM systems. Channel estimation techniques can be classified into non-parametric and parametric ones according to whether channel parameters (the number of channel paths, multipath time delays, etc.) are estimated at the receiver or not. Non-parametric channel estimation was widely investigated in literatures. Nevertheless, in absence of multipath structure, the performance is worse than that of parametric ones because the latter can effectively reduce the signal subspace dimension of channel taps. Parametric channel modeling based channel estimation was put forward in Ref. [6], which utilizes the minimum description length (MDL) principle to estimate the number of paths and then acquires the multipath delays using the

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ESPRIT. Inspired by Ref. [6], a similar approach was raised by introducing the hopping pilots pattern, resulting in a faster normalized mean square error convergence rate [7]. However, these two methods both need eigenvalue decomposition as well as ESPRIT, the computational load was quite heavy. In Ref. [8], a generalized Akaike information criterion (AIC) was exploited to estimate channel parameters through a recursive way. Although the eigenvalue decomposition and ESPRIT are eliminated, the recursion is still of high complexity. The Hannan-Quinn (HQ) information criterion was applied to parametric channel estimation with improved performance as well as low complexity [9]. The high-speed data transmission in a wireless environment potentially induces a sparse multipath fading channel, and then the number of multipath components was smaller than channel length [8]. The sparsity of a channel is measured by ratio of the time duration spanned by the multipaths to the number of taps [10]. Exploiting the sparsity of a multipath channel, a simplified parametric channel estimation method was proposed for OFDM systems. The proposed scheme firstly utilizes the EEF criterion to estimate the channel order and the multipath time delays [11–12]. The channel interpolation is then fulfilled by frequency-domain filtering on the channel impulses over pilot tones. The rest of the article is organized as follows. In Sect. 2, channel model for the OFDM system is described. Sect. 3 develops the proposed simplified parametric channel estimation scheme. Sect. 4 presents the complexity analysis. Simulations are presented in Sect. 5 and the last section comes to the conclusions. Notation: E [⋅ ] represents expectation and Ι stands for the identity matrix.

(⋅)

T

,

(⋅)

transpose, Hermitian transpose operations, respectively.

H

and and

(⋅)

†

denote the

pseudo-inverse

2 Parametric channel model The OFDM system relies upon the following assumptions. 1) The timing and frequency synchronization are perfect at the receiver. 2) The channel impulse response length is smaller than the cyclic prefix (CP) length of the OFDM symbol. 3) The channel is quasi-stationary, which means that the channel is constant within one OFDM symbol duration but

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varies from symbol to symbol. The parametric channel model is constructed by multipath components and is characterized by L

L

l =1

l =1

h(t ,τ ) = ∑ hl ( t ) δ (τ − τ l )= ∑ hl ( t ) δ (τ − ξl Ts )

(1)

where L denotes the number of channel paths, Ts = 1 B is the sampling interval of the receiver with B being the system bandwidth, τ l = ξl Ts is the time delay of the lth path with ξ l an integer, hl ( t ) represents the path gain and δ ( ⋅) stands for the Dirac’s delta function. It is assumed that different channel paths are independent and hl ( t ) is a wide-sense stationary Gaussian process with mean zero and variance normalized to one. A normalized time delay profile is plotted in Fig. 1, which consists of a 6-tap channel with maximum time delay 20 µs .

Fig. 1

The normalized delay profile

Considering an OFDM system, N subcarriers and M symbols constitute each frame. As shown in Fig. 2, the comb pilots are inserted in both time and frequency dimensions. In each OFDM symbol, P = N D pilots are transmitted on equally spaced tones, where D is the number of subcarriers between two adjacent pilots in the frequency dimension. The indexes of pilot tones in frequency dimension are denoted by ni = iD; i = 0,1,..., P − 1 (2) The transmission subcarrier is modulated by a data symbol X n , k and is appended a CP to avoid the intersymbol interference, where n denotes the subcarrier number and k is the symbol number. At the receiver, the CP is stripped off and the received data can be written as Yn , k = H n , k X n , k + θ n , k (3) where θ n , k is a Gaussian noise with mean zero and variance σ 2 . H n , k is the channel gain given by

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QING Hao-bo, et al. / Parametric channel modeling based OFDM channel estimation

L

H n , k = ∑ hl ( kTs ) e

−j

2πnξl N

(4)

l =1

3

correlation matrix of pilot frequency responses and Φk reflects the energy of pilot time-domain impulses. Theorem 1 Φk is a P × P diagonal matrix with the ith diagonal entry  2 σ2 ; i = ξl ; l = 0,1,..., L − 1 σ l + (9) φi =  2 P  σ ; i ≠ ξ ; l = 0,1,..., L − 1 l  P Proof The proof of Theorem 1 is seen in Appendix A. It can be drawn from Theorem 1 that the multiplicity of σ 2 P in the diagonal elements of Φk is P − L , while

σ l2 + σ 2 P corresponds to L. Since σ 2 P is smaller than σ l2 + σ 2 P , the number of channel taps can be Comb pilot pattern of OFDM signal

Fig. 2

3 Simplified parametric channel estimation scheme The proposed method two parts: estimation acquisition of channel composed of the number time delays, while the frequency responses.

conducts channel estimation in of channel parameters and coefficients. The former is of channel paths and multipath latter focuses on the channel

3.1 Principle analysis The pilot tones can be employed to attain the channel frequency responses on pilot subcarriers via the least squares (LS) criterion, Yn , k Hˆ ni , k = i = H ni , k + ε ni , k (5) X ni , k with ε ni , k = θ ni , k X ni , k . The aforementioned equation can be rewritten in vector notation as Hˆ k = H k + εk

(6) T

in which Hˆ k =  Hˆ n0 , k , Hˆ n1 , k ,..., Hˆ n , k  , H k =  H n0 , k , P −1   T

T

 and εk = ε n , k , ε n , k ,..., ε n , k  . 1 P −1   0  The autocorrelation matrix is denoted by R = E Hˆ Hˆ H

H n1 , k ,..., H n

k

{

k

P −1

,k

k

}

(7)

Subsequently, a new matrix is defined as 1 Φk = 2 Gɶ H Rk Gɶ (8) P where Gɶ is a P × P matrix with the ( p, q ) th element

e

− j2 πn p q N

( p, q = 0,1,..., P − 1)

. Physically,

Rk

is the

acquired by counting the multiplicity of the smallest diagonal elements, and multipath time delays are just the indices of the largest diagonal elements. Unfortunately, for practical reasons, there is no access to an accurate Φk since the autocorrelation matrix can only be estimated using the ergodic assumptions as 1 K −1 Rˆ k = ∑ Hˆ k Hˆ kH K k =0

(10)

Consequently, 1 Φˆ k = 2 Gɶ H Rˆ k Gɶ (11) P Due to the finite number of samples, it cannot be guaranteed that Φˆ k is a diagonal matrix, the multiplicity of the smallest diagonal elements is P − L . Therefore, relying on the multiplicity of the smallest diagonal entries of Φk fails with finite number of symbols K. Nevertheless, the characteristic that the

{ξl | l = 0,1,..., L − 1} th diagonal

elements of Φˆ k are larger than the others especially at a high signal-to-noise ratio (SNR) region can still be utilized to estimate the channel parameters. In fact, a similar problem concentrates on this issue in fields of model order selection as well as source number enumeration [12–16]. These references adopt the traditional information theoretic criteria (eg. AIC, MDL and HQ) to detect the multiplicity of noise variance with limited number of samples. In the recent years, the EEF criterion was raised as a model order selection method [11], which is resorted to the theory of EEFs and sufficient statistics. The authors in Ref. [12] proved its consistency and applied it to source enumeration. Motivated by these prior works, this article employs the EEF criterion to determine the number of channel taps in a sparse multipath fading environment.

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{ξl | l = 0,1,..., L − 1} th

3.2 Estimating the number of paths and delays Before elaborating the proposed algorithm, the authors give a brief review of the EEF criterion. The EEF criterion selects the model order via the maximum model. The ith model under the EEF criterion is formulated as   l    l  M EEF ( i ) = li − µi  ln i + 1  u  i − 1 (12) µ µ i     i   where µi is the number of free adjustable parameters,

u ( ⋅) stands for the unit step function with the following form

1; t≥0 u (t ) =  0; t < 0 li is the likelihood ratio defined as

( ) ( ) ˆ p ( x ; θ ,H )

(13)

p x ; θ ,H i 0

(14)

i in which x is the received data vector, θˆ ( ) is the 0 estimated parameters under model Hi and θˆ ( ) is the

estimated parameters under the reference model H0 . Physically, the EEF criterion extends the generalized likelihood ratio test to multiple alternative hypotheses, especially when the alternatives have different numbers of unknown parameters. With similar derivations as in Ref. [12], the EEF criterion can be further expressed as µi =i ( 2 P − i ) + 1 (15)

 i  1 P −1 ɶ   1 P −1   li = −2 K ln ∏ φɶj + ( P − i ) ln  φ j  − P ln  ∑ φɶj   ∑  P − i j =i   P j = 0    j = 0 is the diagonal element φ j

are

sorted index values.

3.3 Channel interpolation The channel interpolation is fulfilled through the frequency responses on pilot tones, which are acquired in Eqs. (5) and (6). The time-domain responses can be calculated using (18) hˆk = Gˆ † Hˆ k ˆ ˆ where G is a P × L matrix with the ( p, q ) th element − j2 πn p ξˆq N

( p = 0,1,..., P − 1; q = 0,1,..., Lˆ − 1) ,

hˆk =  hˆ0, k , 

T

0

where φɶj

diagonal elements of Φˆ k

larger than the others, the multipath delays ξˆl are just the

e

ˆ (i )

li =2 ln

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of Φˆ k

(16) in

descending order. Since the number of channel taps does not excess the number of pilots in sparse multipath fading environment, the number of channel paths can be achieved by the maximum value of Eq. (12) in accordance with the EEF criterion, namely (17) Lˆ = arg max M EEF ( i ) i = 0,1,..., P −1

After estimating the number of paths, we now turn to the multipath delays. The diagonal entries of Φˆ k are extracted firstly and are denoted as φ = [φ0 , φ1 ,..., φP −1 ] . T

Then the indices of the Lˆ largest values of φ are sorted in ascending order. Owing to the fact that the

hˆ1, k ,..., hˆLˆ −1, k  .  According to Eq. (A.7) in Appendix A, the following equation holds 1 (19) Gˆ † = Gˆ H P Thus Eq. (18) is further expressed as 1 (20) hˆk = Gˆ H Hˆ k P With time-domain channel, the frequency-domain channel on all subcarriers are attained as (21) Hˆk = Fhˆk T

in which Hˆk =  Hˆ 0, k , Hˆ 1, k ,..., Hˆ N −1, k  and F is a modified fast Fourier transform (FFT) matrix retaining only those columns associated with the estimated channel tap positions. In other words, F is a N × Lˆ matrix with the ( p, q ) th element e − j2 πpq N ( p = 0,1,..., N − 1; q = 0,1,

)

..., Lˆ − 1 .

3.4 Remarks on the proposed scheme 1) In the system model, path delays are assumed to be sample spaced. That is to say, the channel taps fall at the sampling instants of the receiver. For the nonsample spaced cases, energy leakage is induced, which brings about the error floor. This problem can be settled through oversampled systems or windowing techniques. 2) The proposed scheme utilizes the LS criterion to perform frequency-domain channel estimation on pilot tones. Some better criteria can be applied to improving the performance, such as minimum mean square error (MMSE),

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QING Hao-bo, et al. / Parametric channel modeling based OFDM channel estimation

linear minimum MSE. While carrying out channel interpolation, the time dimension correlation using a finite impulse response filter can be exploited to further enhance the performance. 3) To evaluate the performance of this approach, two metrics are considered: probability of correct estimation and MSE. The former aims to scale the rate that the channel parameters are successfully estimated, the latter measures the estimation accuracy of channel coefficients. Specifically, the MSE is defined as

{

∆MSE = E H n , k − Hˆ n , k

2

}

(22)

Correct estimation indicates that all the channel parameters are perfectly achieved when the following equation holds  L = Lˆ (23)  ξ1 ,ξ 2 ,..., ξ L =ξˆ1 ,ξˆ2 ,..., ξˆL 

4.2

5

Complexity of the hopping pilots based scheme

The steps of the hopping pilots based algorithm [7] are approximately the same as the MDL one [6]. The hopping pilots are exploited to redesign the pilot pattern. Thus the complexity is approximately identical to the MDL method. 4.3

Complexity of the proposed scheme

Before analyzing the complexity, the detailed steps of the proposed algorithm are summarized as follows. Step 1 Estimate the channel frequency responses on the pilot subcarriers via Eq. (5). Step 2 Compute the autocorrelation matrix Rˆ k and Φˆ using Eqs. (10) and (11). k

Step 3

Extract the diagonal elements of Φˆ k and sort

them in descending order as φ = [φ0 , φ1 ,⋯ , φP −1 ]

T

and

T

4 Complexity analysis In this section, the complexity of the algorithm, MDL based scheme [6], hopping pilots based method [7] and HQ based algorithm [9] was analyzed. This section mainly focuses on the dominant parts that affect the computational load. The relatively minor operations are neglected. 4.1

Complexity of the MDL based scheme

The MDL based scheme [6] computes the sample correlation matrix and then performs eigenvalue decomposition. In order to determine the number of channel paths, the eigenvalues are employed to calculate the MDL criterion. Subsequently, the ESPRIT is adopted to acquire the multipath time delays. The main computational burden comes from the calculation of sample correlation matrix and eigenvalue decomposition. The former requires O ( P2 K ) operations since the P ×1 pilot frequency responses are averaged over K frame length. The latter demands O ( P 3 ) operations for a P × P sample correlation matrix. In addition, when performing the ESPRIT, the inverse operation of matrix is necessary, and the singular value decomposition is the most numerically stable way. Therefore the complexity of the MDL based scheme would be at least O ( P 2 K ) + O ( P 3 ) .

φɶ = φɶ0 , φɶ1 ,⋯ , φɶP −1  , respectively. Step 4 Determine the number of channel taps through the EEF criterion Eqs. (12)–(17). Step 5 Sort the indices of the Lˆ largest values of φ in ascending order. The sorted index values correspond to the estimated multipath time delays ξˆl .

Step 6 Perform channel interpolation to recover the frequency dimension responses on all tones by Eqs. (20) and (21).

It is observed that the algorithm needs to estimate the autocorrelation matrix, which calls for O ( P 2 K ) operations. Neither the eigenvalue decomposition nor inverse operation of matrix is required compared to other algorithms, saving much computational cost. Thus the proposal only needs some basic mathematical calculations, i.e. addition/subtraction, multiplication/ division, sorting/comparison. Thereby the conclusion can be drawn that the new method is computationally attractive. 4.4

Complexity of the HQ based scheme

The HQ based algorithm [9] employs the HQ criterion to detect the channel length and the tap positions. The diagonal entries of the energy matrix, instead of its eigenvalues, are adopted to constitute the HQ criterion. Therefore, the eigenvalue decomposition is avoided. Furthermore, it does not involve the inverse operation of matrix, either. In general, the complexity is identical to our

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scheme.

5 Numerical results

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estimation of channel parameters. Meanwhile, the proposal is superior to other algorithms in terms of the probability of correct estimation and the MSE.

In this section, the performance of the scheme is verified through simulations. A 16QAM-OFDM system operates over a bandwidth of 10 MHz with central frequency 5 GHz. The sample period is 0.1 µs . The total bandwidth is divided into 512 subcarriers, among which 64 subbands are CP samples. 64 pilot tones are chosen from the quadrature phase-shift keying constellation with unit power. COST207 typical urban 6-path channel environment with multipath time delays [0, 0.2, 0.5, 1.6, 2.3, 5.0] µs [17] was taken into account. The maximum Doppler frequency 139 Hz is used to represent the mobile environment. To further demonstrate the performance, the MDL criterion [6], hopping pilots [7] and HQ criterion [9] are adopted for performance comparison. The probability of correct estimation and MSE are two key metrics to evaluate the performance of each scheme. 5.1

Performance under varying channel conditions

Firstly, the performance under varying channel conditions is validated, when the number of symbols is fixed at 100. Figs. 3 and 4 illustrate the two metrics versus SNR. Fig. 3 depicts that all the algorithms are able to correctly estimate the channel parameters at high SNR regime.

Fig. 4

MSE of various algorithms vs. SNR

The raw bit error rate (BER) of each algorithm is plotted in Fig. 5. It can be observed that this approach marginally outperforms other algorithms. A nearly 1 dB SNR improving appears at BER of 10 −2 when compared to the worst MDL criterion. If the coding theory, e.g. Turbo, low density parity check code (LDPC), Reed-solomon (RS) code, is adopted in the simulation setup, the performance gap will be promoted to a large extent.

Fig. 5 BER-performance of various algorithms vs. SNR

5.2

Fig. 3 Probability of correct estimation of various algorithms vs. SNR

Nevertheless, the performance deteriorates once the SNR degrades, since channel path with low power will be overwhelmed by the background noise. Similar conclusions can be reached in Fig. 4. Following the decrease of SNR, the performance of parametric channel estimation methods worsens because of inaccurate

Performance with different OFDM symbols

Since the number of symbols affects the accuracy of autocorrelation matrix and further the estimation of channel parameters, this subsection focuses on the impacts of frame length where 20 dB SNR is adopted. As shown in Fig. 6, when the frame length is limited, both our approach and the HQ based scheme are capable of successfully estimating the channel parameters whereas the others fail. It follows from Fig. 7 that once the channel parameters are correctly attained, the MSE of each algorithm converges to

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an ideal constant.

7

demonstrate that the performance of the proposed scheme is significantly enhanced compared to other algorithms. Acknowledgements This work was supported by the National Natural Science Foundation of China (61302083), the National Science and Technology Major Project (2010ZX03003001-004).

Appendix A Proof of Theorem 1 Taking a further look at Eq. (4),

Hk

transformed into H k = Ghk Fig. 6 Probability of correct estimation of various algorithms vs. frame length

(A.1)

in which hk =  h0, k , h1, k ,..., hL −1, k  matrix whose

can be

( p, q ) th

T

and G is a P × L

element is e

− j2 πn p ξq N

( p = 0,1,

..., P − 1; q = 0,1,..., L − 1) . Then the autocorrelation matrix can be derived as Rk = E Hˆ k Hˆ kH = E { H k H kH } + E {εk εkH } = GΨ k G H + σ 2 I P

{

}

where Ψ k = E {hk h

H k

}

(A.2) is a L × L diagonal matrix with

the lth diagonal element ϕl , k = σ l2 since different channel

Fig. 7

MSE of various algorithms vs. frame length

Combined with complexity analysis in the previous section, the conclusion is reached that, compared to the MDL and hopping pilots based methods, the proposed approach and the HQ based algorithm enjoy not only a reduced computational load, but also a better performance. Moreover, under circumstance of identical complexity, the scheme offers a performance enhancement over the HQ based algorithm. In general, the proposed is a favorable choice when conducting parametric channel estimation.

6 Conclusions In this article, a simplified parametric channel estimation approach for OFDM systems is put forward. The proposed scheme firstly estimates the number of channel paths as well as multipath time delays via the EEF criterion. Thereby channel frequency responses can be acquired through these parameters. Our approach does not require EVD or ESPRIT, thus the computational complexity is quite attractive. Numerical results

paths are uncorrelated. Owing to the sparsity of multipath fading channel, a T new vector hɶk =  hɶ0, k , hɶ1, k ,..., hɶP −1, k  is constructed, in which  hi , k ; i = ξl ; l = 0,1,..., L − 1 (A.3) hɶi , k =  0; i ≠ ξl ; l = 0,1,..., L − 1 Accordingly, Eq. (A.1) is equivalent to the following form ɶɶ (A.4) H k = Gh k where Gɶ is a P × P matrix whose ( p, q ) th element is

e

− j2 πn p q N

( p, q = 0,1,..., P − 1) .

Consequently, the autocorrelation matrix in Eq. (A.2) can be transformed as ɶ ɶ Gɶ H + σ 2 I Rk = GΨ (A.5) k P H where Ψɶ = E hɶ hɶ is a P × P diagonal matrix with k

{

k

k

}

the ith diagonal element σ 2 ; i = ξ l ; l = 0,1,..., L − 1 (A.6) ϕɶi , k =  l 0; i ≠ ξl ; l = 0,1,..., L − 1 Since pilots are evenly spaced in the frequency domain, the ( p, q ) th entry in Gɶ can be rewritten as e − j2 πpq P

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according to Eq. (2). Therefore, Gɶ is equivalent to

P P in which P is the P × P FFT matrix whose

( p, q ) th

element is (1

As a result, ɶ ɶ H = Gɶ H Gɶ = PI GG

P )e

− j2 πpq P

( p, q = 0,1,..., P − 1) .

P

(A.7)

Substituting Eqs. (A.5) and (A.7) into Eq. (8), the following derivations are obtained 1 1 ɶ ɶ Gɶ H Gɶ + 1 Gɶ Hσ 2 I Gɶ = Φk = 2 Gɶ H Rk Gɶ = 2 Gɶ H GΨ k P P P P2

σ Ψɶ k + (A.8) IP P Consequently, Φk is a P × P diagonal matrix with 2

the ith diagonal entry given by  2 σ2 ; i = ξl ; l = 0,1,..., L − 1 σ l + φi =  2 P  σ ; i ≠ ξ ; l = 0,1,..., L − 1 l  P

(A.9)

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(Editor: WANG Xu-ying)