Sparse channel estimation in OFDM systems by threshold-based pruning J. Oliver, R. Aravind and K.M.M. Prabhu A threshold-based procedure to estimate sparse channels in an orthogonal frequency division multiplexing (OFDM) system is proposed. An optimal threshold is derived by maximising the probability of correct detection between significant and zero-valued taps estimated by the least squares (LS) estimator. Improved LS estimates are obtained by pruning the LS estimates with the statistically derived threshold.
Introduction: Channel estimation is one of the most important tasks in orthogonal frequency division multiplexing (OFDM) receivers. The purpose of channel estimation is to perform coherent data detection [1]. Procedures for channel estimation require length of the channel impulse response (CIR), i.e. the number of multipath components. However, practical wireless channels are sparse, i.e. only few of the multipath components have significant energy. In sparse channels, not only the length of the CIR, but also the position of significant multipath components will help to improve estimation accuracy [2]. The most significant channel tap (MST) approach is used in [2] to identify the significant channel taps. To apply MST we need channel length. Also in [2], a threshold is suggested based on simulation. A generalised Akaike information criterion (GAIC)-based sparse channel estimation is proposed in [3], which is iterative in nature. The GAIC-based method needs to estimate channel length in order to detect the position of significant taps. A threshold-based method is used in [4] to estimate the number of multipaths. We propose a novel, non-iterative, threshold-based modified least squares (TmLS) algorithm. We derive the threshold by maximising the probability that the threshold can distinguish between significant and zero-valued taps estimated by the LS estimator. Pruning the LS estimates with the threshold results in improved LS estimates. The added advantage of the proposed method over the other methods [2, 3] is that CIR length is not required to detect the position of significant taps. Channel estimation in OFDM: Consider an OFDM system with N subcarriers operating over a Rayleigh fading channel. Assuming perfect synchronisation, the demodulated data samples for a block-type pilot symbol can be expressed as [3] pffiffiffiffi Y ¼ X Fh N þ n ð1Þ where X is a diagonal matrix of pilots, F is an N N unitary discrete Fourier transform matrix and h is the CIR. The noise vector n has the statistics n CN (0, s2nIN). We have considered sample-spaced channels, in which the multipath delays are integers. For non-sample spaced channels (non-integer delays), TmLS may not be useful. However, sparse sample-spaced channels in which only few of the multipath components are significant, may still be realised with oversampled OFDM systems [3] and TmLS can be useful for such channels too. The LS estimator for the CIR is [3] pffiffiffiffi ð2Þ h^ LS ¼ F 1 X 1 Y ¼ h N þ F1 X 1 n
(3). In general, fvig are zero-mean complex Gaussian random variables with variance b/SNR. For M-ary PSK, fvig are independent but for M-QAM they are dependent [3]. Equation (3) reveals two facts. First, the zero-valued taps in the original sparse channel h are now occupied by the noise values from the vector v. By detecting and eliminating these insignificant taps within Lcp , we can improve the MSE of mLS. Secondly, the significant taps in h are corrupted only by noise values from v. Notice the first Lcp values (gˆ ) in the mLS estimator of the sparse p channel h. The estimates, gˆ is, are either vi or hl N þ vl , where i, l ¼ 0, 1, . . . , Lcp 21. The proposed method involves comparing the instantaneous energy of gˆ i to the threshold T. If the instantaneous energy of a tap is less than T, then it is a zero-valued tap, else it is a significant tap. Our aim is to find T such that, on the average, it should be greater than thep instantaneous value jvij2 and less than the instantaneous value jhl N þ vlj2. Mathematically, our goal is to choose T such that n o pffiffiffiffi ð4Þ PT ¼ Pr jvi j2 T , jhl N þ vl j2 is maximum. This is equivalent to minimising the probability of error in the binary p hypothesis testing problem [5]. Note that vi CN (0, b/SNR) and hl N þ vl CN (0, NPl þ b/SNR), where Pl ¼ E [jhlj2] is the average power of the lth channel tap. Take U ¼ jvij2 and W ¼ jhl p N þ vlj2, then U exp (b/SNR) and W exp (NPl þ b/SNR). Since U and W are independent (for BPSK pilots), the required probability in (4) is given by T T PT ¼ 1 exp exp ð5Þ NPl þ b=SNR b=SNR The optimal (according to (4)) T can be obtained by maximising (5) for a particular SNR and is given by b NPl Topt ¼ arg max PT ¼ In 2 þ ð6Þ T SNR b=SNR The value of b for BPSK is 1 and for 16-QAM it is 17/9. The noise variance s2n can be estimated as in [3]. By pruning the LS estimates gˆ i with Topt , the insignificant taps can be eliminated. Evidently, for pruning, estimate of channel length is not required. Simulation results and discussion: The OFDM system we consider has 64 subcarriers with Lcp ¼ 16 samples. The modulation scheme is BPSK as used in [6] for the pilot subcarriers. The channel is assumed to be time invariant over one OFDM symbol. Exponential channel power delay profile is assumed with delay spread of about 1/4 of Lcp , i.e. trms ¼ 4 samples [1]. A three-tap sparse channel h with tap positions randomised between 0 and Lcp 2 1 is used. −5
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where gˆ represents the first Lcp values in the LS estimator and 0 is a zero vector of length N 2 Lcp. The MSE of the mLS estimator is given by MSELcp ¼ (Lcp/N )(b/SNR) [4]. Since in practice L Lcp , the MSE with known true channel length L is MSEL ¼ (L/N )(b/SNR) MSELcp. Also, in sparse channels, only S out of L taps are significant, and by detecting these taps our proposed method can achieve an MSE of (S/N )(b/SNR) , MSEL MSELcp. Proposed sparse channel estimation algorithm: Consider a sparse channel h of length Lcp. The mLS estimate of this sparse channel is given by (3). Let v ¼ F21X21n be a noise vector of length Lcp from
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The mean square error (MSE) of the LS estimator for an individual channel tap is given by MSELS ¼ b/SNR, where SNR ¼ E [jXkj2]/s2n, and b is a signal constellation dependent factor [4]. For samplespaced channels with known length, the modified least squares (mLS) proposed in [1] can be used to improve the LS estimates. Assuming Lcp as channel length, the mLS estimator of the CIR is given by pffiffiffiffi 1 1 g^ F X n h N h^ m ¼ ¼ þ ð3Þ 0 0 0
−25 −30 −35 −40 −45 0
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Fig. 1 MSE performance comparison for different sparse channel estimators
The methods considered for channel estimation are conventional mLS [1], MST [2], GAIC [3] and the proposed TmLS. The threshold used for MST in our simulations is 22 dB below s2n as recommended in [2]. The performance of different estimators is compared in terms of the MSE defined as MSE ¼ (1/N) E [kh 2 hˆ k2]. The MSE of different estimators is averaged over 5000 channel realisations for each SNR and is plotted in
ELECTRONICS LETTERS 19th June 2008 Vol. 44 No. 13
Fig. 1. TmLS has 5 dB lower MSE than mLS at low SNRs and 7 dB lower at high SNRs, whereas methods in [2] and [3] achieve 3.5 and 5 dB in low and high SNRs, respectively. In effect, the proposed method achieves 2 dB improvement compared with existing methods [2] and [3]. Moreover, the bottom curve in Fig. 1 indicates the performance when the exact positions of significant taps are detected. The proposed method achieves this lower bound from the SNR of 12 dB onwards. To further illustrate the performance of TmLS, Fig. 2 shows the average probability of correct detection of the significant taps by employing MST, GAIC and TmLS. It is apparent from Fig. 2 that TmLS has superior detection capabilities when compared to the existing methods. Based on these results we conclude that the proposed TmLS scheme is effective in detecting the position of the significant taps without knowledge of channel length.
Conclusion: We have presented a novel threshold-based mLS (TmLS) scheme for the estimation of slowly time-varying sparse multipath channels. The threshold is derived by maximising the probability that it can distinguish between taps with significant and small energy. The effectiveness of the proposed scheme is demonstrated by comparing its performance with existing schemes. # The Institution of Engineering and Technology 2008 16 April 2008 Electronics Letters online no: 20081089 doi: 10.1049/el:20081089 J. Oliver, R. Aravind and K.M.M. Prabhu (Department of Electrical Engineering, Indian Institute of Technology Madras, Chennai-600 036, India) E-mail:
[email protected] References
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1 Van de Beek, J.J., Edfors, O., Sandell, M., Wilson, S.K., and Borjesson, P.O.: ‘On channel estimation in OFDM systems’. Proc. IEEE VTC, Chicago, IL, USA, 1995, Vol. 2, pp. 815– 819 2 Minn, H., and Bhargava, V.K.: ‘An investigation into time-domain approach for OFDM channel estimation’, IEEE Trans. Broadcast., 2000, 46, (4), pp. 240– 248 3 Raghavendra, M.R., and Giridhar, K.: ‘Improving channel estimation in OFDM systems for sparse multipath channels’, IEEE Signal Process. Lett., 2005, 12, (1), pp. 52– 55 4 Kang, Y., Kim, K., and Park, H.: ‘Efficient DFT-based channel estimation for OFDM systems on multipath channels’, IET Commun., 2007, 1, (2), pp. 197–202 5 Kay, S.M.: ‘Fundamentals of statistical signal Processing, Vol. II, Detection theory’ (Prentice-Hall, 1993) 6 Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications, IEEE Standard 802.11, 2007 (Revision of IEEE Standard 802.11, 1999)
Fig. 2 Average probability of correct detection against SNR for TmLS and existing schemes
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