Channel Estimation methods with low complexity for 3GPP/LTE Moussa Diallo1 and Maryline Hélard2 1
University Cheikh Anta Diop, BP 5005 Dakar Fann Dakar Sénégal
[email protected] 2 INSA Rennes, 20 Avenue des Buttes de Coesmes, CS 70839, 35708 Rennes Cedex3 France
[email protected]
RÉSUMÉ. Les techniques d’estimation de canal, basée sur des symboles pilotes, par passage dans un domaine de tranfert sont très attractives pour les systèmes de télécommunications utilisant l’OFDM. Cependant, elles montrent des limites pour les systèmes de télécommunications, où un ensemble de sous-porteuses de garde, est inséré sur les bords du spectre dans le but d’éviter tout recouvrement spectral avec d’autres applications utilisant des bandes voisines. Ces sous-porteuses de garde ont selon leur nombre tendance à dégrader fortement les performances de ces estimateurs. Nous proposons, dans un premier temps, une optimisation qui permet d’améliorer considérablement les performances de ces estimateurs quels que soit le nombre de porteuses de garde. Dans un second temps, pour de rendre l’estimateur proposé attractif pour les constructeurs, nous avons proposé une technique permettant de réduire leur complexité de réalisation de manière notable. ABSTRACT. OFDM based pilots channel estimation methods with processing into the transform domain appear attractive owing to their capacity to highly reduce the noise component effect. However, in current OFDM systems, null subcarriers are placed at the edge of the spectrum in order to assure isolation from interfering signals in neighboring frequency bands; and the presence of these null carriers may lead, if not taken into account, to serious degradation of the estimated channel responses due to the “border effect” phenomenon. In this paper an improved algorithm based on truncated SVD is proposed in order to correctly support the case of null carriers at border spectrum. A method for optimizing the truncation threshold whatever the system parameters is also proposed. To make the truncated SVD channel estimation method applicable to any SISO or MIMO OFDM system and whatever the system parameters, a complexity reduction algorithm based on the distribution of the power in the transfer matrix (based on DFT or DCT) is proposed. MOTS-CLÉS : systèmes multi porteuses, systèmes multi antennes, Estimation de canal, complexitè, 3GPP/LTE KEYWORDS : OFDM, MIMO, Channel estimation, SVD, complexity, 3GPP/LTE
Received, November 27, 2012 Revised, June 07, 2014 Accepted, October 24, 2014
ARIMA Journal, vol 18 (2014), pp. 93-116
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1. Introduction The use of orthogonal frequency division multiplexing (OFDM) is now generalized in high data rate wireless communication systems. OFDM benefits from it capacity to mitigate inter-symbol interference (ISI) by adding to the OFDM symbol a time guard interval which is longer than the channel impulse response length (or channel delay spread) [1]. Also, frequency-selective channel has been converted to a finite-number of parallel flat channels in the OFDM system owing to the adoption of orthogonal multicarrier technique implemented by fast Fourier transform (FFT). In addition to this, the multicarrier nature of OFDM gives the capability for this technique to overcome the complexity of time equalization method by using a simple frequency equalizer per subcarrier. In coherent OFDM system, channel estimation has to be performed at the receiver side before the equalization that is carried out in the frequency domain (FD). In some papers, as in [2], authors propose a joint estimation of the channel response and the carrier frequency offsets (CFO). But here we focus only on the channel estimation, as in [3][4][5], and complexity reduction. In OFDM pilots based channel estimation, that does not require any knowledge of the statistics on the channels, the least square (LS) estimates of the channel response can be obtained by dividing the demodulated pilots signal by the known pilots symbol one in the FD [5]. But the accuracy of the LS estimates is degraded by the noise component. The optimal FD channel estimation technique is minimum mean square error (MMSE) that however needs the information of channel statistic to perform the auto-covariance matrix of the channel frequency response and signal to noise ratio. In addition to this, the associated computation complexity of the MMSE method is very high [6]. Transform domain channel estimation (TD-CE) methods are considered as one of the most promising alternative because it can provide very good results by significantly reducing the noise component on the LS estimated channel coefficients obtained in the frequency domain. These methods use discrete fourier transform (DFT) or discrete cosine transform (DCT). The DFT based method presents the best result in term of noise reduction. However, a “border effect” may occur [7] when null carriers are placed on the spectrum´s extremities in order to assure isolation from interfering signals in neighboring frequency bands as well as increasing the sampling frequency. This is a real problem because the vast majority of modern multicarrier systems contain null carriers at the spectrum extremities. Discrete cosine transform (DCT) is suggested instead of DFT for mitigating the impact of the “border effect”, owing to its capacity to reduce the high frequency components in the transform domain [5] at the price of a weaker noise reduction [8]. However, when the number of null carriers is important, even the DCT algorithm is not sufficient to reduce significantly the “border effect” [9]. The Karhunen-Loeve transform domain algorithm can also be used as in [10] to perform iteratively the channel smoothing. However the computation complexity of this method is very high. To correctly support the case of null carriers at border spectrum, an improved algorithm based on truncated singular value decomposition (TSVD) of the transform domain matrix is proposed for both DFT and DCT. A method for optimizing the truncation threshold whatever the system parameters is also proposed in this paper. This method allows both mitigation of the “border effect” and reduction of the noise component. The TSVD method presents very good results, but its complexity and latency may cause problems, when the number of subcarrier is large and notably in multi-input multioutput (MIMO) configuration where all the sub-channels between the antenna links have
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to be individually estimated. The combination of MIMO and OFDM technologies (MIMOOFDM) is now largely considered in the new generation of standards for wireless transmission [11] because MIMO system has the potential to obtain a diversity gain and to improve the system capacity [12]. Hence channel estimation method with low complexity for MIMO-OFDM system becomes a challenge. To make the TSVD channel estimation method applicable to MIMO-OFDM system, a complexity reduction algorithm based on the distribution of the power in the transfer matrix is proposed in this paper. The efficiency of this algorithm is demonstrated in 3GPP/LTE system context. The output of this paper is also a set of methods allowing the selection and adaptation of the most appropriate TD channel estimation, depending on the system parameters and the complexity/performance requirements. The paper is organized as follows. Section 2 describes the conventional TD-CE methods (DFT and DCT) and its weakness regarding the “border effect”. Then section 3 is dedicated to the TSVD concept and the optimization of the truncated threshold. Next the complexity of the TSVD channel estimation is studied in section 4. The complexity reduction algorithm and its efficiency are detailed in this last section.
2. CLASSICAL TD-CE The TD-CE based method has been proposed to improve the accuracy of the LS estimated channel response which is degraded by the noise component [5][13].
Figure 1. Smoothing process in TD-CE.
The Fig.1 shows the TD-CE smoothing (noise reduction) process. The frequency channel response, estimated by LS channel estimation method, is first converted into the transform domain by a transfer algorithm based on DCT or DFT. Note that the transform domain corresponds to the time domain in the case of DFT-based algorithm. Then a smoothing filter is applied, assuming that the useful channel power in this domain is concentrated within the first W samples. For a better understanding, note again that with the DFT-based algorithm, the useful power corresponds to the impulse response of the
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channel. After the smoothing, the inverse of the transfer algorithm is applied to return to the frequency domain.
2.1. Comparaison between DFT and DCT 2.1.1. Noise reduction in an ideal context : 2.1.1.1. DFT : The LS estimate is first converted into the time domain by the IDFT (inverse discrete Fourier transform) algorithm. The time domain channel response of the LS estimated channel can then be given by : √
T hIDF n,LS
=
1 N
∑N −1 k=0
Hk,LS ej
2πnk N
(1)
T = hIDF + ξnIDF T n T where ξnIDF T is the noise component in the time domain and hIDF is the IDFT of n the LS estimated channel without noise. The baseband time domain of the effective discrete channel response, also called impulse response, between the transmit antenna and the receive antenna under the multipath fading environments can be expressed as [6] :
h(n) =
L−1 ∑
hl δ(n − τl )
(2)
l=0
with L the number of paths, hl and τl the complex time varying channel coefficient and delay of the l-th path. T Therefore, hIDF can be further developed as follows : n √
T hIDF n
=
√
= The last term of (3)
∑N −1 k=0
e−j
N −1 ∑ k=0
1 N 1 N
2kπτ ∑N −1 ∑L−1 2πkn −j N l )ej N k=0 ( l=0 hl e ∑L−1 ∑N −1 −j 2πk (τl −n) N l=0 hl k=0 e
2πk N (τl −n)
verifies : {
e
−j 2πk N (τl −n)
(3)
N
n = τl
0
otherwise
=
(4)
where τl = 0, 1, ..., L − 1 and n = 0, ..., N − 1. T The time domain channel response of the LS estimate hIDF n,LS can thus be given by : { N hl=n + ξnIDF T n = 0, ..., L − 1 1 IDF T hn,LS = √ (5) N ξnIDF T otherwise Assuming W > L, we retrieve the channel taps as well as a reduction of the noise component by applying the smoothing filter of length W : { N hl=n + ξnIDF T n = 0, ..., W − 1 1 IDF T (6) hn,W = √ N 0 otherwise
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The optimum lenghtfor W is L in order to retrieve all the useful power and maximize the noise reduction. However, as L is not known by the receiver, W is usually set equal to CP (length of the cyclic prefix), which is longer than L in OFDM systems. That is why in the following, the smoothing window size is taken to be equal to CP (W = CP ) [6][13] for DFT based channel estimation. From (6), it can be observed that the DFT based channel estimation method decreases the noise power level by, N ∆(dB) = 10log( ) (7) CP After the smoothing process, a DFT algorithm is used to return in the frequency domain. 2.1.1.2. DCT : The DCT based channel estimator can be realized by replacing IDFT and DFT by DCT and IDCT respectively [5]. The transform domain channel response can be obtained by the following equation after performing DCT to the LS estimate. N hDCT n,LS = Vn
N −1 ∑ k=0
Hk,LS .cos
π(2k + 1)n 2N
(8)
where VnN is the coefficient of DCT which can take two different values, depending on the value of n. { √ 1/N n = 0 N (9) Vn = √ 2/N n ̸= 0 Similarly to the DFT case, hDCT n,LS can be divided into two parts : DCT hDCT + ξnIDCT n,LS = hn
(10)
The DCT method can be considered as the conventional DFT with twice as many points [5][14]. Thus from the multi-path channel characteristics and since the maximum channel taps L is within the cyclic prefix (CP), the transform domain channel response given by (10) would be concentrated in the 2CP first samples of hDCT n,LS . The noise level can then be reduced by using a smoothing filter of size W = 2CP and then (6) becomes : { DCT hn,LS 0 ≤ n ≤ W − 1 DCT hn,W = (11) 0 otherwise From 11, it can be observed that the DCT based channel estimation method decreases the noise power level by, N ∆(dB) = 10log( ) (12) 2CP From 7 and 12 we can conclude that the DCT based method abducts half as much noise as the DFT method. 2.1.2. “Border effect” in a realistic framework : In a realistic framework, only a subset of subcarriers (M ) is modulated among the N due to the insertion of null subcarriers at the spectrum extremities for band isolation from/to interfering signals that occupy neighboring frequency bands[15] as well as for increasing the sampling frequency [16]. The existence of these null subcarriers will directly impact the behavior of the channel estimation in the transform domain.
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2.1.2.1. DFT : For DFT based channel estimation, when M < N , the expression (4) becomes (see Appendix) : N +M −1 n = τl 2 M ∑ 2πk e−j N (τl −n) = (13) −j2π M (τl −n) 1−e 2πN n ̸= τl N −M k=
1−e
2
−j
N
(τl −n)
where τl = 0, 1, ..., L − 1 and n = 0, ..., M − 1. T The channel impulse response hIDF can therefore be rewritten in the following ij,n form : T = √1N × hIDF n ∑ −j2π M (τl −n) N 1−e M.hl=n + L−1 . n
