OFDM symbol detection integrated with channel multipath gains estimation for doubly-selective fading channels

Accepted Manuscript OFDM symbol detection integrated with channel multipath gains estimation for doubly-selective fading channels Wang Yi, Harry Leib PII: DOI: Reference:

S1874-4907(16)30183-5 http://dx.doi.org/10.1016/j.phycom.2016.10.003 PHYCOM 345

To appear in:

Physical Communication

Received date: 4 May 2016 Revised date: 5 September 2016 Accepted date: 22 October 2016 Please cite this article as: W. Yi, H. Leib, OFDM symbol detection integrated with channel multipath gains estimation for doubly-selective fading channels, Physical Communication (2016), http://dx.doi.org/10.1016/j.phycom.2016.10.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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OFDM Symbol Detection Integrated with Channel Multipath Gains Estimation for Doubly-Selective Fading Channels Wang Yi, Harry Leib Abstract Orthogonal Frequency Division Multiplexing (OFDM) is a technique for wideband transmission that is commonly used in modern wireless communication systems because of its good performance over frequency selective channels. However OFDM systems are sensitive to channel time variations resulting in Inter-Carrier Interference (ICI), that without suitable detection methods can degrade performance significantly. Channel State Information (CSI) is essential to various OFDM detection schemes, and its acquisition is a critical factor over time varying channels. This work considers a Kalman filter channel multipath gains estimation technique for time varying environments, integrated with a novel detection scheme for OFDM based on a Sphere Decoding (SD) algorithm derived to exploit the banded structure of the channel matrix. This combined scheme employs decision-feedback from the SD requiring only a low pilot symbol density, and hence improves bandwidth efficiency. Three techniques for integrating the Kalman filter operating in decision-feedback mode, with SD data detection that produces these decisions, are considered in this paper. When compared with other competing schemes, this integrated symbol detection and channel multipath gains estimation approach for OFDM provides performance advantages over time varying channels. Furthermore, it is shown that for moderate Doppler shifts the degradation that carrier phase noise induces in this scheme is small. Index Terms: OFDM, Kalman filtering, Sphere decoding, Channel estimation, Wireless communication.

I. I NTRODUCTION Orthogonal Frequency Division Multiplexing (OFDM) is a commonly used transmission technique for frequency selective channels [1]. Over the years OFDM became a basic technology for broadband services that is used in Digital TV systems [2], WiMAX [3] and 4G LTE-Advanced [4]. Many OFDM systems operate in environments that can be characterized as doubly-selective channels, experiencing frequency selectivity as well as time variations [5] [6]. Symbol detection in OFDM systems over doublyselective channels is a challenging task, since orthogonality between sub-carriers is destroyed when the channel is time varying [7] [8] [9]. For example, at a carrier frequency of 1492 MHz and vehicle speed of 100 km/h the normalized Doppler shift in DVB-T/H (mode 8k) and DAB systems can be as high as 0.2 and 0.14 respectively [7]. Such systems can experience even higher normalized Doppler shifts when used in high speed trains. Hence suitable detection techniques are critical for successful use of OFDM Wang Yi was with the Department of Electrical and Computer Engineering, McGill University, Montreal, Quebec, Canada, H3A 0E9. He is now with Ericsson, Ottawa, Ontario, Canada (email: [email protected]). Harry Leib (corresponding author) is with the Department of Electrical and Computer Engineering, McGill University, Montreal, Quebec, Canada, H3A 0E9 (email: [email protected]).

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in such applications. In a static environment, the channel matrix of an OFDM system in frequency domain is diagonal, and hence data detection can be easily implemented by using a single tap equalizer [7]. Over a time varying channel, however, the OFDM system channel matrix becomes full and introduces InterCarrier Interference (ICI) [9], that without suitable detection techniques can cause significant performance losses [5] [7] [9]. Hence ICI cancellation based demodulation for OFDM systems has been a subject of continuous research over many years [10]. Other approaches for demodulation of OFDM signals include ICI whitening [11], and Minimum Mean Square Error (MMSE) techniques [9]. The ICI in OFDM is mainly contributed by adjacent sub-carriers, and hence the channel matrix can be modelled as banded [7] [12]. Various techniques based on such banded approximation have been proposed to mitigate ICI effects while maintaining a moderate complexity. A serial Zero-Forcing (ZF) equalizer was proposed in [13], where based on a banded channel matrix approximation the original system is decomposed into several sub-systems allowing data detection with reasonable complexity. A similar approach based on the MMSE criterion was considered in [14], where it is integrated with decision-feedback. The banded approximation is also used in [15] to derive block linear and decisionfeedback equalizers for demodulation in OFDM systems over time varying channels. A high performance alternative to the ZF and MMSE approaches is Maximum Likelihood (ML) detection. However the complexity of ML detection, in its exhaustive search form, is exponential in the number of sub-carriers. Employing the banded channel matrix approximation, a Viterbi Algorithm (VA) operating in the frequency domain is considered in [16] for OFDM demodulation. The complexity of the VA is determined by the number of trellis states, that depends on the number of diagonals in the approximated channel matrix. For moderate Doppler rates, approximating the channel matrix by seven diagonals provides diversity gains with a reasonable complexity [16]. In [17] we proposed a data detector for OFDM based on a Sphere Decoding (SD) algorithm designed for a banded matrix approximation of the channel, making it as practical as the VA scheme of [16] while providing important performance gains over the latter. However, the detection techniques of [16] [17], as well as many other demodulation schemes for OFDM, require Channel State Information (CSI). The assumption that CSI is known perfectly at the receiver is not realistic, and in practice it has to be

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acquired. In OFDM systems direct estimation of time varying channel gains is complex because of the large number of sub-carriers [7] [18]. To circumvent this problem, Basis Expansion (BE) techniques, that decompose the time varying channel into a linear combination of time basis functions with coefficients that vary between different OFDM symbols, are often employed [12]. Since the number of basis functions is usually small, the estimation of the BE coefficients is a more feasible task. In [18], a Least Square (LS) channel estimator based on the Karhunen-Loeve (KL) expansion is proposed, and a VA [16] is used for data detection. In [19], a polynomial BE is used with LS estimation of BE coefficients based on pilot symbols. The estimated channel is then used by a successive interference suppression technique for data detection. A BE method known as Generalized Complex Exponential (GCE) is used in [20], with LS, Kalman and Modified Recursive LS (MRLS) estimation based on pilot symbols. It is shown that the Kalman filter converges faster than the MRLS, and both provide good performance. The optimal position of pilot symbols for GCE-BE is studied in [21]. A Kalman filter based on a polynomial BE model is used to estimate the channel in [22]. Data is detected after the time update stage of the Kalman filter, and fed back to the measurement update stage. Another scheme where the state-space equations are derived based on pilot symbols only is presented in [23]. The unknown data is considered as noise, and hence no decision-feedback is required. The use of a Basis Expansion Model (BEM) for OFDM channel estimation employing a Kalman filter switched periodically to operate on pilot symbols and decision-feedback modes is considered in [24]. A joint channel estimation and data detection technique for OFDM over fast time varying channels, that is based on the Space Alternating Generalized Expectation - Maximum A-Posterior Probability (SAGE-MAP) algorithm is considered in [25], where a BE technique employing discrete Legendre basis functions is used. Using complex exponentials, a BE model has been presented in [26] [27] for estimating doubly selective channels in Multiple-Input Multiple-Output (MIMO) systems. In this paper we introduce a joint data detection and channel multipath gains estimation scheme for OFDM systems based on integrating a Kalman filter, operating in decision-feedback mode, with a specially derived SD algorithm providing data decisions. For large Doppler, there is a need for a large pilot density in order to achieve good channel estimation, and hence the throughput is reduced. Employing decision-feedback reduces the number of pilot symbols required for channel tracking, and

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hence increases bandwidth efficiency. The large number of pilots is one of the main factors that limits the throughput in 3G/4G wireless systems, well below of what was envisaged [28]. Hence reducing the required number of pilots is an important improvement for future OFDM systems. We use a KL-BE as in [29] to model the time-varying channel, and employ our new SD algorithm of [17] based on a banded channel matrix approximation because of its good performance and its suitability for practical OFDM systems. Integrating such a detector with a Kalman filter operating in decision-feedback mode for channel tracking presents some intricate issues that are tackled in this paper. In [17] we compared our SD based technique with the VA detector of [16] in terms of performance and complexity assuming that CSI is perfectly known. In the present paper, the comparison is done with [18] where CSI is estimated based on pilot symbols, and also with [25] that considers joint channel estimation and data detection for OFDM over time varying channels. Performance results for LTE channels including effects of carrier phase noise are also presented. In the continuation of this paper, Section II presents the OFDM system with KL-BE modelling of time varying channel multipath gains. Section III presents the state space model for KL-BE coefficients, and the associated Kalman filter. Section IV presents our SD algorithm based on a banded matrix approximation of the channel, named Ordered Partial tree search Sphere Decoder (OPSD). Section V, presents a joint data detection and channel multipath gains estimation scheme based on integrating Kalman filtering with the OPSD using three variations for deriving decision-feedback. Section VI presents Monte-Carlo simulation results for the error rate, and also a comparison with [18] and [25]. Furthermore, error rate results over LTE channels with various normalized Doppler frequencies and carrier phase noise, are also presented. Finally, Section VII presents the conclusions. II. S YSTEM M ODEL

AND

BASIS E XPANSION R EPRESENTATION

OF THE

C HANNEL

In this work, the k, l component of matrix A is denoted as [A]k,l . Consider an OFDM symbol expressed in the continuous time domain as K−1 1  dm [l] exp(j2πfl t), t ∈ [(m − 1)Tsym , mTsym ], sm (t) = √ K l=0 where dm [l], l = 0, ..., K − 1, denote transmitted data symbols on sub-carrier l during the m-th OFDM symbol interval, fl , l = 0, ...K − 1 are sub-carrier frequencies satisfying fl =

l Tsym

to preserve orthog-

onality, and Tsym is the duration of an OFDM symbol excluding the cyclic prefix. The data symbols

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dm [l] are generated with equal probability from a constellation set A (MPSK, MQAM, etc.) of size NA and average energy Es . After adding a cyclic prefix, the OFDM symbol is transmitted over a multipath channel of impulse response h(t, τ ) =

P −1  p=0

hp (t)δ(τ − τp )

where P is the number of multipath components, hp (t) is the time varying gain of path p and τp is its delay. We assume that hp (t) are stationary zero mean circularly symmetric complex Gaussian random processes, that are independent for different p. After sampling at rate K/Tsym and removing the cyclic prefix samples, the discrete time received signal in an OFDM system can be represented as P −1 K−1 1  j2πln −j2πlτp hm,p [n]dm [l] exp( rm [n] = √ ) exp( ) + vm [n], n = 0, · · · , K − 1 K K K p=0 l=0

(1)

where we time index the samples excluding the cyclic prefix, starting with the first one belonging to the OFDM symbol. The variables hm,p [n] = hp ((nTsym /K) + mTsym ) denote the gain of path p during the m-th OFDM symbol interval, τp =

τp , Tsym /K

and vm [n] is uncorrelated zero mean circularly symmetric

complex Gaussian noise samples of variance N0 . Demodulation is performed by using the Fast Fourier Transform (FFT) of rm [n] resulting in K−1 K−1 P −1  j2π(l − k)n 1  −j2πlτp ym [k] = ) exp( ) + wm [k], k = 0, · · · , K − 1 dm [l] hm,p [n] exp( K n=0 l=0 K K p=0

(2)

where wm [k] is uncorrelated zero mean circular complex Gaussian noise of variance N0 . Based on (2), in the frequency domain an OFDM system can be represented as [7, p.288] ym = Cm dm + wm

(3)

where ym = [ym [0], · · · , ym [K −1]]T , dm = [dm [0], · · · , dm [K −1]]T , wm = [wm [0], · · · , wm [K −1]]T , and Cm is channel matrix whose components are given by K−1  1 j2π(l − k)n Hm,l [n] exp( ), k = 0, · · · , K − 1; l = 0, · · · , K − 1 (4) [Cm ]k,l = K K n=0 P −1 p where Hm,l [n] = p=0 hm,p [n] exp( −j2πlτ ). If the channel is time invariant within an OFDM symbol, K then Hm,l [n] do not depend on n, making Cm diagonal. However, in the presence of channel time

variations within an OFDM symbol, Cm is a full matrix introducing ICI.

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The focus of this work is joint estimation of multipath time varying gains hm,p [n] and detection of data symbols dm [l], while assuming as in [19] [22] that the multipath delays τp are known. These delays, that are almost constant over a large number of OFDM symbols [30], change much slower than the multipath gains and in practice corresponding estimates need to be generated much less frequently. Possible techniques for estimating multipath delays in OFDM systems include the ESPRIT algorithm, such as in [31], and the MUSIC algorithm such as in [32]. From [33] we have that the complexity of the basic ESPRIT algorithm in our setting scales as O(K 3 ). However, [33] also proposes a variation of ESPRIT that could lower complexity. We term Nc , the number of basis functions that are used, as the BE order, and denote the corresponding BE coefficients as gq where q = 0, · · · , Nc − 1. The time varying gain of the p-th channel multipath component during the m-th OFDM symbol is approximated as ˜ m,p [n] = h

N c −1  q=0

gm,p,q ψq [n] n = 0, · · · , K − 1

(5)

where ψq [n] is a basis function and gm,p,q is the BE coefficient of the p-th multipath component during the m-th OFDM symbol. From (5) and (4) we have the corresponding approximation of [Cm ]k,l ˜ m ]k,l = [C

K−1  n=0

P −1 j2π(l − k)n −j2πlτp 1 ˜ ) exp( ) hm,p [n] exp( K p=0 K K

K−1 P −1 Nc −1 1   j2π(l − k)n j2πlτp = ) exp( ) gm,p,q ψq [n] exp(− K n=0 p=0 q=0 K K

Nc −1 K−1 P −1  j2π(l − k)n  j2πlτp 1  ) ). ψq [n] exp( gm,p,q exp(− = K q=0 n=0 K K p=0

(6a) (6b) (6c)

Define the matrix Bq with components K−1 [Bq ]k,l = Σn=0 ψq [n] exp(

j2π(l − k)n ), k = 0, · · · , K − 1; l = 0, · · · , K − 1, K

(7)

the matrix F with components [F]l,p = exp(−

j2πlτp ), l = 0, · · · , K − 1; p = 0, · · · , P − 1, K

(8)

and Gm,q = [gm,0,q , · · · , gm,P −1,q ]T as a vector of size P . Let diag{•} denote an operator that constructs a diagonal matrix from a given vector. We can express (6c) in matrix form as N c −1  ˜m = 1 C Bq diag{FGm,q } K q=0

(9)

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In our scheme, the BE coefficients are estimated by a Kalman filter. Assuming that the current transmitted vector dm is known either through decision-feedback or pilot symbols, we express the received vector in terms of BE coefficients, which can be used to construct the state-space equations. Substituting (5) into (2) and using (7) and (8) we have K−1 N −1 c −1 P   1  dm [l] gm,p,q [Bq ]k,l [F]l,p + wm [k] ym [k] = K l=0 q=0 p=0

=

K−1 P −1 Nc −1  1  gm,p,q dm [l][Bq ]k,l [F]l,p + wm [k]. K p=0 q=0

(10)

l=0

Define um,p,q = Bq diag{dm }fp

(11)

as a vector of size K where fp of size K is the pth column of matrix F. In component form we have [um,p,q ]k =

K−1  l=0

Substituting (12) into (10), we have

ym [k] =

dm [l]Bq [k, l]F [l, p], k = 0 · · · K − 1.

P −1 Nc −1 1  gm,p,q [um,p,q ]k + wm [k]. K p=0 q=0

(12)

(13)

In (13), the demodulated received signal at sub-carrier k, ym [k], is represented as a linear combination of the BE coefficients and an AWGN term wm [k]. Let’s define the size Nc vector of BE coefficients for the same multipath component gm,p = [gm,p,0, · · · , gm,p,q , · · · , gm,p,Nc −1, ]T , and then stack gm,p p = 0, · · · , P − 1 in a vector of size Nc P gm = [gTm,0 , · · · , gTm,p , · · · , gTm,P −1 ]T .

(14)

Defining the matrix Km,p = [um,p,0 , · · · , um,p,q , · · · , um,p,Nc −1 ], the observation matrix Km of dimension K × Nc P is given by

Km = [Km,0 , · · · , Km,p , · · · , Km,P −1 ].

(15)

The observation equation for an OFDM system can be obtained from (13), and it is given by ym = Km gm + wm .

(16)

Note that the observation matrix Km depends on the data vector dm through (12), showing that data decision-feedback is required for this scheme. We employ the Karhunen-Loeve (KL) basis because of its MMSE optimality properties [29]. Define the gains for the pth tap as hm,p = [hm,p [0], · · · , hm,p [n], · · · , hm,p [K − 1]]T , and the autocorrelation

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matrix of hm,p as Rh,p (m − m ) = E{hm,p hH m ,p }, where E{•} denotes the expectation operation. For the Jakes’ fading model [34], we have that the nth , nth component of Rh,p (m − m ) is [Rh,p (m − m )]n,n = E{hm,p [n]h∗m ,p [n ]} = σp2 J(2πfd Ts (n − n + K(m − m )))

(17)

where σp2 is the variance of the pth multipath component, J(•) is the zero order Bessel function of first kind, fd is the Doppler frequency, and Ts is the sampling interval. The basis functions of the KL expansion, ψq [n], can be obtained through the eigenvalue decomposition of Rh,p (0) for p = 0, · · · , P − 1

(18) Rh,p (0) = VΛVH where Λ is a diagonal matrix composed of these eigenvalues sorted in descending order, and V is a unitary matrix with the corresponding eigenvectors as columns. In matrix form the KL expansion is [29] hm,p = Vgm,p,K

(19)

where gm,p,K = [gm,p,0, · · · , gm,p,q , · · · , gm,p,K−1]T . Define the matrix Ψ with components [Ψ]n,q = ψq [n], q = 0, · · · , Nc − 1; n = 0, · · · , K − 1,

(20)

that can be constructed by selecting the first Nc columns of V resulting in a matrix of size K ×Nc . Since the eigenvectors are orthonormal to each other, we have ΨH Ψ = INc . From (5), the BE modelled time varying gains for the pth tap can be expressed in matrix form as ˜ m,p = Ψgm,p h

(21)

˜ m,p [0], · · · , ˜hm,p [n], · · · , h ˜ m,p [K−1]]T , g = [gm,p,0, · · · , gm,p,q , · · · , · · · , gm,p,Nc −1 ]T . ˜ m,p = [h where h m,p Defining we have

Rg,p (m − m ) = E{gm,p gH m ,p },

(22)

Rg,p (0) = ΨH Rh,p (0)Ψ = ΨH VΛVH Ψ = [INc 0Nc ×(K−Nc ) ]Λ[INc 0Nc ×(K−Nc ) ]T .

(23)

From (23), we can conclude that Rg,p (0) is a diagonal matrix formed by the Nc most significant eigenvalues of Rh,p (0), and hence the KL BE coefficients of different orders (q = q  ) are uncorrelated. III. C HANNEL M ULTIPATH G AINS E STIMATION U SING

A

K ALMAN F ILTER

In the context of our OFDM system model, the states are the BE coefficients and the observations are the received vectors ym . We model the dynamics of the BE coefficients corresponding to multipath component p as an AR process of order Na gm,p = −

Na  i=1

Ai,p gm−i,p + qm,p ,

(24)

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where Ai,p is the ith order AR coefficient Nc × Nc matrix for the pth path, and qm,p is the driving white noise with covariance matrix Qp of dimension Nc × Nc . The matrices Ai,p i = 1, · · · , Na p = 0, · · · , P − 1 and Qp p = 0, · · · , P − 1 can be obtained recursively by using the multi-channel Levinson algorithm [35]. Using (14) we define the state vector xm = [gTm , · · · , gTm−Na +1 ]T .

(25)

Ai = bdiag{Ai,0 , · · · , Ai,P −1}

(26)

The AR coefficient matrices Ai,p p = 0, · · · , P − 1, are organized as a block diagonal matrix

where bdiag{•} is an operator that generates a block diagonal matrix. From (24) we obtain the transition equation ⎡

where

xm = F xm−1 + Gqm

−A1 −A2 · · · −ANa ⎢ ⎢ I P N 0P N · · · 0 P Nc c c ⎢ ⎢ . .. F =⎢ 0 P Nc ⎢ 0 P Nc ⎢ . .. ⎢ .. . ⎣ 0 P Nc · · · I P Nc 0 P Nc

⎤

⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦

(27)

(28)

qm = [qTm,0 , · · · , qTm,p , · · · , qTm,P −1]T and G = [INc P 0Nc P ×(Na −1)Nc P ]T . From (16) and (25), we obtain the observation equation

ym = [Km 0K×(Na −1)Nc P ] xm + wm 



(29)

Hm

Based on (27) and (29), a Kalman filter can be derived to estimate xm , and with (9) an estimate of the channel matrix in frequency domain can be formed. The Kalman filter is composed of two stages: time update, and measurement update. Define ˆxm/m−1 = E{xm |ym−1 , · · · , y1 } as the prior estimate of xm based on previously received vectors, and xˆm/m = E{xm |ym , · · · , y1 } as the estimate of xm based on all receive vectors including the current one [36]. The time update is given by xˆm/m−1 = F ˆxm−1/m−1

where Pm/m−1

(30)

(31) Pm/m−1 = F Pm−1/m−1 F H + GQGH H H = E{(xm − xˆm/m−1 )(xm − xˆm/m−1 ) }, Pm/m = E{(xm − xˆm/m )(xm − ˆxm/m ) }, and

Q = bdiag{Q0 , · · · , QP −1 }. The measurement update is given by where

xˆm/m = xˆm/m−1 + Lm (ym − Hm ˆxm/m−1 )

(32)

H H (Hm Pm/m−1 Hm + N0 I)−1 Lm = Pm/m−1 Hm

(33)

T T T T T ∗ P−1/−1 = E{x−1 xH −1 } = E{[gm , · · · , gm−Na +1 ] [gm , · · · , gm−Na +1 ] },

(35)

Pm/m = Pm/m−1 − Lm Hm Pm/m−1 and Lm denotes the Kalman gain. Initially, ˆx−1/−1 is set to a zero vector, and

(34)

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where gm is defined in (14), and we also used (25). From (22) we have   E{gm gH m } = bdiag{Rg,0 (m − m ) · · · Rg,P (m − m )}.

(36)

where Rg,k (m − m ) can be calculated from (17),(19),(22).

We also consider the Square Root Kalman Filter (SRKF) because of its robustness to numerical errors. The SRKF works on the same state-space model (27) (29), however it estimates the state vector based on the square root of the error covariance matrix which in general has better numerical properties [36]. The covariance matrix in the time update of the SRKF is given by [36]     H H SH F S m/m−1 m−1/m−1 = tri 0 Q1/2,H GH

(37)

H H where SH m/m−1 and Sm−1/m−1 are upper triangular matrices that satisfy Pm/m−1 = Sm/m−1 Sm/m−1 ,

Pm−1/m−1 = Sm−1/m−1 SH m−1/m−1 , and tri{•} denotes a transformation of a full matrix to triangular implemented through a QR decomposition [36]. The measurement update is given by     1/2 H H + N0 I)1/2,H Km 0 (Hm Pm/m−1 Hm N0 IT = tri , H SH 0 SH SH m/m m/m−1 Hm m/m−1

and the Kalman gain is

H + N0 I)−1/2 . Lm = Km (Hm Pm/m−1 Hm

(38)

(39)

The channel estimator in our work is derived based on the vector state-space model (27) and vector observation (29), resulting in a Vector-State Vector-Observation (VSVO) Kalman filter [23] [37]. Approaches employing VSVO Kalman filtering based on BE techniques have been also considered in [12] [20] [24]. From (27) we have that the state dimension is Na · P · Nc , and from (29) the observation dimension is K. For OFDM we usually have K > Na · P · Nc . The complexity of the Kalman filter in terms of number of operations performed at each iteration is treated in [38, chapt. 7] [39, sec. 3.3.2]. From [38, Table 7.22] and [39, Table 3.1] we have that in our setting the complexity scales as O(K 3 ). Using Newton’s iteration for matrix inversion (Appendix) can speed up the channel estimator. In [23] it is shown that using a scalar observation model for OFDM yields a Vector-State ScalarObservation (VSSO) Kalman filter that lowers overall complexity. This possibility is also mentioned in [39, sec 3.3.2] for independent sequential scalar observations. The application of VSSO Kalman filtering for MIMO-OFDM channel estimation has been considered in [37]. In [40] it is shown that VSVO and VSSO Kalman filtering for OFDM channel estimation are computationally equivalent over Wide-Sense Stationary Uncorrelated-Scattering (WSSUS) channels as considered in our work.

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IV. T HE OPSD

ALGORITHM FOR DATA DETECTION

In this work, we employ a novel technique based on the Sphere Decoding (SD) algorithm, known as Ordered Partial tree search Sphere Decoding (OPSD) [17], for data detection. The SD employs a recursive depth first tree search procedure [41], and the major challenge of applying such a technique to OFDM systems is complexity. As the number of sub-carries increases, the tree search process becomes more complex. In deriving the OPSD algorithm we aimed at reducing complexity by two methods: approximating the channel matrix as banded, and using MMSE V-BLAST ordering [42]. A banded channel matrix introduces a structure in the search tree that can be exploited to reduce complexity. The MMSE V-BLAST ordering arranges the columns of the channel matrix based on a Signal to Interference plus Noise Ratio (SINR) criterion [43], resulting in a reduced number of candidate data points at each level of the search tree, which also reduces complexity. The OPSD algorithm works on an augmented (regularized) version of (3), given by [43]       y C w = d+ 0 αI −αd

 



where α =



Es N0

−1/2

yˇ

ˇ C

(40)

ˇ w

ˇ has always a full rank. For simplicity the and w is the AWGN vector. Note that C

OFDM symbol index is dropped. With MMSE V-BLAST ordering [42] we have the permuted system ˇ (PT d) +w, ˇy = (CP) ˇ

 

¯ C

(41)

d¯

where P is a permutation matrix (PP = I) representing the ordering operation. The column permuted T

ˇ and the permuted data vector is d¯ = PT d. ¯ = CP channel matrix is C ˇ has the same statisThe Maximum Likelihood (ML) regularized solution for (41), assuming that w tics as w, is given by [44] ˆ¯ 2 + (Kα2 r 2 − α2 ||d|| ¯ d ¯ − d)|| ¯ 2 )), dM L = arg min ¯d (||R( max

where rmax is the maximum distance of a point from origin in the constellation set A,     ˇ H CP ˇ ¯ = chol PT C R = chol PT (CH C + α2 I)P ,

(42)

(43)

ˆ¯ is the unconstrained solution to (41) with chol {•} standing for the Cholesky factorization, and d ˆ¯ = (C ¯ H C) ¯ −1 C ¯ H ˇy = PT (CH C + α2 I)−1 CH y. d

(44)

Similarly to the conventional SD [41], the OPSD searches for dM L of (42) in the region specified by ˆ¯ 2 + (Kα2 r 2 − α2 ||d|| ¯ d ¯ − d)|| ¯ 2) < C 2, ||R( max

(45)

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which in component form becomes K−1 K−1 K−1    2 2 ˆ ˆ ¯ ¯ ¯ ¯ ¯ 2) < C 2, ¯ ¯ |[R]i,i (d[i] − d[i]) + [R]i,j (d[j] − d[j])| + (α2 rmax − α2 |d[i]| i=0

j=i+1

(46)

i=0

ˆ¯ of d. ˆ¯ The radius C is initialized large enough such that it ¯ are the components of d, ¯ and d[i] where d[i] ¯ the OPSD searches guarantees to find one possible solution [45]. Starting from row i = K − 1 of R, ¯ and checking if (46) is satisfied. If (46) is satisfied, then i is for dM L by selecting a point from A as d[i] reduced by one. Otherwise, another point of A is selected. This process is represented by a search tree ¯ is associated and a corresponding flow chart can be found in [45]. In the rest of this section, row i of R with level i of the search tree. The OPSD algorithm approximates the channel matrix C as banded of k upper and k lower diagonals, with ordering performed on the first K − (2k + 1) columns. The resulting upper triangular ¯ j] = 0 for matrix is sparse with zero components determined by the permutation P. Furthermore, R[i, i = 0, · · · , K − (2k + 2) and j = K − 1, [17]. There are two metrics that are essential to the OPSD operation, known as the incremental metric at level i  2 K−1     ¯ ˆ¯ + ˆ¯  + (α2 r 2 − α2 d[i] ¯ 2 ), ¯ − d[i]) ¯ − d[j]) ¯ i,j (d[j] i = [R]i,i (d[i] [R] max  

(47)

j=i+1

and the cumulative metric at level i 2 K−1 K−1 K−1       2  2 ˆ ˆ ¯ ¯ ¯ ¯ ¯  ). ¯ ¯ Ci = [R]l,j (d[j] − d[j]) + (α2 rmax − α2 d[l] (48) [R]l,l (d[l] − d[l]) +   l=i j=l+1 l=i ¯ is sparse, there are only a few terms inside the summation Since, the ordered upper triangular matrix R in (47), resulting in complexity reduction. The complexity can be further reduced by pruning the sub-

¯ is represented as a node at level i of trees. In the tree search process, each possible data decision d[i] ¯ is a path connecting these nodes from a tree, that is associated with (47) and (48). A data vector d ¯ i,j = 0 for i < K − (2k + 1) level K − 1 to 0 of the tree, and dM L is one of these paths. Since [R] ¯ − 1] for i < K − (2k + 1). Therefore for and j = K − 1, we have that i is independent of d[K ¯ · · · , d[K ¯ − 2] are the same for two nodes in the tree, then also i are the same i < K − (2k + 1), if d[i], for these two nodes. For any two nodes at level K − (2k + 1) that have the same data decision path ¯ − (2k + 1)], · · · , d[K ¯ − 2], the sub-trees under the node with lager CK−(2k+1) are pruned. Since d[K the incremental metric (47) is nonenegative, dM L will not be a path that includes a branch of the subtree generated by the node with larger CK−(2k+1) . This pruning divides the full search tree into several sub-trees. The search process is done on such selected sub-tree, resulting in complexity reduction.

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The OPSD algorithm fully expands the first 2k + 1 levels of the search tree, and the cumulative ¯ − (2k + metric CK−(2k+1) of each node is obtained by using (47) and (48). The nodes with same d[K ¯ − 2] are grouped, and the group with the smallest CK−(2k+1) is kept. Complex sphere 1)], · · · , d[K decoding is then performed on the surviving nodes, starting from the one with smallest CK−(2k+1) . Rewriting (46) as

i < C 2 − Ci+1 ,



(49)

Ti

we have the recursion

Ti = C 2 − Ci+1 = C 2 − (Ci+2 + i+1 ) = Ti+1 − i+1. Using (47), we can rewrite (49) as 2  K−1     2  ¯ 2 ˆ ˆ ¯ ¯ ¯ ¯ ¯  ) < Ti . ¯ ( d[i] − d[i]) + [ R] ( d[j] − d[j]) − α2 d[i] [ R]  + (α2 rmax  i,i i,j  

(50)

(51)

j=i+1

ˆ¯ − K−1 Defining zi = d[i] j=i+1

¯ i,j [R] ¯ ¯ i,i (d[j] [R]

ˆ ˆ¯ ¯ = ri ejφi , from (51) we have − d[j]) = rˆi ej φi and d[i]

   2 ¯ − zi )2 + (α2 r 2 − α2 d[i] ¯  ) < Ti [R] ¯ i,i (d[i] max  2  2 2 ¯ ) ¯ 2i,i ri ejφi − rˆi ej φˆi  < Ti − α2 (rmax [R] − d[i]

 2 2 ¯ ) ¯ 2i,i (ri2 + rˆi2 −2ri rˆi cos(φi − φˆi )) < Ti −α2 (rmax [R] − d[i]   2  2 2 ¯ ) d[i] T − α (r − 1 i max ri2 + rˆi2 − , cos(φi − φˆi )> 2ri rˆi [R]i,i

(52)

¯ as a component of dM L . It is possible that more than one data resulting in the criterion of selecting d[i] point from A satisfies this criterion. These points are usually stored in a list in ascending order according ˆ¯ (44) is performed by the to i [45]. To speed up the OPSD algorithm, matrix inversion used to obtain d Newton iterative procedure [17] presented in the Appendix. The proposed detection scheme, OPSD, is based on the SD tree search algorithm that is known to haver a variable complexity depending on the channel matrix, and hence for analysis purposes it is treated as a random variable [46] [47]. In [46] the average complexity is found to be often cubic in problem size. In [47] it is found that the average complexity is lower bounded by an exponential function of the problem size, where the exponent coefficient decreases when the SNR increases. Hence for suitable SNR values the SD algorithm becomes competitive with respect to polynomial algorithms, explaining its popularity in practice.

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Although the overall complexity of the OPSD algorithm is essentially random, making it necessary the use of computer simulations for its evaluation, some components have a deterministic complexity that are examined next. From [17] [42] we have that the complexity of MMSE V-BLAST ordering scales as O(K 3 ). The complexity of Cholesky decomposition of a banded Hermitian K × K matrix of one sided band k scales as O(k 2 K) [48]. The complexity of matrix inversion scales as O(K 3 ), however the use of the Newton iterative algorithm (see Appendix) can speed up the process due to its possible parallel implementation. Summarizing the results, we see that the complexity of the OPSD algorithm scales as O(K 3 ). The approximation is due to the random complexity nature of the SD tree search procedure, as mentioned earlier. V. I NTEGRATING OPSD

WITH CHANNEL MULTIPATH GAIN ESTIMATION

Next we consider the integration of Kalman filtering with OPSD resulting in a joint channel multipath gain estimation and data detection scheme. By making use of state vector xm estimates that are provided by the Kalman filter, we consider three variations of employing decision-feedback. A flow chart of these three variations is provided in Fig.1. Several pilot symbols are transmitted at the beginning of each frame for initial training purposes. For each received sample, a time update of the Kalman filter is performed. Using the output from the time update, the transmitted data vector is preliminary detected. Then a measurement update is performed with the decision-feedback data vector. For Variation 1, the steps in the dash box of Fig.1 are skipped. For Variation 2, the steps in the dash box are executed once, and preliminary symbol detection is performed by (44) without permutation, followed by hard decisions on every component. For Variation 3, the steps in the dash box are executed Niter times with all symbol detections performed by the OPSD stage. ˆ m/m−1 to be the estimated channel matrix based on ˆxm/m−1 , and C ˆ m/m the estimated We define C channel matrix based on xˆm/m . Following (25) we further define

and

xˆm/m−1 = E{xm |ym−1 , · · · , y0 } = [ˆgTm/m−1 , · · · , ˆgTm−Na +1/m−1 ]T

(53)

ˆxm/m = E{xm |ym , · · · , y0 } = [ˆgTm/m , · · · , gˆTm−Na +1/m ]T .

(54)

gk,p,0/m−1 , · · · , gˆk,p,Nc−1/m−1 ]T , and In (53) gˆk/m−1 = [ˆgTk,0/m−1 , · · · , gˆTk,P −1/m−1 ]T where ˆgk,p/m−1 = [ˆ gk,p,0/m, · · · , gˆk,p,Nc−1/m ]T . Following (6c) we in (54) gˆk/m = [ˆgTk,0/m , · · · , gˆTk,P −1/m ]T where ˆgk,p/m = [ˆ

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ˆ m/m−1 with components define the matrices C Nc −1 K−1 P −1  1  j2π(l − k)n  j2πlτp ˆ ) ) [Cm/m−1 ]k,l = ψq [n] exp( gˆm,p,q/m−1 exp(− K q=0 n=0 K K p=0

(55)

ˆ m/m with components and C

Nc −1 K−1 P −1  1  j2π(l − k)n  j2πlτp ˆ ψq [n] exp( gˆm,p,q/m exp(− [Cm/m ]k,l = ) ). K q=0 n=0 K K p=0

(56)

ˆ m/m,i employing decision-feedback Following (29) we also define the estimated observation matrix H vectors dm,i i = 0, · · · , Niter . For Variation 1 i = 0, for Variation 2 i = 0, 1, and for Variation 3 i = 0, · · · , Niter . In Variation 2 and Variation 3 the Kalman filter measurement updates are performed iteratively. We define the measurement update results as ˆxm/m,i , where i = 1, 2 for Variation 2, and i = 1, · · · , Niter for Variation 3. The estimated channel matrix based on ˆxm/m,i with components ˆ m/m,i . Analogously to (11) we have u ˆ m,i,p,q = Bq diag{dm,i }fp , and defined in (56) is denoted by C ˆ m,i,p = [ˆ ˆ m,i = [K ˆ m,i,0 , · · · , K ˆ m,i,P −1 ], ˆ m,i,p,q , · · · , u ˆ m,i,p,Nc −1 ], K define: K um,i,p,0, · · · , u ˆ m/m,i = [K ˆ m,i , 0K×(Na −1)Nc P ]. H

(57)

A. Variation 1 This variation implements a simple data decision-feedback procedure, where the channel matrix in the frequency domain is formed from the time update stage based on gˆm/m−1 using (55). The OPSD ˆ m/m−1 to detect the transmitted data vector. The decisionstage uses the estimated channel matrix C feedback vector dm,0 is used to form the observation matrix using (57). The measurement update is then performed using (32) - (34). This algorithm is described below. 1. Initialization of the Kalman Filter (a) ˆx−1/−1 is initialized as a zero vector. (b) P−1/−1 is initialized using (35) (36). 2. Time update of the Kalman Filter ˆxm/m−1 and Pm/m−1 are obtained by using (30) (31), and passed to the data detection stage. 3. Data detection (a) The first Nc P components of vector ˆxm/m−1 , which compose ˆgm/m−1 of (53), are used to ˆ m/m−1 using (55). form an estimate of the channel matrix C

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ˆ m/m−1 , re(b) The OPSD algorithm is used to detect the transmitted data vector employing C sulting in the output dm,0 . The final data decision is dm,dec = dm,0 . ˆ m/m,0 is formed as in (57) aided by (11) using dm,0 . (c) The estimated observation matrix H 4. Measurement update of the Kalman Filter ˆ m/m,0 . ˆxm/m and Pm/m are obtained from (32) - (34) given the received signal vector ym and H Then xˆm/m and Pm/m are forwarded to (30) (31) for the next m + 1th symbol. Following the initialization of the Kalman filter in step 1, channel multipath gain estimation and data detection of every data symbol is performed in steps 2 to 4. B. Variation 2 In Variation 1, xˆm/m is not used by OPSD for the current data detection, despite that in general ˆxm/m provides a better estimate than ˆxm/m−1 . Employing the present observation through the measurement update, the Kalman filter improves the estimate [36]. By making use of a tentative decoder, we could use xˆm/m and improve performance. Such tentative data decisions are obtained by applying hard decisions on the components of the unconstrained solution (44) without permutation, without adding significant complexity. This algorithm is described below. 1. Initialization of the Kalman filter Same as in step 1 in Variation 1. 2. Time Update of the Kalman filter Same as in step 2 in Variation 1. 3. Tentative Data detection (a) Same as step 3 (a) in Variation 1. (b) A tentative data decision is obtained using (44) without permutation by performing hard decisions on each component. Output these tentative decisions as dm,0 . ˆ m/m,0 is obtained as in (57), aided by (11) (c) A tentative estimate of the observation matrix H and using dm,0 . 4. Measurement update of the Kalman Filter using tentative decisions ˆ m/m,0 . The estimate xˆm/m,1 is obtained using (33) (32) given the received signal vector ym and H Then xˆm/m,1 is forwarded to the second data detector. 5. Final data detection (a) The first Nc P components of xˆm/m,1 , which compose ˆgm/m of (54), are used to form an

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ˆ m/m,1 based on ˆxm/m,1 using (56). estimate of the channel matrix C ˆ m/m,1 , resulting in output (b) Final data detection is performed by the OPSD algorithm using C dm,dec . ˆ m/m,1 is obtained by using (57) aided by (11) with (c) The estimated observation matrix H dm,dec . 6. Measurement update of the Kalman Filter The estimate xˆm/m,2 and error covariance matrix Pm/m are obtained using (32) - (34), based ˆ m/m,1 . Then ˆxm/m,2 and Pm/m are on xˆm/m−1 , Pm/m−1 , the received signal vector ym and H forwarded to the time update of the Kalman Filter (step 2) for the next symbol. C. Variation 3 In this variation, the OPSD and the measurement update in the Kalman filter are used iteratively. Different from variations 1 and 2, where the OPSD stage is performed only once based on prior estimate (Variation 1) or tentative post estimates (Variation 2), in this variation the OPSD is used more than once. This algorithm is described below. 1. Initialization of the Kalman filter Same as in step 1 Variation 1. 2. Time Update of the Kalman filter Same as in step 1 Variation 1. 3. Iterative data detection and channel multipath gains estimation ˆ m/m−1 as in step 3(a) of Variation 1. (a) Obtain C (b) Obtain dm,0 as in step 3(b) of Variation 1. ˆ m/m,0 as in step 3(c) of Variation 1. (c) Obtain H (d) for i = 1, · · · , Niter ˆ m/m,i−1 . i. Use the Kalman filter measurement update (33) - (34) to obtain xˆm/m,i given H ˆ m/m,i as in (56) using xˆm/m,i . ii. Obtain C ˆ m/m,i resulting in output dm,i . iii. Use the OPSD for data detection given C ˆ m/m,i as in (57) aided by (11) using dm,i . iv. Obtain H v. Set i = i + 1, and go to i. (e) Output dm,Niter as the final data decision dm,dec . 4. Measurement Update of Kalman filter

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The estimate ˆxm/m,Niter +1 and the error covariance matrix Pm/m are obtained as in (32) - (34), ˆ m/m,N . Then ˆxm/m,N +1 and using xˆm/m−1 , Pm/m−1 , the received signal vector ym and H iter iter Pm/m are forwarded to the time update of the Kalman Filter (step 2) for the next symbol. After initialization, at each OFDM symbol interval, one time update of the Kalman filter and Niter OPSD detection with measurement updates cycles are performed. VI. C OMPUTER S IMULATION R ESULTS Performance of our scheme with all three variations is evaluated by using Monte Carlo simulations. An OFDM system with K = 64 sub-carriers over doubly-selective channels is used in these tests. In our simulations we use k = 1 for OPSD, implying that the channel matrix is approximated as banded of width three (i.e. one additional diagonal on each side of the main one). Increasing k results only in a small performance gain in our system, at the expense of increased complexity. At least 500 channel realizations and 400 bit errors are accumulated to ensure the accuracy of results. Each channel realization spans 56, 140 or 1000 OFDM symbols determined by the frame length used. The sum of sinusoids method for producing Rayleigh fading tap gains [34] was employed to generate fading channel realizations. We first present computer simulation results over the same channels as in [18] for performance comparison. Then we present computer simulation results over LTE channels with carrier phase noise. In this section we define snr =

L Es Ldata N0

and SNR = 10log10 (snr), where Es is the symbol energy, N0 /2

is the two sided noise power spectral density, L is the total frame length in OFDM symbols, and Ldata is the number of OFDM data symbols in a frame. The first L − Ldata symbols in a frame are pilots used for training. We define the overhead efficiency as

Ldata . L

A. Comparison with the VA algorithm In [18] a VA algorithm operating in the frequency domain is used for detection in OFDM systems over fast time varying channels inducing ICI. Hence it is interesting to compare the performance of such a technique with our scheme that employs a SD based detector to perform the same task. Both schemes do not cancel ICI, but rather employ detection techniques that can cope with such interference. A two tap multipath channel (P = 2) following Jakes’ model with exponential power profile exp(−p/P ) , p = 0, · · · , P − 1 σp2 = P −1 exp(−p/P ) p=0

(58)

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is employed in [18], where σp2 is the variance of the pth channel tap. A BPSK OFDM system with 64 carriers is used, where 56 carriers are active. The other 8 subcarriers (4 on each side of the frequency band) are left unused to reduce interference between adjacent bands. The receiver in [18] employs pilot symbols assisted channel estimation, where 4 clusters of pilot symbols (each cluster occupying 3 sub-carriers) are interleaved with data symbols. In our simulations we use the normalized Doppler rates fd Tsym = 0.128, 0.32, which are equivalent to fd Ts = 0.002, 0.005 in [18]. In [18] Ts is the sampling time interval, and for our system since it contains 64 sub-carriers we have Tsym = 64Ts . We initially set L = 1000 and use 10 pilot OFDM symbols at the beginning of each frame for training the Kalman filter to start its operation properly. For fd Tsym = 0.128, a first-order AR process with 6 KL basis functions is used. When fd Tsym = 0.32, we employ a third-order AR process with 3 KL basis functions. We found that for higher normalized Doppler rates, increasing the order of the AR model improves performance. However the multi-channel Levinson algorithm is not able to calculate the matrices Ai,p accurately enough when the number of basis functions increases, and hence we reduced the number of basis functions to 3. Simulation results for Bit Error Rate (BER) are illustrated in Fig.2. Solid lines in Fig.2 represent results for the three variations of our scheme with fd Tsym = 0.128. We see that Variations 2 and 3 provide a better performance than Variation 1. A performance gain close to 4dB at BER = 1.5 × 10−5 is achieved by Variations 2 and 3, showing that ˆxm/m is better when forming an estimate. This improvement, however, comes at the expense of a complexity increase due to the use of a preliminary detection stage in Variations 2 and 3. The preliminary detection in ˆ¯ This detection Variation 2 is done by using (44) followed by hard decisions on each component of d. structure constitutes a linear MMSE detector having a complexity per carrier that scales as K 2 in our system [49, p.560]. The preliminary detection in Variation 3 is done by using the OPSD algorithm that in addition to the unconstrained solution computed through (44) it performs also a tree based search as detailed in Section IV. Hence the complexity of preliminary detection in Variation 3 is higher than in Variation 2. The complexity of OPSD is variable, and has been considered in [17]. The performance improvement of Variation 3 is insignificant over Variation 2. This suggests that for fd Tsym = 0.128 a tentative linear MMSE detector implemented by (44) followed by a quantizer is sufficient to provide performance gains with lower complexity. Variation 3 with Niter = 2 has a similar performance as with

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Niter = 1 indicating that increasing Niter over such channels does not provide additional performance gains. Results for fd Tsymbol = 0.32, represented by dash lines in Fig.2, show significant performance gains for Variation 3 with Niter = 2 over Niter = 1, indicating that using OPSD as a preliminary detector for high Doppler rates helps. However for Variation 3 with Niter = 2 at BER = 7 × 10−4 and for Variation 3 with Niter = 1 at BER = 1.8 × 10−3 we notice an error floor phenomenon. These error floors are due to error propagation when the Kalman filter operates in decisions-feedback mode. The ˆ m/m , resulting in erroneous feedback vectors lead to inaccurate estimates of the observation matrix H poor channel tracking by the Kalman filter. Comparing our BER results with [18], we observe a significant performance improvement of our scheme when fd Tsym = 0.128. The best performance of [18] is summarized in Table I. From Fig.2 and Table I we see that for SNR ≥ 20dB our scheme performs significantly better than the one in [18]. As the SNR increases, the scheme of [18] tends to have an error floor at BER = 7 × 10−4 , while in our scheme, with Variations 1, 2 and 3 no error floors are identified for BER > 10−6 . All of our three variations achieve larger diversity gains than one, and in particular Variations 2 and 3 achieve diversity gains that are close to two. We would like to point out that, for fd Tsym = 0.128, placing 10 training symbols at the beginning of a frame consisting of 1000 OFDM symbols ensures proper operation. Hence our scheme also improves the overhead efficiency (99%) compared with [18] (85.7%). When fd Tsym = 0.32 with L = 1000, our scheme performs worse than [18] at SNR = 20dB, where the later provides BER = 6 × 10−3 . This is because of error propagation effects that degrade performance significantly over such a fast fading channel. Next we address this issue. As a simple alternative to the obvious approach of providing better decision-feedback for overcoming error propagation, we can reduce the frame length. With shorter frames, error propagation affects a reduced number of symbols. In Fig.3 we present simulation results for Variation 3 with Niter = 2 employing frame lengths of L = 140 and L = 56, including 10 and 4 training symbols to ensure a 92.8% overhead efficiency which is higher than that of [18] (85.7%). To avoid numerical errors, a Square Root Kalman Filter (SRKF) [36] is also considered. Both Frame Error Rate (FER) and BER results are presented. We observe that the SRKF provides a slightly better performance in terms of BER and FER when SNR > 30dB. This phenomenon is more obvious when L = 140 indicating that a longer frame could be more vulnerable to numerical errors. Comparison with Fig.2, shows that as the frame length

21

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decreases the performance for fd Tsym = 0.32 improves. With short frames, error propagation affects a reduced number of data symbols, resulting in a reduced degradation in performance. Comparison with [18], where at SNR = 20dB a BER = 6 × 10−3 is achieved, shows that our scheme with L = 140 and L = 56 performs better and also provides a higher overhead efficiency. From [18] we have that the complexity per data symbol of the VA technique for this system is O(210 ). With K = 64 subcarriers the complexity per OFDM symbol becomes O(216 ). The complexity of our scheme for this system is O(K 3 ) = O(218 ). Therefore, the performance advantages and bandwidth efficiency gain, due to the use of a reduced number of pilot symbols, of our scheme comes at the expense of an approximate fourfold increase in complexity. Next we focus on the operation of the Kalman filter when it is employed in decision-feedback mode. We use Variation 3 (Niter = 2) and the SRKF with L = 56. We define two operational modes, named as matched and frozen. In (38), the measurement update requires N0 . In the matched case, when SNR ≥ 40.32dB, the corresponding (true) value of N0 is used in the SRKF. In the frozen case, when SNR ≥ 40.32dB, the SRKF employs the value of N0 corresponding to SNR = 35.32dB and hence it is larger than it is supposed to be. We artificially increase the observation noise assumed by the filter to account for the perturbation in the observation matrix Hm due to decision-feedback errors. Simulation results for BER and FER, as well as the relative number of frames with various number of errors are presented in Fig.4(a)(b). We see that in the matched case, the BER and FER increase when the SNR increases from SNR = 35.32dB to SNR = 42.82dB. This happens because in such a high SNR ˆ m/m due to regime, the Kalman filter tends to rely more on the observation, and hence an erroneous H decision-feedback errors has a more marked effect. In the frozen case we see that the BER and FER are reduced. This artificially increased N0 assumed in the Kalman filter improves the robustness against errors in the observation matrix by reducing the Kalman filter reliance on the observations. Fig.4(b) illustrates the number of error distribution in erroneous frames. Error propagation causes clusters of consecutive errors, leading to an increased number of frames with many errors. For SNR = 35.32dB, we observe that more than 70% of erroneous frames contain only one error, and approximately 10% of erroneous frames contain more than six errors. In the matched case at SNR = 40.32dB, we observe that over 20% of erroneous frames contain more than six errors, indicating a more serious problem of

22

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error propagation. When the SNR is further increased to 42.82dB close to 30% of erroneous frames contain more than six errors. In the frozen case at SNR = 40.32dB, we observe that less than 10% of the erroneous frames have more than six errors, illustrating a less severe error propagation effect that is translated in lower BER and FER. The frozen case at SNR = 42.82dB shows a similar effect. Hence artificially increasing the observation noise variance in the Kalman filter seems to provide a direction for reducing the losses due to decision-feedback errors propagation. B. Performance Over LTE channels Consider now the performance of our scheme over LTE channels. The channel of [18], that has been used in the previous subsection, consists of only two paths. The LTE channel consists of seven paths, and will demonstrate the performance of our scheme in a more severe multipath environment. An OFDM system with 64 sub-carriers employing QPSK modulation is considered. We set L = 140 and use 10 pilot symbols at the beginning of each frame for training. The LTE channel is specified in Table II, where the excessive delay of each tap is normalized by the sampling frequency 30.72 × 106 Hz as in [50]. Setting fd Tsym = 0.02 we provide in Fig.5 simulation results for BER of our system with Variation 3 employing Niter = 2, Variation 1, and OPSD with perfect channel knowledge. We see that Variation 3 provides a better performance than Variation 1 by 0.3dB at BER = 5 × 10−4 . This is a relatively small performance improvement since with such a normalized Doppler frequency, ˆxm/m−1 provides a good enough estimate of xm . Iteratively performing OPSD and measurement update does not improve the estimate of xm significantly. A performance degradation of approximately 1.3dB is observed when comparing Variation 3 employing Niter = 2 with OPSD using perfect CSI. Variation 3 and Variation 1 provide the same diversity gain as the perfect CSI case, which is larger than one. Performance results for our scheme over a faster time varying LTE channel with fd Tsym = 0.128 are illustrated in Fig.6. Over a such rapidly fading channel, Variation 1 looses diversity gains and ˆ m/m−1 based on xˆm/m−1 is suffers from an error floor at BER = 2 × 10−3 , showing that forming C not sufficient to provide an accurate enough channel estimate. Variations 2 and 3 provide significant performance improvements over Variation 1. We see that Variation 2 eliminates the error floor, and provides a performance gain of 3.5dB at BER = 1 × 10−2 over Variation 1. When using Variation 3 with Niter = 1 a performance gain of 1.4dB is achieved at BER = 8 × 10−5 over Variation 2. Using

23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Niter = 2 in Variation 3 provides a performance gain of 0.8dB at BER = 5 × 10−5 over the same variation with Niter = 1. Variation 3 with Niter = 2 employs an extra measurement iteration update and OPSD cycle, reducing decision-feedback errors. We observed that increasing Niter to 3 does not improve performance further. Variations 2 and 3 provide a diversity gain of 1.5 which is larger than obtained at fd Tsym = 0.02 where a diversity gain of only slightly larger than one is achieved. However, the performance loss compared to the perfect CSI case has also increased to 4dB since the tracking error of a faster time varying channel is larger. Next we compare our scheme with [25], that considers joint channel estimation and equalization for OFDM systems over fast time varying channels, employing the SAGE-MAP iterative algorithm in the time-domain. An initial estimate of the channel is provided by a Linear MMSE procedure based on pilot symbols. In addition to an estimate of the channel, the SAGE-MAP algorithm produces also an equalized output of each subcarrier, from which data can be detected. The results for QPSK from Figure 5(b) of [25] are presented in Table I. Notice that in [25] an LTE channel with only three paths is considered, while in our work we employ an LTE (extended ITU EPA) channel with seven multipath components as in [50, p. 271]. Furthermore, the OFDM system in [25] employs 1024 subcarriers, while in our system we use only 64. The Symbol Error Rate (SER) results from [25] were converted to BER by using a factor of 1/2 (i.e BER = SER/2) that provides good accuracy when Gray coding is used with QPSK [51, p. 271]. At a normalized Doppler of 0.02, comparison of Fig. 5 of our paper with the second row of Table I shows that the scheme from [25] performs better than ours by 2.4dB at BER = 3.5 · 10−3 . This advantage decreases to 1.8dB at BER = 1.4 · 10−3 . At BER = 5 · 10−4 the two schemes perform essentially the same. At a normalized Doppler of 0.128, that is larger than 0.08 as considered in [25], comparison of Fig.6 in our paper with the third row of Table I, shows a clear advantage of our technique, since the scheme of [25] suffers from an error floor at BER = 2 · 10−2 , while in our scheme the BER can be reduced to less than 4 · 10−5, with a diversity order larger than one. Hence our scheme provides a performance advantage over that of [25] for fast time varying channels with normalized Doppler larger than 0.08. The complexity of the scheme from [25] scales as O(KP ) per data symbol. Therefore, the complexity per OFDM symbol scales as O(K 2 P ), that is lower than in our scheme that scales as O(K 3 ). Hence the performance advantage of our scheme comes at the expense of complexity increase.

24

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Finally, we consider now phase noise affected LTE channels. The phase noise is modelled as an integrated white noise process [52] and for OFDM can be implemented as [53] n  u[m(Ng + K) + i] φm [n] = φm−1 [K − 1] +

(59)

i=−Ng

where Ng is the cyclic prefix length, u[i] are zero mean independent Gaussian random variables of variance σu2 = 2πβTsym /K with β denoting the phase-noise line-width. From [53, eq.(4)] the discrete time received samples, when impaired by carrier phase noise, become r˜m [n] = eφm [n] rm [n] where in our work rm [n] is given by (1). Hence the effect of phase noise is to multiply the path gains hm,p [n] in (1) by ejφm [n] , resulting in multiplying the time variant channel gains Hm,l [n] of (4) by ejφm [n] . By varying β we test the robustness of our algorithm with different phase noise levels over LTE channels. Simulation results for BER with fd Tsym = 0.02 and β = 4.8Hz, 9.6Hz, 24Hz are presented in Fig.7. Compared to the zero phase noise case of Fig.5, with β = 4.8Hz a loss of 0.5dB is observed at BER = 1.05 × 10−4 for perfect CSI. Variation 3 with Niter = 2 shows a degradation that is less than 0.4dB at BER = 1.1 × 10−4 . Variation 1 suffers a performance loss of approximately 1.3dB at BER = 1.6 × 10−4 . Further increase of the phase noise to β = 9.6Hz increases slightly the loss of Variation 1 and Variation 3 with Niter = 2. With perfect CSI such larger phase noise increases the degradation by another 0.5dB. When β = 24Hz, a loss of 1dB at BER = 1.5 × 10−4 is observed for Variation 3 with Niter = 2 and a loss of 2.3dB is observed at BER = 2.4 × 10−4 for Variation 1. A lager degradation of 3.5dB at BER = 2.3 × 10−4 is observed with perfect CSI. Our scheme shows a certain robustness against phase noise when fd Tsym = 0.02, and Variation 3 with Niter = 2 is least affected as β increases. We see that an additional advantage of performing iterative data detection and measurement update is increased phase noise robustness over moderate time varying channels. Next we present in Fig.8 BER simulation results for our scheme with fd Tsym = 0.128. We observe that in general, over fast fading channels our scheme becomes more sensitive to phase noise. With β = 4.8 Hz, we observe a degradation close to 1 dB compared with the zero phase noise case of Fig.6 for all the simulated variations. With β = 9.6 Hz, a significant performance loss is observed. For the perfect CSI case, a degradation of 3.5 dB is observed at BER = 4 × 10−5 . Variations 2 and 3 lose the diversity gain and show error floor trends. For Variation 2, an error floor at BER = 1 × 10−3 is observed. For Variation 3 with Niter = 1, an error floor at BER = 7 × 10−4 is observed. When Niter = 2 for

25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Variation 3, an error floor at BER = 4 × 10−4 is observed. We conclude that over a fast fading channel, not addressing the phase noise issue would have a significant negative impact on performance. Including phase noise in the state space model could provide better carrier tracking capabilities for the Kalman filter, a subject that is interesting to investigate as a follow up to this work. VII. C ONCLUSIONS This work considered a scheme based on Kalman filter channel multipath gains estimation, integrated with the OPSD algorithm for MIMO detection in OFDM systems. We employed a state-space model for an OFDM system based on the KL basis expansion, that can be used to construct the Kalman filter. Then we integrated the resulting Kalman filter, working in decision-feedback mode, with the OPSD algorithm, that is our simplified version of SD for OFDM systems. We proposed three different methods of providing decision-feedback for this scheme. Monte-Carlo computer simulations were used to evaluate the performance of these methods, revealing an advantage over other competing technique based on VA sequence estimation. Furthermore, for large normalized Doppler shifts, our scheme seems to provide performance advantages also over SAGE-MAP based receivers. We showed that performing iterative data detection by the OPSD algorithm and providing measurement updates to the Kalman filter improves performance with two iterations. We further studied the degradation in the operation of the Kalman filter due to decision-feedback errors, and as a remedy we proposed to artificially increase the observation noise that the filter assumes. This modification lowers the reliance of the Kalman filter on the measurement, and hence reduces the effects of decision-feedback errors. Performance of this scheme for LTE channels was evaluated with different normalized Doppler shifts and phase noise. Without phase noise, our scheme provides good performance over LTE channels even when the Doppler shift is high. In the presence of phase noise, the good performance of our scheme is maintained for a moderate Doppler shift of fd Tsym = 0.02. Performance degrades over a faster fading channel of fd Tsym = 0.128 with phase noise, indicating the need to improve the Kalman filter in the presence of carrier phase noise when the normalized Doppler shift is large. Hence OPSD integrated with a Kalman filter operating in decision-feedback mode is an attractive detection scheme for OFDM over high mobility channels, that offers performance advantages especially in high Doppler environments.

26

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ACKNOWLEDGEMENT The authors wish to thank Dr. Simon Qu from BlackBerry for recommending [28], and for interesting discussions on the effect of pilot symbols on the throughput of 3G/4G wireless communication systems.

A PPENDIX For matrix inversion we employ Newton iterations that allows parallel implementation and hence reduces delay [54] [55]. Consider a complex K × K non-singular matrix M. The Newton iterations ˜ i+1 = (2I − M ˜ i M)M ˜i M

(60)

˜ 0 = a0 MH , produces a sequence of approximate inverses of M, when a0 is chosen properly. Let where M 2 (M) be the smallest and largest eigenvalues of MH M. Then from [54] we have that the best σ12 (M), σK −1

2 ˜ i M, we have Ri+1 = R2i choice is a0 = 2 [σ12 (M) + σK (M)] . Defining the residue series Ri = I − M

showing quadratic convergence. Newton’s method for matrix inversion is masively parallelizable, and offers good numerical stability. It requires O(log(K)) iterations ( that is translated to O(log2 (K)) time units on K 2 parallel calculation units) to provide an accuracy of 2−KO(1) [54]. From [17] [55] we have that using only 30 iterations ensures an un-noticeable error for our applications. R EFERENCES [1] A. Sahin, I. G¨uvenc¸, and H. Arslan, “A survey on multicarrier communications: prototype filters, lattice filters, lattice structures, and implementation aspects,” IEEE Comm. Surveys and Tutorials, vol. 16, no. 3, pp. 1312–1337, 2014. [2] L. Dai, Z. Wang, and Z. Yang, “Next-generation digital television terrestrial broadcasting systems: key technologies, and research trends,” IEEE Comm. Magazine, vol. 50, no. 6, pp. 150–158, June 2012. [3] “IEEE standard for local and metropolitan area networks part 16: air interface for broadband wireless access systems, amendment 3 advanced air interface,” IEEE Standard 802.16m, 2011. [4] “LTE: Evolved universal terrestrial radio access (E-UTRA) and evolved universal terrestrial radio-access network (EUTRAN); overall description,” 3GPP Standard TS36.300, 2011. [5] C. Pan, L. Dai, and Z. Yang, “TDS-OFDM based HDTV transmission over fast fading channels,” IEEE Trans. on Consumer Electronics, vol. 59, no. 1, pp. 16–23, Feb. 2013. [6] T. Hrycak, S. Das, G. Matz, and H. G. Feichtinger, “Practical estimation of rapidly varying channels for OFDM systems,” IEEE Trans. on Comm., vol. 59, no. 11, pp. 3040–3048, Nov. 2011. [7] F. Hlawatsch and G. Matz, Wireless Communications Over Rapidly Time-Varying Channels (Chpt. 7). Academic Press, Burlington MA, USA, 2011. [8] X. Huang and H.-C. Wu, “Intercarrier interference analysis for wireless OFDM in mobile channels,” in IEEE Wireless Comm. and Networking Conference (WCNC 2006), vol. 4, April 2006, pp. 1848 –1853. [9] Y.-S. Choi, P. J. Voltz, and F. A. Cassara, “On channel estimation and detection for multicarrier signals in fast and selective Rayleigh fading channels,” IEEE Trans. on Comm., vol. 49, no. 8, pp. 1375 –1387, Aug 2001.

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[10] N. I. Miridakis and D. D. Vergados, “A survey on the successive interference cancellation performance for singleantenna and multiple-antenna OFDM systems,” IEEE Comm. Surveys and Tutorials, vol. 15, no. 1, pp. 312–335, 2013. [11] H. Wang, D. W. Lin, and T. H. Sang, “OFDM signal detection in doubly selective channels with blockwise whitening of residual intercarrier interference and noise,” IEEE Jour. Sel. Areas in Comm., vol. 30, no. 4, pp. 684–694, May 2012. [12] Z. Tang, R. C. Cannizzaro, G. Leus, and P. Banelli, “Pilot-assisted time-varying channel estimation for OFDM systems,” IEEE Tran. Sig. Proc., vol. 55, no. 5, pp. 2226 – 2238, May 2007. [13] W. G. Jeon, K. H. Chang, and Y. S. Cho, “An equalization technique for Orthogonal Frequency-Division Multiplexing system in time-variant multipath channels,” IEEE Tran. Comm., vol. 47, no. 1, pp. 27–32, Jan 1999. [14] X. Cai and G. Giannakis, “Bounding performance and suppressing intercarrier interference in wireless mobile OFDM,” IEEE Tran. Comm., vol. 51, no. 12, pp. 2047 – 2056, Dec. 2003. [15] L. Rugini, P. Banelli, and G. Leus, “Low-complexity banded equalizers for OFDM systems in Doppler spread channels,” EURASIP Jour. Appl. Sig. Processing, 2006: 067404 (3 August 2006). [16] S. Ohno, “Maximum likelihood inter-carrier interference suppression for wireless OFDM with null subcarriers,” IEEE International Conference on Acoustics, Speech, and Signal Processing. (ICASSP ’05.), vol. 3, pp. iii/849 – iii/852, March 2005. [17] W. Yi and H. Leib, “Sphere Decoding for OFDM systems over Doubly Selective Channel,” in Canadian Conference on Electrical and Computer Engineering (CCECE 2013), Regina, Saskatchewan, Canada, May 2013, DOI:10.1109/CCECE.2013.656747. [18] K. A. D. Teo and S. Ohno, “Pilot-Aided Channel Estimation and Viterbi Equalization for OFDM over DoublySelective Channel,” in IEEE Global Telecomm. Conference (GLOBECOM’06), San Francisco, CA, USA, Dec. 2006, DOI:10.1109/GLOCOM.2006.542. [19] H. Hijazi and L. Ros, “Polynomial Estimation of Time-Varying Multipath Gains With Intercarrier Interference Mitigation in OFDM Systems,” IEEE Trans. on Vehicular Technology, vol. 58, no. 1, pp. 140 –151, Jan. 2009. [20] R. Cannizzaro, P. Banelli, and G. Leus, “Adaptive channel estimation for OFDM systems with Doppler spread,” in IEEE 7th Workshop on Signal Processing Advances in Wireless Communications (SPAWC’06), Cannes, France, July 2006, DOI:10.1109/SPAWC.2006.346485. [21] X. Ma, G. Giannakis, and S. Ohno, “Optimal training for block transmissions over doubly selective wireless fading channels,” IEEE Trans. on Signal Processing, vol. 51, no. 5, pp. 1351 – 1366, May 2003. [22] H. Hijazi and L. Ros, “Joint data QR-detection and Kalman estimation for OFDM time-varying Rayleigh channel complex gains,” IEEE Trans. on Commun., vol. 58, no. 1, pp. 170 –178, Jan. 2010. [23] K. Muralidhar and K. H. Li, “A Low-Complexity Kalman Approach for Channel Estimation in Doubly-Selective OFDM Systems,” IEEE Signal Processing Letters, vol. 16, no. 7, pp. 632 –635, July 2009. [24] P. Banelli, R. C. Cannizzaro, and L. Ruggini, “Data-aided Kalman tracking for channel estimation in Doppler-affected OFDM systems,” in IEEE Int. Conf. Acous. Speech Sig. Proc. ICASP 2007, April 2007, pp. iii/133 – iii/136. [25] H. S¸enol, E. Panairici, and H. V. Poor, “Nondata-aided joint channel estimation and equalization for OFDM systems in very rapidly varying mobile channels,” IEEE Trans. Sig. Proc., vol. 60, no. 8, pp. 4236–4253, Aug 2012. [26] X. Dai, “Optimal training design for linearly time-varying MIMO/OFDM channels modelled by a complex exponential basis expansion,” IET Commun., vol. 1, no. 5, pp. 945–953, Oct. 2007.

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[27] K. Muralidhar and D. Sreedhar, “Comments on ’optimal training design for linearly time-varying MIMO/OFDM channels modelled by a complex exponential basis expansion,” IET Commun., vol. 7, no. 13, p. 1437, Sept. 2013. [28] L. Hanzo, H. Haas, S. Imre, D. O’Brien, M. Rupp, and L. Gyongyosi, “Wireless miths, realities, and futures: From 3G/4G to Optical and Quantum Wireless,” IEEE Proceedings, vol. 100, pp. 1853 – 1888, May 2012. [29] K. A. D. Teo and S. Ohno, “Optimal MMSE finite parameter model for doubly-selective channels,” in IEEE Global Telecomm. Conference (GLOBECOM’05), vol. 6, St. Louis, MO, USA, Dec. 2005, DOI:10.1109/GLOCOM.2005.1578424. [30] O. Simeone and Y. Bar-Ness, “Pilot-based channel estimation for OFDM systems by tracking the delay-subspace,” IEEE Trans. Wireless Commun., vol. 3, no. 1, pp. 315 – 325, Jan. 2004. [31] B. Yang, K. B. Letaief, R. S. Cheng, and Z. Cao, “Channel estimation for OFDM transmission in multipath fading channels based on parametric modeling,” IEEE Tran. Comm., vol. 39, no. 3, pp. 467 – 479, March 2001. [32] M. Oziewiicz, “On the application of MUSIC algorithm to time delay estimation in OFDM channels,” IEEE Tran. Broadc., vol. 51, no. 2, pp. 249 – 255, June 2005. [33] C. Qian, L. Huang, and H. C. So, “Computationally efficient ESPRIT algorithm for direction-of-arrival estimation based on Nystr¨o method,” Singnal Processing (Elsevier Press), vol. 94, pp. 74 – 80, Jan. 2014. [34] Y. Zheng and C. Xiao, “Improved models for the generation of multiple uncorrelated Rayleigh fading waveforms,” IEEE, Comm. Letters, vol. 6, no. 6, pp. 256 –258, Jun 2002. [35] S. M. Kay, Modern Spectral Estimation Theory and Application. Prentice Hall, Englewood Cliffs, NJ, USA, 1999. [36] B. Anderson and J. Moore, Optimal Filtering. Prentice Hall, Englewood Cliffs, NJ, USA, 1979. [37] K. Muralidhar and D. Sreedhar, “Generalized Vector State - Scalar Observation Kalman channel estimator for doublyselective OFDM systems,” in IEEE IEEE Int. Conf. Acoustics, Speech and Sig. Proc (ICASP 2013), vol. 48, May 2013, pp. 4928 – 4932. [38] M. S. Grewal and A. P. Andrews, Kalman Filtering: Theory and practice using MATLAB, 4th ed.

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[46] B. Hasibi and H. Vikalo, “On the sphere-decoding algorithm I: Expected Complexity,” IEEE Trans. Sig. Proc., vol. 53, no. 8, pp. 2806 – 2818, Aug. 2005. [47] J. Jald´e and B. Ottersen, “On the complexity of sphere decoding in digital communications,” IEEE Trans. Sig. Proc., vol. 53, no. 4, pp. 1474 – 1484, Apr. 2005. [48] G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. John Hopkins University Press, Baltimore, USA, 1996. [49] A. Burg and D. Garett, “VLSI implementation of MIMO detection,” in Space-Time Wireless Systems - From array processing to MIMO communications (H. Bolcskei and D. Gesbert and C. B. Papadias and A. -J. van der Veen (editors). Cambridge University Press, 2006, pp. 554 – 574. [50] S. Sesia, I. Toufik, and M. Baker, LTE, The UMTS Long Term Evolution: From Theory to Practice. Wiley Publishing, West Sussex, UK, 2009. [51] J. G. Proakis, Digital Communication, 4th ed. McGraw-Hill, New York, 2001. [52] H. Leib and S. Pasupathy, “Trellis-coded MPSK with reference phase errors,” IEEE Transactions on Communications, vol. 35, no. 9, pp. 888–900, 1987. [53] S. Wu and Y. Bar-Ness, “OFDM systems in the presence of phase noise: consequences and solutions,” IEEE Transactions on Communications, vol. 52, no. 11, pp. 1988–1996, 2004. [54] V. Pan and R. Schreiber, “An Improved Newton Iteration for the Generalized Inverse of a Matrix, with Applications,” SIAM Journal on Scientific and Statistical Computing, vol. 12, no. 5, pp. 1109–1130, 1991. [Online]. Available: http://link.aip.org/link/?SCE/12/1109/1 [55] Y. Wang and H. Leib, “Sphere decoding for MIMO systems with Newton interative matrix inversion,” IEEE Commun. Letters, vol. 17, no. 2, pp. 389 – 392, Feb. 2013.

SNR (dB)

10

15

20

25

30

BER [18] fd Tsym = 0.128 3 · 10−2

1 · 10−2

2.7 · 10−3

1 · 10−3

8 · 10−4

BER [25] fd Tsym = 0.02

3.4 · 10−2

1.2 · 10−2

3.5 · 10−3

1.4 · 10−3

5 · 10−4

BER [25] fd Tsym = 0.08

4.8 · 10−2

3 · 10−2

2.3 · 10−2

2 · 10−2

2 · 10−2

TABLE I P ERFORMANCE OF B ESSEL V ITERBI 3 ALGORITHM FROM [18, F IG . 5], AND BER FOR QPSK FROM [25, F IG . 5( B )].

Tap Number

1

2

3

Excess Delay Tap (Samples)

0

0.9126 2.1504 2.4576 3.3792 5.8368 12.5952

Relative Power (dB)

0.0 -1.0

-2.0

4

-3.0

TABLE II LTE CHANNELS SPECIFICATION [50].

5

-8.0

6

-17.2

7

-20.8

30

0

10 Initialization ˆ −1/−1 x P−1/−1

ym

−1

10 ˆ m/m−1 x Pm/m−1

Time Update

−2

ˆ m/m−1 a) Form Matrix C b) Data Symbol Dection ˆ m/m,0 c) H

10 Eror Rate

ˆ m/m x Pm/m

−3

10 Measurement Update

ˆ m/m−1 x Pm/m−1

a) Measurement Update ˆ m/m,i b) Form Matrix C c) Data Symbol Detection ˆ m/m,i d) H

FER BER L=140 KF L=140 SQT L=56 KF L=56 SQT

−4

10

i=i+1 ddec,m Output

−5

10

Fig. 1. Integrated Kalman filtering and OPSD. Variation 1 skip the steps in dash box. Variation 2 execute once the steps in dash box. Variation 3 execute the steps in dash box Niter times.

10

15

20

25

30

SNR(dB)

35

Fig. 3. BER and FER performance of Variation 3 over the channel of [18] with fd Tsym = 0.32, employing shorter frames.

(a)

−1

10

Error Rate

−3

−2

10

10

−4

10

FER BER Matched Frozen

−5

10

35

−3

36

37

10 BER

38

39 40 SNR(dB)

41

42

43

(b)

fdTsym=0.128 −4

10

f T

Relative Number

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

=0.32

d sym

−5

10

Var 1 Var 2 Var 3 (Niter=1) Var3 (Niter=2)

10

10

15

0.4 0.2 0 1

−6

20

25

30

35.32dB 40.32dB Matched 40.32dB Frozen 42.82dB Matched 42.82dB Frozen

0.6

2

3 4 5 6 Number of Errors per Frame

>6

35

SNR(dB)

Fig. 2. BER performance over the channel of [18], employing frames of 1000 OFDM symbols.

Fig. 4. BER, FER performance and error distribution for Variation 3 over the channel from [18] with fd Tsym = 0.32. (a) Error rate results. (b) Relative number of frames with various number of errors.

31

β=4.8Hz β=9.6Hz β=24Hz Var3 Var1

Var3 (Niter=2) Var1 CSI

−2

−2

10

10

BER

BER

Known CSI

−3

−3

10

−4

10

10

10

−4

15

20

25

SNR(dB)

30

35

Fig. 5. BER performance for LTE channels with fd Tsym = 0.02.

15

20

25

SNR(dB)

30

35

Fig. 7. BER performance for LTE channels with fd Tsym = 0.02 and phase noise.

β=4.8Hz β=9.6Hz Var3 (Niter=2)

−1

−1

10

10

Var3 (N 10

BER

10

−3

10

Var2

−3

10

Var3 (Niter=2) −4

10

−5

10

15

=1)

iter

−2

−2

BER

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Known CSI

Var3 (Niter=1)

−4

10

Var2 Var1 CSI

−5

10 20

25

SNR(dB)

30

35

Fig. 6. BER performance for LTE channels with fd Tsym = 0.128.

15

20

25

SNR(dB)

30

35

Fig. 8. BER performance for LTE channels with fd Tsym = 0.128 and phase noise.

  Wang Yi rreceived the B Bachelor degrree in electriccal and electrronic engineering from Nanyang  Technologgical University, Singaporee  in 2006, and d the M.Eng iin electrical eengineering frrom McGill  Universityy, Montreal, C Canada, in 20 013. From 201 11 to 2013, hee was working in a Blackbeerry funded  research p project on wireless communications at McGill Univeersity. Since 2 2014 he has been a software  developerr with Ericsso on, Ottawa, Caanada. His research intereests include O OFDM commu unications, an nd  adaptive cchannel estim mation.    HARRY  LEEIB  

  Harry Leib b received the e B.Sc. (cum laude) and M.Sc. degrees iin Electrical EEngineering frrom the Technion ‐  Israel Insttitute of Technology, Haifaa, Israel in 197 77 and 1984 rrespectively. In 1987 he reeceived the Ph.D.  degree in Electrical Enggineering from m the University of Torontto, Canada.    During 19 977‐1984 he w was with the Israel Ministrry of Defense,, working in tthe Communication System ms  area. Afteer completingg his Ph.D. stu udies, he was with the Univversity of Torronto as a Posst‐doctoral  Research Associate and as an Assisttant Professor. Since Septeember 1989 h he has been w with the  n Montreal, w where he is no ow as  Departmeent of Electriccal and Computer Engineering at McGill University in a Full Proffessor. He spe ent parts of ssabbatical leaves of absencce at Bell Norrthern Researrch in Ottawa,  Canada (1 1995‐1996), C Communicatio ons Lab of thee Helsinki University of Technology in FFinland (1996), 

and the Department of Electrical Engineering of the École Polytechnique de Montreal (2015‐2016). At  McGill University he teaches undergraduate and graduate courses in Communications, and directs the  research of graduate students. His current research activities are in the areas of Digital Communications,  Wireless Communication Systems, Detection, Estimation, and Information Theory.      Dr. Leib was an Editor for the IEEE Transactions on Communications 2000‐2013, and an Associate Editor  for the IEEE Transactions on Vehicular Technology 2001‐2007. He has been a guest co‐editor for special  issues of the IEEE Journal on Selected Areas in Communication on “Differential and Noncoherent  Wireless Communication”  2003‐2005, and on “Spectrum and Energy Efficient Design of Wireless  Communication Networks” 2012‐2013.