Channel estimation plays an important role in communication systems and, particularly, in the 3GPP Long-Term Evolution (LTE) which aims at continuing the competitiveness of the 3G Universal Mobile Telecommunications System technology. Orthogonal frequency-division Multiple Access (OFDM) is considered as one of the key technologies for the 3GPP LTE to improve the communication quality and capacity of mobile communication system. As the support of high mobility is required in 3GPP LTE systems, the signals at the OFDM receivers are likely to encounter a multi-path, fast time-varying channel environment [1]. Thus, good channel estimation and equalization at the receiver is demanded before the coherent demodulation of the OFDM symbols. In mobile communication, since the radio channel is modelled by some dominant spare paths and is represented by path taps, the channel estimation is to estimate and track the channel taps adaptively and efficiently.

In wideband mobile communications, the pilot-based signal correction scheme has been proven a feasible method for OFDM systems. The 3GPP LTE standard employs a Pilot Symbol-Aided Modulation (PSAM) scheme but does not specify the methods for estimating the channel from the received pilot and data signals. In the 3GPP LTE downlink, pilot symbols, known by both the sender and receiver, are sparsely inserted into the streams of data symbols at pre-specified locations. Hence, the receiver is able to estimate the whole channel response for each OFDM symbol given the observations at the pilot locations. Pilot-symbol-aided channel estimation has been studied [2–4] and the common channel estimation techniques are based on least squares (LS) or linear minimum mean square error (LMMSE) estimation [5]. Note that most pilot-symbol-aided channel estimators, including those mentioned above, work in the frequency domain. LS estimation is the simpler algorithm of the two as it does not use channel correlation information. The LMMSE algorithm makes use of the correlation between subcarriers and channel statistic information to find an optimal estimate in the sense of the minimum mean square error.

In the literature, based on these two basic estimators, various methods are proposed to improve the performance of the channel estimation. As the LS and LMMSE estimators only give the channel estimate at the pilot symbol, most current work on pilot-aided channel estimation considers interpolation filters where channel estimates at known pilot symbols are interpolated to give channel estimates at the unknown data symbols. Since the 3GPP LTE downlink pilot symbols are inserted in a comb pattern in both the time and the frequency domain, the interpolation is a 2D operation. Although some 2D interpolation filters have been proposed [6], presently, interpolation with two cascaded orthogonal 1D filters is preferred in 3GPP LTE. This is because the separation of filtering in time and frequency domains by using two 1D interpolation filters is a good trade-off between complexity and performance. Various 1D interpolation filters have been investigated. Examples are linear interpolation, polynomial interpolation [7], DFT-based interpolation [8], moving window [9] and iterative Wiener filter [10].

From a system point of view, the channel estimation is a state estimation problem, in which the channel is regarded as a dynamic system and the path taps to be estimated are the state of the channel. It is known that Kalman filter (KF) provides the minimum mean square error estimate of the state variables of a linear dynamic system subject to additive Gaussian observation noise [11]. By considering the radio channel as a dynamic process with the path taps as its states, the KF has shown its suitability for channel estimation in the time domain [1]. In the frequency domain, Kalman-based channel estimator in OFDM communication has also been studied [1, 12, 13]. For example, in [1, 12], a modified KF is proposed for OFDM channel estimation where the time-varying channel is modelled as an autoregressive(AR) process and the parameters of the AR process are assumed real and within the range [0.98, 1] for slow-fading channels. However, in the high-mobility environment, these parameters are relative large (e.g. in the 200 km/h environment, they are complex values with magnitudes varying in [0, 1.5]) representing a fast-fading channels.

The difference between the KF in [12] and the one proposed in this article is that the former estimated the parameters of AR by a gradient-based recursive method separately, rather by the linear KF. Whereas, we derive an extended Kalman filter (EKF) for jointly estimating the channel response and the parameters of the AR model simultaneously. In addition, the parameters of the AR model are assumed time-invariant and known *in priori* by solving Yule-Walker equation in [1]. The authors of [13] only considered the comb-type pilot pattern*s* in which some subcarriers are full of pilot symbols without unknown data. As a result, the KF in [13] requires continuous stream of pilot symbols and is not suitable for 3GPP LTE, as the 3GPP LTE employs a scattered pattern where the pilot symbols are distributed sparsely among the data streams.

Although the KF-based channel estimation for LTE uplink has been reported recently [1], there has been no KF-based joint estimation of both time-varying channel taps and the time-correlation coefficients of 3GPP LTE downlink in frequency-time domain. This article focuses on the major challenge of scattered pilot-aided channel estimation and interpolation for a time-varying multipath fast-fading channel in 3GPP LTE downlink. An AR process is used to model the time-varying channel. Both the taps of the multipath and the time-correlation coefficients are jointly estimated by treating the channel as a nonlinear system. Then, a combined estimation and interpolation scheme is present under the EKF framework.

The main contribution of the proposed method is (1) both the time-correlation coefficients and channel taps are estimated simultaneously in the framework of EKF; (2) no assumption on the upper/lower boundaries of the time-correlation coefficients to achieve a good tracking of fast-fading channel in high-mobility scenario; (3) applicable to preamble pilot patterns, comb-type pilot patterns and scattered pilot patterns.

This article is organized as follows: Section “System model” gives an overview of the LTE 3GPP downlink system and formulates its channel estimation problem. In Section “EKF for channel estimation”, an EKF is derived by using a first-order Taylor approximation for the joint estimation of channel taps and time-correlation coefficients at pilot symbols. Section “EKF for channel interpolation” describes the combined estimation and interpolation scheme and summarizes the proposed algorithm. Simulation results of the proposed Kalman interpolation filter are presented and its performance is demonstrated in Section “Simulation results and performance analysis”.

### Notation and terms

Unless specified otherwise, an italic letter (e.g.T, *h*_{
k,n
}) represents a scalar and its bold face lower-case letter represents its corresponding vector (e.g. ${\mathbf{h}}_{k}=\left[{h}_{k,1}{h}_{k,2},{h}_{k,{N}_{p}}\right]$). A bold face upper-case letter (e.g. **A**) represents a matrix. The subscriber k denotes the time index of an OFDM symbol, n denotes the index of subcarriers in the frequency domain, l denotes the l th path of the radio channel. $\left|x\right|$ ($\left|A\right|$) is the element-wise magnitude of a vector x (matrix **A**). **I**_{
N
} is an *N × N* identity matrix. **A**_{
i,j
} denotes the entry at the i th row and the k th column of **A**.

*L* denotes the total number of possible paths in a radio channel, referred to as channel length, *N* denotes the total number of subcarriers, *N*_{
p
} the number of pilot subcarriers, ${g}_{k,l}$ the channel impulse response (CIR) of l th path at k th symbol, referred to as tap, **g**_{
k
} the CIR vector at *k* th symbol time, ${\mathbf{g}}_{k}=\left[{g}_{k,1}\cdots {h}_{k,L}\right]$_{,}*h*_{
k,n
} the channel frequency response (CFR) at k th symbol time and n th subcarrier, ${\overline{\mathbf{h}}}_{k}$ the CFR vector at all subcarriers at k th symbol time, ${\overline{\mathbf{h}}}_{k}=\left[{h}_{k,1}\cdots {h}_{k,N}\right]$, **h**_{
k
} the CFR vector at *N*_{
p
} pilot subcarriers at k th symbol time, ${\mathbf{h}}_{k}=\left[{h}_{k,1}\cdots {h}_{k,{N}_{p}}\right]$_{,}**a**_{
k
} the time-correlation coefficients of CFR at k th symbol time, **x**_{
k
} the vector of transmitted OFDM symbols at pilot subcarriers at k th symbol time and **y**_{
k
} the corresponding received OFDM symbol vector of **x**_{
k
}.