**2. System and channel model**

#### **2.1 System model**

256 Wireless Communications and Networks – Recent Advances

domain. The advantage of the scheme is that there is no loss in information rate, and thus enables higher bandwidth efficiency. However, some useful power must inevitably be allocated to the pilots, and thus resulting in information signal-to-noise ratio (SNR) reduction. Meanwhile, the information sequences are viewed as interference to channel estimation since pilot symbols are superimposed at a low power to the information sequences at the transmitter. The existing ST-based channel estimations are mainly restricted to the case where the channel is linearly time-invariant (LTI), where the channel transfer function can be estimated by using first-order statistics [8]-[13] [17]-[18]. In the latest contributions, J. K. Tugnait [16] extended the conventional ST to time-varying environment where the LTV channels are modeled by complex exponential bases. For the issue of training power allocation, the optimal pilot power has been investigated by [24] for different taps of low-pass filter, and then, the optimization of ST power allocation for LTI channel is

In this paper, a new ST-based channel estimator is proposed for OFDM/MIMO systems over LTV multipath fading channels. The main contributions are twofold. First, the LTV channel coefficients modeled by the truncated discrete Fourier bases (DFB), unlike the existing approaches [1]-[2] [5]-[6] [16], cover multiple OFDM symbols. Then, a two-step channel estimation approach is adopted for LTV channel estimation. Furthermore, a closedform expression of the estimation variance is derived, which provides a guideline for designing the superimposed pilot symbols. We demonstrate by analytical analysis that the estimation variance, unlike that of conventional ST-based schemes [8]-[19], approaches to a fixed lower-bound as the training length increases. Second, for wireless communication systems with a limited transmission power, unlike [10] where the issue of ST power allocation is derived by optimizing the SNR for equalizer design, we provide an optimal solution of ST power allocation with a different point of view by maximizing the lowerbound of channel capacity. Comparatively, the training power allocation scheme [10] can be otherwise considered as a special case compared with the proposed approach. In simulations presented in this paper, we compare the results of our approach with that of the FDM training approaches [5] as latter serves as a "benchmark" in related works. It is shown that the proposed algorithm outperforms that of FDM training, and yields higher

The rest of the paper is organized as follows. Section II presents the system and channel models. In Section III, we estimate the LTV channel coefficients with the proposed two-step channel estimation approach. In Section IV, we derive the closed-form expression of the channel estimation variances. Section V determines the optimal ratio of the ST power to the total transmission power by maximizing the lower-bound of channel capacity. Section VI reports on some simulation experiments in order to test the validity of theoretic results, and

*Notation:* The letter *t* represents the time-domain variable and *k* is the frequency-domain variable. Bold letters denote the matrices and column-vectors, and the superscripts[ ]*<sup>T</sup>* and [ ]*<sup>H</sup>* represent the transpose and conjugate transpose operations,

respectively. , [ ]*k t* denotes the (*k*, *t*) element of the specified matrix.

mathematically analyzed based on equalizer design [15] [19].

transmission efficiency.

we conclude the paper with Section VII.

Consider an MIMO/OFDM system of *N* transmitters or mobile users and a receive array of *M* receive antennas with perfect synchronization. At transmit terminals, an inverse fast Fourier transform (IFFT) is used as a modulator. The modulated outputs are given by

$$\mathbf{X}\_n(i) = \left[ \mathbf{x}\_n(i, 0), \dots, \mathbf{x}\_n(i, t), \dots, \mathbf{x}\_n(i, B - 1) \right]^T = \mathbf{F}^{-1} \mathbf{S}\_n(i) \qquad n = 1, \dots, N \tag{1}$$

where *B* is OFDM symbol-size, ( ) [ ( ,0), ( , ), ( , 1)]*<sup>T</sup> nn n n* **S** *i s i s ik s iB* is the *i* th transmitted symbol of the *n* th transmit antenna. <sup>1</sup> **F** is the IFFT matrix with <sup>1</sup> <sup>2</sup> , [ ] *<sup>j</sup> kt B k t <sup>e</sup>* **F** and <sup>2</sup> *j* 1 . Then, ( ) *<sup>n</sup>* **X** *i* is concatenated by a cyclic-prefix (CP) of length *L* , propagating through the respective channels. At receiver, the received signals of *m* th receive antenna, discarding CP and stacking the received signals ( ) , *<sup>m</sup> y i t t B* 0, 1 , can be written in a vector-form as

$$\mathbf{Y}^{(m)}\left(\mathbf{i}\right) = \left[y^{(m)}(\mathbf{i}, 0), \cdots y^{(m)}(\mathbf{i}, \mathbf{t}), \cdots y^{(m)}(\mathbf{i}, B - 1)\right]^T \qquad m = \mathbf{1}, \cdots M \tag{2}$$

and the received signals ( ) , *<sup>m</sup> y i t* in (2) is given by

$$\begin{aligned} \mathbf{y}^{(m)}(i,t) &= \sum\_{n=1}^{N} \mathbf{X}\_n(i) \otimes \mathbf{h}\_n^{(m)}(i) + \upsilon^{(m)}(i,t) \\\\ &= \sum\_{n=1}^{N} \sum\_{l=0}^{L-1} h\_{n,l}^{(m)}(i,t) \mathbf{x}\_n(i, t-l) + \upsilon^{(m)}(i, t) \qquad t = 0, \dots, B-1 \end{aligned} \tag{3}$$

where ( ) () () ,0 , 1 <sup>1</sup> , , , ,0 *<sup>T</sup> <sup>m</sup> m m <sup>n</sup> <sup>n</sup> n L B L i h it h it* **<sup>h</sup>** is the impulse response vector of the propagating channel from the *n* th transmit to the *m* th receive antenna. The channel coefficients ( ) , , *<sup>m</sup> n l h it* , 0, 1 *l L* is the functions of time variable *t* which will be defined by (6). The notation represents the cyclic convolution and ( ) , *<sup>m</sup> v it* is the additive Gaussian noise.

At receiver, an FFT operation is performed on the vector (2), and the demodulated outputs can be written as

$$\mathbf{U}^{(m)}\begin{pmatrix}i\\ \end{pmatrix} = \left[\boldsymbol{\mu}^{(m)}\begin{pmatrix}i,0\\ \end{pmatrix}, \cdots \boldsymbol{\mu}^{(m)}\begin{pmatrix}i,k\\ \end{pmatrix}, \cdots \boldsymbol{\mu}^{(m)}\begin{pmatrix}i,B-1\\ \end{pmatrix}\right]^T = \mathbf{F}\mathbf{Y}^{(m)}\begin{pmatrix}i\\ \end{pmatrix} \qquad m=1,\cdots,M\ . \tag{4}$$

From (3) and the duality of time and frequency, the FFT demodulated signals in (4) can be written as

$$\begin{aligned} \left(\mu^{(m)}(i,k) = FFT \left\{ \sum\_{n=1}^{N} \sum\_{l=0}^{L-1} h\_{n,l}^{(m)}(i,t) \ge\_n (i, t-l) + \upsilon^{(m)}(i,t) \right\} \\ = \sum\_{n=1}^{N} \sum\_{l=0}^{L-1} FFT \left\{ h\_{n,l}^{(m)}(i,t) \right\} \otimes FFT \left\{ \chi\_n(i,t) \right\} + \overline{\upsilon}^{(m)} \left(i,k\right) \end{aligned} \tag{5}$$

where FFT represents the FFT vector of the specified function and ( )(, ) *<sup>m</sup> v ik* is the frequency-domain noise. Compared with the FFT demodulated signals of OFDM systems with LTI channels, the convolution in (5) between the information sequences and the FFT vectors of time-varying channel coefficients may introduce inter-carrier interference (ICI).

#### **2.2 Channel model**

As mentioned in [1], the coefficients of the time- and frequency-selective channel can be modeled as Fourier basis expansions. Thereafter, this model was intensively investigated and applied in block transmission, channel estimation and equalization (e.g. [2][5]-[6][16]). In this paper, we extend the block-by-block process [2][5]-[6][16] to the case where multiple OFDM symbols are utilized. Consider a time interval or segment*t t* : ( 1) , the channel coefficients in (3) can be approximated by truncated discrete Fourier bases (DFB) within the segment as

$$\begin{aligned} h\_{n,l}^{(m)}\left(t,t\right) &\approx \sum\_{q=0}^{Q} h\_{n,l,q}^{(m)} e^{-j2\pi \left(q-Q/2\right)t/\hbar} \\\\ \lambda &= \sum\_{q=0}^{Q} h\_{n,l,q}^{(m)} \eta\_q\left(t\right) \quad t = (\ell - 1)\Omega, \cdots \ell\Omega \end{aligned} \tag{6}$$

where ( ) , , *m n l <sup>q</sup> h* is a constant coefficient, *Q* represents the basis expansion order that is generally defined as 2 *Qf f d s* [1], *B* is the segment length and is the segment index. Unlike [1]-[2] [5]-[6] [16], the approximation frame covers multiple OFDM symbols, denoted by 1, *i I* , where *I B* ' and *B BL* ' . Since the proposed two-step channel estimation as will be shown in Section III is adopted within one frame, we omit the segment index for simplicity.
