**3. ST-based channel estimation**

In this section, we propose a ST-based two-step approach for LTV channel estimation. In STbased approaches [8]-[19], the pilot symbols are superimposed (arithmetically added) to the information sequences as

$$s\_n(i,k) = c\_n\left(i,k\right) + p\_n\left(i,k\right) \quad k = 0, \cdots, B-1\tag{7}$$

where *c ik <sup>n</sup>* , and *pn i k*, are the information and pilot sequence, respectively. Compared with the FDM/TDM training aided methods [20]-[22], ST requires no additional bandwidth (or time-slot) for transmitting known pilots, and thus offers a higher data rate.

#### **3.1 ST-based channel estimation over one OFDM symbol**

For LTV environment where the channel coefficient ( ) , *m n l h t* is a function of time variable *t*, the vectors ( ) , *m n l FFT h t* in (5) cannot be approximated as a -sequences and, the FFT demodulated signals at the sub-carrier *k* of the *i* th symbol is given by

$$\begin{split} \boldsymbol{u}^{(m)}(\mathbf{i},k) &= \sum\_{n=1}^{N} \sum\_{l=0}^{L-1} FFT\left\{h\_{n,l}^{(m)}\left(\mathbf{i},t\right)\right\} \otimes \mathbf{P}\_{n}\left(\mathbf{i}\right) + \overline{\boldsymbol{v}}^{\cdot(m)}\left(\mathbf{i},k\right) \\ &\approx \sum\_{n=1}^{N} \sum\_{l=0}^{L-1} FFT\left\{\sum\_{q=0}^{Q} \boldsymbol{H}\_{n,l,q}^{(m)} \boldsymbol{\eta}\_{q}\left(t\right)\right\} \otimes \mathbf{P}\_{n}\left(\mathbf{i}\right) + \overline{\boldsymbol{v}}^{\cdot(m)}\left(\mathbf{i},k\right) \\ &= \sum\_{n=1}^{N} \sum\_{q=0}^{Q} \boldsymbol{H}\_{n,q}^{(m)}\left(\mathbf{i},k\right) \boldsymbol{\eta}\_{q}\left(t\_{i}\right) \mathbf{W}\_{q}\left(\mathbf{i},k\right) \otimes \mathbf{P}\_{n}\left(\mathbf{i}\right) + \overline{\boldsymbol{v}}^{\cdot(m)}\left(\mathbf{i},k\right) \end{split} \tag{8}$$

where ( ) <sup>1</sup> ( ) ( ) 1 0 , ' , C , *m m N L <sup>m</sup> n l n l <sup>n</sup> v ik FFT h t i v i k* and *t i BB <sup>i</sup>* 1' 2 . , *<sup>q</sup>* **<sup>W</sup>** *i k* with 0, *k B* 1 is the FFT vector of the complex exponential function (CEF) and, can be written as

$$\mathbf{W}\_{q}\begin{pmatrix}\mathbf{i},\mathbf{0}\end{pmatrix} = \left[w\_{q}\begin{pmatrix}\mathbf{i},\mathbf{0}\end{pmatrix},\cdots,w\_{q}\begin{pmatrix}\mathbf{i},\mathbf{k}\end{pmatrix},\cdots,w\_{q}\begin{pmatrix}\mathbf{i},\mathbf{B}-\mathbf{1}\end{pmatrix}\right]^{\mathrm{T}}\tag{9}$$

$$=\mathbf{F}\left[\eta\_{q}\begin{pmatrix}\mathbf{i}\_{i}-\mathbf{B}/2\end{pmatrix}\Big/\eta\_{q}\begin{pmatrix}\mathbf{i}\_{i}\end{pmatrix},\cdots,\eta\_{q}\begin{pmatrix}\mathbf{i}\_{i}+\mathbf{B}/2-\mathbf{1}\end{pmatrix}\Big/\eta\_{q}\begin{pmatrix}\mathbf{i}\_{i}\end{pmatrix}\right]^{\mathrm{T}}.$$

Notice that , *<sup>q</sup>* **W** *i k* is a cyclic-shifted vector of **W***<sup>q</sup> i*,0 with a shifting length *k* . On the other hand, ICI introduced by the cyclic convolution W, S *q n ik i* depends explicitly on *<sup>q</sup> t* , 0, 1 *t B* . When *q* is not large, the complex exponential functions in (9) are slowly time-varying over an OFDM symbol-duration and, thereby, the principal power or majorlobe of the FFT vector **W***<sup>q</sup> i*,0 may concentrate on its two ends (low frequency tones) with indexes 0,*T* and , 1 *BT B* . Using the major-lobe to approximate the CEF vectors ,0 *<sup>q</sup>* **W** *i q Q* 0,1, , we have

$$\mathbf{W}\_q(i,0) \approx \left[ w\_q(i,0), \cdots w\_q(i,T), 0, \cdots, 0, w\_q(i,B-T), \cdots w\_q(i,B-1) \right]^T\\q = 0, 1, \cdots, Q \tag{10}$$

where*T* is a positive integer.

258 Wireless Communications and Networks – Recent Advances

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).

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)

> ( ) ( ) 2 2 , , , 0

*<sup>Q</sup> m m j qQ t*

*<sup>t</sup>* ( 1) , , 1,2,

In this section, we propose a ST-based two-step approach for LTV channel estimation. In STbased approaches [8]-[19], the pilot symbols are superimposed (arithmetically added) to the

where *c ik <sup>n</sup>* , and *pn i k*, are the information and pilot sequence, respectively. Compared with the FDM/TDM training aided methods [20]-[22], ST requires no additional bandwidth

(or time-slot) for transmitting known pilots, and thus offers a higher data rate.

*n l FFT h t* in (5) cannot be approximated as a

demodulated signals at the sub-carrier *k* of the *i* th symbol is given by

**3.1 ST-based channel estimation over one OFDM symbol**  For LTV environment where the channel coefficient ( )

*s ik c ik nnn* ,,, *p i k* 0, 1 *k B* (7)

, *m*

*n l h t* is a function of time variable *t*,


*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

(6)

,

 ( ) , , 0 *<sup>Q</sup> <sup>m</sup> nlq q*

*h t* 

*q*

**3. ST-based channel estimation** 

information sequences as

the vectors ( )

, *m* *n l nlq q h it h e*

 

**2.2 Channel model** 

within the segment as

where ( ) , , *m*

simplicity.

In general, the FFT vector of the function *<sup>q</sup> t* in (10) may have a great side-lobe that results in a great error. For improving the approximation performance, an intuitional idea is to apply a window function to the received signals in order to reduce the side-lobe leakage. The windowed vector of received signals in (3) of the *i* th symbol is

$$\overline{y}^{(m)}(\mathbf{i},t) = \sum\_{n=1}^{N} h\_n^{(m)}(\mathbf{i},t)\overline{\nu}\_B(\mathbf{t})\mathbf{x}\_n(t-l) + \overline{\nu}^{(m)}(\mathbf{t})\overline{\nu}\_B(\mathbf{t}) \qquad t = 0, \dots, B-1 \tag{11}$$

where *<sup>B</sup> t* is a time-domain windowing function with a length *B*. Performing the FFT demodulated operation on the windowed sequences in (10), the demodulated signals, by (10), can be written by

$$\begin{split} \mu^{(m)}\left(i,k\right) &\approx \sum\_{n=1}^{N} \sum\_{l=0}^{L-1} FFT\left\{ \sum\_{q=0}^{Q} h\_{n,l,q}^{(m)} \eta\_q\left(t\right) \nu\_B\left(t\right) \right\} \otimes \mathbf{P}\_n\left(i\right) + \overline{\boldsymbol{\upsilon}}^{(m)}\left(i,k\right) \\ &= \sum\_{n=1}^{N} \sum\_{q=0}^{Q} H\_{n,q}^{(m)}\left(i,k\right) \eta\_q\left(t\_i\right) \overline{\mathbf{W}}\_q\left(i,k\right) \otimes \mathbf{P}\_n\left(i\right) + \overline{\boldsymbol{\upsilon}}^{(m)}\left(i,k\right) \end{split} \tag{12}$$

where W , *<sup>q</sup> i k* is the CEF vector with the windowing function*<sup>B</sup> t* as

$$\overline{\mathbf{W}}\_{q}(i,k) = \left[\overline{\boldsymbol{w}}\_{q}(i,0), \cdots \overline{\boldsymbol{w}}\_{q}(i,k), \cdots \overline{\boldsymbol{w}}\_{q}(i,B-1)\right]^{\mathrm{T}}$$

$$= \mathbf{F}\Big[\boldsymbol{\nu}\_{B}(t)\boldsymbol{\eta}\_{q}\big(t\_{i}-B/2\big)\big/\boldsymbol{\eta}\_{q}\big(t\_{i}\big), \cdots \boldsymbol{\nu}\_{B}(t)\big(\boldsymbol{\eta}\_{q}\big(t\_{i}+B/2-1\big)\big/\boldsymbol{\eta}\_{q}\big(t\_{i}\right)\big)^{\mathrm{T}}\Big] \tag{13}$$

$$\approx \left[\overline{\boldsymbol{w}}\_{q}\big(i,0\big), \cdots \overline{\boldsymbol{w}}\_{q}\big(i,T\big), 0, \cdots 0, \overline{\boldsymbol{w}}\_{q}\big(i,B-T\big), \cdots \overline{\boldsymbol{w}}\_{q}\big(i,B-1\big)\right]^{\mathrm{T}}.$$

Compared with (10), the approximation of windowing based vector has a much smaller side-lobe with the same index*T* . The experiment studies show that by using a *Kaiser*  function [5], the approximation in (13) of *T* = 2 may capture almost 99% power of 2: 2 1 *<sup>T</sup> FFT t t B t B t B qi i q i* for truncated DFBs when *q B* <sup>10</sup> . Substituting (13) into (12), the FFT demodulated outputs can be approximated by

$$
\begin{split} \boldsymbol{\mu}^{(m)}\left(\boldsymbol{i},\boldsymbol{k}\right) &= \sum\_{n=1}^{N} \sum\_{q=0}^{Q} \boldsymbol{H}\_{n,q}^{(m)}\left(\boldsymbol{i},\boldsymbol{k}\right) \boldsymbol{\eta}\_{q}\left(\boldsymbol{t}\_{i}\right) \bigg[ \sum\_{k'=0}^{T} \overline{\boldsymbol{w}}\_{q}\left(\boldsymbol{i},\boldsymbol{k'}\right) \boldsymbol{p}\_{n}\left(\boldsymbol{i},\boldsymbol{k}-\boldsymbol{k'}\right) + \\ & \sum\_{k'=T}^{1} \overline{\boldsymbol{w}}\_{q}\left(\boldsymbol{i},\boldsymbol{B}-\boldsymbol{k'}\right) \boldsymbol{p}\_{n}\left(\boldsymbol{i},\boldsymbol{k}+\boldsymbol{k'}\right) \bigg] + \overline{\boldsymbol{\nabla}}^{\{m\}}\left(\boldsymbol{i},\boldsymbol{k}\right) \,. \end{split} \tag{14}
$$

The first term of (14) illustrates that 2 1 *T* tones, i.e. *pn n ik k p ik k* , ', , ' should be jointly designed for estimating ( ) , , *<sup>m</sup> H ik n q* . We refer to such 2*T* + 1 consecutive pilot tones as a pilot cluster for differentiating from the isolated tones utilized in the LTI channel estimation [19] [22]-[23]. Denote 1 <sup>Γ</sup> *k k* , as the pilot cluster indexes located at the th pilot symbol and, , *<sup>n</sup> pk T pn k T* as the pilot sequences at the pilot cluster *k* . Since the ST does not entail additional bandwidth, two adjacent pilot-clusters, i.e. *k* and 1 *k* can be placed closed together. The pilot tone distribution is shown in Fig. 1.

Fig. 1. A typical pilot tone distribution. 2 1 *T* consecutive tones are grouped together as one pilot cluster. All pilot clusters are uniformly distributed in frequency domain with each adjacent pilot cluster being closed together.

Then, we focus on ST design. From (14), when the training sequence at each pilot cluster is designed as either a constant modulus sequence, i.e.

$$p\_n(i\_\prime k\_\tau) = p\_n(i\_\prime k\_\tau \pm k^\prime) \quad \tau = 1, \cdots \tau \text{ , } k^\prime = 1, \cdots \text{ } T \tag{15}$$

or a sequence, i.e.

260 Wireless Communications and Networks – Recent Advances

W , ,0 , , , , 1 *<sup>T</sup> qqqq ik w i w ik w iB*

> 

Compared with (10), the approximation of windowing based vector has a much smaller side-lobe with the same index*T* . The experiment studies show that by using a *Kaiser*  function [5], the approximation in (13) of *T* = 2 may capture almost 99% power of

*B qi qi B qi q i*

 

,

*q n*

*N T <sup>Q</sup> m m*

(13) into (12), the FFT demodulated outputs can be approximated by

2 , 2 1 *<sup>T</sup>*

 ,0 , , ,0, 0, , , , 1 *<sup>T</sup> w i w iT w iB T w iB qq q q* .

*q i* for truncated DFBs when *q B* <sup>10</sup> . Substituting

( ) ( )

*u ik H ik t w ik p ik k* 

, , ( , ') , '

, ' , ' ', *<sup>m</sup>*

*w iB k p ik k v ik*

The first term of (14) illustrates that 2 1 *T* tones, i.e. *pn n ik k p ik k* , ', , ' should be

a pilot cluster for differentiating from the isolated tones utilized in the LTI channel

Fig. 1. A typical pilot tone distribution. 2 1 *T* consecutive tones are grouped together as one pilot cluster. All pilot clusters are uniformly distributed in frequency domain with each

*nq q i q n*

 

( )

as the pilot sequences at the pilot cluster *k*

. (14)

, , *<sup>m</sup> H ik n q* . We refer to such 2*T* + 1 consecutive pilot tones as

th pilot

. Since the

can be

 and 1 *k*

1 0 ' 0

*n q k*

estimation [19] [22]-[23]. Denote 1 <sup>Γ</sup> *k k* , as the pilot cluster indexes located at the

*t tB t t tB t* **<sup>F</sup>** (13)

*<sup>B</sup> t* as

> 

where W , *<sup>q</sup> i k* is the CEF vector with the windowing function

2: 2 1 *<sup>T</sup>*

1

'

*pn k T*

placed closed together. The pilot tone distribution is shown in Fig. 1.

ST does not entail additional bandwidth, two adjacent pilot-clusters, i.e. *k*

*k T*

*FFT t t B t B t*

jointly designed for estimating ( )

adjacent pilot cluster being closed together.

 *B qi i*

symbol and, , *<sup>n</sup> pk T*

$$p\_n(i, k\_\tau \pm k') = \begin{cases} p\_n(i, k\_\tau), k'=0\\ 0, \quad \text{otherwise} \end{cases} \quad \tau = 1, \dots, \Gamma, \ k' = 1, \dots, T \quad . \tag{16}$$

Accordingly, the FFT demodulated outputs at pilot cluster *k*can be approximated as

$$\begin{split} \mu^{(m)}(\mathbf{i},k\_{\tau}) &\approx \sum\_{n=1}^{N} \sum\_{q=0}^{Q} H\_{n,q}^{(m)}(\mathbf{i},k\_{\tau}) \eta\_{q}(t\_{i}) \mathbf{g}\_{q}(\mathbf{i},k\_{\tau}) p\_{n}(\mathbf{i},k\_{\tau}) + \overline{\upsilon}^{(m)}(\mathbf{i},k\_{\tau}) \\ &= \sum\_{n=1}^{N} H\_{n}^{(m)}(\mathbf{i},k\_{\tau}) p\_{n}(\mathbf{i},k\_{\tau}) + \overline{\upsilon}^{(m)}(\mathbf{i},k\_{\tau}) \end{split} \tag{17}$$

where <sup>1</sup> ' 0 ' , ( , ') , ' *<sup>T</sup> qq q <sup>k</sup> k T g ik w ik k w ik B k* if the training sequence takes value from (15) and , (, ) *q q g ik w ik* for (16), which are all known, respectively. The channel transfer functions ( ) ' , *<sup>m</sup> H ik <sup>n</sup>* is given by

$$\begin{split} H^{(m)}\_{\
u} \left( \mathbf{i}, \mathbf{k}\_{\tau} \right) &= \sum\_{q=0}^{\underline{Q}} H^{(m)}\_{n,l,q} \left( \mathbf{i}, \mathbf{k}\_{\tau} \right) \eta\_{q} \left( \mathbf{t}\_{i} \right) \mathbf{g}\_{q} \left( \mathbf{i}, \mathbf{k}\_{\tau} \right) \\ = \sum\_{l=0}^{L-1} \sum\_{q=0}^{\underline{Q}} h^{(m)}\_{n,l,q} \eta\_{q} \left( \mathbf{t}\_{i} \right) \mathbf{g}\_{q} \left( \mathbf{i}, \mathbf{k}\_{\tau} \right) \mathbf{e}^{-j \, 2 \pi k\_{\tau} l / \hbar} \approx \sum\_{l=0}^{L-1} h^{(m)}\_{n,l} \left( \mathbf{t}\_{i} \right) \mathbf{e}^{-j \, 2 \pi k\_{\tau} l / \hbar} \,. \end{split} \tag{18}$$

From (17)-(18), we note that ( ) ' , , 1, *<sup>m</sup> H ik <sup>n</sup>* is in fact a LTI system transfer function of which the coefficients are the mid-values of the LTV channel at the *i* th OFDM symbol interval. As a result, the LTV channel estimation can be approximately reduced into that of the LTI channel [22] and [23] by simply designing the ST sequences as (15) or (16).

Let ( ) () () () () 1,0 1, 1 ,0 , 1 ,, , *<sup>T</sup> <sup>m</sup> mm m m ii i i <sup>L</sup> <sup>N</sup> N L it t t t* **<sup>H</sup>** be the channel coefficient vector associated with the *i* th OFDM symbol and stack the FFT demodulated signals at pilot clusters of the *i* th OFDM symbol to form a vector

$$\mathbf{U}^{(m)}\left(i,k\_{1}:k\_{\varGamma^{-}}\right) = \left[u^{(m)}\left(i,k\_{1}\right), \dots, u^{(m)}\left(i,k\_{\varGamma}\right), \dots, u^{(m)}\left(i,k\_{\varGamma^{-}}\right)\right]^{\varGamma}.\tag{19}$$

The received signals at pilot clusters can be thus written as

$$\begin{aligned} \mathbf{U}^{(m)}\left(i,k\_{1}:k\_{\varGamma}\right) &= \underbrace{\mathbf{A}\left(i\right)\mathbf{H}^{(m)}\left(i\right)}\_{\text{desired signal for channel estimation}} + \underbrace{\Xi^{(m)}\left(i,k\_{1}:k\_{\varGamma}\right)}\_{\text{information interference on channel estimation}} \\ &+ \overleftarrow{\mathbf{V}}^{(m)}\left(i,k\_{1}:k\_{\varGamma}\right) \end{aligned} \tag{20}$$

where ( ) <sup>1</sup> , : *<sup>m</sup> ik k* **<sup>V</sup>** is the noise vector in frequencydomain, ( ) ( ) ( ) <sup>Ξ</sup> 1 1 ,: , , , *<sup>T</sup> m mm ik k ik ik* is the interference vector produced by the information sequences with ( ) ( ) <sup>1</sup> , ', , *m m <sup>N</sup> <sup>n</sup> n n ik H ik c ik* , **A***i*

 1,0 , 1, 1 , ,0 , , 1 *L nl N NL* **A A AA A** is a *NL* matrix with the columnvectors

$$\mathbf{A}\begin{pmatrix} n,l \end{pmatrix} = \left[ p\_n(i,k\_1)e^{-j2\pi k\_1 l/\hbar}, \cdots p\_n(i,k\_\tau)e^{-j2\pi k\_\tau l/\hbar}, \cdots p\_n(i,k\_\Gamma)e^{-j2\pi k\_\Gamma l/\hbar} \right]^T. \tag{21}$$

Since the matrix **A***i* is known, when *NL* , the matrix **A***i* is of full column rank, and the channel coefficient vectors can be thus estimated by

$$\hat{\mathbf{H}}^{(m)}\left(\mathbf{i}\right) = \mathbf{A}^{\dagger}\mathbf{U}^{(m)}\left(\mathbf{i}, k\_{\perp} : k\_{\perp} \right)$$

$$\mathbf{I} = \mathbf{H}^{(m)}\left(\mathbf{i}\right) + \mathbf{A}\left(\mathbf{i}\right)^{\dagger}\boldsymbol{\Xi}^{(m)}\left(\mathbf{i}, k\_{\perp} : k\_{\perp} \right) + \mathbf{A}\left(\mathbf{i}\right)^{\dagger}\boldsymbol{\overline{\nabla}}^{(m)}\left(\mathbf{i}, k\_{\perp} : k\_{\perp} \right) \; m = 1, \cdots, M, \mathbf{i} = 1, \cdots \; \mathbf{I} \tag{22}$$

where the superscript ' † ' is the pseudo-inverse operation, and the hat '^' indicates the estimation. From (22), the mainly computational effort is directly proportional to the unknown parameter number *NL* .

Using the specifically designed ST sequences in (15) and (or) (16), the problem of LTV channel estimation for MIMO/OFDM systems can be reduced into that of LTI channel. From (20) and (22), however, we notice that the interference vector due to information sequence can hardly be neglected since the power of data symbol is much larger than the pilot power. For conventional ST based schemes stated in [8]-[13] [17]-[18], first-order statistics are employed to suppress the information sequence interference over multiple training periods in the case that the channel is LTI during the record length. Such arithmetical average process, however, is no longer feasible to the channel assumed in this paper where the channel coefficients are linearly time-variant between consecutive OFDM symbols.

#### **3.2 Channel estimation over multiple OFDM symbols**

In this sub-section, a weighted average approach is developed to suppress the abovementioned information sequence interference over multiple OFDM symbols, and thus overcoming the shortcoming of the existing ST-based approach in estimating the timevariant channels.

 By (22), the LTV channel coefficients can be obtained following the relationship ( ) ( ) , 0 , , *m m Q n l i nlq q i q ht h t* . Taking the LTV channel coefficient estimation of each OFDM symbol ( ) , <sup>ˆ</sup> 1, *<sup>m</sup> n l <sup>i</sup> ti I* by (22) as a temporal result, and form a vector as () () () ,, , <sup>1</sup> <sup>ˆ</sup> ˆ ˆ , *mm m <sup>T</sup> nl nl n l <sup>I</sup>* **h** *t t* , we thus have

 

*<sup>T</sup> j klB j klB j k lB nl p ik e p ik e p ik e nnn* 

 ( ) † ( ) <sup>1</sup> <sup>ˆ</sup> , : *m m i ik k* **H AU**

where the superscript ' † ' is the pseudo-inverse operation, and the hat '^' indicates the estimation. From (22), the mainly computational effort is directly proportional to the

Using the specifically designed ST sequences in (15) and (or) (16), the problem of LTV channel estimation for MIMO/OFDM systems can be reduced into that of LTI channel. From (20) and (22), however, we notice that the interference vector due to information sequence can hardly be neglected since the power of data symbol is much larger than the pilot power. For conventional ST based schemes stated in [8]-[13] [17]-[18], first-order statistics are employed to suppress the information sequence interference over multiple training periods in the case that the channel is LTI during the record length. Such arithmetical average process, however, is no longer feasible to the channel assumed in this paper where the channel coefficients are linearly time-variant between consecutive OFDM

In this sub-section, a weighted average approach is developed to suppress the abovementioned information sequence interference over multiple OFDM symbols, and thus overcoming the shortcoming of the existing ST-based approach in estimating the time-

By (22), the LTV channel coefficients can be obtained following the

. Taking the LTV channel coefficient estimation of

*n l <sup>i</sup> ti I* by (22) as a temporal result, and form a vector as

**A V**

<sup>1</sup> 22 2 A, , , , , , <sup>1</sup>

 ( ) † † ( ) ( ) <sup>Ξ</sup> 1 1 , : , : *mm m i i ik k i ik k* **H A** 

<sup>1</sup> , ', , *m m <sup>N</sup> <sup>n</sup> n n ik H ik c ik*

is the noise vector in frequency-

. (21)

is the interference vector produced by

 

 

*NL* , the matrix **A***i* is of full column rank, and

, **A***i*

 

 

*m M* 1, , *i I* 1, (22)

*NL* matrix with the column-

where ( ) <sup>1</sup> , : *<sup>m</sup> ik k* **<sup>V</sup>**

vectors

symbols.

variant channels.

relationship ( ) ( ) , 0 , , *m m Q n l i nlq q i q ht h t*

*nl nl n l <sup>I</sup>* **h** *t t* , we thus have

each OFDM symbol ( )

 () () () ,, , <sup>1</sup> <sup>ˆ</sup> ˆ ˆ , *mm m <sup>T</sup>*

Since the matrix **A***i* is known, when

unknown parameter number *NL* .

domain, ( ) ( ) ( ) <sup>Ξ</sup> 1 1 ,: , , ,

*<sup>T</sup> m mm ik k ik ik*

1,0 , 1, 1 , ,0 , , 1 *L nl N NL* **A A AA A** is a

the channel coefficient vectors can be thus estimated by

**3.2 Channel estimation over multiple OFDM symbols** 

, <sup>ˆ</sup> 1, *<sup>m</sup>*

the information sequences with ( ) ( )

$$\hat{\mathbf{h}}\_{n,l}^{(m)} = \mathbf{p} \hat{\mathbf{h}}\_{n,l,q}^{(m)}$$

$$= \begin{bmatrix} e^{j2\pi(0-Q/2)t\_1/\Omega} & \cdots & e^{j2\pi(Q-Q/2)t\_1/\Omega} \\ \vdots & \ddots & \vdots \\ e^{j2\pi(0-Q/2)t\_1/\Omega} & \cdots & e^{j2\pi(Q-Q/2)t\_1/\Omega} \end{bmatrix} \begin{bmatrix} \hat{h}\_{n,l,0}^{(m)} \\ \vdots \\ \hat{h}\_{n,l,Q}^{(m)} \end{bmatrix} \qquad n = \mathbf{1}, \cdots N, \ l = 0, \cdots L - 1$$

where () () () () , , , ,0 , , , , ˆ ˆ ˆˆ , , *<sup>T</sup> m m mm nlq nl nlq nlQ h hh* **<sup>h</sup>** is estimation of the complex exponential coefficients vector modeling the LTV channel, **<sup>η</sup>** is a *I Q* <sup>1</sup> matrix with 2 2 , <sup>e</sup> *<sup>i</sup> j qQ t q i* **η** . Thus, when 1 *I Q* , the matrix **η** is of full column rank, and the basis expansion model coefficients can be computed by

$$
\hat{\mathbf{h}}\_{n,l,q}^{\{m\}} = \mathfrak{n}^{\dagger} \hat{\mathbf{h}}\_{n,l}^{\{m\}} \quad n = 1, \cdots N \text{ , } l = 0, \cdots L - 1 \text{ .} \tag{24}
$$

Substituting *t i BB <sup>i</sup>* 1' 2 into the matrix **η** , we obtain the pseudo-inverse matrix as

$$\left[\mathbf{n}^{\dagger}\right]\_{i,q} = \left. e^{-j2\pi \left(q - Q/2\right) \left((i-1)\mathcal{B}^{\flat} + \mathcal{B}/2\right)} \right| \mathbf{\hat{I}}\,. \tag{25}$$

By (23)-(25), the modeling coefficients (6) can be computed by

$$
\hat{\boldsymbol{h}}\_{n,l,q}^{(m)} = \sum\_{i=1}^{l} e^{-j2\pi (q-Q/2) \left( (i-1)\mathbb{B}^\* + \mathcal{S}/2 \right) \left\{ \Omega \right\}\_{n,l}^{(m)}} \hat{\boldsymbol{h}}\_{n,l}^{(m)} \left( \mathbf{i} \right) \Big/ \boldsymbol{I} \,. \tag{26}
$$

In fact, (26) is estimated over multiple OFDM symbols with a weighted average function of 2 2 e *<sup>i</sup> j qQ t I* .

 Compared with the conventional ST strategies, the proposed channel estimation is composed of two steps: First, with specially designed ST signals in (15) and (16), channel estimation can be reduced into that of LTI channel, and we are allowed to estimate the channel coefficients during each OFDM symbol as temporal results. Second, the temporal channel estimates are further enhanced over multiple OFDM symbols by using a weighted average procedure. That is, not only the target OFDM symbol, but also the OFDM symbols over the whole frame are invoked for channel estimation. Similar to the first-order statistics of LTI case [8]-[13] [17]-[18], it is thus anticipated that the weighted average estimation may also exhibit a considerable performance improvement for the LTV channels over a long frame .

#### **4. Channel estimation analysis**

In this section, we analyze the performance of the channel estimator proposed in Section III and derive a closed-form expression of the channel estimation variance which can be, in turn, used for ST power allocation. Before going further, we make the following assumptions:

(H1) The information sequence*c ik <sup>n</sup>* , is zero-mean, finite-alphabet, i.i.d., and equipowered with the power <sup>2</sup> *<sup>c</sup>* .

(H2) The additive noise ( ) , *<sup>m</sup> v it* is white, uncorrelated with*c ik <sup>n</sup>* , , with <sup>2</sup> ( ) <sup>2</sup> , *<sup>m</sup> E v it v* .

(H3) The LTV channel coefficients ( ) , *m n l* **h** are complex Gaussian variables, and statistically independent for different values of *n* and *l*.

From (22)-(26), the mean square error (MSE) of channel estimation is given by

$$MSE^{(m)} = E\left\{ \sum\_{n=1}^{N} \sum\_{l=0}^{L-1} \left\| h\_{n,l}^{(m)}(i,t) - \sum\_{q=0}^{Q} \hat{h}\_{n,l,q}^{(m)} \eta\_q(t) \right\|^2 \right\}$$

$$= E\left\{ \sum\_{n=1}^{N} \sum\_{l=0}^{L-1} \left\| h\_{n,l}^{(m)}(i,t) - \sum\_{q=0}^{Q} h\_{n,l,q}^{(m)} \eta\_q(t) + \sum\_{q=0}^{Q} h\_{n,l,q}^{(m)} \eta\_q(t) - \sum\_{q=0}^{Q} \hat{h}\_{n,l,q}^{(m)} \eta\_q(t) \right\|^2 \right\} \tag{27}$$

where is the Euclidean norm. In (27), the first error term <sup>1</sup> ( ) ( ) 1 0 , <sup>0</sup> , , , *N L m m <sup>Q</sup> n l n l nlq q q h it h t* is caused by the orthonormal basis expansion model in (6), which is referred to as the channel modeling error. The second error term <sup>1</sup> () () 10 0 , , , , *NL Q m m* ˆ *nl q nlq nlq q q h th t* is duo to the information interference to channel estimation (22) and additive noise. Explicitly, two error signals are mutually independent. Herein, we do not elaborate the topic of channel modeling error and focus on channel estimation error, which is mainly produced by the interference of information sequence. By (H2), the MSE of the estimation in one OFDM symbol can be written as

$$\begin{split} MSE^{(m)}(i) &= \frac{1}{\left(Q+1\right)NL} E\left\{ \sum\_{n=1}^{N} \sum\_{l=0}^{l-1} \sum\_{q=0}^{Q} \left\| \boldsymbol{\mu}\_{n,l,q}^{(m)} \boldsymbol{\eta}\_{q} \left(t\right) - \hat{\boldsymbol{\mu}}\_{n,l,q}^{(m)} \boldsymbol{\eta}\_{q} \left(t\right) \right\|^{2} \right\} \\ &= \underbrace{\frac{1}{\left(Q+1\right)NL} \text{tr}\left\{ \mathbf{A}\left(i\right)^{\mathsf{T}} E\left\{ \Xi^{(m)}\left(i,k\_{1}:k\_{\mathsf{T}}\right) \left(\Xi^{(m)}\left(i,k\_{1}:k\_{\mathsf{T}}\right)\right)^{H} \right\} \left| \left(\mathbf{A}\left(i\right)^{\mathsf{T}}\right)^{H} \right\rangle}\_{\text{estimation variance due to information sequence interference}} \\ &+ \underbrace{\frac{1}{\left(Q+1\right)NL} \text{tr}\left\{ \mathbf{A}\left(i\right)^{\mathsf{T}} E\left\{ \widetilde{\mathbf{V}}^{(m)}\left(i,k\_{1}:k\_{\mathsf{T}}\right) \left(\widetilde{\mathbf{V}}^{(m)}\left(i,k\_{1}:k\_{\mathsf{T}}\right)\right)^{H} \right\} \left| \left(\mathbf{A}\left(i\right)^{\mathsf{T}}\right)^{H} \right\rangle}\_{\text{estimation variance due to additive noise}}. \end{split} \tag{28}$$

For zero-mean white noise, we have

$$\mathbb{E}\left\{\overline{\mathbf{V}}^{(m)}\left(\mathbf{i},k\_{I}:k\_{I^{\prime}}\right)\Big(\overline{\mathbf{V}}^{(m)}\left(\mathbf{i},k\_{I}:k\_{I^{\prime}}\right)\Big)^{H}\right\}=\sigma\_{v}^{2}\mathbf{I}\_{I^{\prime}}\,.\tag{29}$$

Invoking the assumption (H1), information sequence interference ( ) ( ) <sup>1</sup> , ' , , , 1, *m m <sup>N</sup> <sup>n</sup> n n ik H ik c ik* is approximately Gaussian distributed for a large . Therefore, the channel estimation variance due to information sequence interference can be obtained as

$$E\left|\Xi^{\{\mathfrak{m}\}}\left(i,k\_{1}:k\_{\varGamma}\right)\left(\Xi^{\{\mathfrak{m}\}}\left(i,k\_{1}:k\_{\varGamma}\right)\right)^{H}\right|=\frac{\sigma\_{c}^{2}}{\Gamma^{2}}\sum\_{\mathfrak{r}=0}^{\Gamma-1}\sum\_{n=1}^{N}\sum\_{l=0}^{L-1}\left|H\_{\mathfrak{n}}^{\{\mathfrak{m}\}}\left(i,k\_{\varGamma}\right)\right|^{2}=\frac{\rho\_{c}}{\Gamma}\sum\_{\mathfrak{r}=0}^{\Gamma-1}\sum\_{n=1}^{N}\sum\_{l=0}^{L-1}\left|h\_{n,l}^{\{\mathfrak{m}\}}e^{-2\pi k\_{\varGamma}l/\hbar}\right|^{2}.\tag{30}$$

Substituting (29) and (30) into (28), we have

264 Wireless Communications and Networks – Recent Advances

(H1) The information sequence*c ik <sup>n</sup>* , is zero-mean, finite-alphabet, i.i.d., and equi-

(H2) The additive noise ( ) , *<sup>m</sup> v it* is white, uncorrelated with*c ik <sup>n</sup>* , ,

*n l* **h** are complex Gaussian variables, and statistically

2

2

(27)

(28)

.

*<sup>v</sup>* . (29)

<sup>2</sup> <sup>1</sup>

, , , ,

† † ( ) ( ) 1 1

† † ( ) ( ) 1 1

*<sup>H</sup> <sup>H</sup> m m i E ik k ik k i*

*<sup>H</sup> <sup>H</sup> m m i E ik k ik k i*

*nlq nlq q q*

 

 

 

 

, , ,

ˆ ,

*n l nlq q*

1 0 0

( ) ( ) ( ) ( ) , , , , , , ,

where is the Euclidean norm. In (27), the first error

model in (6), which is referred to as the channel modeling error. The second error term

 is duo to the information interference to channel estimation (22) and additive noise. Explicitly, two error signals are mutually independent. Herein, we do not elaborate the topic of channel modeling error and focus on channel estimation error, which is mainly produced by the interference of information sequence. By

 

10 0 1 ˆ

*nlq MSE i E h th t*

 estimation variance due to information sequence interference

 estimation variance due to additive noise

 **A A** 

> ( ) ( ) <sup>2</sup> V ,: V ,: I 1 1 *<sup>H</sup> m m E ik k ik k*

Invoking the assumption (H1), information sequence interference

is approximately Gaussian distributed for a

 **A A** 

tr <sup>Ξ</sup> , : <sup>Ξ</sup> , : <sup>1</sup>

<sup>1</sup> tr V , : V , : <sup>1</sup>

( ) () ()

*def N L <sup>Q</sup> <sup>m</sup> m m*

ˆ ,

is caused by the orthonormal basis expansion

*n l nlq qqq nlq nlq*

 

, *m*

From (22)-(26), the mean square error (MSE) of channel estimation is given by

1 ( ) ( ) ( )

*N L <sup>Q</sup> <sup>m</sup> m m*

1 0 0 0 0

 

*n l q q q E h it h t h t h t*

 

(H2), the MSE of the estimation in one OFDM symbol can be written as

1

*Q NL*

*N L QQQ mm m m*

*n l q MSE E h i t h t*

powered with the power <sup>2</sup>

 .

with <sup>2</sup> ( ) <sup>2</sup> , *<sup>m</sup> E v it*

*<sup>c</sup>* .

*v*

independent for different values of *n* and *l*.

1

term <sup>1</sup> ( ) ( ) 1 0 , <sup>0</sup> , , , *N L m m <sup>Q</sup> n l n l nlq q q*

1

*Q NL*

*Q NL*

For zero-mean white noise, we have

 ( ) ( ) <sup>1</sup> , ' , , , 1, *m m <sup>N</sup> <sup>n</sup> n n ik H ik c ik*

 

10 0 , , , , *NL Q m m* ˆ *nl q nlq nlq q q h th t* 

*h it h t*

<sup>1</sup> () ()

(H3) The LTV channel coefficients ( )

$$MSE^{(m)}(i) = \frac{1}{(Q+1)NL} \left(\sigma\_v^2 + \frac{\sigma\_c^2}{\Gamma} \sum\_{\tau=0}^{\Gamma-1} \sum\_{n=1}^N \sum\_{l=0}^{L-1} \left| h\_{n,l}^{(m)} e^{-2\pi k\_l l \beta} \right|^2 \right) \text{tr}\left[ \left(\mathbf{A}(i)\right)^H \mathbf{A}(i) \right]^{-1} . \tag{31}$$

Apparently, the channel estimation performance depends crucially on the matrix **A***i* . The optimal estimation or minimum MSE (MMSE) estimation may require *<sup>H</sup>* **AA I** *i i* where is a constant. From (21), we adopt the training sequence as , , 0, 1 *n p p ik k B* as (15), the above MMSE condition can be well satisfied. We thus have

$$\text{tr}\left[\left(\mathbf{A}(i)\right)^{H}\mathbf{A}(i)\right] = \Gamma \sigma\_p^2 \mathbf{I}\_{N\mathcal{L}(Q+1)}\,. \tag{32}$$

Substituting (32) into (31), the MSE of channel estimation over one OFDM symbol can be derived as

$$MSE^{(m)}\left(i\right) = \frac{\sigma\_c^2}{\Gamma^2 \sigma\_p^2} \sum\_{\tau=0}^{\Gamma-1} \sum\_{n=1}^N \sum\_{l=0}^{L-1} \left| h\_{n,l}^{\{m\}} e^{-2\pi k\_r l \beta b} \right|^2 + \frac{\sigma\_v^2}{\Gamma \sigma\_p^2}.\tag{33}$$

It is seen that the first term of (33) is the estimation variance due to information interference, and depends upon the channel transfer functions. We thus define the normalized variance as

$$\text{MMSE}^{(m)}(i) = \frac{\sigma\_c^2}{\Gamma^2 \sigma\_p^2} \sum\_{\tau=0}^{\Gamma-1} \sum\_{n=1}^{N} \sum\_{l=0}^{L-1} \left| \hbar\_{n,l}^{(m)} e^{-2\pi k\_\tau l/\hbar} \right|^2 \left/ \left| \overline{\hbar}^{(m)}(i) \right|^2 \tag{34}$$

where <sup>2</sup> <sup>2</sup> ( ) 1 1 ( ) <sup>2</sup> 0 10 , *m N L m klB n l n l i e NL* . Following the definition of (34), we obtain the normalized variance as

$$\text{NNMSE}^{(m)}(i) = \frac{\sigma\_c^2}{\Gamma^2 \sigma\_p^2} \sum\_{\tau=0}^{I-1} \sum\_{n=1}^N \left| \boldsymbol{h}\_{n,l}^{(m)} e^{-2\pi k\_\tau |l|B} \right|^2 \left/ \left| \overline{\boldsymbol{h}}^{(m)}(i) \right|^2 = \frac{\text{NL}}{\Gamma} \frac{\sigma\_c^2}{\sigma\_p^2}.\tag{35}$$

From (35), we can find that the estimation variance due to the information interference is directly proportional to the information-to-pilot power ratio 2 2 *c <sup>p</sup>* , thereby resulting in an inaccurate solution for the general case that 2 2 *<sup>c</sup> <sup>p</sup>* .

Then, we analyze the channel estimation performance of the weighted average approach over multiple OFDM symbols (the whole frame ). Define the vectors A 1 ,... *<sup>T</sup>* **A A** ,

 () () ( ) Ξ Ξ 1 1 1, : , <sup>Ξ</sup> , : *<sup>T</sup> m m <sup>m</sup> kk kk* and () () ( ) V V 1, : ,...V , : 1 1 *<sup>T</sup> m m <sup>m</sup> kk kk* , the MSE of the weighted average channel estimator over multiple OFDM symbols is given by

$$MSE^{(m)} = \frac{1}{(Q+1)NL} \text{tr} \left\{ \mathbf{n}^{\dagger} \boldsymbol{E} \left\{ \left\| \mathbf{A}^{\dagger} \boldsymbol{\Sigma}^{(m)} + \mathbf{A}^{\dagger} \mathbf{V}^{(m)} \right\|^{2} \right\} \middle| \left( \mathbf{n}^{\dagger} \right)^{H} \right\} = \frac{1}{I} \sum\_{i=1}^{I} MSE^{(m)} \left( i \right) \text{tr} \left\{ \left\| \mathbf{n}^{\dagger} \left( \boldsymbol{\eta}^{\dagger} \right)^{H} \right\} \left( \left\| \boldsymbol{\Theta} \right\|^{2} \right)^{H} \right\} \tag{36}$$

Note that the column vectors of the matrix **η** in (23) are in fact the FFT vectors of a *I I* matrix, we thus have ( 1) *<sup>H</sup>***η η** *<sup>I</sup> <sup>Q</sup>* **<sup>I</sup>** and <sup>1</sup> tr <sup>1</sup> *<sup>H</sup> Q I* **η η** . Substituting (33) into (36), the MSE of channel estimation over multiple OFDM symbols is given by

$$MSE^{(m)} = \frac{\left(Q + 1\right)\sigma\_c^2}{\left.\prod^2 \sigma\_p^2\right|} \sum\_{\tau=0}^{I-1} \sum\_{n=1}^N \sum\_{l=0}^{L-1} \left| h\_{n,l}^{\{m\}} e^{-2\pi k\_r l/\hbar} \right|^2 + \frac{\left(Q + 1\right)\sigma\_v^2}{\left.\prod^2 \sigma\_p^2\right|}\tag{37}$$

In (37), the second term is caused by information sequence interference, which may become the dominant component of the channel estimation variance for the general case of 2 2 *<sup>c</sup> <sup>p</sup>* , especially for large SNRs. Therefore, we solely consider information sequence effect. Similar to (34)-(35), we derive the normalized variance due to information interference by removing the channel gain as

$$NMSE^{(m)} = \frac{\left(Q + 1\right) \sigma\_c^2}{\varprod^2 \sigma\_p^2} \sum\_{\tau=0}^{I-1} \sum\_{n=1}^N \sum\_{l=0}^{L-1} \left| h\_{n,l}^{(m)} e^{-2\pi k\_r l/\hbar} \right|^2 \left/ \left| \overline{h}^{(m)} \right|^2 \tag{38}$$

where 2 2 ( ) ( ) 1 *m m I <sup>i</sup> i I* . It follows that

$$\text{NMMSE}^{(m)} = \frac{\sigma\_c^2}{\sigma\_p^2} \frac{\text{NL}\left(Q+1\right)}{\text{II}} \approx \frac{\sigma\_c^2}{\sigma\_p^2} \frac{\text{NL}\left(Q+1\right)}{\Omega} \frac{B}{\Gamma} = \frac{\sigma\_c^2}{\sigma\_p^2} \frac{\text{NL}\left(Q+1\right)}{\partial \Omega} \tag{39}$$

where *B* is the training ratio of one OFDM symbol. For conventional ST-based LTI schemes where isolated pilots are exploited for channel estimation [8]-[13] [17]-[18], we have 1 . However, for estimating the LTV channels addressed in this paper, pilot clusters, instead of isolated pilot tones, are exploited. Thus, the corresponding training ratio yields 1 (2 1) *T* . From (39), the normalized variance is directly proportional to the information-pilot power ratio 2 2 *c <sup>p</sup>* , the training ratio and the ratio of unknown parameter number *NL Q* 1 over the frame length .

Compared with the variances of channel estimation over one OFDM symbol as in (33)-(35), the estimation variances of the weighted average estimator(37)-(39) is significantly reduced owing to the fact that *I Q* 1 1 . Theoretically, the weighted average operation can be considered as an effective approach in estimating LTV channel, where the information sequence interference can be effectively suppressed over multiple OFDM symbols. As stated in conventional ST-based LTI schemes [8]-[13] [17]-[18], channel estimation performance can be improved along with the increment of the recorded frame length , i.e. the estimation variance approaches to zero as . This can be easily comprehended that larger frame length means more observation samples, and hence lowers the MSE level. From the LTV channel model (6), however, we note that as the frame length is increased, the corresponding truncated DFB requires a larger order *Q* to model the LTV channel (maintain a tight channel model), and the least order should be satisfied 2 *Q ff d s* , where *df* and *sf* are the Doppler frequency and sampling rate, respectively. Consequently, as the frame length increases, the LTV channel estimation variance (39) approaches to a fixed lower-bound associate with the system Doppler frequency as well as the information to pilot power ratio. This is quite different from the existing ST-based channel estimation approaches [8]-[19].

According to the theoretic analysis in (37)-(39), the proposed two-step LTV channel estimator achieves a significant improvement over multiple OFDM symbols compared with that of block-by-block process (33)-(35). However, as the frame length is increased, the estimation variances approach to a fixed lower-bound. Further enhancement of the channel estimation should resort to increasing the ST power <sup>2</sup> *<sup>p</sup>* . For wireless communication systems with a limited transmission power, however, an increased ST power allocation reduces the data power <sup>2</sup> *<sup>c</sup>* , leading to SER degradation. Accordingly, in the analysis presented in the next section, the ratio of ST power allocation is determined by maximizing the lower bound of the average channel capacity.
