**5. Analysis of ST power allocation and system capacity**

In this section, we consider the issue of ST power allocation where the lower bound of the average channel capacity is maximized and then mathematically derived for the proposed two-step channel estimator.

Define the ST power allocation factor

266 Wireless Communications and Networks – Recent Advances

and () () ( ) V V 1, : ,...V , : 1 1

the MSE of the weighted average channel estimator over multiple OFDM symbols is given

*MSE*( ) *<sup>m</sup>* <sup>2</sup> †† † † () () ( ) † †

Note that the column vectors of the matrix **η** in (23) are in fact the FFT vectors of

*<sup>H</sup>***η η** *<sup>I</sup> <sup>Q</sup>* **<sup>I</sup>** and <sup>1</sup>

2 2 1 1 <sup>2</sup> ( ) ( ) <sup>2</sup>

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

effect. Similar to (34)-(35), we derive the normalized variance due to information

*p n l*

*NL Q NL Q <sup>B</sup> NL Q NMSE*

22 2 *<sup>m</sup>* 11 1 *cc c pp p*

schemes where isolated pilots are exploited for channel estimation [8]-[13] [17]-[18], we

clusters, instead of isolated pilot tones, are exploited. Thus, the corresponding training ratio

*<sup>p</sup>* , the training ratio

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

. However, for estimating the LTV channels addressed in this paper,

*<sup>Q</sup> NMSE <sup>e</sup>*

 *c* 

parameter number *NL Q* 1 over the frame length .

 

1 1 *N L <sup>m</sup> c v <sup>m</sup> klB n l p p n l*

0 10

*Q Q MSE <sup>e</sup>*

2 2 ,

*<sup>p</sup>* , especially for large SNRs. Therefore, we solely consider information sequence

 <sup>2</sup> 1 1 <sup>2</sup> <sup>2</sup> ( ) ( ) <sup>2</sup> ( ) 2 , 0 10 1 *N L m m <sup>c</sup> <sup>m</sup> klB*

*n l*

22 2

*B* is the training ratio of one OFDM symbol. For conventional ST-based LTI

1 (2 1) *T* . From (39), the normalized variance is directly proportional to the

tr Ξ V tr

*E MSE i*

1 1

*Q NL I* 

(36), the MSE of channel estimation over multiple OFDM symbols is given by

 

*<sup>T</sup> m m <sup>m</sup> kk kk* 

**η η** . Substituting (33) into

,

1

*H H <sup>I</sup> m m <sup>m</sup> i*

**<sup>η</sup> A A <sup>η</sup> η η** . (36)

tr <sup>1</sup> *<sup>H</sup> Q I*

(37)

(38)

 

   

> pilot

and the ratio of unknown

(39)

 

 

 () () ( ) Ξ Ξ 1 1 1, : , <sup>Ξ</sup> , : *<sup>T</sup> m m <sup>m</sup> kk kk* 

1

a *I I* matrix, we thus have ( 1)

interference by removing the channel gain as

*<sup>i</sup> i I* . It follows that

( )

where 2 2 ( ) ( )

1 *m m I*

information-pilot power ratio 2 2

by

of 2 2 *<sup>c</sup>* 

where 

yields

have 1 

$$\beta = \frac{E\left[\left|p\_n(k)\right|^2\right]}{E\left[\left|p\_n(k)\right|^2\right] + E\left[\left|c\_n(k)\right|^2\right]} = \frac{\sigma\_p^2}{\sigma\_p^2 + \sigma\_c^2} \tag{40}$$

For a fixed SNR or transmitted power budget, higher implies smaller effective SNR at the receiver due to decreased power in the information sequence but higher channel estimation accuracy. Having removed ST sequence, we obtain the received signals in a vector-form as

$$\begin{aligned} \overline{\mathbf{U}}^{(m)}(i) &= \left[ \overline{u}^{(m)}(i,0), \dots, \overline{u}^{(m)}(i,k), \dots, \overline{u}^{(m)}(i,B-1) \right]^T \\ &= \underbrace{\sum\_{n=1}^{N} \hat{\mathbf{H}}\_{n}^{(m)}(i) \mathbf{C}\_{n}(i)}\_{\text{desired signals:} - \mathbf{A}^{(m)}(i)} + \underbrace{\sum\_{n=1}^{N} \Delta \hat{\mathbf{H}}\_{n}^{(m)}(i) \left[ \mathbf{P}\_{n}(i) + \mathbf{C}\_{n}(i) \right]}\_{\text{interference to information signal recovery:} - \mathbf{H}^{(m)}(i)} + \underbrace{\overline{\mathbf{V}}^{(m)}(i)}\_{\text{(41)}} \end{aligned} \tag{41}$$

with the received signals ( ) , , 0, 1 *<sup>m</sup> u ik k B* in (41) as

$$\overline{u}^{(m)}\left(i,k\right) = \lambda^{(m)}\left(i,k\right) + \mu^{(m)}\left(i,k\right) + \overline{v}^{(m)}\left(i,k\right)$$

$$\overline{u} = \sum\_{n=1}^{N} \hat{H}\_{n}^{\prime(m)}\left(i,k\right)c\_{n}\left(i,k\right) + \sum\_{n=1}^{N} \Delta \hat{H}\_{n}^{\prime(m)}\left(i,k\right) \left[p\_{n}\left(i,k\right) + c\_{n}\left(i,k\right)\right] + \overline{v}^{(m)}\left(i,k\right) \tag{42}$$

where ( ) ( ) ( ) 1 ˆ ˆ ', ', ', *m mm <sup>N</sup> n nn <sup>n</sup> H ik H ik H ik* is the estimation error due to information interference as well as additive noise. Using the proposed two-step estimator (23)-(26), the channel estimation variance can be smoothed over multiple OFDM symbols, and approaches to a small fixed lower bound. The estimated vector ˆ ( ) H' *<sup>m</sup> <sup>n</sup> i* as well as the error vector ˆ ( ) H' *<sup>m</sup> <sup>n</sup> i* , therefore, can be thus approximated to be of the similar characteristics of distribution as that of ( ) H' *<sup>m</sup> <sup>n</sup> i* . Consequently, following the assumption (H1)-(H3), the interference vector ( ) *<sup>m</sup>* **μ** *i* is approximately white for a large symbol-size *B*, and independent of the noise vector ( ) V *<sup>m</sup> i* . Similar to the procedure of (29)-(30), the covariance matrix of ( ) *<sup>m</sup>* **μ** *i* and ( ) λ *<sup>m</sup> i* can be obtained as

$$\operatorname{var}\left(\lambda^{(m)}(i)\right) = E\left| \left(\lambda^{(m)}(i)\right)^H \lambda^{(m)}(i) \right| = \sigma\_{\text{fi}}^2 \sigma\_p^2 \mathbf{I} \tag{43}$$

 <sup>H</sup> ( ) () () <sup>222</sup> var <sup>ˆ</sup> <sup>I</sup> *<sup>H</sup> m mm p c iE i i* **μ μμ** (44)

where <sup>H</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup> ( ) 1 1 ( ) <sup>2</sup> <sup>ˆ</sup> 0 10 , <sup>ˆ</sup> *<sup>m</sup> B NL <sup>m</sup> kl B n l k nl i e NLB* , and <sup>H</sup> <sup>2</sup> <sup>2</sup> 1 1 ( ) <sup>2</sup> <sup>ˆ</sup> 0 10 , <sup>ˆ</sup> *B NL <sup>m</sup> kl B n l k nl e NLB* . Since the ST power allocation factor is derived within each isolated OFDM symbol, we neglect the symbol-index *i* for simplicity. A lower bound on the OFDM channel capacity with channel estimation error has been derived in [20]-[21] for uniform pilot distribution. Such expression can readily be extended to issue of ST where the pilots are spread over the whole frequency band. Therefore, the lower bound of the average channel capacity for an ST-based OFDM system can be obtained by summing over all the subcarriers, i.e.,

$$\mathbf{C}^{(m)} \ge \overline{\mathbf{C}}^{(m)} = \frac{1}{B} \sum\_{k=0}^{B-1} E\left\{ \log \left[ 1 + \frac{\sigma\_c^2}{\left( \sigma\_p^2 + \sigma\_c^2 \right) \sigma\_{\Lambda \hat{\mathbf{H}}}^2 \left/ \sigma\_{\hat{\mathbf{H}}}^2 + \sigma\_v^2 \right/ \sigma\_{\hat{\mathbf{H}}}^2} \right] \right\} \tag{45}$$

For the sake of simplicity, we assume the transmission power satisfies that 2 2 1 *p c* . By (40), we thus have <sup>2</sup> *<sup>p</sup>* and <sup>2</sup> 1 *<sup>c</sup>* . Considering that the normalized MSE of the proposed two-step channel estimator is sufficiently small and approaches to a fixed lower bound (37)-(39), it allows us to make the approximation of HH H <sup>H</sup> 2222 ˆˆ ˆ ( ) *<sup>m</sup> NMSE* . As a result, ( ) C *<sup>m</sup>* in (45) can be approximated as

$$\mathbf{C}^{(m)} \approx \frac{1}{B} \sum\_{k=0}^{B-1} E\left\{ \log \left[ 1 + \frac{\sigma\_c^2}{\sigma\_{\rm A\rm H}^2 / \sigma\_{\rm H}^2 + \sigma\_v^2 / \sigma\_{\rm H}^2} \right] \right\}$$

$$= \frac{1}{B} \sum\_{k=0}^{B-1} E\left\{ \log \left[ 1 + \frac{1-\beta}{(1-\beta)(Q+1)\,\mathrm{NL} / \beta\mathrm{I}\,\mathrm{I}\,\mathrm{T} + (Q+1)\,\sigma\_v^2 / \beta\mathrm{I}\,\mathrm{I}\,\sigma\_{\rm H}^2 + \sigma\_v^2 / \sigma\_{\rm H}^2} \right] \right\} \tag{46}$$

$$= \log \left( 1 + \frac{(1-\beta)\beta\mathrm{I}\,\mathrm{I}}{\beta \left[ 1\,\mathrm{I}\,\mathrm{V} / \mathfrak{R}\_{\rm SNR} - (Q+1)\,\mathrm{NL} \right] + (Q+1)\,\mathrm{NL} \left( \mathrm{I} / \mathfrak{R}\_{\rm SNR} + 1 \right)} \right)$$

where SNR 22 2 2 22 *H <sup>p</sup> c v Hv* . In fact, the averaged channel capacity of (46) is a log-function of , which is a monotonically increasing function. Therefore, the lower-bound of ( ) C *<sup>m</sup>* with respect to can be achieved by maximizing the following function

$$\chi^{(m)}\left(\boldsymbol{\beta}\right) = \frac{\boldsymbol{\beta}\left(1-\boldsymbol{\beta}\right)}{\alpha\_1\boldsymbol{\beta}+\alpha\_2} = \frac{\left(1-\boldsymbol{\beta}\right)\boldsymbol{\beta}}{\boldsymbol{\beta}\left[1/\mathfrak{R}\_{\text{SNR}} - \left(Q+1\right)\operatorname{NL}\left/I\Gamma\right] + \left(Q+1\right)\operatorname{NL}\left(1/\mathfrak{R}\_{\text{SNR}} + 1\right)/I\Gamma} . \tag{47}$$

where

268 Wireless Communications and Networks – Recent Advances

() () () () ,,,, *m m mm u ik ik ik v ik*

 

ˆ ˆ ', , ', , , , *N N mm m nn n n n*

interference as well as additive noise. Using the proposed two-step estimator (23)-(26), the channel estimation variance can be smoothed over multiple OFDM symbols, and

interference vector ( ) *<sup>m</sup>* **μ** *i* is approximately white for a large symbol-size *B*, and independent of the noise vector ( ) V *<sup>m</sup> i* . Similar to the procedure of (29)-(30), the covariance

> <sup>H</sup> ( ) () () 2 2 var <sup>ˆ</sup> λ λλ I *<sup>H</sup> m mm*

<sup>H</sup> ( ) () () <sup>222</sup> var <sup>ˆ</sup> <sup>I</sup>

derived within each isolated OFDM symbol, we neglect the symbol-index *i* for simplicity. A lower bound on the OFDM channel capacity with channel estimation error has been derived in [20]-[21] for uniform pilot distribution. Such expression can readily be extended to issue of ST where the pilots are spread over the whole frequency band. Therefore, the lower bound of the average channel capacity for an ST-based OFDM system can be obtained by

1 2

 

 

proposed two-step channel estimator is sufficiently small and approaches to a fixed lower

For the sake of simplicity, we assume the transmission power satisfies that 2 2 1

bound (37)-(39), it allows us to make the approximation of HH H <sup>H</sup>

*<sup>B</sup> m m <sup>c</sup>*

*E*

*<sup>H</sup> m mm*

*n l k nl i e NLB*

,

*<sup>p</sup> iE i i*

*p c iE i i*

. Since the ST power allocation factor is

*H ikc ik H ik p ik c ik v ik*

( ) ( ) ( )

*n nn <sup>n</sup> H ik H ik H ik* is the estimation error due to information

*<sup>n</sup> i* , therefore, can be thus approximated to be of the similar characteristics of

*<sup>n</sup> i* . Consequently, following the assumption (H1)-(H3), the

 

**μ μμ** (44)

HH H

0 ˆˆ ˆ

*k p c v*

2 22 2 22

 

(45)

. Considering that the normalized MSE of the

2222 

(42)

*<sup>n</sup> i* as well as the error

(43)

 *p c* . By

ˆˆ ˆ ( ) *<sup>m</sup> NMSE* .

approaches to a small fixed lower bound. The estimated vector ˆ ( ) H' *<sup>m</sup>*

1 1

*n n*

where ( ) ( ) ( ) 1 ˆ ˆ ', ', ', *m mm N*

matrix of ( ) *<sup>m</sup>* **μ** *i* and ( ) λ *<sup>m</sup> i* can be obtained as

<sup>2</sup> <sup>2</sup> <sup>2</sup> ( ) 1 1 ( ) <sup>2</sup> <sup>ˆ</sup> 0 10 , <sup>ˆ</sup> *<sup>m</sup> B NL <sup>m</sup> kl B*

*n l k nl e NLB*

<sup>1</sup> C C log 1

*B*

 and <sup>2</sup> 1 *<sup>c</sup>*

<sup>2</sup> <sup>2</sup> 1 1 ( ) <sup>2</sup> <sup>ˆ</sup> 0 10 , <sup>ˆ</sup> *B NL <sup>m</sup> kl B*

() ()

 *<sup>p</sup>* 

As a result, ( ) C *<sup>m</sup>* in (45) can be approximated as

summing over all the subcarriers, i.e.,

vector ˆ ( ) H' *<sup>m</sup>*

where <sup>H</sup>

(40), we thus have <sup>2</sup>

and <sup>H</sup>

distribution as that of ( ) H' *<sup>m</sup>*

$$\alpha\_1 = \mathbf{1} / \Re\_{\text{SNR}} - (Q+1) \text{NL} / \text{I} \Gamma \text{ , } \alpha\_2 = (Q+1) \text{NL} \left( \mathbf{1} / \Re\_{\text{SNR}} + \mathbf{1} \right) \text{I} \Gamma \text{ .} \tag{48}$$

Setting the first derivation of ( ) *<sup>m</sup>* with respect to to zero, we obtain (after some manipulations) a quadratic equation in , i.e.

$$
\beta^2 + \frac{2a\_2}{a\_1}\beta - \frac{a\_2}{a\_1} = 0 \,\, . \tag{49}
$$

Consequently, the global maximum of ( ) *<sup>m</sup>* can be obtained when

$$\beta = \frac{\sqrt{\left(1/\mathfrak{R}\_{\text{SNR}} + 1\right)\left(I\Gamma\left/NL\left(Q+1\right)\mathfrak{R}\_{\text{SNR}} + 1\right\mathfrak{R}\_{\text{SNR}}\right)} - \left(Q+1\right)NL\left(1\middle/\mathfrak{R}\_{\text{SNR}} + 1\right)}{I\Gamma\left/\mathfrak{R}\_{\text{SNR}} - \left(Q+1\right)NL} \ . \tag{50}$$

As will be shown in simulations, an increase in the training power allocation factor does not necessarily improve the overall system performance since a larger implies a better channel estimation while substantially scarifying the effective received signal SNR at the same time.

## **6. Simulations**

We assume the MIMO/OFDM system with *N* = 2 and *M* = 4. The symbol-size is *B* = 1024 and the transmitted data *s ik <sup>n</sup>* , is 8-PSK signals with symbol rate <sup>7</sup> 10 *sf* /second. Before transmission, the transmitted data are coded by 1/2 convolutional coding and block interleaving over one OFDM symbol. The channel is assumed to be *L* = 10 taps and, the

coefficients ( ) , ( ) *<sup>m</sup> n l h t* are generated as low-pass, Gaussian and zero mean random processes and uncorrelated for different values of *n* and *l*. The multi-path intensity profile is chosen to be *l l* exp /10 0, 1 *l L* . The Doppler spectra are <sup>2</sup> ( ) *<sup>n</sup> f f f* for *<sup>n</sup> f f* , where *nf* is the Doppler frequency of the *n*th user, otherwise, *f* 0 . CPlength is chosen to be 32 to avoid inter-symbol interferences. The additive noise is a Gaussian and white random process with a zero mean.

#### *Test Case 1. Channel Estimation*

We run simulations with the Doppler frequency *nf* 300Hz that corresponds to the maximum mobility speed of 162 km/h as the users operate at carrier frequency of 2GHz. In order to model the LTV channel, the frame is designed as *B B* ' 128 CP-length 128 135168 , i.e. each frame consists of 128 OFDM symbols. During the frame, the channel variation is *n s f f* 4.1. Over the frame , we utilize truncated DFB of order 10 2 *Q ff d s* to model the LTV channel coefficients. In order to estimate the MIMO/OFDM channels, the superimposed pilots are designed according to (15) with the pilot power 2 2 0.2 *<sup>p</sup> <sup>c</sup>* . Fig.2 depicts the LTV channel coefficient estimation over the frame . It is clearly observed that although the channel coefficient is accurately estimated during the centre part of the frame, the outmost samples over the whole frame still exhibit errors. A possible explanation is that as the Fourier basis expansions in (6) are truncated, and an effect similar to the Gibbs phenomenon, together with spectral leakages, will lead to some errors at the beginning and the end of the frame. This may be a common problem for the proceeding literature [1]-[2] [5]-[6] [16] that employing basis expansions to model the LTV channels. To solve the problem, the frames are designed to be partially overlapped, e.g. the frames are designed as ( 1) ' *B t* , 2,3, , where is a positive integer. By the frame-overlap, the channel at the beginning and the end of one frame can be modeled and estimated from the neighboring frames.

To further evaluate the new channel estimator, we use the mean square errors to measure the channel estimation performance by

$$\begin{aligned} MSE\_n^{(m)} &= \sum\_{i=1}^{\Omega B^{+}} MSE\_n^{(m)} \left( i \right) \Big/ \left( \Omega / B^{\circ} \right) = \\ \frac{B^{+} \Omega^{\circ}}{\Omega} \sum\_{i=1}^{\Omega B^{+}} E \left| \sum\_{t=0}^{B-1} \sum\_{l=0}^{L-1} \left| h\_{n,l}^{(m)} \left( i, t \right) - \sum\_{q=0}^{Q} \hat{h}\_{n,l,q}^{(m)} \mathbf{e}^{\{j2\pi(q-Qf2)t\}} \right|^{2} \right| & \left. \begin{aligned} \left( \mathbf{51} \right)^{\circ} \\ \left| \mathbf{B} L \right| \left| h\_{n,l}^{(m)} \left( i, t \right) \right|^{2} \end{aligned} \right| \end{aligned} \tag{51}$$

where ( ) , , ˆ *m n l <sup>q</sup> h* is the channel coefficient estimation.

We firstly test the two-step channel estimator under the different pilot powers and different channel coefficient numbers to verify the channel estimation variance analysis. The LTV channel is the same as that in Fig.2. As shown in Fig.3, the MSE of the channel estimation approach are almost independent of the additive noises, especially as SNR>5dB. This is consistent with the channel estimation analysis (38)-(39) where the additive noise has been

and uncorrelated for different values of *n* and *l*. The multi-path intensity profile is chosen to

for *<sup>n</sup> f f* , where *nf* is the Doppler frequency of the *n*th user, otherwise, *f* 0 . CPlength is chosen to be 32 to avoid inter-symbol interferences. The additive noise is a

We run simulations with the Doppler frequency *nf* 300Hz that corresponds to the maximum mobility speed of 162 km/h as the users operate at carrier frequency of 2GHz. In order to model the LTV channel, the frame is designed as *B B* ' 128 CP-length 128 135168 , i.e. each frame consists of 128 OFDM symbols. During the frame, the channel variation is *n s f f* 4.1. Over the frame , we utilize truncated DFB of order 10 2 *Q ff d s* to model the LTV channel coefficients. In order to estimate the MIMO/OFDM channels, the superimposed pilots are designed according

estimation over the frame . It is clearly observed that although the channel coefficient is accurately estimated during the centre part of the frame, the outmost samples over the whole frame still exhibit errors. A possible explanation is that as the Fourier basis expansions in (6) are truncated, and an effect similar to the Gibbs phenomenon, together with spectral leakages, will lead to some errors at the beginning and the end of the frame. This may be a common problem for the proceeding literature [1]-[2] [5]-[6] [16] that employing basis expansions to model the LTV channels. To solve the problem, the frames are designed to be partially overlapped, e.g. the frames are designed as ( 1) ' *B t* , 2,3, , where is a positive integer. By the frame-overlap, the channel at the beginning and the end of one frame can be modeled and estimated

To further evaluate the new channel estimator, we use the mean square errors to measure

' ( ) ( )

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

*n l nlq n l*

'

1

2 '

1 00 0

*i tl q*

*n l <sup>q</sup> h* is the channel coefficient estimation.

*MSE MSE i B*

 ˆ , e , *<sup>B</sup> B L <sup>Q</sup> m m j qQ t <sup>m</sup>*

 

We firstly test the two-step channel estimator under the different pilot powers and different channel coefficient numbers to verify the channel estimation variance analysis. The LTV channel is the same as that in Fig.2. As shown in Fig.3, the MSE of the channel estimation approach are almost independent of the additive noises, especially as SNR>5dB. This is consistent with the channel estimation analysis (38)-(39) where the additive noise has been

*<sup>B</sup> E h it h BL h i t*

*B m m n n i*

*l l* exp /10 0, 1 *l L* . The Doppler spectra are <sup>2</sup> ( ) *<sup>n</sup> f*

*n l h t* are generated as low-pass, Gaussian and zero mean random processes

*<sup>c</sup>* . Fig.2 depicts the LTV channel coefficient

*f f*

(51)

coefficients ( )

be , ( ) *<sup>m</sup>*

*Test Case 1. Channel Estimation* 

from the neighboring frames.

'

where ( ) , , ˆ *m*

the channel estimation performance by

to (15) with the pilot power 2 2 0.2

*<sup>p</sup>*

Gaussian and white random process with a zero mean.

greatly suppressed by the weighted average procedure. Thus, the estimation errors depend mainly on the information- pilot power ratio as well as the system unknowns *NL* . This is rather different from the FDM training based schemes [20]-[23].

Fig. 2. One tap coefficient of the LTV channel and the estimation over the frame <sup>7</sup> 135168 10 13.52 ms.

Fig. 3. MSE of the weighted average estimation versus SNR for the LTV channel of *nf* 300Hz and 13.52 ms under the different pilot powers and different unknown parameters.

We then compare the proposed two-step channel estimation scheme with the conventional ST-based methods [8]-[13] [17]-[18] under different Doppler frequencies. In the conventional ST scheme, the LTV channel is firstly estimated from the LTI assumption at each OFDM symbol, and then all the estimations from the frame are averaged to confront the information sequence interferences. It shows clearly in Fig. 4 that for the LTI channel of *nf* 0Hz, both the conventional ST and the weighted average estimator exhibit the similar performance. In addition, the estimation performance can be improved with the increment of the frame or average length. However, when the channel involved in simulations is timevarying, the channel estimation performance of the conventional ST-based schemes is degraded extensively. The simulation reveals the shortcoming of the conventional ST in estimating the LTV channels. On the contrary, the MSE level is reduced by the weighted average process (23)-(26) for the LTV channels of *nf* 100Hz, 300Hz with *T* 2 (one pilot cluster is composed of 2 1 5 *T* pilots). We also observe that the MSE approaches to a constant as the increment of the frame length, i.e. the lower-bound that associated with the given Doppler frequency.

Fig. 3. MSE of the weighted average estimation versus SNR for the LTV channel

parameters.

given Doppler frequency.

of *nf* 300Hz and 13.52 ms under the different pilot powers and different unknown

We then compare the proposed two-step channel estimation scheme with the conventional ST-based methods [8]-[13] [17]-[18] under different Doppler frequencies. In the conventional ST scheme, the LTV channel is firstly estimated from the LTI assumption at each OFDM symbol, and then all the estimations from the frame are averaged to confront the information sequence interferences. It shows clearly in Fig. 4 that for the LTI channel of *nf* 0Hz, both the conventional ST and the weighted average estimator exhibit the similar performance. In addition, the estimation performance can be improved with the increment of the frame or average length. However, when the channel involved in simulations is timevarying, the channel estimation performance of the conventional ST-based schemes is degraded extensively. The simulation reveals the shortcoming of the conventional ST in estimating the LTV channels. On the contrary, the MSE level is reduced by the weighted average process (23)-(26) for the LTV channels of *nf* 100Hz, 300Hz with *T* 2 (one pilot cluster is composed of 2 1 5 *T* pilots). We also observe that the MSE approaches to a constant as the increment of the frame length, i.e. the lower-bound that associated with the

Fig. 4. MSE versus frame or average length under the different Doppler frequencies of the LTV channel with 2 2 0.25 *<sup>p</sup> <sup>c</sup>* , SNR = 20dB.

From Fig. 4, we observe that channel estimation performance would be degraded as the increment of mobile users' speed (or corresponding system Doppler shift). To further enhance the channel estimation performance of the systems with a limited pilot power while suffering from a high Doppler shift, an iterative decision feedback (DF) approach can be adopted at the receiver. Explicitly, the iterative method can be considered as a twofold process. First, the information sequences are recovered by a hard detector [5] based on the LTV channel estimation in Section III. Second, the recovered data symbols are removed from the received signals to cancel the information sequence interference and, thus to enhance the channel estimation performance. Fig. 5 depicts the performance between the weighted average scheme and the iterative DF estimator in terms of channel MSE. For a fairness of comparison, we also simulate the MSE of the FDM training-based channel estimator [5] as latter serves as a "benchmark" in related works. For estimating the MIMO/OFDM channels, 40 pilot clusters with (2 1) 200 *T* known pilot symbols which are subject to the proposed pilot specifications in (15) are used in one OFDM symbol. That is, approximately 10% total bandwidth is assigned for pilot tones. Comparatively, as shown in Fig.5, the iterative DF estimation exhibits a more significant improvement than that of weighted average estimation, and outperforms the FDM channel estimator [5] by using a small pilot power of 2 2 0.25 *<sup>p</sup> <sup>c</sup>* , which conforms that the information sequence interferences can be effectively cancelled by iterative DF procedure. Moreover, it should be noted that since the superimposed pilots are spread over the entire band, the proposed ST-based channel estimator is also feasible to estimate the channel with a very long delay spread, i.e. clusterbased channel.

Fig. 5. MSE versus SNR for different estimators for the LTV channel of 300 *nf* Hz, 20 *NL* .

To further validate the effectiveness of the DF scheme, we also provide the channel estimation MSE of the DF method versus the iteration numbers under SNR = 15dB. Fig. 6 shows that the iterative DF method is feasible for a wide range of system Doppler spreads. Obviously, the enhancement of the iterative DF is at the cost of an increment in computational complexity that is directly proportional to the iteration number. However, as is shown in Fig. 6 that the iterative DF approach converges to the steady-state performance by only a few iterations, the overall computational complexity will be acceptable for many wireless communication systems.

Fig. 6. MSE of the iterative DF channel estimation versus iterations for the LTV channel of different system Doppler spreads when SNR=15dB.

#### *Test Case 2. Training Power Allocation*

274 Wireless Communications and Networks – Recent Advances

proposed pilot specifications in (15) are used in one OFDM symbol. That is, approximately 10% total bandwidth is assigned for pilot tones. Comparatively, as shown in Fig.5, the iterative DF estimation exhibits a more significant improvement than that of weighted average estimation, and outperforms the FDM channel estimator [5] by using a small pilot

effectively cancelled by iterative DF procedure. Moreover, it should be noted that since the superimposed pilots are spread over the entire band, the proposed ST-based channel estimator is also feasible to estimate the channel with a very long delay spread, i.e. cluster-

Fig. 5. MSE versus SNR for different estimators for the LTV channel of 300 *nf* Hz, 20 *NL* .

To further validate the effectiveness of the DF scheme, we also provide the channel estimation MSE of the DF method versus the iteration numbers under SNR = 15dB. Fig. 6 shows that the iterative DF method is feasible for a wide range of system Doppler spreads. Obviously, the enhancement of the iterative DF is at the cost of an increment in computational complexity that is directly proportional to the iteration number. However, as is shown in Fig. 6 that the iterative DF approach converges to the steady-state performance by only a few iterations, the overall computational complexity will be acceptable for many

*<sup>c</sup>* , which conforms that the information sequence interferences can be

power of 2 2 0.25 *<sup>p</sup>*

based channel.

wireless communication systems.

As aforementioned, for wireless communication systems with a limited transmission power, some useful power must inevitably be allocated to the superimposed pilots, and thus resulting in the received signal SNR reduction. Herein, we carry out several experiments to assess the effect of ST power allocation factor on the lower-bound of the average channel capacity for different SNRs.

Fig. 7 shows the effect of different value of training power allocation factor on the lower bound of the average channel capacity for received signal SNR = 10 and 20 dB, respectively. It is seen that the average channel capacity decreases with the increment of . It reveals that although higher implies that higher fraction of transmitted power is allocated to training leading to more accurate channel estimates, the received signal SNR is substantially decreased, resulting in potential decrement of the average channel capacity. In addition, we further simulate the approximated in order to test the validity of theoretic results in (50). It can be seen that the approximation of is almost consistent with that of the actual results.

Fig. 7. Lower bound of the average channel capacity versus different values of ST power allocation factor under SNR = 0dB, 10dB and 20dB, respectively.

Fig. 8 shows the plots of the optimal value of training power allocation factor versus received SNR for different frame length. It is observed that the increment of SNR leading to a corresponding increase in the optimal ST power allocation factor. This can be easily comprehended that according to (41), the effective interference is composed of two factors, i.e. the bias of channel estimation and the additive noise. That is, for large SNRs, higher is required to improve the channel estimation performance, thus leading to a reduction of the effective interference. Conversely, when SNR is small, improving the channel estimation accuracy has a small effect in reducing the effective interference. On the other hand, we notice that decreases as the frame length increases but approximately unvaried when is sufficiently large, i.e. is almost unchanged when 192 *I* . This result arises because we have theoretically analyzed in Section III that the estimation variance approaches to a fixed lower bound that can be only improved by increasing ST power allocation when the frame length is large enough. Therefore, the power allocated to the training sequence can be reduced with no loss in channel estimation performance when the frame length is increased, but finally approaches to a fixed lower bound associate with the channel estimation variance when is sufficiently large. This is somewhat different from those presented in [10].

Fig. 8. Optimal ST power allocation factor of the proposed weighted average channel estimator versus SNR for different frame lengths.
