**1. Introduction**

In the past few years, Multiple Input Multiple Output (MIMO) system received lots of attentions, since it is capable in providing higher spectrum efficiency, as well as better transmission reliability. This improvement is brought by the multiple antennas at both sides of transmission. Since there are additional parallel sub-channels in spatial domain, system can not only increase the reliability by spatial diversity technology, but also provide higher data rate utilizing spatial multiplexing (see [1],[2], etc.). Orthogonal Frequency Division Multiplexing (OFDM) can also accommodate high data rate requirement by providing frequency multiplexing gain. To fully utilize the benefits of both technologies, the combination of the above two, MIMO-OFDM, has been employed in many wireless communication systems and protocols, such as WLAN [3] and LTE systems [4].

The introduction of MIMO-OFDM raises plenty challenges in channel estimation and measurement. Transmitted signals are reflected and scattered, resulting in a multipath spread in the received signals. Moreover, the transmitters, receivers, and reflecting or scattering objects are moving, which means that channels change rapidly over time [5]. Inter-Channel interference may also bring a destructive effect in transmission, which should be cancelled by accurate channel measurement or estimation.

As an important application of MIMO-OFDM technology, WLAN is suitable in providing high data rate service in hotspot area, such as office buildings, airports, libraries, stations, hospitals and restaurants. This means lots of MIMO-OFDM applications (such as WLAN) take place in indoor situations, where both transmitters and receivers are surrounded by mobile and static scatters. Different from outdoor scenario, there are some unique characteristics of indoor scenario. More scatters result in larger influence by multipath effect; higher density of users and overlap between different access points bring larger interference. Because of these differences in channel parameters, it is crucial to obtain a better understanding of indoor channels. Statistics such as delay spread, Doppler spread, angle spread and path loss must be estimated by detail channel measurement. This requirement rises the interests in indoor channel measurement in the past few years.

Channel measurement or estimation schemes can be divided into two major categories, blind and non-blind. Blind channel estimation method requires large data and can only exploit the statistical behavior of channel, hence, suffers a lot in fast fading channels.

On contrast, non-blind method utilizes pre-determined information in both transmitters and receivers, to measure the channel. One of the most frequently used methods is the pilot/data aided channel measurement/estimation. The method can be further divided into two sub-methods considering the resources occupied by pilots with each resource unit. In the first sub-method, pilots occupy the whole resource unit, for example, an OFDM training symbol. This type of pilot arrangement is largely used for pure purpose of channel measurement without the request of communication. In real-time communication, it is only suitable for slow channel variation and for burst type data transmission schemes. In the second sub-method, pilots only occupy part of the resource unit; the other part of the unit is allocated to data. This pilot arrangement is capable to provide real-time communication which takes place in time and frequency varying channels. However, a linear interpolation or higher order polynomial fitting should be applied to recover the whole channel, which will certainly cause some errors.

Recently, there are plenty works considering indoor channel measurement and estimations, with non-blind pilot/data aided method, each of which focuses on different aspects of the issue. In [6], authors introduced a detail design of a MIMO channel sounder. In their measurement process, they used a PN sequence as the probing signal (pilot), which occupy the whole frequency and time resources. Their measurement took place in Seoul railway station, and they used the results to illustrate the characteristics of delay and Doppler spread of indoor channel. In [7], authors provide a PC-FPGA design in solving a similar problem for WLAN system. Instead of occupying the whole channel resource, the PN sequences only insert in certain parts of the resource unit. In this way, channel measurement and transmission can be carried out simultaneously. The effect of polarized antennas has also been considered in [8]. Wireless situations include non-line-of-sight, propagation along the corridor and propagation over a metallic ceiling.

Other important applications include several scenarios such as: near-ground indoor channel aiming military or emergency usage [9], and indoor channel model for wireless sensor network and internet of things [10]. Pilot signal design is flexible. Instead of a PN sequence, other pseudolite signals are available, too. Also, the kinds of carrier signals are variable, such as an OFDM signal [7] or a GPS-based signal [11].

In addition, some rough estimations of channel parameters are provided. Most of these works based on the assumption that indoor channels follow the Ricean distribution. The most important parameter of Ricean distribution for indoor channels is the K-factor, which represents the ratio between the average power of deterministic and random components of the channel. In [12], a two-moment method of the Ricean K-factor is provided theoretically. Experimental results can be found in [13], which gives an application for the two-moment estimation of the Ricean K-factor in wideband indoor channels at 3.7 GHz. Although the Ricean distribution can only provide an unclear view of the channel, it is convenient and low-complexity estimation; thus can be applied in scenarios that require only partial channel state information.

#### **2. Measurement schemes**

230 Wireless Communications and Networks – Recent Advances

Channel measurement or estimation schemes can be divided into two major categories, blind and non-blind. Blind channel estimation method requires large data and can only

On contrast, non-blind method utilizes pre-determined information in both transmitters and receivers, to measure the channel. One of the most frequently used methods is the pilot/data aided channel measurement/estimation. The method can be further divided into two sub-methods considering the resources occupied by pilots with each resource unit. In the first sub-method, pilots occupy the whole resource unit, for example, an OFDM training symbol. This type of pilot arrangement is largely used for pure purpose of channel measurement without the request of communication. In real-time communication, it is only suitable for slow channel variation and for burst type data transmission schemes. In the second sub-method, pilots only occupy part of the resource unit; the other part of the unit is allocated to data. This pilot arrangement is capable to provide real-time communication which takes place in time and frequency varying channels. However, a linear interpolation or higher order polynomial fitting should be applied to recover the whole channel, which

Recently, there are plenty works considering indoor channel measurement and estimations, with non-blind pilot/data aided method, each of which focuses on different aspects of the issue. In [6], authors introduced a detail design of a MIMO channel sounder. In their measurement process, they used a PN sequence as the probing signal (pilot), which occupy the whole frequency and time resources. Their measurement took place in Seoul railway station, and they used the results to illustrate the characteristics of delay and Doppler spread of indoor channel. In [7], authors provide a PC-FPGA design in solving a similar problem for WLAN system. Instead of occupying the whole channel resource, the PN sequences only insert in certain parts of the resource unit. In this way, channel measurement and transmission can be carried out simultaneously. The effect of polarized antennas has also been considered in [8]. Wireless situations include non-line-of-sight, propagation along

Other important applications include several scenarios such as: near-ground indoor channel aiming military or emergency usage [9], and indoor channel model for wireless sensor network and internet of things [10]. Pilot signal design is flexible. Instead of a PN sequence, other pseudolite signals are available, too. Also, the kinds of carrier signals are variable,

In addition, some rough estimations of channel parameters are provided. Most of these works based on the assumption that indoor channels follow the Ricean distribution. The most important parameter of Ricean distribution for indoor channels is the K-factor, which represents the ratio between the average power of deterministic and random components of the channel. In [12], a two-moment method of the Ricean K-factor is provided theoretically. Experimental results can be found in [13], which gives an application for the two-moment estimation of the Ricean K-factor in wideband indoor channels at 3.7 GHz. Although the Ricean distribution can only provide an unclear view of the channel, it is convenient and low-complexity estimation; thus can be applied in

exploit the statistical behavior of channel, hence, suffers a lot in fast fading channels.

will certainly cause some errors.

the corridor and propagation over a metallic ceiling.

such as an OFDM signal [7] or a GPS-based signal [11].

scenarios that require only partial channel state information.

As is introduced before, channel measurement schemes can be divided into two major categories, blind and non-blind. Since blind measurement and estimation is much less reliable, we only discuss non-blind pilot/data aided schemes. Two of the most frequently used schemes are discussed: measurement based on PN sequence which occupies a whole resource unit, and measurement based on OFDM pilot who occupies only part of an OFDM symbol.

#### **2.1 Indoor channel model**

The low-pass time-variant channel impulse response (CIR) is denoted as *h t* (;) , which represents the response of the channel at time *t* due to an impulse applied at time *t* .Then transmission can be expressed as:

$$\log(t,\tau) = \sum\_{n=0}^{N\_{\text{CR}}-1} h(n,\tau) \mathbf{x}(t-n,\tau) + w(t,\tau)\_k \tag{1}$$

Where *x* is the transmit signal, and *y* is the transmit signal, *N*CIR is the duration of the CIR, *w* is the noise sequence.

By taking the Fourier transform of *h t* (;) , we can obtain the time-variant channel transfer function

$$H(f;t) = \int\_{-\infty}^{+\infty} h(\tau;t)e^{-j2\pi f\tau}d\tau\tag{2}$$

On the assumption that the scattering of the channel at two different delays is uncorrelated, the autocorrelation function of *h t* (;) can be defined as:

$$\frac{1}{2}E\left[h^\*(\tau\_1;t)h(\tau\_2;t+\Delta t)\right] = \rho\_h(\tau\_1;\Delta t)\delta(\tau\_1-\tau\_2) \tag{3}$$

If we let 0 *t* , the resulting autocorrelation function ( ) ( ;0) *h h* is called delay power profile of the channel. The range of values of over which ( ) *h* is essentially nonzero is called the multipath spread of the channel. Then the scattering function of the channel is defined as:

$$S(\tau;\lambda) = \int\_{-\infty}^{+\infty} \rho\_{\text{li}}(\tau;\Delta t) e^{-j2\pi\Delta t} d\Delta t \tag{4}$$

By taking the integration of *S*(; ) , we obtain the Doppler power spectrum of the channel as:

$$S(\mathcal{A}) = \int\_{-\infty}^{+\infty} S(\tau; \mathcal{A}) d\tau \tag{5}$$

The range of values of over which *S*( ) is essentially nonzero is called the Doppler spread of the channel.

Indoor channel conditions are much more complex than that of outdoor channel. There are plenty kinds of scattering figures, such as walls, tables, etc. People indoor can also act as scattering figures, and the movements of cell phones caused by this bring about worse channel conditions. As a result, angle of arrival, multipath spread and scattering factor of indoor channels are different from those of outdoor channels. Consequently, measurement schemes should be redesign carefully according to the above distingue characteristics, so as to to fulfill the needs of indoor channel measurements.

It is worth noticing that channels of phone calls made in indoor conditions are mixtures of both indoor and outdoor channels. Large scale fading, small scale fading and shadow fading should be equally considered in such channels. Each of these factors can cause a large channel capacity decrease.

#### **2.2 Channel measurement using PN sequence**

In this scheme, Pseudo-Noise (PN) Sequence is used as a probing signal. To authors' best knowledge, the most widely used binary PN sequence is the Maximum-Length-Shift-Register (MLSR) sequence. The length of MLSR sequence is PN 2 1 *<sup>m</sup> N* bits. And one of the possible generators of this sequence is an m-stage linear feedback shift register (see [14]). As a result, MLSR sequence is periodic with period n. Within each period, there are <sup>1</sup> 2*<sup>m</sup>* ones and <sup>1</sup> 2*<sup>m</sup>* zeros.

One of the most important characteristics of the periodic PN sequence is its sharp autocorrelation. Consider an PN sequence *<sup>k</sup> x* , we have :

$$\sum\_{k=0}^{N\_{\rm PN}-1} \mathbf{x}\_{k+m} \mathbf{x}\_k = \begin{cases} N\_{\rm PN}, m = 0, \pm N\_{\rm PN}, \pm 2N\_{\rm PN}, \dots \\ 0, & \text{Otherwise} \end{cases} \tag{6}$$

If *N*PN 1 , it is approximate that:

$$\frac{1}{N\_{\rm PN}} \sum\_{k=0}^{N\_{\rm PN}-1} \mathbf{x}\_{k+m} \mathbf{x}\_k \approx \sum\_{i=-\infty}^{+\infty} \delta\_{m-iN\_{\rm PN}} \tag{7}$$

We can represent the received signal *<sup>k</sup> y* as the convolution of transmitted signal *<sup>k</sup> x* and the CIR *<sup>k</sup> h* as:

$$\mathbf{y}\_k = \sum\_{n=0}^{N\_{\text{QR}}-1} h\_n \mathbf{x}\_{k-n} + w\_k \tag{8}$$

The transmitter use PN sequence as the transmitting data *<sup>k</sup> x* . If the generators of PN sequence in both transmitter and receiver are synchronous, at each time slot, receiver is aware of the transmitting PN sequence. Hence, receiver can obtain the CIR by doing a crosscorrelating between *<sup>k</sup> y* and *<sup>k</sup> x* :

Indoor channel conditions are much more complex than that of outdoor channel. There are plenty kinds of scattering figures, such as walls, tables, etc. People indoor can also act as scattering figures, and the movements of cell phones caused by this bring about worse channel conditions. As a result, angle of arrival, multipath spread and scattering factor of indoor channels are different from those of outdoor channels. Consequently, measurement schemes should be redesign carefully according to the above distingue characteristics, so as

It is worth noticing that channels of phone calls made in indoor conditions are mixtures of both indoor and outdoor channels. Large scale fading, small scale fading and shadow fading should be equally considered in such channels. Each of these factors can cause a large

In this scheme, Pseudo-Noise (PN) Sequence is used as a probing signal. To authors' best knowledge, the most widely used binary PN sequence is the Maximum-Length-Shift-Register (MLSR) sequence. The length of MLSR sequence is PN 2 1 *<sup>m</sup> N* bits. And one of the possible generators of this sequence is an m-stage linear feedback shift register (see [14]). As a result, MLSR sequence is periodic with period n. Within each period, there are <sup>1</sup> 2*<sup>m</sup>*

One of the most important characteristics of the periodic PN sequence is its sharp auto-

0, Others

We can represent the received signal *<sup>k</sup> y* as the convolution of transmitted signal *<sup>k</sup> x* and the

*k nkn k*

The transmitter use PN sequence as the transmitting data *<sup>k</sup> x* . If the generators of PN sequence in both transmitter and receiver are synchronous, at each time slot, receiver is aware of the transmitting PN sequence. Hence, receiver can obtain the CIR by doing a cross-

PN PN PN

*Nm N N*

*k m k m iN*

, 0, , 2 ,

(6)

PN

(7)

(8)

is essentially nonzero is called the Doppler

The range of values of

channel capacity decrease.

ones and <sup>1</sup> 2*<sup>m</sup>* zeros.

CIR *<sup>k</sup> h* as:

spread of the channel.

to to fulfill the needs of indoor channel measurements.

**2.2 Channel measurement using PN sequence** 

correlation. Consider an PN sequence *<sup>k</sup> x* , we have :

*N*

PN 1

0

*k*

If *N*PN 1 , it is approximate that:

correlating between *<sup>k</sup> y* and *<sup>k</sup> x* :

*kmk*

PN

PN 0 1 *<sup>N</sup>*

1

*k i x x*

CIR 1

*N*

0

*n y hx w* 

*x x*

*N*

over which *S*( )

$$\begin{split} \hat{h}\_{k} &= \mathbf{N}\_{\text{PN}} \sum\_{m=0}^{N\_{\text{PN}}-1} \mathbf{y}\_{m+k} \mathbf{x}\_{m} \\ &= \frac{1}{N\_{\text{PN}}} \sum\_{m=0}^{N\_{\text{PN}}-1} \left( \sum\_{n=0}^{N\_{\text{CP}}-1} l\_{n} \mathbf{x}\_{m+k-n} + w\_{m+k} \right) \mathbf{x}\_{m} \\ &= \sum\_{n=0}^{N\_{\text{CP}}-1} h\_{n} \frac{1}{N\_{\text{PN}}} \sum\_{m=0}^{N\_{\text{PN}}-1} \mathbf{x}\_{m+k-n} \mathbf{x}\_{m} + \frac{1}{N\_{\text{PN}}} \sum\_{k=0}^{N\_{\text{PN}}-1} w\_{m+k} \mathbf{x}\_{m} \\ &= \sum\_{n=0}^{N\_{\text{CP}}-1} h\_{n} \sum\_{i=-\infty}^{+\infty} \delta\_{k-n-iN\_{\text{PN}}} + \frac{1}{N\_{\text{PN}}} \sum\_{k=0}^{N\_{\text{PN}}-1} w\_{m+k} \mathbf{x}\_{m} \\ &= \sum\_{i=-\infty}^{+\infty} l\_{k-iN\_{\text{PN}}} + w'\_{m} \end{split} \tag{9}$$

It can be seen that the result of cross-correlating is a summation of noise *w* and the periodic extension of *<sup>k</sup> <sup>h</sup>* . If *N N* CIR PN , a period of <sup>ˆ</sup> *<sup>k</sup> h* can be used as the estimate of *<sup>k</sup> h* [7].

There are two drawbacks in PN sequence measurement. First, the PN sequence takes up a great amount of time and frequency resources (most of the time, all the transmitting resources). This results in a great loss of throughput, as well as a channel mismatch caused by the delay between channel measurement and data transmission. Second, the accuracy of synchronizers in both sides should be in a high level, which raises the cost of equipments for channel measurement.

#### **2.3 Channel measurement using OFDM pilot**

Unlike measurement based on PN sequence, the pre-determined data of measurement based on OFDM pilot only occupied a relatively small percentage of time and frequency resources. Channel segments located on the pilots can be measured directly and correctly. However, other channel segments can only be estimated indirectly with some interpolations. Although the pilot-based measurement can only give an imperfect result, it provides a possibility of transmitting data and measuring channel simultaneously. This characteristic is crucial for real systems with limited feedback, such as WLAN, WiMax, LTE and LTE-A systems. There are two major problems in this scheme: how to design the pilot pattern and how to interpolate with discrete channel value on both time and frequency domain.

#### **2.3.1 OFDM pilot pattern**

OFDM pilots may be inserted in both time and frequency resources. A pilot pattern refers to the places where pilots are inserted in every OFDM symbol. An effective pilot pattern needs to be designed carefully in both frequency and time domains.

In frequency domain, according to the Nyquist sampling theorem, if we want to capture the variation of channel, the frequency space *Df* between pilots should be small enough:

$$D\_f \le \frac{1}{\tau\_{\text{max}} \Delta f} \tag{10}$$

where max represents the maximum delay of channel, and *f* denotes the frequency space between subcarriers.

If the above condition is not satisfied, the channel estimation cannot sample the accurate channel, since channel fading in frequency cannot be detected fast enough.

In time domain, spacing between pilots inserted in the same frequency is determined by the coherence time. In order to capture the variation of channel, the time space of pilots *Dt* must be correlated with coherence time, and must be small enough:

$$D\_t \le \frac{1}{2f\_{doppler}T\_f} \tag{11}$$

Where *doppler f* represents the maximum Doppler spread of channel, and *Tf* denotes the duration of each OFDM symbol.

However, it is worth pointing out that pilot allocation is a tradeoff of many factors in real systems. These include channel estimation accuracy, spectral efficiency of the system, wasted energy in unnecessary pilot symbols, and fading process not being sampled sufficiently. As a result, there is no optimal pilot pattern for all the channels, as fading process are varied.

Another important element of pilot pattern is the power allocation. Power is equally allocated to pilots and data symbols in regular cases. However, the accuracy of channel estimation can increase greatly with the power allocated to pilots. Considering the total power constraint, this will result in a decline of data symbols' SNR. Hence, another tradeoff between channel estimation and transmission capacity has to be evaluated.

There is a lack of pilots at the edges of OFDM symbols, which leads to a much higher error rate in such places. One simple but less effective way is to place more pilots at the edge. The drawback of this scheme is obvious: it reduces the frequency efficiency of systems. Some other scheme utilizes periodic behavior of the Fourier Transform, and establishes certain correlations between the beginning and the end of OFDM symbols. Simulations are reported to verify the effectiveness of this scheme.

#### **2.3.2 Measurements on pilots**

Channel segments locating on the pilots can be measured directly by some well-known algorithm, such as Least Square (LS) and Linear Minimum Mean Square Error (LMMSE).

Both LS and LMMSE algorithm aim to minimize a parameter: min{( - ) ( - )} *<sup>H</sup> k kk k kk y x h y x h* .

Using LS algorithm, we have:

$$\tilde{\mu}\_{\mathbf{k}}^{\rm LS} = \mathbf{x}\_{k}^{\rm 1} y\_{k} = h\_{k} + \mathbf{x}\_{k}^{\rm 1} w\_{k} = \mathbb{I}\frac{y\_{k}^{1}}{\mathbf{x}\_{k}^{1}} \Big/ \frac{y\_{k}^{2}}{\mathbf{x}\_{k}^{2}} \Big. \cdots \Big/ \frac{y\_{k}^{M\_{k}}}{\mathbf{x}\_{k}^{M\_{k}}} \Big\|^{T} \tag{12}$$

where *Mk* is the length of transmit and receive signal.

Using LMMSE algorithm, we have:

234 Wireless Communications and Networks – Recent Advances

If the above condition is not satisfied, the channel estimation cannot sample the accurate

In time domain, spacing between pilots inserted in the same frequency is determined by the coherence time. In order to capture the variation of channel, the time space of pilots *Dt*

1

*doppler f*

(11)

*k kk k kk y x h y x h* .

*f T*

Where *doppler f* represents the maximum Doppler spread of channel, and *Tf* denotes the

However, it is worth pointing out that pilot allocation is a tradeoff of many factors in real systems. These include channel estimation accuracy, spectral efficiency of the system, wasted energy in unnecessary pilot symbols, and fading process not being sampled sufficiently. As a result, there is no optimal pilot pattern for all the channels, as fading

Another important element of pilot pattern is the power allocation. Power is equally allocated to pilots and data symbols in regular cases. However, the accuracy of channel estimation can increase greatly with the power allocated to pilots. Considering the total power constraint, this will result in a decline of data symbols' SNR. Hence, another tradeoff

There is a lack of pilots at the edges of OFDM symbols, which leads to a much higher error rate in such places. One simple but less effective way is to place more pilots at the edge. The drawback of this scheme is obvious: it reduces the frequency efficiency of systems. Some other scheme utilizes periodic behavior of the Fourier Transform, and establishes certain correlations between the beginning and the end of OFDM symbols. Simulations are reported

Channel segments locating on the pilots can be measured directly by some well-known algorithm, such as Least Square (LS) and Linear Minimum Mean Square Error (LMMSE).

1 2

*k k k*

*x x x* (12)

*k k M k k k T*

Both LS and LMMSE algorithm aim to minimize a parameter: min{( - ) ( - )} *<sup>H</sup>*

<sup>k</sup> 1 2 = [ , , , ]

*y y <sup>y</sup> h xy h xw*

*kk k k k M*

LS -1 -1

where *Mk* is the length of transmit and receive signal.

2 *<sup>t</sup>*

*D*

between channel estimation and transmission capacity has to be evaluated.

channel, since channel fading in frequency cannot be detected fast enough.

must be correlated with coherence time, and must be small enough:

represents the maximum delay of channel, and *f* denotes the frequency space

where max 

between subcarriers.

duration of each OFDM symbol.

to verify the effectiveness of this scheme.

**2.3.2 Measurements on pilots**

Using LS algorithm, we have:

process are varied.

$$\tilde{h}\_k^{\text{LMMSE}} = \mathbf{R}\_{\text{H}\_k\text{H}\_k} (\mathbf{R}\_{\text{H}\_k\text{H}\_k} + \frac{\beta}{\text{SNR}} \mathbf{I})^{-1} \tilde{\mathbf{H}}\_k^{\text{LS}} \tag{13}$$

where k \* H H {H H } *<sup>k</sup>* **<sup>R</sup>** *<sup>E</sup> k k* autocorrelation of channel. and 2 2 *Ex E x k k* 1 / .

Comparing both LS algorithm and LMMSE algorithm, we can draw the following conclusion. LS algorithm is much easier to realize and apply. LS algorithm only needs one discrete divider to estimate the channels on all the pilots. Moreover, statistical information about channel and noise are not necessary while employing LS algorithm. However, the accuracy of LS algorithm is very sensitive to the noise and synchronization errors.

On the other hand, LMMSE algorithm can be seen as a filtering on the estimation result on LS algorithm. And this filtering is based on the MMSE criteria. It can be proven that, under the same MSE conditions, estimations results of LMMSE algorithm provide a larger gain than that of LS algorithm. Drawbacks of LMMSE algorithm are also obvious. Due to the inversion operation of matrices, complexity of LMMSE algorithm is relatively high. Furthermore, LMMSE algorithm requires knowing the statistical information of channel and noise in prior, which is unrealistic in applications.

While taking the errors of estimated H H*<sup>k</sup>* <sup>k</sup> **R** into account, the MSE and SNR gains provided

by LMMSE algorithm are marginally larger than that of LS algorithm. Considering the tradeoff between complexity and performance, LS algorithm may be a better solution than LMMSE algorithm.

### **2.3.3 Interpolations**

By applying LS or LMMSE algorithm, one can easily obtain the CIR of piloted channel segments. However, we still have no idea of channel segments not occupied with pilots. In order to obtain CIR of these segments, interpolations should be used.

The best interpolation algorithm may be 2-D Wiener filtering, since it can cancel noise as much as possible, in both frequency and time domain. However, in order to decide the weights of Wiener filtering, channel statistics must be known. Moreover, the complexity brought by matrix inversion increases gigantically with data in pilots. All of the above prevent the usage of 2-D Wiener filtering in real systems.

Some achievable interpolations include: cascade 1-D Wiener filtering, Lagrange interpolation, and transform domain interpolation.

Cascade 1-D Wiener filtering tries to realize a 2-D Wiener filtering by cascading two 1-D Wiener filtering. The complexity of cascade 1-D Wiener filtering is much less than 2-D Wiener filtering, while the performance only decreases a little bit. There two kinds of cascade 1-D Wiener filtering, in respects of interpolation order of frequency and time domain.

In frequency domain, the major interpolations include Lagrange interpolation, LMMSE interpolation, transform domain interpolation, DFT based interpolation, and low-pass filtering interpolation. In time domain, the available schemes are LMMSE interpolation, Lagrange interpolation.

#### **2.3.3.1 Two dimensions LMMSE interpolations**

Assume that the estimated channel matrix of all the piloted subcarriers is ', ' *H ni n i* , { ', '} P , where P is the set of positions of all the pilots, ' *n* is the index in frequency domain, ' *i* is the index in time domain. We have:

$$
\tilde{H}\_{n^{\circ},i^{\circ}} = \frac{Y\_{n^{\circ},i^{\circ}}}{X\_{n^{\circ},i^{\circ}}} = H\_{n^{\circ},i^{\circ}} + \frac{N\_{n^{\circ},i^{\circ}}}{X\_{n^{\circ},i^{\circ}}} \,\forall\{n^{\circ},i^{\circ}\} \in \mathcal{P} \tag{14}
$$

Then, estimate the channel parameters by two dimensions interpolation filtering:

$$
\hat{H}\_{n,i} = \sum\_{\{n^\*, i^\*\} \neq \Gamma\_{n,i}} w\_{n^\*, i^\*, n, i} \tilde{H}\_{n^\*, i^\*} \quad \Gamma\_{n, i} \subseteq \mathbb{P} \tag{15}
$$

Where *wn i ni* ', ', , are the weights of interpolation filter, , <sup>ˆ</sup> *Hn i* is the estimated channel, *n i*, is the number of used pilots. The number of weights in the filter is *Ntap n i*, .

Applying MSE criteria, the MSE *n i*, *J* in subcarrier ( ,) *n i* is:

$$\begin{aligned} \boldsymbol{\varepsilon}\_{n,i} &= \boldsymbol{H}\_{n,i} - \hat{\boldsymbol{H}}\_{n,i} \\ \boldsymbol{J}\_{n,i} &= \boldsymbol{E} \{ \left| \boldsymbol{\varepsilon}\_{n,i} \right|^2 \} \end{aligned} \tag{16}$$

A filter following the MMSE criteria is a two dimensions Wiener filter. According to the orthogonality of such a filter, we have:

$$E\left\{\varepsilon\_{n,i}\tilde{H}\_{n",i"}^\*\right\} = 0, \forall \{n^n, i^n\} \in \Gamma\_{n,i} \tag{17}$$

Where ( '', '') *n i* represents the positions of pilots while channel estimation is conducted.

The Wiener-Hopf equation can be derived from (17), which follows:

$$E\{H\_{n,i}\tilde{H}\_{n^\*,i^\*}^\*\} = \sum\_{\{n^\*,i^\*\} \in \Gamma\_{n,i}} w\_{n^\*,i^\*,n,i} E\{\tilde{H}\_{n^\*,i^\*} \tilde{H}\_{n^\*,i^\*}^\*\}, \forall \{n^\*,i^\*\} \in \Gamma\_{n,i} \tag{18}$$

Define the correlation function as:

$$\theta\_{n-n^\*,i-i^\*} = \mathcal{E}\left\{ H\_{n,i} \tilde{H}\_{n^\*,i^\*}^\* \right\} \tag{19}$$

If the mean value of noise is zero, and is independent to the transmission data, the correlation can be further expressed as:

$$\theta\_{n-n^\*,i-i^\*} = \mathbb{E}\left\{ H\_{n,i} \tilde{H}\_{n^\*,i^\*}^\* \right\} \tag{20}$$

Define the right part of (18) as the autocorrelation of channels at pilots:

$$\sigma \rho\_{n^\circ - n^\circ, i^\circ - i^\circ} = E\left\{ \tilde{H}\_{n^\circ, i^\circ} \tilde{H}\_{n^\circ, i^\circ}^\* \right\} \tag{21}$$

According to (14), it follows:

236 Wireless Communications and Networks – Recent Advances

Assume that the estimated channel matrix of all the piloted subcarriers is ', ' *H ni n i* , { ', '} P , where P is the set of positions of all the pilots, ' *n* is the index in frequency domain, ' *i* is the

', ' ', '

, { ', '} P *n i n i*

(14)

(15)

(17)

*n n i i ni n i* , ,, *EH H* (19)

*n n i i ni n i* , ,, *EH H* (20)

*n n i i ni n i* ' ,' *EH H*', ' , (21)

(18)

*Hn i* is the estimated channel, *n i*, is

(16)

', ' ', '

, ', ', , ', ' .

ˆ , P

,,, 2

A filter following the MMSE criteria is a two dimensions Wiener filter. According to the

, , , 0, { '', ''} *E H ni ni n i n i*

, , ', ', , ', ' , ,

\*

\*

\*

*ni n i n i ni n i n i n i*

If the mean value of noise is zero, and is independent to the transmission data, the

Where ( '', '') *n i* represents the positions of pilots while channel estimation is conducted.

{ } { }, { '', ''}

\* \*

*EH H w EH H n i*

*n i*

*H H*

{ } *ni ni ni*

ˆ

, ,

*ni ni*

*J E*

*n i n i Y N H H n i X X*

', ' ', '

.

the number of used pilots. The number of weights in the filter is *Ntap n i*, .

\*

The Wiener-Hopf equation can be derived from (17), which follows:

.

Define the right part of (18) as the autocorrelation of channels at pilots:

', '

*n i*

', '

*n i H wH* 

Where *wn i ni* ', ', , are the weights of interpolation filter, , <sup>ˆ</sup>

Applying MSE criteria, the MSE *n i*, *J* in subcarrier ( ,) *n i* is:

orthogonality of such a filter, we have:

Define the correlation function as:

correlation can be further expressed as:

Then, estimate the channel parameters by two dimensions interpolation filtering:

*n i n i n i ni n i ni*

*n i n i*

**2.3.3.1 Two dimensions LMMSE interpolations** 

index in time domain. We have:

$$\begin{split} \rho\_{n^{\circ}-n^{\circ},i^{\circ}-i^{\circ}} &= \theta\_{n^{\circ}-n^{\circ},i^{\circ}-i^{\circ}} + \frac{\sigma^{2}}{E\left\{ \left| \mathbf{X}\_{n^{\circ},i^{\circ}} \right|^{2} \right\}} \delta\_{n^{\circ}-n^{\circ},i^{\circ}-i^{\circ}} \\ &= \theta\_{n^{\circ}-n^{\circ},i^{\circ}-i^{\circ}} + \frac{1}{\text{SNR}} \delta\_{n^{\circ}-n^{\circ},i^{\circ}-i^{\circ}} \end{split} \tag{22}$$

Where <sup>2</sup> *E Xn i* ', ' is the average power of pilot symbols.

Equation (22) shows that the correlation function depends on the distance between the position of channel being estimated ( ,) *n i* and the positions of pilots employed in the estimation process ( '', '') *n i* . And the autocorrelation function depends on the distances between pilots.

Substituting (20) and (21) into (18), we have:

$$\boldsymbol{\Theta}\_{n,i}^{\mathrm{T}} = \mathbf{w}\_{n,i}^{\mathrm{T}} \boldsymbol{\Phi} \tag{23}$$

Where **Φ** is the *N N tap tap* autocorrelation matrix, *n i*, **θ** is the correlation vector with length *Ntap* , and **w***n i*, is the parameter vector of filter with length *Ntap* . Therefore, the parameter of the 2-D Wiener filter is:

$$\mathbf{w}\_{n,i}^{\mathrm{T}} = \boldsymbol{\Theta}\_{n,i}^{\mathrm{T}} \boldsymbol{\upPhi}^{-1} \tag{24}$$

The full estimated channel matrix can be expressed as:

$$
\hat{H}\_{n,i} = \mathbf{w}\_{n,i}^T \tilde{\mathbf{H}}\_{n,i} \tag{25}
$$

In conclusion, the design of such a filter is to decide its parameters **w***n i*, , which can be derived by the correlation function *nnii* , and average SNR. Unfortunately, the correlation of channel cannot be achieved accurately in real systems. Hence, approximate models with typical multipath delay profile ( ) and Doppler power spectrum ( ) *S f fD <sup>D</sup>* are employed.

2-D Wiener filtering is the optimal interpolation scheme in respect of MMSE; it can obtain optimal performance theoretically. However, its requirement of prior statistic knowledge of channel matrix, as well as the complexity of matrix inversion, makes it almost impossible to apply in real systems.

#### **2.3.3.2 Interpolations in frequency domain**

A 2-D interpolation can be form by a cascade of two 1-D interpolations. By appropriate designs, such a cascade can largely reduce the complexity while maintaining a good performance. The order of interpolation should be taken into consideration. We propose to interpolate firstly in frequency domain, then to conduct the time domain interpolation. The reason is that frequency-time scheme can start once the piloted OFDM symbols receive, while time-frequency scheme has to wait for the arrivals of all the symbols in one frame or subframe before the interpolation begins. As a result, frequency-time scheme can decrease the delay of channel measurement, hence provides more effective interpolation.

Interpolations in frequency domain aim to obtain all the channel function respond (CFR) <sup>C</sup> H , according to the measured CFR ˆ Hp in each piloted subcarrier:

$$
\hat{\mathbf{H}}\_{\text{C}} = \mathbf{w} \bullet \mathbf{\tilde{H}}\_{\text{p}} \tag{26}
$$

where **w** is the frequency domain interpolation matrix, and is the channel vector of piloted subcarriers obtained by (12) or (13). We have:

$$\mathbf{\tilde{H}\_p = \mathbf{X}\_{pp}^{-1} \mathbf{Y}\_p = \mathbf{H}\_p + \mathbf{X}\_{pp}^{-1} \mathbf{N}\_p = \mathbf{H}\_p + \mathbf{\tilde{n}} \tag{27}$$

#### *2.3.3.2.1 LMMSE interpolation*

Projecting equation (23) on frequency domain, we obtain the optimal interpolation parameter vector **w** as:

$$\mathbf{R}\_{\tilde{\mathbf{H}}\_{\mathrm{p}}\tilde{\mathbf{H}}\_{\mathrm{p}}}\mathbf{w}^\* = \mathbf{R}\_{\tilde{\mathbf{H}}\_{\mathrm{p}}\mathbf{H}\_{\mathrm{C}}} \tag{28}$$

where p p \* H H p p **<sup>R</sup>** *<sup>E</sup>*{H H } represents the autocorrelation matrix of estimation channel segments with pilots HP , p C \* H H P C **<sup>R</sup>** *<sup>E</sup>*{H H } denotes the correlation matrix of estimation channel segments with pilots HP and the real channel being interpolated HC .

If H Hp p **R** is invertible, **w** can be expressed as:

$$\mathbf{w} = \mathbf{R}\_{\mathbf{H}\_{\mathrm{C}}\hat{\mathbf{H}}\_{\mathrm{P}}} \mathbf{R}\_{\hat{\mathbf{H}}\_{\mathrm{P}}\hat{\mathbf{H}}\_{\mathrm{P}}}^{\mathrm{-1}} \tag{29}$$

Combining (12) and (27), it follows:

$$\begin{aligned} \mathbf{R}\_{\|\|} &= E\{\|\mathbf{\tilde{m}}^{\ast}\|\} = E\{\mathbf{X}\_{\mathrm{p}}^{-1}\mathbf{N}\_{\mathrm{p}} \cdot \mathbf{X}\_{\mathrm{p}}^{\ast}(\mathbf{X}\_{\mathrm{p}}^{-1})^{\ast}\} \\ &= \sigma\_n^2 E\{\mathbf{X}\_{\mathrm{p}}^{-1} \cdot \mathbf{I} \cdot (\mathbf{X}\_{\mathrm{p}}^{-1})^{\ast}\} = \sigma\_n^2 E\{(\mathbf{X}\_{\mathrm{p}}^{\ast}\mathbf{X}\_{\mathrm{p}})^{-1}\} \\ &= \mathbf{I}\_{N\_r} E\{\mathbf{1} \,/\left|\,\mathbf{X}\_{\mathrm{k}}^{\mathrm{p}}\right|^2\} \sigma\_n^2 \end{aligned} \tag{30}$$

where <sup>p</sup> *Xk* is the constellation point of piloted channel, and <sup>2</sup> *<sup>n</sup>* is the power of noise. Substituting (12) into (29), we have:

$$\mathbf{w} = \mathbf{R}\_{\mathrm{H}\_{\mathrm{C}}\mathrm{H}\_{\mathrm{p}}} \left( \mathbf{R}\_{\mathrm{H}\_{\mathrm{p}}\mathrm{H}\_{\mathrm{p}}} + \mathbf{I}\_{\mathrm{N}\_{\mathrm{p}}} E \{ 1 \;/ \left| X\_{k}^{\mathrm{p}} \right|^{2} \} \sigma\_{n}^{2} \right)^{-1} \tag{31}$$

where H HC p **R** and H Hp p **R** are the ideal correlation and autocorrelation matrices. Further representing the CFRs of channels with their CIRs, it is obvious that H h C CL L **= F** and H= h P PL L **F** (where hL is the discrete CIR, **F**CL and **F**PL are the corresponding DFT transform matrices). This converts (31) as followed:

subframe before the interpolation begins. As a result, frequency-time scheme can decrease

Interpolations in frequency domain aim to obtain all the channel function respond (CFR)

where **w** is the frequency domain interpolation matrix, and is the channel vector of piloted

Projecting equation (23) on frequency domain, we obtain the optimal interpolation

p p p C \*

\*

nn pp pp 2 1 1\* 2 \* 1 p p p p

**R X (X )**

2 p 2

*E E*

*n n*

Cp pp p

{1 / } *<sup>p</sup>*

*E X*

**I**

where <sup>p</sup> *Xk* is the constellation point of piloted channel, and <sup>2</sup>

*N n k*

*E E*

channel segments with pilots HP and the real channel being interpolated HC .

H H p p **<sup>R</sup>** *<sup>E</sup>*{H H } represents the autocorrelation matrix of estimation channel

Cp pp -1

\* 1 \* 1\*

 

HH HH ( { *N n* 1 / }) *E Xk*

representing the CFRs of channels with their CIRs, it is obvious that H h C CL L **= F** and H= h P PL L **F** (where hL is the discrete CIR, **F**CL and **F**PL are the corresponding DFT transform

**R** and H Hp p **R** are the ideal correlation and autocorrelation matrices. Further

**X I X XX**

{nn } { N N } { ( ) } {( ) }

C p H H <sup>ˆ</sup> **<sup>w</sup>** (26)


HH HH **R w =R** (28)

HH HH **wR R** (29)

(30)

*<sup>n</sup>* is the power of noise.

H H P C **<sup>R</sup>** *<sup>E</sup>*{H H } denotes the correlation matrix of estimation

2 p 2 1

**w R R +I** (31)

the delay of channel measurement, hence provides more effective interpolation.

<sup>ˆ</sup> Hp in each piloted subcarrier:

<sup>C</sup> H , according to the measured CFR

*2.3.3.2.1 LMMSE interpolation* 

segments with pilots HP , p C

Combining (12) and (27), it follows:

Substituting (12) into (29), we have:

matrices). This converts (31) as followed:

parameter vector **w** as:

where p p

where H HC p

If H Hp p

subcarriers obtained by (12) or (13). We have:

\*

**R** is invertible, **w** can be expressed as:

$$\mathbf{w} = \mathbf{F}\_{\text{CL}} \mathbf{R}\_{\text{h}\_{\text{L}} \text{h}\_{\text{L}}} \mathbf{F}\_{\text{PL}}^{\*} \{ \mathbf{F}\_{\text{PL}} \mathbf{R}\_{\text{h}\_{\text{L}} \text{h}\_{\text{L}}} \mathbf{F}\_{\text{PL}}^{\*} + \mathbf{I}\_{\text{N}\_{\text{p}}} \mathrm{E} \{ 1 \/ \left| X\_{\text{k}}^{\text{p}} \right|^{2} \} \sigma\_{n}^{2} \}^{-1} \tag{32}$$

Applying Parseval Theorem, which certifies that the power in frequency domain equals that of time domain, we draw the following conclusion:

L L L L p L L L L L L p L L L L p 2 p 2 \* \* 1 CL h h PL PL h h PL h h 2 p 2 \* \* 1 CL h h PL PL h h PL \* C C 2 2 p p 2 \* \* 1 CL h h PL PL h h PL 2 p \* C C {1 / } ( ) ( ) {1/ } ( ) (H H ) {1/ } { } ( ) { } (H H ) *k n N k n N k k n N k E X trace E X E E X EX EX E* **w FR F FR F I R FR F FR F I FR F FR F I** (33)

In special cases where QPSK modulation and equal power allocation are adopted, the interpolation is:

$$\mathbf{w} = \mathbf{F}\_{\text{CL}} \overline{\mathbf{R}}\_{\text{h}\_{\text{h}} \text{h}\_{\text{h}}} \mathbf{F}\_{\text{PL}}^{\*} (\mathbf{F}\_{\text{PL}} \overline{\mathbf{R}}\_{\text{h}\_{\text{h}} \text{h}\_{\text{h}}} \mathbf{F}\_{\text{PL}}^{\*} + \mathbf{I}\_{N\_{p}} \frac{1}{SNR})^{-1} \tag{34}$$
 
$$\text{where } \text{SNR} = \frac{P\_{r}}{\sigma\_{n}^{2}} = \frac{E\{\left|\mathbf{X}\_{k}\right|^{2}\} E\{\mathbf{H}\_{\text{C}}^{\*} \mathbf{H}\_{\text{C}}\}}{\sigma\_{n}^{2}} = \frac{E\{\left|\mathbf{X}\_{k}^{p}\right|^{2}\} E\{\mathbf{H}\_{\text{C}}^{\*} \mathbf{H}\_{\text{C}}\}}{\sigma\_{n}^{2}}.$$

#### *2.3.3.2.2 Lagrange interpolation*

Lagrange interpolation is widely used and easy to implement. It is a group of interpolation algorithms, including linear interpolation, Gaussian interpolation, cubic interpolation, etc. Lagrange interpolation is suitable for both frequency and time domain interpolation. However, the disadvantage of Lagrange interpolation is obvious. It is unable to cancel the noise.

Linear interpolation in frequency domain utilizes each pair of adjacent piloted channel segments to obtain the channel function within them. The interpolation process follows the following equation:

$$
\hat{H}(l+d) = \left(1 - \frac{d}{D}\right) \tilde{H}\_p(l) + \frac{d}{D} \tilde{H}\_p(l+D), 1 \le d \le D-1 \tag{35}
$$

where D is the interval between two adjacent pilots, ( ) *H l <sup>P</sup>* and ( ) *HlD <sup>P</sup>* are the corresponding channel estimation results.

Gaussian interpolation in frequency domain employs the measured channels of three adjacent pilots, which can be represented as followed:

$$
\hat{H}(\mathbf{x}) = \begin{cases}
\tilde{H}\_p(l\_{j-1}) \frac{\mathbf{x} - l\_j}{l\_{j-1} - l\_j} \frac{\mathbf{x} - l\_{j+1}}{l\_{j-1} - l\_{j+1}} + \tilde{H}\_p(l\_j) \frac{\mathbf{x} - l\_{j-1}}{l\_j - l\_{j-1}} \frac{\mathbf{x} - l\_{j+1}}{l\_j - l\_{j+1}} \\
+ \tilde{H}\_p(l\_{j+1}) \frac{\mathbf{x} - l\_{j-1}}{l\_{j+1} - l\_{j-1}} \frac{\mathbf{x} - l\_j}{l\_{j+1} - l\_j}, (l\_j \neq \mathbf{K}\_{\text{max}}) \\
\tilde{H}\_p(l\_{j-2}) \frac{\mathbf{x} - l\_{j-1}}{l\_{j-2} - l\_{j-1}} \frac{\mathbf{x} - l\_j}{l\_{j-2} - l\_j} + \tilde{H}\_p(l\_{j-1}) \frac{\mathbf{x} - l\_{j-2}}{l\_{j-1} - l\_{j-2}} \frac{\mathbf{x} - l\_j}{l\_{j-1} - l\_j} \\
+ \tilde{H}\_p(l\_j) \frac{\mathbf{x} - l\_{j-2}}{l\_j - l\_{j-2}} \frac{\mathbf{x} - l\_{j-1}}{l\_j - l\_{j-1}}, (l\_j = \mathbf{K}\_{\text{max}})
\end{cases} \tag{36}$$

Where *<sup>K</sup>*max denotes the maximum position of pilots, 1 ( ) *H l P j* , () *H l <sup>P</sup> <sup>j</sup>* , and 1 ( ) *H l P j* are the channel measurement results of three used pilots, and *j j* <sup>1</sup> *l xl* .

Cubic interpolation further increase the number of used pilots onto four. The expression for interpolation is showed as followed:

$$\begin{split} \hat{H}(\mathbf{x}) &= \tilde{H}\_{p}(l\_{j-2}) \frac{\mathbf{x} - l\_{j-1}}{l\_{j-2} - l\_{j-1}} \frac{\mathbf{x} - l\_{j}}{l\_{j-2} - l\_{j}} \frac{\mathbf{x} - l\_{j+1}}{l\_{j-2} - l\_{j+1}} + \tilde{H}\_{p}(l\_{j-1}) \frac{\mathbf{x} - l\_{j-2}}{l\_{j-1} - l\_{j-2}} \frac{\mathbf{x} - l\_{j}}{l\_{j-1} - l\_{j}} \frac{\mathbf{x} - l\_{j+1}}{l\_{j-1} - l\_{j+1}} \\ &+ \tilde{H}\_{p}(l\_{j}) \frac{\mathbf{x} - l\_{j-2}}{l\_{j} - l\_{j-2}} \frac{\mathbf{x} - l\_{j-1}}{l\_{j} - l\_{j-1}} \frac{\mathbf{x} - l\_{j+1}}{l\_{j} - l\_{j+1}} + \tilde{H}\_{p}(l\_{j+1}) \frac{\mathbf{x} - l\_{j-2}}{l\_{j+1} - l\_{j-2}} \frac{\mathbf{x} - l\_{j-1}}{l\_{j+1} - l\_{j-1}} \frac{\mathbf{x} - l\_{j}}{l\_{j+1} - l\_{j}} \\ & \left( l\_{j} \neq \mathcal{K}\_{\max}, l\_{j} \neq d, l\_{j-1} < \mathbf{x} < l\_{j} \right) \end{split} \tag{37}$$

All the above schemes are simple to apply in real systems. However, they all introduce certain level of noise, and yield effect of error floor. This can be eliminated by employing a low pass filter after interpolation.

#### *2.3.3.2.3 Transform domain interpolation*

The basic idea of transform domain interpolation is to reduce the complexity by conducting interpolation in various transform domains. The most widely used kind of transform domain interpolation is based on DFT.

The fundamental principle of DFT based interpolation is: in process of signal processing, zeroizing in time domain is equivalent to interpolating in frequency domain. If a sequence of *N* points has *N N* p zeros in the end, its Fourier transform values at the positions of multiples of *N*<sup>p</sup> are the same as the counterparts of Fourier transform of sequence formed by the former *N*<sup>p</sup> points. On the other hand, the Fourier transform values not at the positions of multiples of *N*<sup>p</sup> consist of linear combinations of the Fourier transform of truncated sequence.

After receiving the information of piloted channels, DFT based interpolation conducts IFFT of length *N*<sup>p</sup> . Then the interpolation zeroizes the transformed sequence into a *N* pointed sequence. Finally, transform the sequence into frequency domain by a *N* points FFT.

The zeroizing can be conducted as followed:

1 max

,( ) *j j j*

Where *<sup>K</sup>*max denotes the maximum position of pilots, 1 ( ) *H l P j* , () *H l <sup>P</sup> <sup>j</sup>* , and 1 ( ) *H l P j* are the

Cubic interpolation further increase the number of used pilots onto four. The expression for

( ) ,( )

*j j p j j j jj j*

2 1

( ) ( )

1

*xl xl H l l K l ll l*

( ) ( )

*p j p j*

*H l H l*

1 11

*p j p j*

2 1

*H l H l*

*jj jj xl xl*

2 1

211 2 1 1 211 1 21 1

*xl xl xl xl xl xl*

*jj jj jj j jj jj*

 

*ll llll l l l l l*

*jjj jjj*

All the above schemes are simple to apply in real systems. However, they all introduce certain level of noise, and yield effect of error floor. This can be eliminated by employing a

The basic idea of transform domain interpolation is to reduce the complexity by conducting interpolation in various transform domains. The most widely used kind of transform

The fundamental principle of DFT based interpolation is: in process of signal processing, zeroizing in time domain is equivalent to interpolating in frequency domain. If a sequence of *N* points has *N N* p zeros in the end, its Fourier transform values at the positions of multiples of *N*<sup>p</sup> are the same as the counterparts of Fourier transform of sequence formed by the former *N*<sup>p</sup> points. On the other hand, the Fourier transform values not at the positions of multiples of *N*<sup>p</sup> consist of linear combinations of the Fourier transform of

After receiving the information of piloted channels, DFT based interpolation conducts IFFT of length *N*<sup>p</sup> . Then the interpolation zeroizes the transformed sequence into a *N* pointed

sequence. Finally, transform the sequence into frequency domain by a *N* points FFT.

*ll ll* 

2 1

1

( )

channel measurement results of three used pilots, and *j j* <sup>1</sup> *l xl* .

ˆ () ( ) ( )

*Hx H l H l*

*p j p j*

( ) ( )

*p j p j*

*H l H l*

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

*l K l dl x l*

low pass filter after interpolation.

*2.3.3.2.3 Transform domain interpolation* 

domain interpolation is based on DFT.

The zeroizing can be conducted as followed:

truncated sequence.

*j jj j*

*p j*

*H l*

 

ˆ ( )

*H x*

interpolation is showed as followed:

1 1 1

(36)

(37)

1 11 1 1

*j jj j jj jj*

*l ll l l l l l*

*xl xl xl xl*

*j j j j*

1 2

*xl xl xl xl*

*j j j j*

2 12 1 21

*l K*

*l ll l l ll l*

*j jj j j jj j*

max

11 21

1

*j*

*l*

2 12 2 1 1 21 1 1

*j j j jj j j j j jj j*

*xl xl xl xl xl xl*

*l l l ll l l l l ll l*

*j jj j jj*

$$
\hat{h}\_N(i) = \begin{cases}
 \tilde{h}\_{N\_\mathrm{P}}(i) & 0 \le i < N\_\mathrm{P} \nmid 2 \\
 0 & N\_\mathrm{P} \nmid 2 \le i < N - N\_\mathrm{P} \nmid 2 \\
 \tilde{h}\_{N\_\mathrm{P}}(i - (N - N\_\mathrm{P})) & N - N\_\mathrm{P} \nmid 2 \le i \le N - 1
\end{cases}
\tag{38}
$$

Considering the influences of noise and inter-channel interference, introducing a low pass filtering before zeroizing can effective increase the accuracy of channel measurement. The block diagram of the above procedure is showed in Fig. 1.

Fig. 1. DFT and low-pass filter based interpolation.

Since the most complex calculations in this kind of interpolation are FFT and IFFT, the complexity of DFT based interpolation is much lower than others. However, the performance will drop largely, if the multipath spread is not a multiple of sampling period.

#### **2.3.3.3 Interpolations in time domain**

After frequency interpolation is done, we can launch the interpolation in time domain. The interpolation can also be expressed as a interpolation matrix as followed:

$$
\hat{\mathbf{H}}\_{\rm Ct} = \mathbf{w}\_{\rm t} \cdot \tilde{\mathbf{H}}\_{\rm pt} \tag{39}
$$

Where **w**<sup>t</sup> is the time domain interpolation matrix, Hpt denotes the CFR of channel segments on the pilots, Ct H represents the CFR for al <sup>ˆ</sup> l the channel segments. By assuming H =H +n pt pt , we consider the impact of AWGN in the interpolation.

#### *2.3.3.3.1 LMMSE interpolation*

LMMES interpolation in time domain is similar to that in frequency domain. The only difference is that we project equation (23) into time domain, so that it follows:

$$\mathbf{w}\_{\mathbf{t}} = \mathbf{R}\_{\mathbf{H}\_{\rm Ci}\tilde{\mathbf{H}}\_{\rm pt}} \mathbf{R}\_{\mathbf{\tilde{H}}\_{\rm pt}\tilde{\mathbf{H}}\_{\rm pt}}^{\mathbf{1}} \tag{40}$$

where pt pt \* H H pt pt **<sup>R</sup>** *<sup>E</sup>*{H H } represents the autocorrelation matrix of estimation channel segments with pilots HPt , Ct pt \* H H Ct pt **<sup>R</sup>** *<sup>E</sup>*{H H } denotes the correlation matrix of estimation channel segments with pilots HPt and the real channel being interpolated HCt .

Following the steps of derivation for frequency LMMSE interpolation, the interpolation matrix of time LMMSE interpolation can be simplified as below:

$$\mathbf{w}\_{\rm t} = \mathbf{R}\_{\rm H\_{\rm Cr}H\_{\rm pt}} \left( \mathbf{R}\_{\rm H\_{\rm pt}H\_{\rm pt}} + \frac{\beta}{SNR} \mathbf{I}\_{N\_{\rm pt}} \right)^{-1} \tag{41}$$

We consider a special case where QPSK modulation and average power allocation are used. Then the correlation between adjacent pilots *i* and '' *i* is:

$$R\_{\rm H\_{Cr}H\_{pt}}\left(i,i''\right) = f\_0(2\pi f\_{D\_{\rm max}}\left(i-i''\right)T\_s\left(\right) \tag{42}$$

Where *<sup>D</sup>*max *f* is the maximum Doppler spread, ' *Ts* is the interval between two symbols, and <sup>0</sup>*J x*( ) represents the first zero-order Bessel function.

However, the previous frequency interpolation and the corresponding filtering cause changes in the noise power of each subcarrier. Therefore, the signal-to-noise-ratio of each channel segment no longer equals the original value. An adjustment was proposed based on the MSE after the frequency interpolation.

Assume that channel frequency response after frequency LMMSE interpolation is:

$$\mathbf{H}\_f^n = \left[ H\_0^{(n)} + \tilde{w}\_0^{(n)}, \dots, H\_i^{(n)} + \tilde{w}\_i^{(n)}, \dots, H\_{L-1}^{(n)} + \tilde{w}\_{L-1}^{(n)} \right]^\mathrm{T} \tag{43}$$

Where *<sup>L</sup>* is the number of OFDM symbols in each frame, ( ) *<sup>n</sup> Hi* represents the channel frequency response of ith OFDM symbol in nth subcarrier, ( ) *<sup>n</sup> wi* denotes the corresponding residual noise. Hence, variance of ( ) *<sup>n</sup> wi* is equivalent to the MSE after interpolation.

$$MSE\_{\text{LMMSE},n} = \left[ \mathbf{R}\_{\text{H}\_{\text{C}}\text{H}\_{\text{C}}} - \mathbf{R}\_{\text{H}\_{\text{C}}\text{H}\_{\text{p}}} \left( \mathbf{R}\_{\text{H}\_{\text{p}}\text{H}\_{\text{p}}} + \mathbf{I}\_{\text{N}\_{\text{p}}} E \{ 1 \, \middle| \, \mathbf{X}\_{\text{k}}^{\text{p}} \right) \sigma\_{n}^{2} \right]^{-1} \mathbf{R}\_{\text{H}\_{\text{C}}\text{H}\_{\text{p}}}^{\text{H}} \}\_{\text{inv}} \\ n = 0, 1, 2, \cdots, N - 1 \tag{44}$$

Where N is the number of subcarriers waiting for measured.

As a result, time domain LMMSE interpolation should be optimized according to the variance of noise. The interpolation matrix is then:

$$\mathbf{w}\_{\rm t} = \mathbf{R}\_{\rm H\_{\rm Ci}\rm H\_{\rm pt}} \left( \mathbf{R}\_{\rm H\_{\rm pt}\rm H\_{\rm pt}} + \text{diag} (\sigma\_0^2 \cdots \prime, \sigma\_i^2 \cdots \prime, \sigma\_{\rm L-1}^2) \right)^{-1} \tag{45}$$

Where <sup>2</sup> *<sup>i</sup>* represents the variance of residual noise ( ) *<sup>n</sup> wi* .

To reduce the complexity, channel segments in the same OFDM symbol can utilize their average noise variance in the interpolation, and an approximate interpolation matrix can be expressed as followed:

$$\mathbf{w}\_{\rm t} = \mathbf{R}\_{\rm H\_{\rm Cl}\rm H\_{\rm pt}} \left( \mathbf{R}\_{\rm H\_{\rm pt}\rm H\_{\rm pt}} + \frac{1}{N} \sum\_{i=0}^{N-1} \sigma\_0^2(i) \mathbf{I}\_L \right)^{-1} \tag{46}$$

#### *2.3.3.3.2 Lagrange interpolations*

Lagrange interpolations in time domain are almost the same as those in frequency domain. The only difference is channel response of piloted channel segments. One can refer to previous sections for details.

#### **3. Applications**

Here we show some useful and easily implemented examples to illustrate the indoor channel measurement. Measurement based on PN sequence, as well as OFDM pilot, will be discussed.

We consider a special case where QPSK modulation and average power allocation are used.

H HCt pt max <sup>0</sup> *R ii J f i i T* ( , ) (2 ( ) ') 

Where *<sup>D</sup>*max *f* is the maximum Doppler spread, ' *Ts* is the interval between two symbols, and

However, the previous frequency interpolation and the corresponding filtering cause changes in the noise power of each subcarrier. Therefore, the signal-to-noise-ratio of each channel segment no longer equals the original value. An adjustment was proposed based on

> () () () () () () <sup>T</sup> H [ ,, ,, ] 0 0 1 1 *n nn nn n n*

Where *<sup>L</sup>* is the number of OFDM symbols in each frame, ( ) *<sup>n</sup> Hi* represents the channel frequency response of ith OFDM symbol in nth subcarrier, ( ) *<sup>n</sup> wi* denotes the corresponding

*<sup>f</sup> Hw Hw H w i i L L* (43)

2 p 21H

*<sup>n</sup> nn n N* **R R R +I <sup>R</sup>** (44)

 *i L* **wR R** (45)

1

0 <sup>1</sup> ( () ) *N*

*i*

*N*

2 2 21

2 1

*i*

**wR R <sup>I</sup>** (46)

*L*

Assume that channel frequency response after frequency LMMSE interpolation is:

residual noise. Hence, variance of ( ) *<sup>n</sup> wi* is equivalent to the MSE after interpolation.

CC Cp pp p C p

, HH HH HH [ ({1/ } ) ] H H , 0,1,2, , 1 *MSELMMSE n <sup>N</sup> E Xk*

As a result, time domain LMMSE interpolation should be optimized according to the

t HH HH <sup>0</sup> <sup>1</sup> ( diag( , , , , ))

To reduce the complexity, channel segments in the same OFDM symbol can utilize their average noise variance in the interpolation, and an approximate interpolation matrix can be

Lagrange interpolations in time domain are almost the same as those in frequency domain. The only difference is channel response of piloted channel segments. One can refer to

Here we show some useful and easily implemented examples to illustrate the indoor channel measurement. Measurement based on PN sequence, as well as OFDM pilot, will be discussed.

Ct pt pt pt

t HH HH 0

*D s* (42)

Then the correlation between adjacent pilots *i* and '' *i* is:

<sup>0</sup>*J x*( ) represents the first zero-order Bessel function.

Where N is the number of subcarriers waiting for measured.

Ct pt pt pt

*<sup>i</sup>* represents the variance of residual noise ( ) *<sup>n</sup> wi* .

variance of noise. The interpolation matrix is then:

Where <sup>2</sup> 

expressed as followed:

*2.3.3.3.2 Lagrange interpolations* 

previous sections for details.

**3. Applications** 

the MSE after the frequency interpolation.
