**3.2.2.1 Autocorrelation matrix** h hL L **R**

Since the real channels are time-varying, it is impossible to obtain the accurate autocorrelation of channels. The most widely used scheme is to estimate the approximate autocorrelation through some known channel models. It is well-known that two of the most important factors in wireless channel models are multipath spread and Doppler spread. While in the frequency domain, we mainly consider the influence of multipath spread, and propose a simple but useful construction scheme for wireless channels as followed.

The CIR of such a multipath channel is showed as followed:

$$\ln(\tau) = \sum\_{l=0}^{N\_l - 1} \mathbf{h}\_l \mathcal{S}(\tau - \tau\_l) \tag{48}$$

Where *<sup>l</sup>* and h*<sup>l</sup>* are the delay and amplitude of the *th l* path. *N*<sup>L</sup> denotes the max number of taps. represents the impulse function.

Define L {0,1 1} *N*<sup>L</sup> . Define <sup>1</sup> <sup>h</sup> *<sup>N</sup> <sup>C</sup>* , h 0, { 1} *<sup>l</sup>* <sup>L</sup> *lN N* as the multipath amplitude vector, N as subcarriers in each OFDM symbol.

Within digital baseband, we assume that the discrete delay as:

$$
\tau\_I = \frac{lT\_s}{N}, l \in \mathcal{L} \tag{49}
$$

Where is the length of an OFDM symbol.

Further assume that power <sup>2</sup> *<sup>l</sup>* of independent Rayleigh-distributed tap h*l* is fading exponentially with time constant *<sup>d</sup>* :

$$
\sigma\_l^2 \sim e^{-\frac{l}{\tau\_d}}, l \in \mathcal{L} \tag{50}
$$

Then the normalized CIR autocorrelation can be expressed as:

$$\overline{\mathbf{R}}\_{\mathbf{h}\_{\mathrm{L}}\mathbf{h}\_{\mathrm{L}}} = \frac{\mathbf{R}\_{\mathbf{h}\_{\mathrm{L}}\mathbf{h}\_{\mathrm{L}}}}{\left\| \mathbf{R}\_{\mathbf{h}\_{\mathrm{L}}\mathbf{h}\_{\mathrm{L}}} \right\|} = \frac{\operatorname{diag}(\sigma\_{0}^{2} \cdots \sigma\_{\mathbf{N}\_{\mathrm{L}} - 1}^{2})}{\left\| \mathbf{R}\_{\mathbf{h}\_{\mathrm{L}}\mathbf{h}\_{\mathrm{L}}} \right\|} = \frac{\operatorname{diag}(\sigma\_{0}^{2} \cdots \sigma\_{\mathbf{N}\_{\mathrm{L}} - 1}^{2})}{\sum\_{i} \sigma\_{i}^{2}} \tag{51}$$

Base on the above derivation, we need to determine *N*<sup>L</sup> and *<sup>d</sup>* to obtain h hL L **R** . The number of available taps *N*<sup>L</sup> can be same as the length of cyclic prefix (CP), with the purpose of simplification. Yet such a simplification is reasonable, since the multipath spread is less than the length of CP in most of the time. The multipath spread can be estimated with real-time scheme, so as to refine the channel model, as well as the autocorrelation h hL L **R** .

One of the possible schemes to estimate the multipath spread is provided as followed:


$$
\hat{\mathbf{h}}\_{\text{pow}}^{s} = \left| \hat{\mathbf{h}}\_{\text{s}} \right|^{2} \qquad \mathbf{s} = \mathbf{1}\_{\text{\textdegree L}} \mathbf{2}\_{\text{\textdegree L}} \cdots \mathbf{N}\_{\text{L}}^{\text{max}} \tag{52}
$$

Where pow <sup>ˆ</sup>*<sup>s</sup> <sup>h</sup>* denotes the square of amplitude for the s-th element in L ˆ h .

Then obtain a decision object *Ks* as followed;

$$K\_s = \frac{\left(\sum\_{j=s-9}^{s} \hat{h}\_{\text{pow}}^j\right) / \left(2 \times 10\right)}{\left(\sum\_{k=s+1}^{N\_\text{max}^\text{max}}\right) / \left(2 \times \left(N\_\text{L}^{\text{max}} - s\right)\right)}\tag{53}$$

**Step 3.** Find a value of s by the following procedure:

Decrease the value of s from max <sup>L</sup> *N* 15 to 1 with a step of 5. Take the first value of s that satisfies 2.55 *Ks* as the estimate multipath spread. One can refer to [15] for the reason of choosing 2.55 as the threshold.

After determining the multipath spread, one can obtain h hL L **R** by following previous derivation.

#### **3.2.2.2 Signal-to-noise-ratio**

SNR value may be measured or estimated in other blocks of receiver. If it is not, the following estimation scheme can be applied.

Denote <sup>L</sup> L L 1/2 <sup>h</sup> h h **R R** , PL PL hL **F R** . Do a singular value decomposition on PL , so that \* PL **USV** . Project estimated channel matrix Hp and real channel matrix H as p

\* U Hp and \* U Hp . The element in the project of real channel \* U Hp tends to zero when the singular value of PL is zero. But things are different in \* U Hp . Since we have \* \* -1 \* \* -1 U H U (H + N )=U H U N p p pp p p pp p **X X** , it is clear that when the last elements of \* U Hp are zeros, the corresponding elements of \* U Hp reflect the impact of noise. As a result, we can estimate the noise power by these elements.

Let ps p sp s { *NNN NN* 1},1 be the range of index, *N*<sup>p</sup> denotes the number of pilots, *N*<sup>s</sup> represents the number of zero singular value in PL . Then the estimated noise power is <sup>2</sup> \* .s p <sup>2</sup> <sup>s</sup> 1 *pn* H *N* **<sup>U</sup>** , and the corresponding signal power is 2 <sup>p</sup> <sup>2</sup> p 1 *ps n* <sup>H</sup> *<sup>p</sup> <sup>N</sup>* .

Assume that the SNR is constant within adjacent k pilots, then an average SNR can be obtain as followed:

$$\overline{SNR} = \frac{\sum\_{i=1}^{k} \tilde{p}\_{n\_i}}{\sum\_{i=1}^{k} \tilde{p}\_{s\_i}} \tag{54}$$

Since the last element in \* **U**.s p H rarely contains signal information, it is the most suitable one for SNR estimation. Therefore, we can simplify the process by setting <sup>s</sup> *N* 1 .

#### **3.2.2.3 Inversion of matrix**

250 Wireless Communications and Networks – Recent Advances

<sup>2</sup> ~ ,L *<sup>d</sup> l <sup>l</sup> e l* 

L L L L

hh 0 1 0 1

2 2 2 2

( )( ) *N N*

 *diag*

(50)

to obtain h hL L **R** . The number

h , and set max *N N* L p as

(53)

*i i*

ˆ

ˆ h .

*<sup>s</sup> h sN* (52)

<sup>L</sup> *N* 15 to 1 with a step of 5. Take the first value of s that

(51)

h h 2

of available taps *N*<sup>L</sup> can be same as the length of cyclic prefix (CP), with the purpose of simplification. Yet such a simplification is reasonable, since the multipath spread is less than the length of CP in most of the time. The multipath spread can be estimated with real-time

**Step 1.** Measure the channel matrix of piloted segments Hp in a symbol, with LS

pow <sup>L</sup> <sup>ˆ</sup> <sup>ˆ</sup> h 1,2, *<sup>s</sup>*

> pow 9

satisfies 2.55 *Ks* as the estimate multipath spread. One can refer to [15] for the reason of

After determining the multipath spread, one can obtain h hL L **R** by following previous

SNR value may be measured or estimated in other blocks of receiver. If it is not, the

<sup>h</sup> h h **R R** , PL PL hL **F R** . Do a singular value decomposition on PL , so that \* PL **USV** . Project estimated channel matrix Hp and real channel matrix H as p

*<sup>s</sup> <sup>j</sup>*

*h*

pow L

ˆ ( ) /(2 ( ))

*h Ns*

ˆ ( ) /(2 10)

<sup>2</sup> max

max

L L L L

scheme, so as to refine the channel model, as well as the autocorrelation h hL L **R** .

algorithm. Take a *N*<sup>p</sup> points IFFT to obtain the rough CIR <sup>L</sup>

<sup>ˆ</sup>*<sup>s</sup> <sup>h</sup>* denotes the square of amplitude for the s-th element in L

max L

*k s*

*<sup>s</sup> <sup>N</sup>*

*K*

**Step 3.** Find a value of s by the following procedure:

Decrease the value of s from max

choosing 2.55 as the threshold.

**3.2.2.2 Signal-to-noise-ratio** 

1/2

following estimation scheme can be applied.

1

*j s*

*k*

ˆ*<sup>s</sup> h* as:

One of the possible schemes to estimate the multipath spread is provided as followed:

*diag*

h h h h

**R R**

Then the normalized CIR autocorrelation can be expressed as:

Base on the above derivation, we need to determine *N*<sup>L</sup> and *<sup>d</sup>*

**R**

L L

the max length of multipath spread.

Then obtain a decision object *Ks* as followed;

**Step 2.** Define a parameter pow

Where pow

derivation.

Denote <sup>L</sup> L L

**R**

It is clear from equation (34) that in order to obtain the interpolation matrix **w** , a *N*p order matrix inversion operation must be conducted. The overhead will be very large. Fortunately, instead of the entire matrix, we only need several discrete h hL L **R** matrices. Therefore, if we apply discrete average SNR in equation (34), the parameter of interpolation matrix **w** will be discrete. We can pre-design the discrete range of **w** , and save it in a table. Then the real-time calculation is simplified as a looking up in a table, according to the measured h hL L **R** and SNR.

Specifically, we can adapt a look-up table which cuts the SNR range into several intervals. Each SNR interval combines with a corresponding multipath spreadˆ . Each of such pairs jointly determines a pre-designed **w** . With this scheme, the complexity of matrix inversion in real-time process is converted to the design of look-up table. Since the look-up table is generated off-line, real-time calculation burden for LMMSE interpolation is largely reduced.

#### **3.2.3 Design of time domain interpolation**

According to LTE standardization, each transmission time interval (TTI) is of length 1ms, which is the exact length of a subframe. Consequently, mobile stations process date in units of subframe. When time domain interpolation is conducting, there are at most four pilots in each subframe. As a result, the reference of time domain interpolation of LTE system is at most four estimated channel segments. Two of the most widely used schemes in time domain interpolation are LMMSE interpolation and linear interpolation. The detailed procedures of these two interpolations are presented in previous sections, so we only provide some simulation results to illustrate the advantages and disadvantages of each scheme.

The following simulation considers a unban macro scenario, in which the bandwidth is 10MHz, center frequency is 2GHz and noise is AWGN. Fig. 10 shows the MSE performances of both LMMSE and linear interpolations under different MS speeds.

Fig. 10. (a) MSE performances of LMMSE and linear interpolations under MS speed 1m/s.

Fig. 10. (b) MSE performances of LMMSE and linear interpolations under MS speed 30m/s.

The following conclusions can be inferred from the simulation results.

When the speed of MS is small, correspondingly small Doppler spread, LMMSE interpolation can save 4 dB SNR while achieving the same MSE performance of linear interpolation. However, when the speed (as well as the Doppler spread) of MS increases to a relatively high level, performances of LMMSE and linear schemes become very close. This means that the large overhead spent on LMMSE outputs marginal gains on the performance. When the errors of Doppler spread estimations are taken into account, the MSE performance of LMMSE scheme may even be worse than that of linear interpolation. Consequently, after considering the tradeoff between performance and complexity, we propose to use a simple linear interpolation in time domain.
