- Research
- Open Access

# Optimal pilot symbol power allocation under time-variant channels

- Michal Šimko
^{1}Email author, - Qi Wang
^{1}and - Markus Rupp
^{1}

**2012**:225

https://doi.org/10.1186/1687-1499-2012-225

© Simko et al.; licensee Springer. 2012

**Received:**15 February 2012**Accepted:**12 June 2012**Published:**20 July 2012

## Abstract

Nowadays, most wireless communication systems employ coherent demodulation on the receiver side. Under this circumstance, part of the available transmission resource is reserved and utilized for channel estimation, referred to as pilot symbols. In recent standards, for example long-term evolution (LTE), a certain adjustment is allowed for the power radiated on the pilot symbols. This additional degree of freedom creates space for a further optimization of the system performance. In this article, we consider an orthogonal frequency division multiplexing system and investigate how to distribute the available power between data symbols and pilot symbols under transmissions over time-variant channels so that the overall throughput is maximized. We choose the post-equalization signal-to-interference and noise ratio as the cost function and solve the problem analytically. Simulation results obtained by the Vienna LTE simulator are consistent with the analytical results. With an optimal power distribution between data and pilot symbols, a throughput increase of around 10% can be achieved compared to a system with evenly distributed power between data and pilot symbols.

## Keywords

- Mean Square Error
- Orthogonal Frequency Division Multiplex
- Channel Estimate
- Power Allocation
- Multiple Input Multiple Output

## Introduction

Nowadays, most wireless communication systems are based on orthogonal frequency division multiplexing (OFDM), for example worldwide inter-operability for microwave access (WiMAX) and long-term evolution (LTE). At the receiver side, coherent detection is employed where channel estimation is required. For the purpose of channel estimation, known symbols are inserted into the transmitted data stream. These so-called pilot symbols consume available resources like bandwidth and power. Some standards, e.g., LTE, allow to assign different power levels to the data and pilot subcarriers, which makes room for system optimization. A power increase at the pilot subcarriers results in a more reliable channel estimate [1] which implies higher throughput; however, the power available for the data subcarriers is decreased given a constant sum power constraint. Therefore, it is necessary to find an optimal power allocation between the pilot and data subcarriers which delivers a maximized system performance.

- 1.
In Section “Channel estimation” ahead, we show that the channel estimation error becomes saturated with the increasing Doppler spread. Therefore, a power boost at the pilots does not necessary lead to a better channel estimate.

- 2.
A high Doppler spread destroys the desired orthogonality between subcarriers. The system performance becomes limited by the simple and low-cost channel estimation based on individual subcarriers. Therefore, a more sophisticated channel estimation scheme is required.

### Related study

In order to optimize the pilot symbol power allocation under time-varying channels, a model that takes into account the pilot power adjustment, receiver structure, and channel estimation error at the same time, is needed. It has been shown by simulation that the channel capacity strongly depends on the power that is assigned to the pilot symbols [2]. Authors of [3] showed the impact of different power allocations on the bit error ratio (BER). However, their analysis was based on the signal-to-noise ratio (SNR); only an approximation of the impact of imperfect channel knowledge on BER was provided for a simple binary phase-shift keying modulation. In [4], an optimal pilot symbol allocation was derived analytically for phase-shift keying modulation of order two and four, using BER as the optimization criterion. In [5], the optimal pilot symbol power in multiple input multiple output (MIMO) systems was derived based on a lower bound for capacity. Authors of [6] investigated power allocations between pilot and data symbols for MIMO systems using the post-equalization signal-to-interference and noise ratio (SINR) as an optimization function. However, they only approximated the SINR expression; only a linear minimum mean square error (LMMSE) channel estimator was considered.

In [1], the authors derived the optimal power distribution between pilot and data symbols for time-invariant channels under imperfect channel knowledge. The optimal distribution of power turned out to be independent of the SNR and channel realizations. In [7], this study was extended to multi eNodeB scenarios where the interference from neighboring eNodeBs was included. Due to the LTE pilot symbol design, the pilot symbols from neighboring eNodeBs are overlapping with the data symbols in the eNodeB of interest, which complicates the optimization problem.

### Contribution

In this article, we consider zero forcing (ZF) equalizers under imperfect channel knowledge in a time-variant scenario and develop an analytical model for the post-equalization SINR. In order to answer the question, how to distribute the available power between data and pilot symbols, we choose the post-equalization SINR as the cost function which implies a maximization of the system throughput. Contributions of this article are:

● We deliver an optimal pilot symbol power adjustment in MIMO OFDM systems under time-variant channels.

● A post-equalization SINR expression is derived for a ZF receiver under realistic, imperfect channel knowledge.

● The analytical derivation of mean squared error (MSE) performance is provided for least squares (LS) channel estimators utilizing a two-dimensional linear interpolation in the time-frequency grid.

● Simulation results with an LTE compliant simulator [8, 9] confirm our theoretical analysis.^{
a
}

The remainder of the article is organized as follows. In the following section, we introduce a mathematical model for MIMO OFDM transmissions. In Section “Post-equalization SINR”, we derive the post-equalization SINR expression for ZF equalizers with imperfect channel knowledge. The channel estimators that are involved in this study as well as their MSEs are briefly discussed in Section “Channel estimation”. In Section “Power allocation”, we formulate the optimization problem for optimal pilot symbol power allocation. Finally, we present LTE simulation results in Section “Simulation results” and conclude the article in Section “Conclusion”.

## Transmission model

**Most important variables**

Variable | Dimension | Description |
---|---|---|

${\stackrel{~}{\mathbf{y}}}_{{n}_{\text{r}}}$ | $\left(\right)close="">{\mathbb{C}}^{{N}_{\text{sub}}\times 1}$ | Received symbol vector at antenna |

${\stackrel{~}{\mathbf{H}}}_{{n}_{\text{r}},{n}_{\text{t}}}$ | $\left(\right)close="">{\mathbb{C}}^{{N}_{\text{sub}}\times {N}_{\text{sub}}}$ | Channel matrix between antennas |

$\left(\right)close="">{\text{x}}_{{n}_{\text{t}}}$ | $\left(\right)close="">{\mathbb{C}}^{{N}_{\text{sub}}\times 1}$ | Transmit symbol vector at antenna |

$\left(\right)close="">{\mathbf{x}}_{\text{p},{n}_{\text{t}}}$ | $\left(\right)close="">{\mathbb{P}}^{{N}_{\text{p}}\times 1}$ | Pilot symbols vector at antenna |

$\left(\right)close="">{\mathbf{x}}_{\text{d},{n}_{\text{t}}}$ | $\left(\right)close="">{\mathbb{C}}^{{N}_{\text{d}}\times 1}$ | Transmit data vector at antenna |

${\stackrel{~}{\mathbf{n}}}_{{n}_{\text{r}}}$ | $\left(\right)close="">{\mathbb{C}}^{{N}_{\text{sub}}\times 1}$ | Additive noise at antenna |

| $\left(\right)close="">{\mathbb{C}}^{{N}_{\text{r}}\times 1}$ | Received symbol vector at subcarrier |

| $\left(\right)close="">{\mathbb{C}}^{{N}_{\text{r}}\times {N}_{\text{t}}}$ | Channel matrix between subcarriers |

| $\left(\right)close="">{\mathbb{C}}^{{N}_{\text{t}}\times {N}_{\text{l}}}$ | Precoding matrix at subcarrier |

| $\left(\right)close="">{\mathbb{C}}^{{N}_{\text{r}}\times {N}_{\text{l}}}$ | Effective channel matrix |

| $\left(\right)close="">{\mathbb{D}}^{{N}_{\text{l}}\times 1}$ | Transmit data at subcarrier |

| $\left(\right)close="">{\mathbb{C}}^{{N}_{\text{r}}\times 1}$ | Additive noise at subcarrier |

| $\left(\right)close="">{\mathbb{C}}^{{N}_{\text{r}}\times {N}_{\text{t}}}$ | Channel estimation error at subcarrier |

$\left(\right)close="">{\sigma}_{s}^{2}$ | $\mathbb{R}$ | Transmit power of one layer |

$\left(\right)close="">{\sigma}_{\text{p}}^{2}$ | $\mathbb{R}$ | Transmit pilot symbol power |

$\left(\right)close="">{\sigma}_{x}^{2}$ | $\mathbb{R}$ | Transmit data power |

$\left(\right)close="">{\sigma}_{\text{ICI}}^{2}$ | $\mathbb{R}$ | ICI power |

$\left(\right)close="">{\sigma}_{n}^{2}$ | $\mathbb{R}$ | Noise power |

$\left(\right)close="">{\sigma}_{e}^{2}$ | $\mathbb{R}$ | Channel estimation error variance |

| $\mathbb{R}$ | Channel saturation coefficient |

| $\mathbb{R}$ | Carrier frequency |

| $\mathbb{R}$ | Maximal user velocity |

| $\mathbb{R}$ | Maximal Doppler frequency |

| $\mathbb{R}$ | OFDM symbol duration |

| $\mathbb{N}$ | Number of pilot symbols |

| $\mathbb{N}$ | Number of data symbols |

| $\mathbb{N}$ | Number of layers |

| $\mathbb{N}$ | Number of transmit antennas |

| $\mathbb{N}$ | Number of receive antennas |

| $\mathbb{N}$ | Number of subcarriers |

| $\mathbb{R}$ | Offset between power of pilot symbols and data symbols |

| $\mathbb{R}$ | Post-equalization SINR at layer |

*N*

_{s}=7 OFDM symbols; with the extended cyclic prefix length,

*N*

_{s}=6. In LTE, the subcarrier spacing is fixed to 15 kHz. Twelve adjacent subcarriers in one slot are grouped into a so-called resource block. The number of resource blocks in an LTE slot ranges from 6 up to 100, corresponding to a bandwidth from 1.4 up to 20 MHz. Figure 1 displays the pilot symbol pattern in LTE. The colored squares correspond to pilot symbols. If there is a pilot symbol inserted on a specific position in the time-frequency grid at one transmit antenna, the remaining antennas on that position remain silent. In Figure 1, crosses correspond to these silent positions. Such pilot symbol pattern allows to estimate a MIMO channel as multiple, individual single input single output (SISO) channels as long as the spatial correlation is neglected.

*n*

_{r}th receive antenna can be written as

where ${\stackrel{~}{\mathbf{H}}}_{{n}_{\mathrm{t}},{n}_{\mathrm{r}}}\in {\mathbb{C}}^{{N}_{\mathrm{sub}}\times {N}_{\mathrm{sub}}}$ represents the channel matrix in the frequency domain between the *n*_{t}th transmit and *n*_{r}th receive antennas. The transmitted signal vector is referred to as $\left(\right)close="">{\mathbf{x}}_{{n}_{\mathrm{t}}}$, the received signal vector as ${\stackrel{~}{\mathbf{y}}}_{{n}_{\mathrm{r}}}$. The vector ${\stackrel{~}{\mathbf{n}}}_{{n}_{\mathrm{r}}}\in {\mathbb{C}}^{{N}_{\mathrm{sub}}\times 1}$ is additive white zero mean Gaussian noise with variance $\left(\right)close="">{\sigma}_{n}^{2}$ on antenna *n*_{r}. In case of a time-invariant channel, the channel matrix ${\stackrel{~}{\mathbf{H}}}_{{n}_{\mathrm{t}},{n}_{\mathrm{r}}}$ appears as a diagonal matrix, whereas a time-variant channel forces the channel matrix ${\stackrel{~}{\mathbf{H}}}_{{n}_{\mathrm{t}},{n}_{\mathrm{r}}}$ to become non-diagonal. These non-diagonal elements indicate that the subcarriers are not orthogonal anymore, leading to the so-called intercarrier interference (ICI).

*n*

_{t}th transmit antenna placed by a suitable permutation matrix

**P**

*N*

_{sub}entries, corresponding to the number of non-zero subcarriers. Let us denote the number of pilot symbols and the number of precoded data symbols by

*N*

_{p}and

*N*

_{d}, respectively. On subcarrier

*k*of the data symbol vector $\left(\right)close="">{\mathbf{x}}_{\mathrm{d},{n}_{\mathrm{t}}}$, the precoding process can be described as

where $\left(\right)close="">{x}_{\mathrm{d},{n}_{\mathrm{t}},k}$ is a precoded data symbol at the *n*_{t}th transmit antenna port and the *k* th subcarrier, $\left(\right)close="">{\mathbf{W}}_{k}\in {\mathbb{C}}^{{N}_{\mathrm{t}}\times {N}_{\mathrm{l}}}$ is a unitary precoding matrix at the *k* th subcarrier and $\left(\right)close="">{s}_{{n}_{\mathrm{l}},k}\in {\mathbb{D}}^{1\times 1}$ is the data symbol of the *n*_{l}th layer at the *k* th subcarrier. Here, $\mathbb{D}$ is the set of available modulation alphabets. In LTE, three different sets can be used, namely 4 quadrature amplitude modulation (QAM), 16 QAM, and 64 QAM. In order to obtain data symbol vectors $\left(\right)close="">{\mathbf{x}}_{\mathrm{d},{n}_{\mathrm{t}}}$, one has to stack data symbols $\left(\right)close="">{x}_{\mathrm{d},{n}_{\mathrm{t}},k}$ obtained via Equation (3) at a specific antenna *n*_{t}into a vector.

*k*th and

*m*th subcarriers. The MIMO channel matrix

**H**

_{k,m}contains appropriately ordered elements of matrices ${\stackrel{~}{\mathbf{H}}}_{{n}_{\mathrm{t}},{n}_{\mathrm{r}}}$ located in the

*k*th row and

*m*th column. Matrix

**W**

_{ k }is a unitary precoding matrix of size

*N*

_{t}×

*N*

_{l}. In LTE, the precoding matrix can be chosen from a finite set of precoding matrices [10]. The vector

**s**

_{ k }consists of the data symbols of all layers at the

*k*th subcarrier. Vector

**n**

_{ k }represents additive white zero mean Gaussian noise with variance $\left(\right)close="">{\sigma}_{n}^{2}$ at subcarrier

*k*. We denote the effective channel matrix by

Furthermore, the average power transmitted on each of the *N*_{
l
} layers is denoted by $\left(\right)close="">{\sigma}_{s}^{2}$. The total power transmitted on one data position is $\left(\right)close="">{\sigma}_{\mathrm{d}}^{2}$, while that on one pilot position is $\left(\right)close="">{\sigma}_{\mathrm{p}}^{2}$.

### Example

where *N*_{d} is the number of data symbols and *N*_{p}the number of pilot symbols.

## Post-equalization SINR

In this section, we consider a time-variant scenario and derive an analytical expression for the post-equalization SINR of a MIMO system using a ZF equalizer based on imperfect channel knowledge.

**s**

_{ k }is given as

*l*th layer given as [11]

**e**

_{ l }is an

*N*

_{l}×1 zero vector with a one on the

*l*th element. This vector extracts the signal on the corresponding layer after the equalizer. The variable $\left(\right)close="">{\sigma}_{\mathrm{ICI}}^{2}$ represents the ICI power, that is given as

_{0}(·) denotes the zeroth-order Bessel function, representing the channel time autocorrelation function. Variable

*f*

_{ d }represents the maximal Doppler frequency and

*T*

_{ s }the OFDM symbol duration. The maximal Doppler frequency can be obtained by the following expression

*v*

_{ max }is the maximal user velocity,

*f*

_{ c }the carrier frequency, and

*c*

_{0}the speed of light.Let us proceed to the case of imperfect channel knowledge. We define the perfect channel as the channel estimate plus the error matrix due to the imperfect channel estimation

**E**

_{k,k}are random variables, statistically independent of each other, each with variance $\left(\right)close="">{\sigma}_{e}^{2}$. Inserting Equation (14) in Equation (4), the input–output relation changes to

**E**

_{k,k}is unknown at the receiver, the ZF solution is given again by Equation (9), but channel matrix

**H**

_{k,k}is replaced by its estimate ${\widehat{\mathbf{H}}}_{k,k}$, which is known at the receiver

*l*th layer

Note that in practice variables $\left(\right)close="">{\sigma}_{s}^{2},{\sigma}_{\mathrm{d}}^{2}$, $\left(\right)close="">{\sigma}_{\mathrm{ICI}}^{2}$, and $\left(\right)close="">{\sigma}_{n}^{2}$ need to be replaced by their estimates.

## Channel estimation

In this section, we present state-of-the-art channel estimators and derive analytical expressions for their MSE. Due to the orthogonal pilot symbol pattern utilized in LTE, the MIMO channel can be estimated as *N*_{t}*N*_{r}individual SISO channels. To ease the reading, we thus simplify the notation in the following section and omit the antenna indices.

### LS channel estimation

The channel estimates at the data positions have to be obtained using a two-dimensional interpolation. In this study, we restrict ourselves to a two-dimensional linear interpolation. At each data position, the three closest pilot symbols are located and the channel estimate is obtained by spanning a plane defined by the channel estimate at the closest pilot symbol’s positions.

^{ b }Let us denote the channel estimate at the

*j*th data position by ${\u0125}_{\mathrm{d},j}$ and the channel estimate at the

*i*th pilot position by ${\u0125}_{\mathrm{p},i}$. Note that we use

*i*th and

*j*th positions as general description of location in the time-frequency grid:

where ${\mathcal{P}}_{j}$ denotes a set of the three nearest pilot symbol positions to the data position *j*, that span a plane. The weight *w*_{j,i} is a real number which implies how much the channel estimate at the *j* th data position is influenced by the channel estimate at the *i* th pilot position. The weight *w*_{j,i}depends on the distance of the symbols in the time-frequency grid. The sum over *w*_{j,i} over $i\in {\mathcal{P}}_{j}$ is one, namely $\sum _{i\in {\mathcal{P}}_{j}}{w}_{j,i}=1$. Note that due to the linear interpolation/extrapolation by a plane, some weight can become negative.

It the following, we discuss how to obtain the weighting factor *w*_{j,i} of Equation (22). First of all, we define a vector **p**_{
i
}, whose entries are pilot positions of the *i* th pilot in the time-frequency grid, namely $\left(\right)close="">{\mathbf{p}}_{i}={\left[{f}_{i},{t}_{i}\right]}^{\mathrm{T}},i=1,2,\dots ,{N}_{\mathrm{p}}$, where the scalar *f*_{
i
} is a frequency index and *t*_{
i
} a time index. Similarly, we denote the position of the *j* th data symbol in the time-frequency grid by a vector **d**_{
j
}.

#### Example

*h*

_{d,1}is located within the plane spanned by the channel estimates at the first three pilot symbols,

*h*

_{p,1},

*h*

_{p,2}, and

*h*

_{p,3}. The plane spanned by these three channel estimates is defined as follows

*a*and

*b*are real scalars. For a general solution, Equation (23) can be reformulated as

Comparing Equation (22) with Equation (24), using a linear interpolation, we recognize that *w*_{1,1}=1−*a*−*b*, *w*_{1,2}=*a*, and *w*_{1,3}=*b*.

The last step can be justified by the fact that the channel estimate at the pilot position can be represented as the true channel superimposed by some estimation error, and furthermore that this estimation error is uncorrelated with the channel value at the data position. The coefficient $\left(\right)close="">{R}_{j,i}=\mathbb{E}\left\{{h}_{\mathrm{d},j}^{\ast}{h}_{\mathrm{p},i}\right\}$ denotes the correlation between the channels at the *j* th data symbol and the *i* th pilot symbol positions.

*n*

_{est,i}is equal to $\left(\right)close="">{\sigma}_{n}^{2}+{\sigma}_{\mathrm{ICI}}^{2}$. We obtain

*n*

_{est,i}is statistically independent of the channel coefficients. To summarize, we obtain from Equation (25)

*w*

_{j,i}and the autocorrelation matrix. The dependency of the autocorrelation matrix can also be regarded as dependency on the maximum Doppler spread (or equivalently the user velocity). The second part of the MSE depends on $\left(\right)close="">{\sigma}_{n}^{2}+{\sigma}_{\mathrm{ICI}}^{2}$. This term is on the other hand independent of the autocorrelation matrix. This can be stated analytically as follows

*c*

_{ e }is a scalar depending on the weights

*w*

_{j,i}, given as

*c*

_{ e }is obtained as arithmetical average over all data symbol positions. Its depends only on the pilot symbols pattern. In case of a two-dimensional linear interpolation with LTE pilot symbol pattern using an LS channel estimator, we find

*c*

_{ e }=0.6623 evaluating Equation (30). The constant

*d*is a scalar depending on the weights

*w*

_{j,i}as well as the autocorrelation matrix. In the following text, it is referred to as channel saturation

*d*, given as

*d*over user velocity for different antenna configurations. Its value grows with increasing user velocity. Note that the channel saturation coefficient

*d*in Equation (29) causes the error floor in the MSE of the LS channel estimator. This saturation originates from a low correlation over time. Figure 4 shows the simulated and theoretical MSE of an LS channel estimator. An excellent agreement can be found.

### LMMSE channel estimation

where the matrix $\left(\right)close="">{\mathbf{R}}_{{\mathbf{h}}_{\mathrm{p}},{\mathbf{h}}_{\mathrm{p}}}=\mathbb{E}\left\{{\mathbf{h}}_{\mathrm{p}}{\mathbf{h}}_{\mathrm{p}}^{\mathrm{H}}\right\}\in {\mathbb{C}}^{{N}_{\mathrm{p}}\times {N}_{\mathrm{p}}}$ denotes the channel autocorrelation matrix at the pilot symbols, and the matrix $\left(\right)close="">{\mathbf{R}}_{{\mathbf{h}}_{\mathrm{d}},{\mathbf{h}}_{\mathrm{p}}}=\mathbb{E}\left\{{\mathbf{h}}_{\mathrm{d}}{\mathbf{h}}_{\mathrm{p}}^{\mathrm{H}}\right\}\in {\mathbb{C}}^{{N}_{\mathrm{d}}\times {N}_{\mathrm{p}}}$ is the channel cross-correlation matrix.

*N*

_{d}is the number of data symbols. Figure 5 depicts the simulated (solid line) and the analytical (dashed line) MSE of an LMMSE channel estimator at different user velocities. The simulated MSE and the analytically derived MSE show nearly perfect identity. Considering this figure, it may be believed that a similar decomposition is possible as in the case of an LS channel estimator. However, this is not the case. The MSE of an LMMSE channel estimator can only be approximated by

*c*

_{ e }and the channel saturation coefficient

*d*can be obtained by the following method. Let us assume that there is no noise and insert $\left(\right)close="">{\sigma}_{n}^{2}=0$ and $\left(\right)close="">{\sigma}_{\mathrm{ICI}}^{2}=0$ into Equation (36), providing us with an MSE value for infinitely high SNR, i.e., in the point of the MSE saturation. Such saturation is expressed by the value of

*d*. Figure 3 shows the value of the channel saturation coefficient

*d*over user velocity for 1×1 and 4×4 systems for LTE pilot symbol pattern. In case of a 2×2 system using the pilot symbol pattern defined in LTE, the channel saturation coefficient

*d*is the same as in a 1×1 case, since the amount of the pilot symbols is the same and an LMMSE estimator performs identically. The value of

*c*

_{ e }can be obtained by calculating the true MSE and solving $\left(\right)close="">{\sigma}_{e}^{2}={c}_{e}\left({\sigma}_{n}^{2}+{\sigma}_{\mathrm{ICI}}^{2}\right)+d$.

## Power allocation

In this section, we analytically derive an optimal power distribution among pilot and data symbols in high velocity scenarios. As a cost function, we choose the post-equalization SINR in Equation (19). Although the provided results are shown in the context of the current LTE standard, the presented concept can be applied to any MIMO OFDM-based system.

*N*

_{p}and

*N*

_{d}represent the number of pilot symbols and data symbols in a subframe, respectively. In order to describe the interconnection between $\left(\right)close="">{c}_{\mathrm{p}}^{2}$ and $\left(\right)close="">{c}_{\mathrm{d}}^{2}$, we introduce a variable

*p*

_{off}which is the power offset between the power of the pilot symbols and the data symbols, denoted by

*p*

_{off}:

*l*with adjusted power of the pilot symbols

*p*

_{off}:

*f*(

*p*

_{off}) is given as

*d*and is given as

Note that Equation (45) is independent of channel realization, noise variance, ICI power and even user velocity. It is the same as in [1], only the constant *c*_{
e
}has a different value due to the distinct performance of the channel estimators. Let us focus on the term $\stackrel{~}{d}$ in Equation (44). This term is always positive. It thus becomes obvious that it causes the overall limitation of the post-equalization SINR. Even if the power allocation function would be very small, the expression in the brackets of the denominator in Equation (44) will not be smaller than the value of $\stackrel{~}{d}$. This term is dependent on the noise variance, ICI power and the factor *d*, which is dependent on the user velocity. This term not only causes limitations in terms of post-equalization SINR, but when it becomes larger than the power allocation function, it also causes the post-equalization to be less sensitive to the variable *p*_{off}.

Before, we find the minimum of the power allocation function, let us discuss the expected solution. Intuitively, we would expect that once we have reached the saturation in the MSE of a channel estimator, it does not pay off to increase the power radiated at the pilot symbols, since the MSE will not become better. However, consider for a moment Equation (43), especially the term ${\stackrel{~}{\sigma}}_{e}^{2}{\sigma}_{\mathrm{d}}^{2}{c}_{\mathrm{d}}^{2}$ that corresponds to the interlayer interference caused by the channel estimation error. By increasing the power radiated at the data symbols, the interlayer interference is also increased. Therefore, even if the saturation of a channel estimator is reached, it might not be beneficial to decrease the power radiated at the pilot symbols [15].

*p*

_{off,opt}that maximizes the post-equalization SINR while keeping the overall transmit power constant

*f*(

*p*

_{off}) in the denominator of Equation (44) has to be minimized. The minimum of the power allocation function

*f*(

*p*

_{off}) can be found by simple differentiation, resulting in

*N*

_{d},

*N*

_{p}, and

*c*

_{ e }are provided in Table 2. Note, that although

*N*

_{d}and

*N*

_{p}depend on the utilized bandwidth, the minimum of

*f*(

*p*

_{off}) is independent of it, since

*N*

_{d}and

*N*

_{p}scale with the same constant with increasing bandwidth and actually only their ratio is what matters. The value of

*c*

_{ e }is different for four transmit antennas due to the lower number of pilot symbols at the third and fourth antennas. The last row of Table 2 presents the optimal values of

*p*

_{off,opt}for different numbers of transmit antennas and an ITU VehA [16] type channel model.

**Values of the parameters of** f **(** p_{
off
}**) for different number of transmit antennas for 1.4 MHz bandwidth, ITU PedA[**[16]**] channel model, LS, and LMMSE channel estimators**

Parameter | Tx = 1 | Tx = 2 | Tx = 4 |
---|---|---|---|

| 960 | 912 | 864 |

| 48 | 96 | 144 |

LS | |||

| 0.6623 | 0.6971 | 0.7359 |

| ≈5.61 | ≈4.11 | ≈3.22 |

LMMSE | |||

| ≈1.27 | ≈-0.35 | ≈-1.35 |

## Simulation results

## Conclusion

**Simulator settings for fast fading simulations**

Parameter | Value |
---|---|

Bandwidth | 1.4 MHz |

Number of transmit antennas | 1, 2, 4 |

Number of receive antennas | 1, 2, 4 |

Receiver type | ZF |

Transmission mode | Open-loop spatial multiplexing |

Channel type | ITU VehA [16] |

MCS | Adaptive |

*p*

_{off}, that maximizes the post-equalization SINR. Detailed values are listed in Table 2. The simulation results show an excellent match with the analytical solution shown in Section “Power allocation”.Using the optimal value of power offsets between pilot and data symbols results in throughput maximization. Moving away from this value, a throughput loss can be observed. However, this loss is usually not severe and the relatively broad maximum indicates a high robustness against inaccurate power distribution between pilot and data symbols.

A negative value of the variable *p*_{off}(in dB) corresponds to the reduction of the power radiated at the pilot symbols and increasing power radiated at the data symbols. Such negative value is optimal in case of four transmit antennas applying an LMMSE estimator. This kind of channel estimator is of superb performance and therefore requires less power to obtain a high-quality channel estimate.

Considering a single transmit antenna with an LS channel estimator, the optimal value of *p*_{off,opt}=5.61 dB may be considered rather high. However, due to the low number of pilot symbols compared to the number of data symbols, the difference in terms of energy is much lower than in terms of power.

*p*

_{off}=0

*dB*. In Figure 8, we observe that using optimal power distribution results in throughput gains up to 10%. Table 4 summarizes throughput gain in percent of a system using optimal power distribution between data and pilot symbols compared to one that does not. For the LMMSE channel estimators, only a small gain can be observed because the values of

*p*

_{off,opt}are close to 1 (0 dB). Therefore, a system with equal power radiated at the pilot and data symbols is already close to optimal.

**Throughput gain when using optimal power distribution between data and pilot symbols for various number of transmit antennas and LS and LMMSE channel estimators**

Parameter | Tx = 1 | Tx = 2 | Tx = 4 |
---|---|---|---|

LS | |||

Throughput gain | 10% | 10% | 7% |

LMMSE | |||

Throughput gain | 0.5% | 0.1% | 0.7% |

In this article, we answer the question of how to distribute power between pilot and data symbols in a way, that maximizes the overall performance of an OFDM MIMO system under time-variant channels. For this purpose, we made use of the post-equalization SINR with imperfect channel knowledge. Furthermore, we generalized the solution to the power distribution problem for time-variant channels. Simulation results obtained by the Vienna LTE simulator confirm our analytical solution. We also provide scripts, that allow to reproduce all results shown in this article. By adjusting the pilot symbol power individually, the solution from this article allows to increase the throughput of a transmission system by up to 10%.

## Endnotes

^{a}We are devoted to provide reproducible results. Thus, following our previous work, all data, tools, as well implementations needed to reproduce the results of this article can be downloaded from our homepage [17].^{b}Note that the three nearest pilot symbols cannot be located on the same subcarrier or within the same OFDM symbol. Would it be the case, they would not span any plane. In LTE, this is not of concern due to the defined pilot symbol pattern.

## Declarations

### Acknowledgements

The authors would like to thank the LTE research group and in particular Prof. Christoph Mecklenbräuker and Prof. Paulo S. R. Diniz for continuous support and lively discussions. This study was funded by the Christian Doppler Laboratory for Wireless Technologies for Sustainable Mobility, KATHREIN-Werke KG, and A1 Telekom Austria AG. The financial support by the Federal Ministry of Economy, Family and Youth and the National Foundation for Research, Technology and Development is gratefully acknowledged.

## Authors’ Affiliations

## References

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