- Open Access
Optimal pilot symbol power allocation under time-variant channels
© Simko et al.; licensee Springer. 2012
- Received: 15 February 2012
- Accepted: 12 June 2012
- Published: 20 July 2012
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.
- Mean Square Error
- Orthogonal Frequency Division Multiplex
- Channel Estimate
- Power Allocation
- Multiple Input Multiple Output
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  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.
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.
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.
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 . Authors of  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 , 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 , the optimal pilot symbol power in multiple input multiple output (MIMO) systems was derived based on a lower bound for capacity. Authors of  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 , 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 , 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.
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.
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”.
Most important variables
Received symbol vector at antenna nr
Channel matrix between antennas ntand nrin frequency domain
Transmit symbol vector at antenna nt
Pilot symbols vector at antenna nt
Transmit data vector at antenna nt
Additive noise at antenna nr
Received symbol vector at subcarrier k
Channel matrix between subcarriers k and l
Precoding matrix at subcarrier k
Effective channel matrix Hk,kW k
Transmit data at subcarrier k
Additive noise at subcarrier k
Channel estimation error at subcarrier k
Transmit power of one layer
Transmit pilot symbol power
Transmit data power
Channel estimation error variance
Channel saturation coefficient
Maximal user velocity
Maximal Doppler frequency
OFDM symbol duration
Number of pilot symbols
Number of data symbols
Number of layers
Number of transmit antennas
Number of receive antennas
Number of subcarriers
Offset between power of pilot symbols and data symbols
Post-equalization SINR at layer l
where represents the channel matrix in the frequency domain between the ntth transmit and nrth receive antennas. The transmitted signal vector is referred to as , the received signal vector as . The vector is additive white zero mean Gaussian noise with variance on antenna nr. In case of a time-invariant channel, the channel matrix appears as a diagonal matrix, whereas a time-variant channel forces the channel matrix to become non-diagonal. These non-diagonal elements indicate that the subcarriers are not orthogonal anymore, leading to the so-called intercarrier interference (ICI).
where is a precoded data symbol at the ntth transmit antenna port and the k th subcarrier, is a unitary precoding matrix at the k th subcarrier and is the data symbol of the nlth layer at the k th subcarrier. Here, 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 , one has to stack data symbols obtained via Equation (3) at a specific antenna ntinto a vector.
Furthermore, the average power transmitted on each of the N l layers is denoted by . The total power transmitted on one data position is , while that on one pilot position is .
where Nd is the number of data symbols and Npthe number of pilot symbols.
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.
Note that in practice variables , , and need to be replaced by their estimates.
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 NtNrindividual 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.
where denotes a set of the three nearest pilot symbol positions to the data position j, that span a plane. The weight wj,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 wj,idepends on the distance of the symbols in the time-frequency grid. The sum over wj,i over is one, namely . 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 wj,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 , 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 .
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 denotes the correlation between the channels at the j th data symbol and the i th pilot symbol positions.
LMMSE channel estimation
where the matrix denotes the channel autocorrelation matrix at the pilot symbols, and the matrix is the channel cross-correlation matrix.
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.
Note that Equation (45) is independent of channel realization, noise variance, ICI power and even user velocity. It is the same as in , only the constant c e has a different value due to the distinct performance of the channel estimators. Let us focus on the term 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 . 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 poff.
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 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 .
Values of the parameters of f ( p off ) for different number of transmit antennas for 1.4 MHz bandwidth, ITU PedA[] channel model, LS, and LMMSE channel estimators
Tx = 1
Tx = 2
Tx = 4
Simulator settings for fast fading simulations
Number of transmit antennas
1, 2, 4
Number of receive antennas
1, 2, 4
Open-loop spatial multiplexing
ITU VehA 
A negative value of the variable poff(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 poff,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.
Throughput gain when using optimal power distribution between data and pilot symbols for various number of transmit antennas and LS and LMMSE channel estimators
Tx = 1
Tx = 2
Tx = 4
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%.
aWe 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 .bNote 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.
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.
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