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Optimal pilot symbol power allocation under time-variant channels
EURASIP Journal on Wireless Communications and Networking volume 2012, Article number: 225 (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.
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 the meanwhile, high mobility users are considered to be supported by standardization organizations due to the increasing demand of high-speed data connections on the move. LTE, for example, aims to serve users moving up to 500 km/h. At such velocities, the aforementioned power distribution problem becomes even more challenging, mainly because of two reasons:
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”.
In this section, we briefly point out the key aspects in the LTE standard that are relevant to this article and introduce a transmission model suitable for our further derivation. The most important variables used in this article and their description are summarized in Table 1.
In the time domain, the LTE signal consists of frames with a duration of 10 ms. Each frame is split into ten equally long subframes and each subframe into two equally long slots with a duration of 0.5 ms. With the normal cyclic prefix length, each slot consists of Ns=7 OFDM symbols; with the extended cyclic prefix length, Ns=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.
A received OFDM symbol in the frequency domain at the nrth receive antenna can be written as
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).
Specifically, the vector in Equation (1) comprises the precoded data symbols and the pilot symbols from the set of all possible pilot symbols defined in LTE, at the ntth transmit antenna placed by a suitable permutation matrix P
The vector has Nsub 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 Np and Nd, respectively. On subcarrier k of the data symbol vector , the precoding process can be described as
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.
For the derivation of the post-equalization SINR, we will use a MIMO input–output relation at the subcarrier level, given as:
Matrix denotes the MIMO channel matrix between the k th and m th subcarriers. The MIMO channel matrix Hk,m contains appropriately ordered elements of matrices located in the k th row and m th column. Matrix W k is a unitary precoding matrix of size Nt×Nl. In LTE, the precoding matrix can be chosen from a finite set of precoding matrices . 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 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 . The total power transmitted on one data position is , while that on one pilot position is .
When the power is evenly distributed between the data and pilot symbols, there 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.
If perfect channel knowledge available at the equalizer, the ZF estimate of the data symbol s k is given as
where the vector e l is an Nl×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 represents the ICI power, that is given as
Authors of  derived the ICI power leaking from neighboring subcarriers due to the loss of orthogonality between subcarriers. For a typical assumption of the Jake’s spectrum, the ICI power can be expressed as
where J0(·) 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
where v max is the maximal user velocity, f c the carrier frequency, and c0the 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
Since the channel estimation error matrix Ek,kis unknown at the receiver, the ZF solution is given again by Equation (9), but channel matrix Hk,k is replaced by its estimate , which is known at the receiver
with matrix being equal to . The symbol error after the ZF equalizer is given as
From Equation (17), we can compute the MSE matrix assuming that the estimation error matrix and the effective channel matrix are statistically independent
with being the MSE of the channel estimator. Equation (18) directly leads to the SINR on the l th layer
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 LS channel estimate at the pilot symbol positions is obtained by solving the following minimization problem
where the matrix is assumed to be diagonal, representing the channel only at the pilot symbol positions. Obviously, such estimator ignores the non-diagonal elements in the channel matrix and treats the potential ICI as noise. More details on this topic can be found in . The resulting MSE at the pilot symbol positions is given as
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.
Figure 2 shows a channel estimate at 50 km/h of an LS estimator with a two-dimensional linear interpolation. The vertical lines represents pilot symbols. It can be observed, that the channel estimates at the data positions are obtained by spanning planes defined by the three nearest pilot symbols.
A channel estimate at an arbitrary data position using a linear interpolation is given by a weighted sum of the three nearest channel estimates (in the Euclidean sense of the time-frequency grid) at pilot positions, that span a plane.bLet us denote the channel estimate at the j th data position by and the channel estimate at the i th pilot position by . Note that we use i th and j th positions as general description of location in the time-frequency grid:
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 .
We assume that the channel of the first data symbol hd,1is located within the plane spanned by the channel estimates at the first three pilot symbols, hp,1, hp,2, and hp,3. The plane spanned by these three channel estimates is defined as follows
where a and b are real scalars. For a general solution, Equation (23) can be reformulated as
In the following, we evaluate the performance of such a channel estimator by analytically deriving its theoretical MSE at the data positions. Given the definition of the MSE
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.
Let us proceed to the last term of Equation (25), in which we insert Equation (22) and consequently the equality , which states that the channel estimate at the pilot symbol position is given as the true channel superimposed by an estimation error. Note that due to Equation (8), it can be shown  that the MSE of the LS channel estimator at the pilot symbol positions is identical to the noise power for time-invariant channels. In the case of time-variant channels, we have to consider also the ICI power. Therefore, the power of nest,i is equal to . We obtain
In the last step, we used the fact that the estimation error nest,i is statistically independent of the channel coefficients. To summarize, we obtain from Equation (25)
It is noticed that one part of Equation (28) is independent of , as it only depends on the weights wj,iand 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 . This term is on the other hand independent of the autocorrelation matrix. This can be stated analytically as follows
where c e is a scalar depending on the weights wj,i, given as
The value of 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 wj,i as well as the autocorrelation matrix. In the following text, it is referred to as channel saturation d, given as
Figure 3 shows the value of the channel saturation coefficient 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
The LMMSE channel estimator requires knowledge of the second-order statistics of the channel and the noise. It can be shown that the LMMSE channel estimate is obtained by multiplying the LS estimate with a filtering matrix 
In order to find the LMMSE filtering matrix, the MSE
has to be minimized, leading to
where the matrix denotes the channel autocorrelation matrix at the pilot symbols, and the matrix is the channel cross-correlation matrix.
After a straightforward manipulation, the average MSE at the data subcarriers is expressed as
where Nd 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
where the scalar coefficient 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 and 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 .
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.
We introduce power adjusting factors and for the pilot and data symbols, respectively. Since the overall transmit power remains constant, that can be expressed by
where Np and Nd represent the number of pilot symbols and data symbols in a subframe, respectively. In order to describe the interconnection between and , we introduce a variable poffwhich is the power offset between the power of the pilot symbols and the data symbols, denoted by
Therefore, and can be expressed with respect to poff:
For the aforementioned channel estimators, we increase the power at the pilot symbol by , the noise and ICI-dependent parts of the MSEs in Equation (29) and (37) decrease by the same factor . Therefore, we have
Plugging the variables and into Equation (19), we obtain the SINR expression at layer l with adjusted power of the pilot symbols
for which the power allocation function f(poff) is given as
The constant is proportional to the channel saturation coefficient 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 , 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 .
Let us proceed to the definition of the optimization problem. The target is to find an optimal value of poff,opt that maximizes the post-equalization SINR while keeping the overall transmit power constant
In order to maximize the post-equalization SINR, the power allocation function f(poff) in the denominator of Equation (44) has to be minimized. The minimum of the power allocation function f(poff) can be found by simple differentiation, resulting in
Figure 6 shows an example of the power allocation function for LS and LMMSE channel estimators for various numbers of transmit antennas. We observe, that all of these functions have minimum points. This minimum point corresponds to the maximization of the post-equalization SINR. Typical values of parameters Nd, Np, and c e are provided in Table 2. Note, that although Nd and Np depend on the utilized bandwidth, the minimum of f(poff) is independent of it, since Ndand Npscale 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 poff,optfor different numbers of transmit antennas and an ITU VehA  type channel model.
In this section, we present simulation results and discuss the performance of LTE transmission systems with different pilot symbol powers under time-variant channels. All results are obtained with the LTE Link Level Simulator version “r1089” [8, 9], which can be downloaded from http://www.nt.tuwien.ac.at/ltesimulator. All data, tools, and scripts are available online  in order to allow other researchers to reproduce the results shown in this article. The structure of the utilized Vienna LTE Link Level simulator is shown in Figure 7. Note that the simulator performs all routines according to the standard  and it includes blocks like frequency offset compensation , timing offset compensation , channel estimation , and more.
Table 3 presents the most important simulator settings. Since what is to be observed is the optimal value of pilot symbol power adjustment, an SNR of 10 dB has been chosen.
Simulation results showing throughput performance for 1×1, 2×2, and 4×4 antenna configurations are shown in Figure 8 for LS and LMMSE channel estimators. In this example, we set the user velocity to 100 km/h and SNR = 10 dB. Little arrows always indicate the theoretically derived optimal value of the variable poff, 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 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.
An OFDM transmission system that does not utilize different power levels for its pilot and data symbols, corresponds to poff=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 poff,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.
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.
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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.
The authors declare that they have no competing interests.
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Šimko, M., Wang, Q. & Rupp, M. Optimal pilot symbol power allocation under time-variant channels. J Wireless Com Network 2012, 225 (2012). https://doi.org/10.1186/1687-1499-2012-225
- Mean Square Error
- Orthogonal Frequency Division Multiplex
- Channel Estimate
- Power Allocation
- Multiple Input Multiple Output