Combined channel estimation and PAPR reduction technique for MIMO–OFDM systems with null subcarriers

Pilot placement and power distribution

To determine pilot sets and power distributions to the pilot subcarriers, we modify the algorithm in[11] to accommodate multiple antennas while guaranteeing that the designed pilot sets are disjoint from each transmit antenna. The main objective of disjoint pilot sequences in each transmit antenna is to ensure appropriate separation of pilot sequences at the receiver.

Pilot set for the first transmit antenna is obtained from the designed optimal preamble with semidefinite programming (SDP) by iterative removal of a certain number N _m , of subcarriers with minimum power symmetrically about the active subcarrier set, followed by optimization of the remaining subcarriers as in[11, 14].

Once the pilot set for the first transmit antenna is found, it is excluded from the active subcarrier set and pilot set for the second transmit antenna is obtained from the remaining active subcarriers by repeating the iterative algorithm. The algorithm is executed until pilot sets for all N _t transmit antennas are obtained.

In the algorithm, once the set${\mathcal{K}}_{p_i}$ is obtained, it is excluded from the remaining active subcarriers${\mathcal{K}}_r\setminus {\mathcal{K}}_{p_i}$. This assures the unique pilot placement of the optimized pilot set from each transmit antenna. That is, the locations of the pilot symbols for each transmit antenna are distinct from the pilot set of any other transmit antenna within the active subcarrier band. Note that the symmetrical removal of N _m subcarriers after every optimization guarantee the symmetrical placement of pilot sets about the center of the signal band.

When the algorithm exits, we obtain the pilot positions and the normalized pilot powers for each antenna. To optimally distribute power between pilot symbols and data subcarriers, we can also modify the method in[11, 14] depending on the data transmission scheme. If one adopts OFDMA for data transmission, the method in[11] can directly be applied, while for space time block coding or other schemes, the method in[11] calls for some modifications.

It should noted that, for a given set of antennas, we did not consider the order intuitively, we simply considered the antenna index from 1 to N _t and design pilot set for each antenna, one by one for all transmit antennas. Generally, the order of the optimization with respect to transmit antennas is not critical to the final performance. However, total number of active subcarriers$\left|{\mathcal{K}}_s\right|$, number of transmit antennas N _t , and pilot symbols N _p , may affect the final performance. For systems with large number of transmit antennas, it is likely possible for the pilot symbols of the first designed antenna to have better performance than the last one. In such situation, to obtain better performance, pilot rearrangement between transmit antennas or some other existing selection algorithms can be utilized.

The proposed convex optimization design can be modified and be applied in the systems where the prior information of the channel is available. In[19], it has been shown that the optimization problem is still convex when spatial channels are considered. This suggests that, even though the scheme in[19] is different from the proposed method as it design training or preamble symbols for multiple transmitted OFDM symbols, the proposed design may also be adopted in spatially correlated channels.

PAPR reduction using phase information of the pilot symbols and TR techniques

For an OFDM symbol that consists of unused subcarriers, pilot and data subcarriers, phase information of the pilot tones together with a certain number of unused subcarriers can be utilized to mitigate the problem of high PAPR in OFDM systems. Careful design of the phase information to the pilot symbols can substantially minimize the peak levels of the time domain OFDM signals[21]. In line with the phase information of the pilot symbols, tone reservation (TR) technique which makes use of some reserved or unused subcarriers and insert dummy symbols that simultaneously minimize the peak levels of the sampled time domain OFDM signals can be utilized.

In this article, a method that mixes TR technique and phase information of the pilot symbols to efficiently reduce the PAPR of the MIMO–OFDM signals is presented. First, transmitters that reduce the PAPR by utilizing phase information of the pilot symbols is introduced. We adopt the techniques proposed in[21] to design phase information of the pilot symbols primarily dedicated for channel estimation (see Section IV and[3]) to reduce the PAPR of an OFDM symbol. Then, TR technique is employed to design dummy symbols that effectively reduce the PAPR. The optimal power distribution and phase information of the dummy symbols are determined by the solution of a convex optimization problem.

where${\mathit{X}}_{d_i}$,${\mathit{X}}_{p_i}$ and${\mathit{X}}_{v_i}$ are the vectors containing data, pilot, and dummy symbols, respectively. In the following sections, we will discuss these techniques used to counteract the PAPR problem.

Pilot phase design for PAPR reduction

For channel or carrier frequency offset estimation, it is sufficient to design the placement and power of each pilot symbol, and there are no particular requirements for the phase information. Thus, we can utilize phase information of the pilot symbols to enhance a reasonable PAPR reduction. We consider a set of frequency domain pilot symbols designed for channel estimation of the received OFDM symbol from the i th transmit antenna and design plausible phase information that lower the peak amplitudes of the pilot symbols in time domain.

where ϕ _i is an N _p ×1 vector containing phase information of the pilot symbols from the i th transmit antenna link.

which is the minimization of the maximum amplitude of the time domain signals${\mathit{x}}_{p_i}$.

Phase design to reduce PAPR is a non-convex and nonlinear optimization problem[21]. The non-convex optimization problem is addressed in[22], where exhaustive search method is employed to design phase information of the pilot symbols. However, the exhaustive search scheme becomes computationally prohibitive especially for pilot sets with a large number of subcarriers. Furthermore, the performance of the scheme depends on the searching granularity. In[21], phase information of the pilot symbols are obtained by the CE optimization techniques. Compared to the exhaustive search method, the algorithm in[21] converges fast to the near optimal solution. Due to its high convergence rate, the scheme has a potential to make practical design of phases for different applications. Thus, we resort to the later approach and slightly modify it to be used for the design of pilot phase information when multiple transmit antennas are employed, while limiting the number of iterations without improvement.

Algorithm 1. CE algorithm

1: Initialize${\widehat{\mu }}_0$ and${\widehat{\sigma }}_0$ and the CE parameters α,β and ρ.

2: Initialize$\mathit{\Psi }\left({\varphi }^{\ast }\right)=\left|\right|{\mathit{x}}_p^0|{|}_{\infty }$ and set the iteration counter t = 0 and$t^{\prime }=0$.

3: while ($t^{\prime }<\mathcal{Z}$ and$t<\mathcal{J}$ and$\underset{k}{\text{max}}\left({\widehat{\sigma }}_{k,t}\right)<\epsilon$)

{

4: Generate random sample ϕ₁,…,ϕ _M from the distributions$\mathcal{N}({\widehat{\mu }}_{k,t-1},{\widehat{\sigma }}_{k,t-1}^2)$.

5: Compute Ψ(ϕ) and order the sample in such a way that Ψ(ϕ¹)≤Ψ(ϕ²)…≤Ψ(ϕ ^M )

6: Select$\mathcal{I}$ indices of the best performing samples, and calculate mean and standard deviation.${\mu }_{k,t}=\frac{1}{M_{\ell }}\sum _{i\in \mathcal{I}}{\varphi }_{k,i}$ and${\sigma }_{k,t}^2=\frac{1}{M_{\ell }}\sum _{i\in \mathcal{I}}{({\varphi }_{k,i}-{\mu }_{k,t})}^2$

7: Smoothen the mean and standard deviation of the best performing samples using${\widehat{\mu }}_{k,t}=\alpha {\mu }_{k,t}+(1-\alpha ){\widehat{\mu }}_{k,t-1}$ and${\widehat{\sigma }}_{k,t}=\alpha {\sigma }_{k,t}+(1-\alpha ){\widehat{\sigma }}_{k,n-1}$ for k = 0,…,N _p −1.

8: if Ψ(ϕ^∗)<Ψ(ϕ¹) then

9:$t^{\prime }\leftarrow t^{\prime }+1$

10: else

11: Ψ(ϕ^∗)=Ψ(ϕ¹) and$t^{\prime }=0$

12: end if

13: Increment$t\leftarrow t+1$

14: }

The pseudocode of the CE method is as presented in Algorithm 5.2.1. The algorithm starts by initializing the mean and standard deviation as μ₀ and σ₀, respectively. The best objective function value is initialized to$\left|\right|{\mathit{x}}_p^0|{|}_{\infty }$, which is the peak amplitude of the zero phase pilot symbols and the iteration counters t (total number of iterations) and$t^{\prime }$ (iterations without improvement) are initialized to zero (see steps 1–2 in the pseudocode). The main loop includes the two main tasks that every CE method must perform, namely, the generation of a sample and the updating of the parameters associated with the chosen probability distribution. Step 4 generates M sample vectors${\left\{{\Phi }_m\right\}}_{m=1}^M$ of length N _p using a family of normal probability density functions (PDF)$\mathcal{N}({\mu }_k,{\sigma }_k^2)$ for k = 0,…,N _p −1.

The ordering of the sample in step 5 is such that the best observation is placed in the first position of the list and the worst is placed in the last position. The mean and standard deviation values calculated in step 6 corresponds to the variables of the top M _ℓ solutions in the current sample. Note that, a fixed number of the best performing samples M _ℓ are referred to as the elite samples expressed as M _ℓ =ρM. In step 7, we obtain the smoothed mean${\widehat{\mu }}_{k,t}$ and standard deviation${\widehat{\sigma }}_{k,t}$ by using some fixed smoothing parameter α where 0<α<1.

Steps 9 and 11 update the best solution found and reset the counter of the number of iterations without improvement, respectively. The global iteration counter is updated in step 13. Note that, the mean${\widehat{\mu }}_{k,t}$ converges to ϕ^∗ and the standard deviation${\widehat{\sigma }}_{k,t}$ to the zero. In brief, we obtain a degenerated PDF with all mass concentrated in the vicinity of the vector ϕ^∗.

At each stage t of the CE procedure we simulate a sample ϕ from a$\mathcal{N}({\widehat{\mu }}_{t-1},{\widehat{\sigma }}_{t-1}^2)$ distribution, and update${\widehat{\mu }}_t$ and${\widehat{\sigma }}_t$ of the best samples M _ℓ .

The pseudocode in Algorithm 5.2.1 is faster than the CE method proposed in[21], as it limits the number iterations without improvements. The algorithm can be utilized to design pilot phase information for all transmit antennas by repeating the same procedure for each set of pilot symbols .

TR for PAPR reduction

TR technique is one of the promising approach for PAPR reduction in OFDM systems[5, 23–25], where the transmitter mitigates the PAPR problem by sending dummy symbols (i.e., symbols not conveying information) in some reserved subcarriers[26]. The advantages of TR-based schemes is that there is no specific PAPR reduction information that needs to be communicated to the receiver. However, one problem with TR techniques lies on the computationally efficient determination of dummy symbols that effectively minimizes the PAPR[23]. The amount of PAPR reduction depends on some factors such as location of the dummy symbols, number of dummy symbols, and allowed power on these dummy symbols.

Some issues to be considered before using the TR techniques include PAPR reduction capacity, power increase in transmit signals and the loss in data rate. In the proposed TR technique, reduction in PAPR is achieved at the expense of increasing the total transmitted power of an OFDM symbol.

Note that, for simplicity we consider that, a same set of null subcarriers is reserved for PAPR reduction on different transmit antenna link. However, power and phase information are different for each transmit antenna link, i.e.,${\mathcal{K}}_v={\mathcal{K}}_{v_1}={\mathcal{K}}_{v_2}=\dots ={\mathcal{K}}_{v_{N_t}}$, and${\mathit{X}}_{v_1}\ne {\mathit{X}}_{v_2}\ne \dots \ne {\mathit{X}}_{v_{N_t}}$.

For an OFDM symbol with N _G number of null edge subcarriers or guard subcarriers, we need to select a set of${\mathcal{K}}_v$ subcarriers and optimally allocate the power as well as phase information to ensure that the designed dummy symbols${\mathit{X}}_{v_i}$, significantly reduces the PAPR of each transmit antenna link.

Pilot symbol design

The minimum number of pilot symbols needed per OFDM symbol depend on the channel length, L (see[1, 3, 8, 12] and the references therein). For the multipath channel to be estimated, the number of pilot symbols need to be at least equal to the length of the channel impulse response, i.e., N _p ≥ L. The general assumption is that, taps of the channel impulse response beyond L samples are insignificant[8], which is similar to the assumption made in OFDM to justify that no ISI occurs. Thus in our simulations we use N _p = L.

To design disjoint pilot symbols to multiple transmit antennas, we construct a composite pilot sequence with index sets$\left\{{\mathcal{K}}_{p_i}\right\}$ having N _t N _p subcarriers with significant pilot power and reasonable position. The pilot set for the first antenna is obtained as in[11, 14], then by utilizing our proposed algorithm in Section IV, which exclude the designed pilot set from the preamble and repeat the same procedure for the remaining subcarriers to obtain pilot sets for all N _t transmit antennas.

In the proposed design, total power allocated to the pilot symbols of each transmit antenna link is normalized to 1. By normalizing the power of pilot symbols, one can easily distribute the power when total power dedicated to the pilot symbols is given.

The analytical solution for optimal pilot power distribution in[13] can also be adopted, for a given set of arbitrary pilot placement,[13] is a prominent candidate for OFDM systems with large number of subcarriers. In[13] to obtain the placement of the pilot symbols, fifth-order parameterization is employed. However, the fifth-order parameterization like the cubic parameterization in[12] requires contiguous subcarriers except for the central DC subcarriers. This limits the adoption of[12, 13] in designing disjoint pilot symbols for OFDM systems with multiple transmit antennas.

Figure1 shows the designed disjoint pilot set for four transmit antennas when N _p = 8. For all transmit antennas, pilot power and location are well distributed within the in-band region which promises better estimation of the channels even at the edge of the band.

Figure2 compares our proposed pilot symbols and the partially equi-spaced pilot (PEP) symbols proposed in[3] for N _p = 16. In the two designs, total power dedicated to the pilot symbols is the same. For our proposed design power allocated to the edge pilot symbols is slightly lower than that of the mid pilot symbols, however the difference is not as large as in the PEP design. In[3], it is stated that, the power of pilot symbols decreases when the pilot symbols are close to the null/virtual carriers zone due to the fact that there are less data carriers, this might be true, however the power allocated to these subcarriers need to be significant, otherwise the problem of channel estimation via the extrapolation of the edge subcarriers will still persist[32].

Comparison of MSE performance

We evaluate the MSE performance of the designed pilots symbols with respect to the existing designs. First, we compare each of the designed pilot set with the existing IEEE 802.16e standard pilot symbols separately, i.e., SISO–OFDM mode. The aim is to observe the performance of the designed pilot symbols in each antenna with respect to the standard one to ensure that each designed pilot set has better performance. A noteworthy fact is that, when some SISO–OFDM methods are adopted in MIMO–OFDM pilot designs the performance of some designed pilot set deteriorates with increased number of transmit antennas. This implies that, only a few pilot sets yield a significant performance.

In Figure3, the normalized channel estimate MSE of the designed disjoint pilot symbols in Figure1 is compared with the existing standard which places the eight subcarriers at {±13,±38,±63,±88}. The total pilot power for each antenna is taken to be N _p , for both the standard (equi-spaced, equi-powered pilot symbols) and our proposed method. From the plot it is clear that each individual antenna outperforms the conventional standard design. The standard pilot design does a poor job of estimating channel at the subcarriers near the guard band, due to lack of the pilot subcarriers at the edge of the OFDM symbol in IEEE 802.16e standard pilots. As a result, the estimation via extrapolation for the edge subcarriers results in a higher error[15]. A possible solution would be to increase the number of pilot subcarriers at the edge subcarriers as proposed in[33]. However, this would lower the spectral efficiency of the system. Our proposed design illuminates the improvement obtained by rearranging the pilot symbols without any addition of pilot subcarriers at the edge as suggested in[33]. This clarify that the equi-spaced and equi-powered pilot symbols are suboptimal for an OFDM system with null subcarriers.

In Figure4, we make a comparison of the channel estimate MSE of each active subcarrier symbol for the designed disjoint pilot symbols in Figure2. The proposed design produces MSE with small variation across all of the data subcarriers. The plot show that our proposed design outperforms the PEP for some antennas. The PEP design produces a better MSE performance for some data subcarriers around the middle of the active band. However, the design does a poor job of estimating channel at the subcarriers near the guard band. This is not due to lack of the pilot subcarriers at the edge of OFDM symbols but insignificant power allocated to the pilot symbols close to the null subcarrier zone. This further suggests that both pilot powers and placements need to be carefully considered in the design.

To further demonstrate the potential of our proposed design, we make a comparison of the average channel estimate MSE vs channel length L. To obtain the channel MSE of our proposed design as well as the PEP scheme, we varied the channel length L, from 1 to 16. Figure5 presents the average channel MSE. The proposed optimized pilot symbols exhibit better (lower) channel MSE than the PEP symbols. This clearly demonstrates the efficiency of our proposed design.

We also compare the average MSE for different signal-to-noise ratio (SNR). Figure6 depicts the average MSE versus SNR for N _p = 16 for different channel length L. From the plot it is clear that when L = 1 and L = 4 the MSE performance of the PEP design is comparable to the proposed design. However, for L = 16 the proposed design outperforms the PEP, which further corroborates the effectiveness of our proposed design over the PEP method.

BER performance

In this section, we explore the bit error rate (BER) performance gains that could be realized if the pilot symbols in IEEE 802.16e are designed to conform with the proposed method. We demonstrate the efficacy of our proposed method by comparing the BER performance of the proposed design, PEP and the standard pilots. The frequency selective channel with L=N _p taps is considered. Each channel tap is i.i.d. complex Gaussian with zero mean and the exponential power delay profile is given by the vector ρ=[ρ₀…ρ_L−1] where${\rho }_l=\mathcal{C}{e}^{-l/2}$, and$\mathcal{C}$ is a constant selected such that$\sum _{l=0}^{L-1}{\rho }_l=1$. For MIMO case, to transmit an OFDM symbol, we adopt the well known Alamouti STBC for data subcarriers${\mathcal{K}}_{d_i}$, given by${\mathcal{K}}_{d_i}\subseteq {\mathcal{K}}_s\setminus ({\mathcal{K}}_{p_1}\cup {\mathcal{K}}_{p_2}\dots \cup {\mathcal{K}}_{p_{N_t}})$.

Figures7 and8 depict the BER performance of QPSK signals for both SISO and MIMO systems. For a SISO system with N _p =8, we make a comparison of the proposed, PEP and the standard design. For the SISO case, the results in Figure7 show identical BER performance of our proposed design and the PEP. Both PEP and the proposed design outperform the standard pilots of IEEE 802.16e. However, when two transmit antennas are considered, the proposed design provides better BER performance over the PEP design. Figure8 verifies that, for N _p =16, the proposed design provides improved BER performance over PEP for both SISO and MIMO cases. This performance gap is a result of the PEP design having insignificant power distribution to the pilot symbols at the edge of the active subcarrier band which leads to poor estimate of the channels.

PAPR designs

To demonstrate the effectiveness of the designed TR and phase information of the pilot tones in PAPR reduction, we adopt the disjoint pilot tones in Figure2, primarily used for channel estimations in MIMO–OFDM systems.

The PAPR CCDF of an uncoded OFDM signal are computed with an oversampling factor of$\mathcal{L}=8$ and are obtained by considering 5×10³ independent OFDM symbols. The data subcarriers are loaded with QPSK signals and we consider an OFDM system with N _t = 3, N _p = 16 and the number of dummy symbols N _v = 16.

Figure9 depicts the CCDF curves of the PAPR reduction for the combined (pilot phase and TR) technique, TR technique and the original (data and pilot tones with no phase information) signals. The results show relatively lower PAPR for the OFDM symbols that utilize dummy symbols to reduce the PAPR over the data and pilot tones without any modifications. By combining phase information of the pilot tones with dummy symbols, the PAPR can further be reduced. From the figure it is clear that, there is slight improvement in PAPR reduction for the combined TR and phase information as compared to the TR technique only. This verifies the potential of our proposed combined technique in minimizing the PAPR. It should be noted that, this small improvement in PAPR due to pilot phase information is attained without any addition of resources. The total power allocated to the pilot symbols in each transmitter link is taken to be equals N _p . For system that loads more power to the pilot symbols, the significant improvement in PAPR due to phase information can be revealed. The proposed position of the dummy symbol is not necessarily optimal. Our method can adopt any available optimal location and optimize the power and phase of the dummy symbols to obtain the best possible performance. Addition of the number of dummy symbols may lead to a lower PAPR, but this will increase the total transmit power per OFDM symbol.