Multi-segmental OFDM signals equalization with piecewise linear channel model over rapidly time-varying channels
- Li Alex Li^{1},
- Hua Wei^{1},
- Yao Yao^{1},
- Gong Chen^{1},
- Weiwei Ling^{1},
- Jiang Du^{1} and
- Yao Huang^{1}Email author
https://doi.org/10.1186/s13638-017-0977-1
© The Author(s) 2017
Received: 21 April 2017
Accepted: 27 October 2017
Published: 14 November 2017
Abstract
For rapidly time-varying channels, the performance of (orthogonal frequency division multiplexing) OFDM systems with the conventional one tap equalizer will be significantly degraded. Because the orthogonality between subcarriers is destroyed, the conventional way to combat the inter-carrier interference (ICI) is employing the banded minimum mean square error (MMSE) equalizer, which can save computational efforts introduced by a large number of subcarriers. However, the width of the banded channel matrix is mainly determined by the normalized Doppler frequency in the sense that with the high Doppler frequency the complexity of equalization for one OFDM block will significantly increase with the band width D. In order to reduce the equalization complexity, the authors proposed multi-segmental OFDM signal equalization method with piecewise linear model (PLM) to approximate the time variations and mitigate the corresponding ICI. Its complexity is significantly reduced with the small segments. Furthermore, an alternative MMSE method with the iterative rank-1 matrix updates is proposed to further reduce the complexity. We also derive the theoretical pre-equalized and equalized signal to interference ratio (SIR) for different normalized Doppler frequencies and segment numbers, which implies that the larger segment number can achieve the better performance. Simulation results demonstrate that the proposed method outperforms the conventional banded MMSE equalizer and the partial fast Fourier transform (FFT) method in terms of bit error rate (BER) with almost the same complexity.
Keywords
1 Introduction
Orthogonal frequency division multiplexing (OFDM) is a very promising modulation technique to achieve high spectral efficiency. However, the inter-carrier interference (ICI) will be introduced by the non-orthogonality between subcarriers that interfere with each other. A large number of different algorithms have been studied in the last decade for ICI cancellation, e.g., [1–3], which take advantage of the banded channel matrix to remove the ICI from neighbouring subcarriers sequentially. Additionally, some techniques in [4, 5] exploit the sparsity of the banded matrix to design the OFDM block equalizers. Also, several pre-equalized methods have been proposed in [6–10] to reduce the time variations and obtain a quasi-diagonal channel matrix as convectional OFDM systems over the slowly time-varying channels. Some decision feedback equalization techniques are also compatible with OFDM systems [11–16]. According to the previous literature, we can observe that the equalizers with the banded channel matrix can be efficiently performed in sequential or block manners. Additionally, the overall complexity will approximately scale with O(D ^{2}) [2–4]. Note that the algorithm proposed in [9, 17] can reach a very low complexity, which linearly increases with the number of subcarriers. But the performance loss is too significant at high normalized Doppler frequency regime. After the equalization, the power of the desired subcarrier that spreads across the neighbouring subcarrier are aggregated, and the ICI is also mitigated.
In this paper, we proposed a multi-segmental OFDM signal equalization method in rapidly time-varying channels. Unlike the methods in [3, 4, 15, 16, 18], the conventional banded channel matrix is not adopted for equalization. Motivated by [8, 19–21], a new form of multi-segmental OFDM signals equalizer with the PLM is derived for ICI suppression. Unlike the previous literature, we do not assume that the channels remain constant during each sample duration and neither exploit real oversampling benefits as in [19], in which more samples are obtained around each subcarrier. Additionally, we approximate the time variations by the PLM between two segments rather than that between two OFDM blocks in [20]. Although the multi-segmental operations can effectively reduce the time variations, the additional irreducible interference will be introduced with the large segments number M. In other words, the SIR performance will be gradually saturated with the increasing number of segments M. In [21], the authors proposed a similar method to approximate the time variations by the differences between real channel coefficients in each symbol duration and the midpoint channel coefficients in each segment. However, it is very difficult to obtain the accurate channel coefficients in each symbol duration for the rapidly time-varying channels. Furthermore, the authors in [22] approximate the time variations of OFDM systems by a Taylor expansion. In [23], a simple and efficient polynomial surface channel estimation technique is exploited to reduce ICI in a low complexity. The algorithm proposed in [24] decomposes the ICI caused by the time variations into a simple inter-symbol interference and a low ICI. Although these methods can substantially improve the performance of OFDM systems in high Doppler scenarios with lower complexity, the requirement of the relatively large matrix inversion operations and addition memory is still unavoidable. In contract, the proposed algorithm is more flexible in terms of complexity with different segment numbers and can save a large amount of memory compared to the methods that use the matrix inversion.
The contribution of this paper is summarized as (1) a new system model of multi-segmental OFDM signals with PLM is formulated, (2) a MMSE equalizer for the system model is presented, (3) an alternative MMSE method with the iterative rank-1 matrix updates is proposed, (4) efficient computation of the channel coefficients for the equalizers is presented, and (5) the theoretical performance on pre-equalized and equalized SIR is derived.
The paper is organized as follows. Section 2 states the new system model for multi-segmental OFDM signals with PLM. Section 3 presents the proposed equalization methods for multi-segmental OFDM signals and their complexity is also discussed. In Section 4, the SIR performance is analyzed. The simulation results are given in Section 5, and Section 6 draws the conclusions.
2 System model
2.1 Conventional model
where the quantity L denotes the length of multipath channels. The guard interval is assumed to be equal to the maximum delay spread in the sense that the multipath effects is perfectly removed for the conventional OFDM systems.
where the quantities m _{1} and m _{2} denote the start and the end symbol duration at the mth segment, respectively. If the CFR H _{ k }(n) in (3) is constant over time, the equation can be reduced to the conventional receive OFDM signal model over the very slowly time-varying channel, i.e., y _{ d }=s _{ d } H _{ d }+v _{ d }. Otherwise, the desired subcarrier s _{ d } will be interfered by the other subcarriers s _{ k,k≠d } due to the time variation of the channels. Thus, we employ the piecewise linear model (PLM) that approximates time variation by the channel slopes in each segment.
2.2 Piecewise linear model
Note that the channel slopes are computed with the perfect CSI. Although the perfect CSI is used, the channel coefficients mismatch between the PLM and the perfect channel coefficients still exists and will be more significant with the increasing normalized Doppler frequency. However, the proposed model will be more accurate than the ones used in [8, 20], which approximate the channels by the midpoint CIR with the slopes between two consecutive OFDM symbols rather than the two segments or only the midpoint CIR in one particular segment.
where the diagonal matrix of the CFR is defined as \(\mathbf {H}_{k} = \mathcal {D}\left (\left [H_{k}(1),H_{k}(2),\ldots,H_{k}(M)\right ]^{T}\right)\), ε _{ k }=[I _{ k }(1),I _{ k }(2),…,I _{ k }(M)]^{ T }, \(\boldsymbol {\nu }_{k}=\left [\nu _{k}(1),\nu _{k}(2),\ldots,\nu _{k}(M)\right ]^{T}\!, \boldsymbol {\Phi }_{k}(0)=\mathcal {D}\left ([\alpha _{k}(1_{0}), \alpha _{k}(2_{0}),\ldots,\alpha _{k}(M_{0})]^{T}\right)\) for the first region, and the vector ν k′ the second slope matrix Φ _{ k }(1) are defined accordingly.
3 Equalization of multi-segmental OFDM signals with piecewise linear model
Firstly, we will present two equalization methods including the conventional minimum mean square error (MMSE) equalizer and the modified one with iterative rank-1 matrix updates (IRU). Secondly, an appropriate rank value selection will be discussed to further reduced the complexity of the MMSE equalizers.
3.1 Convectional MMSE equalizer
3.2 Modified MMSE equalizer with iterative rank-1 matrix updates
where \(g_{k} = \frac {1}{1+\text {tr}\left (\mathbf {B}^{-1}_{k}\mathbf {u}_{k} \mathbf {u}_{k}^{H}\right)}\), and the symbol tr(·) denotes the trace of the matrix. Additionally, \(\mathbf {B}^{-1}_{0}=\left (\frac {\sigma ^{2}}{M}\mathbf {I}_{M}\right)^{-1}\). It can be observed that the only direct computation of the matrix inversion is \(\mathbf {B}^{-1}_{0}\) for the initialization of (20). However, the complexity of the larger number of iterative updates will be higher than the conventional matrix inversion using Cholesky factorization [26]. Note that the performance modified MMSE equalizer is not comparable to the convectional one, but its performance will be significantly improved with ICI cancellation and the complexity is still very low. More details about the exact complexity can be found in subsection 3.5.
3.3 Efficient computation of u _{ k }
The notation ⊙ denotes the element-wise multiplication. Note that the IFFT of Υ, Ψ, and Ψ ^{′} can be pre-computed before the transmission, and C=U ^{ H } U.
3.4 Low-rank approximation of the matrix inversion
- 1.
Select the subcarriers with the peak power P(f _{ k }) and \(k \in \mathcal {F}_{1}\).
- 2.
Extract the main lobes and side lobes according to \(\mathcal {F}_{1}\), so the extracted indices for one particular lobe are \(\mathcal {L}_{0}=\{d-(i+1)M,d-(i+1)M+1,\ldots,d-iM\}\) for k<d and \(\mathcal {L}_{1}=\{d+iM,d+iM+1,\ldots,d+(i+1)M\}\) for k>d.
- 3.
Set the minimum target power \(P^{\ast }_{f}\).
- 4.
Search the subcarrier, the power \(P_{f_{k}}\) of which is larger than the minimum target power \(P^{\ast }_{f}\) in the zig-zag manner within \(\mathcal {L}_{0}\) or \(\mathcal {L}_{1}\). Put the desired subcarrier index in \(\mathcal {R}\).
- 5.
Repeat step (4) for other lobes.
In other words, the number of matrices summation and the number of updates required in (14) and (19) are reduced.
3.5 Complexity analysis
In this part, we have discussed the complexity of the algorithms required in each step, which is evaluated by the complex multiplications (CMs).
3.5.1 MMSE equalizer with efficient computation of u _{ k } and low-rank approximation
The computational complexity of MMSE equalizer for the dth subcarrier is primarily determined by the computation of autocorrelation matrix \(\phantom {\dot {i}\!}\mathbf {R}_{\mathbf {y}_{d}}\) and its inversion, whose complexity scale with \(O(M^{3})+O(\log N_{s}+3ML)+O\left (\frac {\vert \mathcal {R}\vert }{2}(M^{2}-M)\right)\). For each subcarrier equalization, the matrix inversion using Cholesky factorization requires O(M ^{3}) CMs. The computation of u _{ k } needs one FFT and three matrix vector multiplications, whose complexity is upper bounded by O(logN _{ s }+3ML) CMs. To obtain the autocorrelation matrix \(\phantom {\dot {i}\!}\mathbf {R}_{\mathbf {y}_{d}}\), the complexity is determined by the size of chosen subcarriers \(\vert \mathcal {R} \vert \) and the autocorrelation matrix \(\mathbf {u}_{k}\mathbf {u}^{H}_{k}\), which is around \(O\left (\frac {\vert \mathcal {R}\vert }{2}\right)\).
3.5.2 Modified MMSE equalizer with iterative rank-1 matrix updates
As discussed in [26], the computational complexity of the modified matrix inversion will be significantly increasing with the number of dimensions, i.e., \(\vert \mathcal {R}\vert \). Hence, the size of \(\mathcal {R}\) is limited to 3. In other words, the autocorrelation matrices \(\mathbf {u}_{k}\mathbf {u}^{H}_{k}, k = d,d-1,d+1\) are used to yield the approximate matrix inversion, whose complexity scales with \(O\left (\vert \mathcal {R}\vert ^{3} M^{2}\right)\). For further complexity reduction, k=d. Its complexity reduces to O(M ^{2}).
3.5.3 Complexity comparison between different equalization algorithms
Complexity comparison between different equalization algorithms
Algorithm | Complex multiplications |
---|---|
Full-MMSE [29] | \(O\left (N_{s}^{3}+N_{s}^{2}\right)\) |
Conventional banded MMSE [3] | O(8D ^{3}+12D ^{2}+6D+1) |
Partial FFT+MMSE [8] | \(O\left (M^{3}+3\vert \mathcal {R}\vert M^{2}\right)\) |
Partial FFT+RLS [8] | O(6M ^{2}+2M+2) |
MMSE+PLM | \(O\left (M^{3}+3\vert \mathcal {R}\vert M^{2}\right)\) |
Modified MMSE equalizer with ICI cancellation \(\vert \mathcal {R}\vert =1\) | O(2M ^{2}+2M) |
Modified MMSE equalizer with ICI cancellation \(\vert \mathcal {R}\vert =3\) | \(O\left (2\vert \mathcal {R}\vert ^{3}M^{2}+2M\right)\) |
4 Performance analysis in signal to interference ratio
In this section, we will present signal to interference ratio (SIR) analysis for the pre-equalized and equalized cases, which indicate the different behaviours of the different segment numbers on the SIR. The first case is the upper bound, which is based on the pre-equalized SIR at low normalized Doppler frequencies. The other case is for the equalized SIR with different M at a wide range of normalized Doppler frequencies.
4.1 Case I: pre-equalized signal to interference ratio analysis for multi-segmental OFDM signals
In this part, the theoretical pre-equalized SIR is derived for the slowly time-varying channels.
4.1.1 Derivation of the power of subcarriers
where i=k−d. Hence, Eq. (35) yields the power of the desired subcarrier and the interference caused by the other subcarriers with k=d, k≠d, respectively.
Hence, with very large number of subcarriers and segments, the interference is approximately bounded by the number of segments M. This is because the time variations can be omitted by the large M.
4.1.2 The upper bound of the SIR for slowly time-varying channels
4.2 Case II: equalized signal to interference ratio analysis for multi-segmental OFDM signals
where \(\text {var}\left \lbrace w^{\ast }_{d}(m_{1}) \right \rbrace =1\), \(\text {var}\left \lbrace f_{k}(m_{1}) \right \rbrace =\frac {1}{M^{2}}\text {sinc}^{2}\left (\frac {i\pi }{M}\right)\), \(\text {cov}\left \lbrace w^{\ast }_{d}(m_{1}),w_{d}(m_{2}) \right \rbrace =J_{0}\left (2\pi f_{d}(m_{1}-m_{2})\frac {N_{s}}{M}\right)\), and \(\text {cov} \left \lbrace f^{\ast }_{k}(m_{1}),f_{k}(m_{2}) \right \rbrace =\frac {1}{M^{2}}\text {sinc}^{2}\left (\frac {i\pi }{M}\right)J_{0}\left (2\pi f_{d}(m_{1}-m_{2})\frac {N_{s}}{M}\right)\). If the midpoint CFRs between two different segments are uncorrelated and the segment number M is very small, the final expression of the first term of the interference will be given by
The above equation is simple and can be numerically computed. The second interference term can be evaluated by
4.3 Discussion on the pre-equalized and equalized SIR
4.3.1 Case I
4.3.2 Case II
5 Simulation results
In this section, we have presented the simulation results of the proposed equalization method with the piecewise linear model. We assume practical simulation parameters as follows: the carrier frequency f _{ c }=1800 MHz, the subcarrier spacing Δ f=976.5 Hz with 512 subcarriers, and the OFDM symbol duration T is about 1 ms. The binary phase shift keying (BPSK) is employed to investigate the bit error rate (BER) performance. The channel coefficients are generated by Jakes’ model, and the exponential power delay profile (0,−4.3429,−8.6859 dB) is employed. Additionally, the channel coefficients between different paths are independent identically distributed (i.i.d) with the same maximum normalized Doppler frequency F _{ d } T _{ s }. The power of the multipath channels is normalized to unit.
5.1 BER performance against normalized doppler frequencies of MMSE with PLM and conventional partial FFT
5.2 BER performance against SNR of MMSE with PLM, conventional partial FFT, and banded MMSE equalizer
5.3 BER performance against normalized Doppler frequencies of MMSE with PLM, conventional partial FFT, and banded MMSE equalizer
5.4 BER performance against normalized Doppler frequencies of MMSE with PLM and modified MMSE equalizer with ICI cancellation
6 Conclusions
In this paper, we have investigated the multi-segmental OFDM signal equalizer with PLM, which can improve the BER performance with negligible complexity increases compared to the partial FFT method. We have also proposed the modified version of the equalizer, which can significantly save the computational efforts. Additionally, the theoretical pre-equalized and equalized SIR performance have been given in closed-form mathematical expressions with the aid of the simple numerical evaluation. In the future, the proposed method could be extended to the MIMO applications to suppress the ICI and inter-antenna interference simultaneously.
7 Appendix A: derivation of the piecewise linear model
where the quantities \(J^{\prime }_{n}(x) = \sqrt {\frac {\pi }{2x}}J_{n+\frac {1}{2}}(x)\) and J _{ n }(x) denote the spherical Bessel function of the first kind and Bessel function of the first kind, respectively. For i=0, i.e., x=0, \(J^{\prime }_{0}(x)=1\), \(J^{\prime }_{1}(x)=0\). The second term of the right hand side of (54) can be derived as (51).
7.1 Appendix B: proof of uncorrelated terms between \(\mathbf {w}^{\star H}_{d}\mathbf {f}_{k}\) and \(\mathbf {w}^{\star H}_{d}\mathbf {g}_{k}\)
Declarations
Funding
This work is supported by the Scientific Research Foundation of CUIT (KYTZ201501, KYTZ201502, KYTZ201701), the Sichuan Provincial Department of Science and Technology Innovation and R&D projects in Science and Technology Support Program (2015RZ0060), the Department of Human Resources and Social Security of Sichuan, Scientific Innovation Team projects (15ZA0118,2016Z003,16ZB0210), the National Natural Science Foundation of China (61201094,61601065), and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.
Authors’ contributions
LAL and HW wrote the paper and completed the derivations of the algorithms. YY, GC, and WL derived the SINR analysis and complexity analysis. YH and JD wrote the code for the performance simulation. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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