Frequencydomain equalizers with zero restoration for zeropadded block transmission with high SNR
 Gert Cuypers^{1}Email author and
 Marc Moonen^{1}
https://doi.org/10.1186/s1363801605917
© Cuypers and Moonen. 2016
Received: 25 July 2015
Accepted: 27 March 2016
Published: 14 April 2016
Abstract
This paper focuses on the equalization of block transmissions with zero pad (ZP). If the channel impulse response length does not exceed the ZP length, it is possible to construct zeroforcing equalizers (ZFEs). Improved performance may be achieved using a minimum mean squared error (MMSE) equalizer. However, these equalizers are computationally intensive when a timedomain implementation is used. While the frequencydomain implementation of a ZFE has a lower complexity, it is prone to—potentially infinite—noise enhancement in the presence of spectral zeros. The MMSE equalizer in the frequency domain performs better by limiting the noise enhancement but still loses all information stored at the spectral zeros. This paper presents a method to exploit the redundancy of the padding to recover this lost information, leading to two new frequencydomain equalizers, a ZFE and an MMSElike equalizer. These two equalizers are evaluated through simulations. They have a performance close to the timedomain equalizers, while maintaining the low complexity of the original frequencydomain equalizers. The equalizers are especially useful for scenarios with a high signaltonoise ratio (SNR), where the performance is not limited by the presence of noise, but by the absence of the information stored in the spectral zeros. In practice, this means an SNR >30 dB. With minor modifications, the equalizers can also be applied if the ZP is replaced by a unique word (UW).
Keywords
1 Introduction
The idea of using a discrete Fourier transform (DFT) to (de)modulate data on carriers—referred to as tones or subcarriers—goes back a long time [1]. The transmission is organized in blocks and operates as follows: each tone is assigned a complex data symbol corresponding to a constellation point. These complex data symbols for all tones are packed together in a vector which is converted to the time domain by means of an inverse DFT (IDFT). The resulting timedomain waveform can be seen as a summation of orthogonal signals—one for each tone—hence the name orthogonal frequency division multiplexing (OFDM). This waveform is now transmitted over the channel. The received samples are stored in a vector which is converted back to the frequency domain using a DFT. Unfortunately, the channel causes intersymbol interference and destroys the orthogonality between the tones, complicating the equalization. Both problems can be solved by introducing a cyclic prefix (CP): a block of samples at the end of the timedomain waveform is copied and added in front of the waveform prior to transmission [2]. At the receiver side, the samples pertaining to the CP are discarded. The resulting effect is that the linear channel convolution now appears to be circular. Accordingly, the corresponding channel matrix in the frequency domain becomes a diagonal one and the received symbols on each tone only depend on the transmitted symbols on that tone. Equalization can now easily be done by multiplying with the inverse of the diagonal channel matrix. A good overview is given in [3–5].
Later on, it was realized that this equalization technique is not limited to OFDM but also applicable to singlecarrier block transmissions (SC), see [6], who observes that this idea was already suggested two decades earlier [7].
Since one can be converted into the other by merely adding an orthogonal precoding, OFDM and SC have many similarities and mathematical techniques for one of them can usually be adapted for use with the other. The differences mostly concern practical implementation issues: because SC systems generally have a lower peaktoaverage power ratio (PAPR) and a better robustness against nonlinearities or carrier offset [6, 8, 9], their demands on the analog hardware are lower than those of OFDM. Adaptive OFDM, on the other hand, has the advantage of allowing bit loading and power loading of each subcarrier according to its quality [9].
Furthermore, adding a CP is not the only way to convert the linear channel convolution into a circular one. This can also be obtained by adding a zero pad (ZP) or unique word (UW) to the waveforms in the time domain. Specifically, a CP system relies on the overlapsave (OLS) technique, while a ZP system uses overlapadd (OLA) to obtain a circular convolution. Although CP and ZP systems appear to be interchangeable, there are important differences. More specifically, if the channel frequency response exhibits spectral zeros on or zeros near the unit circle, zeroforcing equalization may be impossible for a CP system or may lead to severe noise enhancement [10, 11]. A CPOFDM system can take this into account by discarding the affected subcarriers if the channel is known at the transmitter or by using forward error correction (FEC) codes if it is not [8, 11]. The signal projection scheme from [12] offers a fundamental solution but is prohibitive in complexity, while the precoding technique from [13] requires the channel to remain constant over a period of multiple blocks. For a CPSC system with a zeroforcing equalizer (ZFE), the presence of spectral zeros in the channel response is even worse, as the enhanced noise spreads out over the entire timedomain block. The classical solution is to limit the noise enhancement by using a minimum mean square error (MMSE) equalizer instead [14]. The information stored in the subcarriers corresponding to the zeros, however, will still be lost.
For a ZP system, on the other hand, zeroforcing equalization is always possible [11, 15]. Unfortunately, using the OLA technique to implement the ZFE in the frequency domain introduces the same vulnerability to spectral zeros in the channel response as in CP systems. This results in an illconditioned problem and again the need to revert to an MMSE equalizer. Again, the information stored in the subcarriers corresponding to the zeros is still lost. A decision feedback equalizer (DFE) on sample level might solve this problem. However, due to the processing in the time domain, its implementation will be much less efficient than the frequencydomain techniques mentioned before.
This paper presents two new frequencydomain equalizers for ZP transmission which can recover the information lost in the spectral zeros, while maintaining a complexity, comparable to frequencydomain solutions. They are applicable to OFDM as well as SC transmissions. The equalizers are analysed through simulation for several realistic scenarios, including situations where the performance is not limited by the signaltonoise ratio (SNR), but by the missing information of the spectral zero. With minor modifications, they can also be used for systems using a UW.
The paper is organized is as follows: in Section 2, the system model and existing equalization techniques are presented. In Section 3, the new frequencydomain equalizers are introduced. In Section 4, simulations results are presented. Finally, in Section 5, conclusions are summarized.
2 System model and equalizers
The following conventions are followed: bold lowercase letters denote vectors and bold uppercase letters denote matrices. Normal uppercase letters denote constants. (.)^{ T }, \((.)^{\mathcal {H}}\), (.)^{ † } and E{.} denote transpose, Hermitian, pseudoinverse and expected value, respectively, and A(:,[l _{1}…l _{k}]) selects columns l _{1}…l _{k} of A. The DFT, IDFT, zero and identity matrix of size K are represented by \(\boldsymbol {\mathcal {F}}_{\mathrm {K}}\), \(\boldsymbol {\mathcal {I}}_{\mathrm {K}}\), 0 _{K} and I _{K}, respectively. Occasionally, for clarity, the dimensions of a matrix are indicated as [.]_{K×L}.
with H the linear convolution matrix of h and n additive noise. Without loss of generality, it is assumed that the noise is zeromean and white with variance \(\sigma _{\mathrm {n}}^{2}\), i.e. \(E\{\mathbf {n} \mathbf {n}^{\mathcal {H}}\}=\sigma _{\mathrm {n}}^{2}\mathbf {I}_{\textrm {N+L}}\).
2.1 Time domain equalization
Here, \(\mathbf {H}^{\mathcal {H}} \mathbf {H}\) is a fullrank N×N matrix and hence always invertible. This is referred to as the ZFETD (timedomain ZFE) where ‘timedomain’ refers to the absence of a transformation to the frequency domain (see Section 2.2). It is the optimal ZFE because it has the smallest noise enhancement.
The calculation of these equalizers has a complexity of \(\mathcal {O}(N^{2})\) flops (using structured matrix inversion formulae) while calculating \(\hat {\mathbf {x}}=\mathbf {Wy}\) requires no more than \(\mathcal {O}(N^{2})\) flops per block [16].
2.2 Frequencydomain equalization based on matrix folding
The calculation of the inversion of the diagonal matrix in this equation requires \(\mathcal {O}(N)\) flops, whereas the calculation of \(\hat {\mathbf {x}}\) requires only \(\mathcal {O}(N\log (N))\) flops per block, if the (I)DFT operation is implemented using the (inverse) fast Fourier transform. Obviously, an important saving is to be made compared to the ZFETD, especially for large values of N. However, this speedup comes at a cost: if the channel has spectral zeros or zeros near the unit circle, the inversion of Λ _{f} may be impossible or lead to very large values and hence a significant noise enhancement.
where the regularization term \(\left (\frac {N+L}{N}\right)\) reflects the difference in noise shaping due to G _{f}.
2.3 Frequencydomain equalization based on matrix extension
The calculation of the inversion of the diagonal matrix in this equation requires \(\mathcal {O}(M)\) flops, whereas the calculation of \(\hat {\mathbf {x}}\) requires \(\mathcal {O}(M\log (M))\) flops per block. Note that, in the absence of spectral zeros, there exists a clever method to express the optimal ZFE, W _{ZFETD} in terms of W _{ZFEFDEXT} and a correction term of low complexity [19].
It is remarkable that the assured possibility of zeroforcing equalization is lost in going from Eq. (1) to Eq. (12), because the number of equations is unchanged (contrary to Eq. (5), where information is truly discarded). The problem arises in Eq. (10) when the matrix H _{e} is not of full rank. For the solution in the time domain, however, H _{e} does not need to be of full rank because the last L elements of x _{ZP} are known to be zero. In the frequency domain, this additional information is unavailable. In the next section, this problem is solved by feeding back this known information and fixing the possible ‘gaps’ in Λ _{e}, thereby restoring the information that was lost due to the spectral zeros.
3 Improved frequencydomain equalization
In Section 2.3, the ZP part of the equalized signal is discarded by the multiplication with \(\left [ \begin {array}{ccc} \mathbf {I}_{\mathrm {N}} &\bigg & \mathbf {O} \end {array} \right ] \)in Eqs. (12) and (13). The equalized ZP, however, contains information that can help to improve the equalization of the useful signal.
3.1 Frequencydomain ZFE with zero restoration
in which Λ _{nz} holds the nonzero diagonal elements of Λ _{e}, Λ _{ ε } holds the (closeto) zero elements and F _{A} _{(V×N)}, and F _{B} _{(V×L)}, F _{C} _{(K×N)} and F _{D} _{(K×L)} are submatrices of F. Because \(\mathbf {F}^{\mathcal {H}} \mathbf {F} = \mathbf {I}\), specific relations hold between these submatrices, e.g. \(\mathbf {F}_{\mathrm {B}}^{\mathcal {H}} \mathbf {F}_{\mathrm {A}}=\mathbf {F}_{\mathrm {D}}^{\mathcal {H}} \mathbf {F}_{\mathrm {C}}\) and \(\mathbf {F}_{\mathrm {A}}^{\mathcal {H}} \mathbf {F}_{\mathrm {A}}+\mathbf {F}_{\mathrm {C}}^{\mathcal {H}} \mathbf {F}_{\mathrm {C}}=\mathbf {I}\).
Note that for high SNR values, this converges to the MMSEsolution of Eq. (13).
with \(\mathbf {Q}=(\mathbf {F}_{\mathrm {D}} \mathbf {F}_{\mathrm {D}}^{\mathcal {H}})^{1} \mathbf {F}_{\mathrm {D}} \mathbf {F}_{\mathrm {B}}^{\mathcal {H}}=\left (\mathrm {F}_{\mathrm {D}}^{\mathcal {H}}\right)^{\dag } \mathbf {F}_{\mathrm {B}}^{\mathcal {H}}\).
So the ZFEZR is indeed a ZFE, capable of recovering the lost information. Note that this derivation holds even if the diagonal elements of Λ _{ ε } are not exactly equal to zero, which is more common in realworld situations.
Note that this procedure involves calculating the pseudoinverse of \(\mathrm {F}_{\mathrm {D}}^{\mathcal {H}}\), which is poorly conditioned if the corresponding columns of the IDFT matrix are close to each other. In other words, the restoration of the information lost in spectral (closeto) zeros works best if these spectral (closeto) zeros are not clustered.
which can easily be shown to satisfy Eq. (2) as well. Note that Eq. (23) assumes that K=P, because F_{A} needs to be square. Under this assumption, the solutions of Eqs. (21) and (23) are mathematically identical, because of the uniqueness of \(\mathbf {F}_{\mathrm {A}}^{1}\). Regarding the computational complexity, however, they are very different. The evaluation of Eq. (23) requires the inversion of F _{A}. Taking into account the Vandermonde structure of this matrix, this requires 5N ^{2}/2 operations [20], which typically is much higher than the evaluation of Eq. (21), as will be shown later. Note that the same author also offered a similar solution, which was the first linear equalization scheme to achieve maximum multipath diversity over singleinput singleoutput wireless links [21].
3.2 Frequencydomain MMSElike equalization with zero restoration
As noted before, frequencydomain MMSE equalizers can avoid dramatic noise enhancement due to an illconditioned channel matrix. However, they too do not offer any solution for the loss of information associated with a (closeto) zero on the diagonal of Λ _{ c }. This can be solved using a structure similar to the ZFEZR presented in the previous section.
with exactly the same Q as in Eq. (21).
For a theoretical analysis of these two equalizers, we refer to the Appendix.
3.3 Remarks
Before proceeding to the simulation results, some additional remarks are in place:
3.3.1 Threshold
Practical channel knowledge is imperfect because it is based on estimates and hence spectral zeros will mostly be identified as spectral closetozeros. Therefore, a decision needs to be made as to which elements of Λ _{e} should effectively be treated as zero, e.g. based on a threshold. This can be set at some percentage of the root mean square value of h or through some other method. In any case, it needs to be such that the number of zeros K is not larger than P. A tradeoff needs to be made: setting a low threshold and identifying too few spectral closetozeros can lead to noise enhancement. Setting a high threshold can lead to too many spectral closetozeros, all of whose received information is discarded.
In the case of reasonably high values of the SNR (e.g. >30 dB), elements of Λ _{e} with an absolute value below σ _{n} should be treated as zeros, because they have a major contribution to the MSE. In the case of nonwhite noise, the threshold should also take into account the SNR level at each subcarrier.
3.3.2 Condition
Related to the previous point, it should be avoided to assign consecutive elements of Λ _{e} as zeros. This would lead to an LS system based on consecutive rows of a DFT submatrix, which can be poorly conditioned.
Both points should lead to some heuristic algorithm to determine the selection of the elements which will be treated as zeros, e.g. the approach used in the simulations section is to only retain the ‘lowest’ of a set of consecutive zeros.
Extensive simulations have shown that treating one element of Λ _{e} as a zero generally does no harm and has the potential to improve the performance significantly. Keeping in mind that spectral zeros are relatively rare, and multiple spectral zeros are even rarer, designating the lowest element of Λ _{e} as a zero suffices for most ‘difficult’ channels. Even if this tone was carrying useful information, this can easily be recovered from the information in the other tones. On the other hand, in the extreme case, where K=P, Eq. (18) is no longer a LS system and will be modeling the noise as well as reconstruct the information stored in the spectral zeros. This should only be considered for relatively high values of the SNR.
3.3.3 UW
The zerorestoration equalizers can also be adapted to UWbased transmission, either by subtracting the contribution of the UW at the receiver and proceeding as in the ZP case or by replacing \(\hat {\mathbf {x}}_{\text {ZP}}\) by \(\hat {\mathbf {x}}_{\text {UW}}\) and modifying the LS system such that the last P samples of \(\hat {\mathbf {x}}_{\text {UW}}\) equal the UW.
3.3.4 Implementation
Note that, while in Eq. (15) the last K columns of \(\mathbf {F}^{\mathcal {H}}\) are effectively multiplied by coefficients equal to zero, even nonzero coefficients may be used here, because these are then absorbed by q anyway. It is therefore also possible to choose coefficients equal to \(\mathbf {\Lambda }_{\epsilon }^{1}\). This implies that the ZFEZR can be implemented as a postprocessing of the ZFEFDEXT.
A practical implementation could therefore start with the evaluation of —without the premultiplication with \(\left [ \begin {array}{ccc} \mathbf {I}_{\mathrm {N}} &\bigg & \mathbf {O} \end {array} \right ]\)—as an alternative for \({\hat {\mathbf {x}}}_{\text {ZP}_{\text {temp}}}\). Next, the LS system from Eq. (18) is solved, which is typically very small. The calculation of q requires \((N+L+\frac {K}{3})K^{2}\) flops ([22], p. 238) and the calculation of \({\hat {\mathbf {x}}}_{\text {ZP}_{\text {corr}}}\) requires another (2K−1)N flops to multiply q and \(\mathbf {F}^{\mathcal {H}}_{\mathrm {C}}\). The overall computational complexity is thus equal to the complexity of Eq. (12) plus approximately \(\mathcal {O}((K^{2}+2K)N)\) flops and is generally significantly smaller than the complexity of the timedomain equalizers of Eqs. (3) and (4). Likewise, the MMSEZR can be implemented as a postprocessing of the MMSEFDEXT. Based on the measurement of the actual channel and the SNR, it can be decided to use the postprocessing step or not.
Comparing the complexity for block size N, ZPlength P and K spectral zeros (M=N+P)
Equalizer  Calculating W  Evaluating \(\hat {\textbf {x}}\) 

W _{ZFETD}, W _{MMSETD}  \(\mathcal {O}(N^{2})\)  \(\mathcal {O}(N^{2})\) 
W _{ZFEFDFOLD}, W _{MMSEFDFOLD}  \(\mathcal {O}(N)\)  \(\mathcal {O}(N\log (N))\) 
W _{ZFEFDEXT}, W _{MMSEFDEXT}  \(\mathcal {O}(M)\)  \(\mathcal {O}(M \log (M))\) 
W _{ZFEZR}, W _{MMSEZR}  \(\mathcal {O}(M)\)  \(\mathcal {O}(M \log (M))\) 
+\(\left (M+\frac {K}{3}\right)K^{2}\)  + (2K−1)N 

(per channel update) evaluating Eq. (13), except for the DFT operations: 2M complex multiplications, M complex additions

to evaluate \(\hat {\mathbf {x}}\):

FFT: \(\frac {1}{2}M \log (M)\) complex multiplications, \(\frac {1}{2}M \log (M)\) complex additions

M complex multiplications

IFFT: \(\frac {1}{2}M \log (M)\) complex multiplications, \(\frac {1}{2}M \log (M)\) complex additions


(per channel update) calculate \(\mathbf {F}_{\mathrm {D}} \mathbf {F}_{\mathrm {D}}^{\mathcal {H}}\), needed for q: P complex multiplications, P−1 complex additions

to evaluate \(\hat {\mathbf {x}}\):

calculating q: P complex multiplications, P−1 complex additions

scaling and adding the complex exponential pertaining to the missing zero: N complex multiplications, N complex additions.

For N=48 and P=16, this means that the number of complex multiplications for a channel update, assuming Λ _{f} is known, increases from 128 to 144 (+ 13 %). The number of complex multiplications to evaluate \(\hat {\mathbf {x}}\) it increases from 448 to 510 (+ 14 %). The figures for the additions are nearly identical.
4 Simulations and discussion
To illustrate the performance of the ZR technique, simulations have been done based on two channels found in literature as well as for the general case of Rayleigh fading. The section is concluded with a more general comparison of equalization techniques using ITU channels.
The OFDM system in the original papers is replaced by ZPSC with M=64, P=L and N=64−P. Note that h _{1}(n) has a spectral closetozero at the 30th subcarrier and h _{2}(n) has a spectral zero as well as two spectral closetozeros. The noise is white Gaussian with different variances to match the SNR. The BER performance is simulated using a Monte Carlo method with 100,000 random blocks, assuming a 16QAM system.

SCZP, using MMSETD, ZFETD, MMSEZR, ZFEZR, MMSEFDEXT, ZFEFDEXT and MMSEFDFOLD

SCCP, using an MMSEstyle FD equalizer

OFDMZP, using MMSEFDEXT and MMSEZR.

OFDMCP, using an MMSEstyle FD equalizer without the final IDFT step. This is the typical OFDM operation.
The best performance for SCZP systems is again obtained using the MMSETD and ZFETD, followed by the MMSEZR and ZFEZR and the other equalizers. It is interesting to see that the SCZP with MMSEFDFOLD has the same performance as SCCP, which makes sense because the operations on the useful signal are identical; only the noise distribution is different. The OFDM schemes systematically perform worse than the corresponding SC alternative. This result is pessimistic, though, because no bit loading is being used. Nevertheless, the OFDMMMSEZR again performs better than the OFDMMMSEFDEXT.
5 Conclusions
Two new frequencydomain equalizers have been proposed to equalize zeropadded OFDM and SCblock transmissions over a channel exhibiting spectral zeros or spectral closetozeros. Both exploit the redundancy of the ZP to restore the lost information. The first technique is shown to have the zero forcing property; the second one is very similar to an MMSE equalizer. A particularly interesting feature is that these equalizers can be implemented by adding a postprocessing to the output of a classical frequencydomain equalizer, typically requiring only very limited additional resources. Simulations show a performance comparable to timedomain equalizers, at a computational complexity comparable to the original frequencydomain equalizers. The equalizers can easily be modified to work with unique wording as well.
6 Appendix
6.1 Theoretical analysis of the performance
The MSE has a contribution stemming from the inexact channel equalization and a noise contribution. The first contribution obviously equals zero for any ZFE.
6.2 ZFE
This MSE has a complex dependency on the noise of the subcarriers corresponding to Λ _{nz} and the correlation between the submatrices of F. However, it is clear that it lacks the detrimental term in \(\boldsymbol {\Lambda }_{\epsilon }^{1}\). As the diagonal elements of Λ _{ ε } approach zero, the MSE of the ZFETDEXT increases unboundedly while the MSE of the ZFEZR is not influenced. Therefore, the latter has a superior performance in the presence of spectral (closeto) zeros.
6.3 MMSE
The MSE of MMSE equalizers has a contribution stemming from the inexact channel equalization as well as a noise contribution. Both will now be analysed for the MMSEFDEXT and the MMSEZR.
6.3.1 Signal contribution
From Eq. (32), it is seen that part of the information is irrevocably lost due to Eq. (34), leading to an irreducible error. If K=1 (only one spectral zero), this MSE contribution is identical for all elements of the block and equal to M ^{−1}; otherwise, it is dependent on the correlation between the rows of F _{C}.
in which the approximation is valid for high SNR values, i.e. small σ _{n}.
Summarizing: in case the elements of Λ _{ ε } are (closeto) zero, the MSE due to inexact channel equalization decreases as the SNR increases. For the MMSEFDEXT, this MSE hits a lower bound for low noise levels. In the case of the MMSEZR, there is no such lower bound, implying that this equalizer has more benefit from a higher SNR. The reason for this difference is that the information stored at the corresponding subcarriers is irrevocably lost for the MMSEFDEXT, while it can be recovered by the MMSEZR, and the quality of this recovered signal improves as the SNR increases.
6.3.2 Noise contribution
It is not trivial to give a quantitative description of this contribution to the MSE. Obviously, in the case of a spectral zero, this MSE contribution will be larger than for the case of the MMSEFDEXT. However, with increasing SNR, i.e. decreasing values of σ _{n}, this contribution will go down rapidly. For spectral closetozeros, the situation is even more favourable, especially at higher SNR values, because the dominant contributions from Λ _{ ε }, as described above, are not present here.
When looking at both the signal and the noise contributions to the MSE, it can be concluded that above a certain SNR threshold, the MMSEZR is expected to perform better than the MMSEFDEXT. Simulations have shown that for realistic scenarios this SNR threshold is typically somewhere between 10 and 30 dB.
Declarations
Acknowledgements
The authors are with the Department of Electrical Engineering (ESAT) and iMinds Future Health department, KU Leuven, Belgium. This research was carried out at ESAT, KU Leuven, in the frame of KU Leuven Research Council CoE EF/05/006 ‘Optimization in Engineering’ (OPTEC) and PFV/10/002 (OPTEC), Concerted Research Action GOAMaNet, the Belgian Programme on Interuniversity Attraction Poles initiated by the Belgian Federal Science Policy Office IUAP P7/19 ‘Dynamical systems, control and optimization’ (DYSCO) 20122016 and IWT Project ‘PHysical layer and Access Node TEchnology Revolutions: enabling the next generation broadband network’ (PHANTER).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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