 Research
 Open Access
 Published:
Perfect reconstruction of signal—a new polynomial matrix inverse approach
EURASIP Journal on Wireless Communications and Networking volume 2018, Article number: 107 (2018)
Abstract
The paper outlines a new approach to the signal reconstruction process in multivariable wireless communications tasks. A new solution is proposed using the socalled Smith factorization, which is efficiently used in the synthesis of control systems described by polynomial matrix notation. In particular, the socalled polynomial Sinverse is used, which, together with the applied degrees of freedom, creates a potential for the improvement of the operation of wireless data communications systems comprising different numbers of inputs/antennas and outputs/antennas. Simulations performed in the Matlab environment indicate the practical applicability of the proposed solution.
Introduction
In mobile wireless communications, increasing attention is paid to the quality and quantity of data transmitted in a given unit of time. A higher capacity of the radio channel of MIMO (multiinput/multioutput) systems, i.e., systems comprising multiple inputs/antennas and multiple outputs/antennas, is gradually replacing the traditional SISO (singleinput/singleoutput) approach [1]. This is confirmed by the widely used WiMAX, WiFi 802.11n, DVBT, or LTE/LTE advanced standards, the majority of which use the OFDM (orthogonal frequency division multiplexing) technology [2–5]. Therefore, an increased capacity of these systems requires the use of a large number of subcarriers and a parallel data transmission mechanism. An intriguing alternative can be therefore seen in systems based on different numbers of transmitting and receiving antennas, where, unlike in the SISO and square MIMO systems (identical numbers of inputs and outputs), the socalled nonuniqueness is present, thus creating viable possibilities of improving the efficiency of the nonsquare wireless communications systems. It should be emphasized that the drawback observed here in the form of interchannel interference (ICI) is eliminated by applying the socalled SVD (singular value decomposition) form in the signal reconstruction process [6–8], dedicated solely to the traditional analysis based on a parameter matrix calculus [9–11]. On the other hand, in the approach based on the polynomial matrix calculus [12–14], the aforementioned dysfunction is eliminated by using the PSVD (polynomial SVD) [15–17]. Unfortunately, all mentioned approaches to the signal perfect reconstruction [18–20], also those including the Smith decomposition method [21], have not so far included the socalled degrees of freedom [22]. Therefore, they were solely related to a certain “optimal” solution associated with the application of the minimumnorm/leastsquares inverses to an Eigen matrix obtained from the factorization process [11]. What is important is that even though the above methods involve an infinite number of pairs of precoderequalizer, our degrees of freedom should be understood in terms of usage of the different inverses to Eigen matrix under a unique precoderequalizer pair. It will be shown that former cases are quite inappropriate for the polynomial matrix description [16], and the Smith factorization method proposed here outperforms the classical solutions remarkably.
Method
In this paper, a new analytical solution to the signal perfect reconstruction is presented. Not only does this approach in discretetime domain, dedicated to nonsquare systems, eliminates parasitic effects in the form of ICI and ISI (intersymbol interference), but also it efficiently uses the highly expected mechanism of nonuniqueness, which considerably improves the “robustness” of wireless communications. The approach proposed herein is based on the socalled Smith form of nonsquare polynomial matrices, which is the foundation of the polynomial Sinverse [22, 23].
System representation
We carry out the analysis of the wireless data communications system with N_{T}transmitter antennas and N_{R}receiver antennas described by the (discrete) polynomial matrix in the form of \(\mathbf {\underline {C}}\left ({z^{1}}\right)\) [16]
where (L_{c}1) is the order of the FIR (finite impulse response) matrix \(\mathbf {{\underline C}}\left ({z^{1}}\right)\).
The deterministic signal reconstruction process is performed here in accordance with the difference equation
where the vector of transmitted signals S(t) and the vector of received signals R(t) have the dimensions N_{T} and N_{R}, respectively (see Section 4). Note that the symbol t denotes a discretetime domain, whereas q^{−1} is the backward shift operator corresponding to z^{−1} one.
In the new perfect reconstruction approach presented in this paper, a number of different inverses are used; they are described in detail in the next section.
Inverses of nonsquare polynomial matrices
Due to the nonsquare form of \(\mathbf {\underline {{C}}}_{N_{\mathrm {R}}\times N_{\mathrm {T}}}\left ({z^{1}}\right)\), the authors suggest using new inverses of nonsquare polynomial matrices [22–25] in the signal reconstruction process [22]. We start with the classical minimumnorm right and leastsquares left inverses known as Tinverses in the polynomial case.
Tinverses
For the polynomial matrix \(\mathbf {{\underline {C}}}\left (q^{1}\right)=\mathbf {\underline {c}_{0}} + \mathbf {\underline {c}_{1}}q^{1} +\ldots +\mathbf {{\underline {c}}_{m}}q^{m}\) of full normal rank, the unique minimumnorm right Tinverse is defined as
while the unique leastsquares left Tinverse is in the following form
τinverses
The nonunique right τinverse of the polynomial matrix \(\mathbf {{\underline {C}}}\left (q^{1}\right)\) is defined as (N_{R}<N_{T})
where polynomial matrices \(\mathbf {\underline {\beta }} \left (q^{1}\right)\) and \(\mathbf {\underline {\beta }_{s}} \left (q^{1}\right)\) are defined in References [22, 23]. On the other hand, the nonunique left τinverse takes the following form (N_{R}>N_{T})
The aforementioned forms \(\left [\mathbf {{\underline {\beta }}_{s}} \left (q^{1}\right)\right ]_{0}^{\mathrm {R}}\) and \(\left [\mathbf {{\underline {\beta }}_{s}} \left (q^{1}\right)\right ]_{0}^{\mathrm {L}}\) stand for the minimumnorm right and leastsquares left Tinverses of polynomial matrix \(\mathbf {{\underline {\beta }}_{s}} \left (q^{1}\right)\), respectively, while \(\mathbf {I}_{N_{\mathrm {T}}}\) is the identity N_{T}matrix.
σinverses
A generalization of the polynomial τinverses is the socalled right
and left
nonunique σinverses implementing the degrees of freedom in the form of an arbitrary matrix polynomial \(\mathbf {{\underline {\beta }}} \left (q^{1}\right)\).
It should be emphasized that the new forms of polynomial right and left σinverses (including also the parameter cases) are given in References [24, 26] as follows:
Crucial nonunique Sinverses are presented below. They are effectively used when designing robust communications systems.
Sinverses
Nonunique polynomial Sinverses are associated with the socalled Smith factorization of the polynomial matrix \(\mathbf {{\underline {C}}} \left (q^{1}\right)\) to obtain
where \(\mathbf {{\underline {U}}}\left (q^{1}\right)\) and \(\mathbf {{\underline {V}}}\left (q^{1}\right)\) are unimodular polynomial matrices, and the unique matrix polynomial \(\mathbf {{\underline {\Sigma }}}\left (q^{1}\right)\) of dimension N_{R}×N_{T} includes the eigenvalues of \(\mathbf {\underline C}\left (q^{1}\right)\). The right and left Sinverses are defined as
respectively, where (non)unique right and left inverses of the polynomial matrix \(\mathbf {{\underline {\Sigma }}}\left (q^{1}\right)\) include the degrees of freedom. Note that the parameter counterpart of the Sinverse strictly dedicated to statespace systems has been given in Reference [25].
Remark 1
It should be noted that all of the abovementioned inverses are reduced to the regular one \(\mathbf {\underline {C}}^{1}\left (q^{1}\right)\) in case of N_{R}=N_{T}.
New approach to signal reconstruction process
MIMO wireless communications systems, including multiple transmitter and receiver antennas, are becoming more and more common, and they even replace traditional SISO solutions by offering high transmission/reception channel capacity improvement. Multivariable systems ensure not only an increase in capacity but also, importantly, improvement without loss of the required technological parameters of the received signal. An intriguing case here is an approach implementing different numbers of transmitting and receiving antennas. In such nonsquare systems, we can find the nonuniqueness of the obtained solution, which has a positive impact on the whole signal reconstruction/recovery process. By selecting appropriate degrees of freedom of inverses, we can considerably influence the robustness and energy of the received signal (in the case of the control theory see Reference [26]). In the authors’ opinion, the new method can eliminate the parasitic impact of the natural environment in the context of the applied inverses of nonsquare polynomial matrices. Such operations directly improve the signal transmission rate while maintaining approved quality standards. Of course, the entire signal perfect reconstruction process only occurs in case of N_{R}>N_{T}, since we have full information about the transmitted signal.
It is important that the proposed approach to perfect reconstruction of signal is based on the polynomial matrix calculus. The solutions used so far were based on the parameter matrix calculus, using unique inverses with the socalled Hermitian conjugates of certain (full rank) nonsquare matrices [27]. Unfortunately, this calculus does not include the aforementioned degrees of freedom, thus making it considerably more difficult to adjust to the detrimental impact of the environment on the data transmission process.
Remark 2
It should be emphasized that the unique right and left inverses including the Hermitian conjugates are not applicable in the timedomain signal perfect reconstruction approach presented here [23].
To illustrate the discussed problems, let us analyze the stochastic process of perfect reconstruction of signal and rewrite Eq. (2) to the form
where ζ(t) is the uncorrelated zeromean Gaussian white noise at (discrete) time t.
Then, let us perform the nonunique Smith factorization of \(\mathbf {\underline {C}}\left (q^{1}\right)\) and, at the same time, eliminate ICI and ISI parasitic effects
where the polynomial matrices \(\mathbf {{\underline {U}}}\left (q^{1}\right)\) and \(\mathbf {{\underline {V}}}\left (q^{1}\right)\) are the “equalizer” and the “precoder,” respectively [22].
After using the Sinverse, the perfect reconstruction of signal for the selected N_{R}>N_{T} takes the following form
where the symbol “L” stands for the (non)unique left inverse of matrix polynomial \(\mathbf {{\underline {\Sigma }}}\left (q^{1}\right)\).
Of course, Eq. (16) can be rewritten in the following form
with \(\mathbf {S}^{\prime }(t)=\mathbf {S}(t)+\mathbf {{\underline {V}}}^{1}\left (q^{1}\right)\mathbf {{\underline {\Sigma }}}^{\mathrm {L}}\left (q^{1}\right){\mathbf {\underline {U}}}^{1}\left (q^{1}\right)\mathbf {\zeta } (t)\) being a stochastic N_{T}input vector.
Taking into account the above considerations, for N_{R}>N_{T}, we obtain [28]
where the polynomial matrices \(\mathbf {\underline {M}}\left (q^{1}\right)\) and \(\mathbf {\underline {D}}\left (q^{1}\right)\) include significant degrees of freedom and transmission zeros (if any in the \(\mathbf {\underline {C}}\left (q^{1}\right)\) [23]), respectively. In case of absence of transmission zeros, we have \(\mathbf {\underline {D}}\left (q^{1}\right)=\mathbf {I}_{N_{\mathrm {T}}}\).
Finally, based on the pilot knowledge, the optimal degrees of freedom of \(\mathbf {\underline {M}}\left (q^{1}\right)\) are chosen according to the square performance index
where N denotes the number of samples.
Remark 3
\(\mathbf {{\underline {\Sigma }}}^{\mathrm {L}}\left (q^{1}\right)\) can also be obtained by using polynomial σinverses in two parallel forms
where \( \left [\mathbf {{\underline {\beta }}} \left (q^{1}\right)\right ]_{0}^{\mathrm {L}}=\left [\mathbf {{\underline {\beta }}}^{\mathrm {T}} \left (q^{1}\right)\mathbf {{\underline {\beta }}} \left (q^{1}\right)\right ]^{1} \mathbf {{\underline {\beta }}}^{\mathrm {T}} \left (q^{1}\right)\) while \(\mathbf {{\underline {\beta }}} \left (q^{1}\right)\) includes the degrees of freedom [24].
Remark 4
If we apply the unique left Tinverse directly to \(\mathbf {\underline {C}} \left (q^{1}\right)\) or \(\mathbf {\underline {\Sigma }} \left (q^{1}\right)\), we will obtain no degrees of freedom.
Remark 5
It should be pointed out that in square systems, i.e., systems having equal numbers of transmitting and receiving antennas, there are also no degrees of freedom. Therefore, the optimization cannot efficiently eliminate the impact of noise on the whole signal reconstruction process.
Remark 6
We start our optimization task in the Matlab environment with the degrees of freedom included in the parameter matrix; a more general case is based on the matrix polynomial.
Remark 7
Now, it is clear that in the deterministic case, we immediately obtain the perfect reconstruction of signal according to the following formula
where \(\mathbf {\underline {C}}^{\mathrm {L}}\left (q^{1}\right)\) denotes the abovementioned polynomial matrix Sinverse of full normal rank \(\mathbf {\underline {C}}\left (q^{1}\right)\).
Remark 8
It should be emphasized that in the deterministic case of the signal perfect reconstruction as mentioned in Remark 7, the left inverse of \(\mathbf {\underline {C}}\left (q^{1}\right)\) is not determined; due to the elimination of ISI and ICI drawbacks, the Sinverse has been applied. Therefore, assuming that \(\mathbf {R}(t)=\mathbf {\underline {C}}\left (q^{1}\right)\mathbf {S}(t)\), the stochastic recovery task as presented in Eq. (16) can be rewritten in the following form
where \(\mathbf {\underline {\Sigma }}^{\text {L1}}\left (q^{1}\right) \neq \mathbf {\underline {\Sigma }}^{\text {L2}}\left (q^{1}\right)\), in general, under any \(\mathbf {\underline {\Sigma }}^{\mathrm {L}}\left (q^{1}\right)\).
Remark 9
Since our new polynomial method of perfect signal reconstruction does not correspond to the MoorePenrose inverse, we must consider them separately.
Remark 10
The aforementioned signal recovery may be impossible after using the unique Tinverses in case of nonsquare systems (previously known as minimumnorm right/leastsquares left inverses); these inverses may significantly destabilize the whole signal reconstruction process due to the existence of socalled unstable control zeros [22]. The whole signal reconstruction process is always destabilized in case of the existence of unstable transmission zeros which are the modes of the fundamental system matrix \(\mathbf {\underline {C}}\left (q^{1}\right)\) [22, 23].
Remark 11
Note that the entire task of signal reconstruction should be understood in terms of an adaptive process, where the said degrees of freedom are selected cyclically with a period adjusted by the designer.
What is important is that the solution based on the polynomial matrix calculus (along with nonzero degrees of freedom, i.e., at \(\mathbf {\underline {M}}\left (q^{1}\right)\neq \mathbf 0\)), is a new approach so far unknown in the field of the modern signal reconstruction. The application of left inverses in Eq. (16) improves the capacity/robustness of the wireless communications network in terms of the elimination of the parasitic impact of noise. Of course, it is possible by choosing appropriate components/degrees of freedom of matrix \(\mathbf {{\underline {M}}}\left (q^{1}\right)\) of Eq. (18) according to the criterion (19). The same can be achieved as a result of applying nonunique type τ and σinverses. An adequate selection of the degrees of freedom \(\mathbf {{\underline {\beta }}_{s}} \left (q^{1}\right)\) and \(\mathbf {{\underline {\beta }}} \left (q^{1}\right)\) that are not relative to the propagation environment \(\mathbf {{\underline {C}}}\left (q^{1}\right)\) can provide a greater degree of independence of the parasitic effects (see Eqs. (20) and (21)). Finally, we can strongly note that an alternative to the applied Smith decomposition can be offered by the use of the PSVD method implementing the nonzero degrees of freedom, whose derivation is either based on the use of the PEVD (polynomial EVD) approach [29, 30], or one that is obtained in a direct manner [31]. This intriguing proposal is briefly described in the next section.
PSVD vs. Smith decompositionbased approach
In Reference [16], the authors applied a successful polynomial singular value decomposition. As stated earlier, the current methods in the wireless telecommunications studies use zero degrees of freedom. Hence, the idea of the use of a new nonzero degrees of freedom was conceived with the purpose of limiting the impact of the noise on the process of data transfer. The same paradigm can be applied in the signal reconstruction process based on the PSVD method, which is worth further research. However, the methods basing on SVD and PSVD decompositions cannot be directly compared with the new method applied for signal recovery. The reason for this can be associated with different dynamic parameters of the propagation environments derived as a result of using SVD and PSVD on one hand and a method applying polynomial Sinverse on the other hand. In the former case, we merely obtain an approximation of the dynamic properties of the propagation environment, whereas the latter approach implies that an accurate dynamics of this environment is obtained. Appendix offers an outline of the working characteristics of the signal reconstruction methods for a nonsquare system comprising two transmitting antennas and four receiving ones.
The paradigm of the new intriguing method is analytically confirmed by a more general first polynomial part of the next section containing simulation examples, whereas the second parameter part contains the results of complex optimization runs using the genetic algorithm mechanism.
Results and discussion
Let us analyze the wireless telecommunications system with no transmission zeros, including two transmitter antennas N_{T}=2 and three receiver antennas N_{R}=3. Assuming that the matrix obtained by the pilot identification and describing the dynamics of the parasitic impact of the environment on the signal reconstruction process takes the following form
After using the Smith factorization of \(\mathbf {{\underline {C}}}\left (q^{1}\right)\), we obtain
and containing no transmission zeros
Now, in accordance with the signal perfect reconstruction as presented in Eq. (16), for the received signal R^{′}(t)=[R1′(t)R2′(t)R3′(t)]^{T} (due to its complexity, vector R^{′}(t) blurred by a zeromean white noise ζ(t) is not given here), we obtain the vector of the transmitted signal
Finally, for the determined value ζ(t)=[0.1 − 0.2 0.2]^{T} and the degrees of freedom of matrix \({\mathbf {\underline {M}}}(q) =\!\!\begin {array}{*{20}c} \left [ \! {\begin {array}{*{20}c} {\frac {{343712}}{{841\left ({\text {1996}{q}^{2}  2023 {q} + 521} \right)}}} & {\frac {{\text { 14944(38348} {q}^{2} + 15109 {q})}}{{707281\left ({\text {1996} {q}^{2}  2023{q} + 521} \right)}}} \end {array}}\!\! \right ]\end {array}\!\!= \![\underline M_{1}(q)\,\underline M_{2}(q)]^{\mathrm {T}}\!\!,\) selected as a result of analytical calculations, the reconstructed vector S(t) is S(t)=[3+3i − 3−i]^{T} (corresponding to the two points of 16QAM constellation of transmitted signal S(t)). Note that in our simulation example, there is \(\mathbf {\underline {\Sigma }}^{\mathrm {L}}\left (q^{1}\right) = \mathbf {\underline {\Sigma }}^{\text {L1}}\left (q^{1}\right) = \mathbf {\underline {\Sigma }}^{\text {L2}}\left (q^{1}\right)\), see Remark 8.
To better describe the advantages of the method proposed, complex tests were carried out by using the authors’ OFDM technology simulator running in the Matlab environment [28]. For this purpose, 103776 bits of random input data from 64QAM constellation were transferred by means of the IQmodulated signal through the single carrier system with the matrix \(\mathbf {{\underline {C}}}\left (q^{1}\right)\) as in Eq. (24). For the assumed rigorous tolerance, the special parameter matrices \(\mathbf {{\underline {M}}}\left (q^{1}\right)\) were obtained using the genetic algorithm according to the performance index (19). Thus, we have different degrees of freedom for each of the SNRs, not presented in this paper due to space limitation. It is evident that the new method outperforms the classical one, where zero degrees of freedom associated with the application of minimumnorm/leastsquares inverses to \(\mathbf {\underline {\Sigma }}\left (q^{1}\right)\) a polynomial matrix can be find. This statement is confirmed in Fig. 1.
In addition, two simulation tests were performed covering propagation environments described by the following matrices
and
respectively.
Figure 2 presents the results obtained for single carrier system given by Eq. (29), whereas Fig. 3 for single carrier system as in Eq. (30).
Conclusions
In this paper, the new approach to the process of perfect reconstruction of signals is presented. The new solution is based on polynomial matrix calculus, mainly the socalled left Sinverse of the polynomial matrix. Errors generated in the process of signal reconstruction are compensated by the appropriate selection of components/degrees of freedom of the nonzero matrix \(\mathbf {\underline {M}}\left (q^{1}\right)\). Simulation tests carried out in the Matlab environment have indicated a considerable implementation potential of the innovative approach proposed in this paper to the tasks of efficient signal recovery in nonsquare MIMO telecommunications systems. It should be emphasized that the new method still outperforms the typical one in case of presence of noise with uniform distribution.
Appendix
Method based on PSVD
The matrix applied to described the dynamics of the propagation environment assumes the form
Following PSVD factorization, we receive
where
and \(\mathbf {V}^{\dagger }(q)=\mathbf {V}^{\mathrm {H}}(1/q^{*})= \left [ \begin {array}{ll} 3q^{2}&\qquad 0 \\ \;\;0& \thinspace 3q^{1} \\ \end {array} \right ]\).
Evidently, U(q)U^{†}(q)=U^{†}(q)U(q)=2.5I_{4} and V(q)V^{†}(q)=V^{†}(q)V(q)=9I_{2}, where I_{ n } denotes nidentity matrix, and both U(q) and V(q) are paraunitary matrices.
Let us consider a deterministic process of perfect signal reconstruction
where R(t) and S(t) are the vectors of the received and transmitted signals, respectively. By consideration of Eq. (I.2) and application of the precoder V(q) and equalizer U^{†}(q) structures (for an example see Reference [15]), we receive
Unfortunately, R^{′}(t)≠R(t).
Method based on SVD
By the analogy to the case of PSVD for R(t)=CS(t), where C is a parameter matrix, we receive
where V and U^{†} denote the precoder and equalizer structures, respectively, fulfilling the condition of unitarity.
In this case, also R^{′}(t)≠R(t).
Method based on polynomial Sinverse
Taking into consideration that
where \(\mathbf {{\underline {U}}}\left (q^{1}\right)\) and \(\mathbf {{\underline {V}}}\left (q^{1}\right)\) are unimodular matrices obtained as a result of applying Smith factorization, Eq. (I.3) can be written in the following form
By solving Eq. (III.7) in respect to S(t), we receive the actual errorfree transmitted signal S(t) in accordance with the relation
where superscript “L” denotes every nonunique left inverse of matrix polynomial \(\mathbf {{\underline {\Sigma }}}^{\mathrm {L}}\left (q^{1}\right)\).
Finally, let us remark that we cannot directly compare the two methods of signal recovery, i.e., approaches involving respective (P)SVD and Smith factorization mechanisms. The reason for this was associated, for example, with the lack of causation phenomenon in the precoder and/or equalizer structures (see Eq. (I.4), where, e.g., \(\mathbf {U}(q) = \left [ \begin {array}{lrrr} 0.5 & q & 0.5 & q^{2} \\ \thinspace q^{1}& 0.5&\thinspace q^{2}&0.5 \\ 0.5& q&\thinspace 0.5&\thinspace q^{2} \\ \thinspace q^{1}& 0.5&q^{2}&\thinspace 0.5 \\ \end {array} \right ]\), whereas for the case when, e.g., q^{2} constitutes a double feedforward, i.e., we have y(t)=u(t+2)).
Abbreviations
 BER:

Bit error rate
 FIR:

Finite impulse response
 ICI:

Interchannel interference
 ISI:

Intersymbol interference
 MIMO:

Multiinput/multioutput
 OFDM:

Orthogonal frequency division multiplexing
 PEVD:

Polynomial eigenvalue decomposition
 PSVD:

Polynomial singular value decomposition
 SISO:

Singleinput/singleoutput
 SNR:

Signaltonoise ratio
References
 1
A Goldsmith, SA Jafar, N Jindal, S Vishwanath, Capacity limits of MIMO channels. IEEE J. Sel. Areas Commun.21(5), 684–702 (2003).
 2
H Shu, EP Simon, L Ros, On the use of tracking loops for lowcomplexity multipath channel estimation in OFDM systems. Signal Process.117:, 174–187 (2015).
 3
Z Zhao, M Schellmann, X Gong, Q Wang, R Böhnke, Y Guo, Pulse shaping design for OFDM systems. EURASIP J. Wirel. Commun. Netw.2017(74) (2017). https://jwcneurasipjournals.springeropen.com/track/pdf/10.1186/s1363801708498.
 4
X Ji, Z Bao, Ch Xu, Power minimization for OFDM modulated twoway amplifyandforward relay wireless sensor networks. EURASIP J. Wirel. Commun. Netw.2017(70) (2017). https://jwcneurasipjournals.springeropen.com/track/pdf/10.1186/s1363801708489.
 5
D Mattera, M Tanda, Optimum singletap persubcarrier equalization for OFDM/OQAM systems. Digit. Signal Process.49:, 148–161 (2016).
 6
M Nagahara, Discrete signal reconstruction by sum of absolute values. IEEE Signal Process. Lett.22(10), 1575–1579 (2015).
 7
O Nibouche, J Jiang, Palmprint matching using feature points and SVD factorisation. Digit. Signal Process.23(4), 1154–1162 (2013).
 8
QH Spencer, A Lee Swindlehurst, M Haardt, Zeroforcing methods for downlink spatial multiplexing in multiuser MIMO channels. IEEE Trans. Signal Process.52(2), 461–471 (2004).
 9
EKS Au, S Jin, MR McKay, WH Mo, Analytical performance of MIMOSVD systems in ricean fading channels with channel estimation error and feedback delay. IEEE Trans. Wirel. Commun.7(4), 1315–1325 (2008).
 10
D Sellathambi, J Srinivasan, S Rajaram, in Proc. of the International Conference on Design and Manufacturing (IConDM’2013), Procedia Engineering, 64. Hardware implementation of adaptive SVD beamforming algorithm for MIMOOFDM systems (ElsevierChennai, 2013), pp. 84–93.
 11
M Sadek, A Tarighat, AH Sayed, A leakagebased precoding scheme for downlink multiuser MIMO channels. IEEE Trans. Wirel. Commun.6(5), 1711–1721 (2007).
 12
WQ Wang, H Shao, J Cai, MIMO antenna array design with polynomial factorization. International Journal of Antennas and Propagation, Special Issue: MIMO Antenna Design and Channel Modeling. 358413:, 9 (2013). https://www.hindawi.com/journals/ijap/2013/358413/.
 13
J Lebrun, P Comon, Blind algebraic identification of communication channels: symbolic solution algorithms. AAECC. 17(6), 471–485 (2006).
 14
GH Norton, On the annihilator ideal of an inverse form. AAECC. 28(1), 31–78 (2017).
 15
IA Akhlaghi, H Khoshbin, A novel method for singular value decomposition of polynomial matrices and ICI cancellation in a frequencyselective MIMO channel. Int. J. Tomogr. Stat.11(S09), 83–98 (2009).
 16
N Moret, A Tonello, S Weiss, in Proc. of the 73rd IEEE Vehicular Technology Conference (VTC’2011). MIMO precoding for filter bank modulation systems based on PSVD (Budapest, pp. 1–5.
 17
H ZamiriJafarian, M Rajabzadeh, in Proc. of the 67th IEEE Vehicular Technology Conference (VTC’2008). A polynomial matrix SVD approach for time domain broadband beamforming in MIMOOFDM systemsMarina Bay, pp. 802–806.
 18
L Li, G Gu, Design of optimal zeroforcing precoders for MIMO channels via optimal full information control. IEEE Trans. Signal Process.53(8), 3238–3246 (2005).
 19
S Li, J Zhang, L Chai, in Proc. of the 20th European Signal Processing Conference (EUSIPCO’2012). Noncausal zeroforcing precoding subject to individual channel power constraintsBucharest, pp. 1623–1627.
 20
A Scaglione, S Barbarossa, GB Giannakis, Filterbank transceivers optimizing information rate in block transmissions over dispersive channels. IEEE Trans. Inf. Theory. 45(3), 1019–1032 (1999).
 21
AV Krishna, Hari KVS, Filter bank precoding for FIR equalization in highrate MIMO communications. IEEE Trans. Signal Process.54(5), 1645–1652 (2006).
 22
WP Hunek, Towards a General Theory of Control Zeros for LTI MIMO SystemsOpole University of Technology Press, Opole, 2011).
 23
WP Hunek, KJ Latawiec, A study on new right/left inverses of nonsquare polynomial matrices. Int. J. Appl. Math. Comput. Sci.21(2), 331–348 (2011).
 24
WP Hunek, KJ Latawiec, R Stanisławski, M Łukaniszyn, P Dzierwa, in Proc. of the 18th IEEE International Conference on Methods and Models in Automation and Robotics (MMAR’2013). A new form of a σinverse for nonsquare polynomial matrices (Miedzyzdroje, pp. 282–286.
 25
WP Hunek, in Proc. of the 20th IEEE International Conference on System Theory, Control and Computing (ICSTCC’2016). New SVDbased matrix Hinverse vs. minimumenergy perfect control design for statespace LTI MIMO systemsSinaia, pp. 14–19.
 26
WP Hunek, in Proc. of the 8th IFAC Symposium on Robust Control Design (ROCOND’2015). An application of new polynomial matrix σinverse in minimumenergy design of robust minimum variance control for nonsquare LTI MIMO systemsBratislava, pp. 149–153.
 27
Y Zhang, Y Li, M Gao, Robust adaptive beamforming based on the effectiveness of reconstruction. Signal Process.120:, 572–579 (2016).
 28
P Majewski, Research towards increasing the capacity of wireless data communication using inverses of nonsquare polynomial matricesPhD thesis, Opole University of Technology, Opole, 2017).
 29
JG McWhirter, PD Baxter, T Cooper, S Redif, J Foster, An EVD algorithm for paraHermitian polynomial matrices. IEEE Trans. Signal Process.55(5), 2158–2169 (2007).
 30
S Weiss, AP Millar, RW Stewart, in Proc. of the 18th European Signal Processing Conference (EUSIPCO’2010). Inversion of parahermitian matrices (Aalborg, pp. 447–451.
 31
JG McWhirter, in Proc. of the 18th European Signal Processing Conference (EUSIPCO’2010). An algorithm for polynomial matrix SVD based on generalised Kogbetliantz transformationsAalborg, pp. 457–461.
Acknowledgements
Invaluable comments from the anonymous reviewers are gratefully acknowledged.
Funding
This work was funded by the Department of Electrical, Control and Computer Engineering, Opole University of Technology, Poland.
Author information
Affiliations
Contributions
Both authors contributed to the work. Both authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
Both authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
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.
About this article
Cite this article
Hunek, W.P., Majewski, P. Perfect reconstruction of signal—a new polynomial matrix inverse approach. J Wireless Com Network 2018, 107 (2018). https://doi.org/10.1186/s1363801811225
Received:
Accepted:
Published:
Keywords
 Perfect signal reconstruction
 Polynomial matrix approach
 Polynomial matrix inverses
 Nonsquare MIMO systems
 Communications systems