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Polynomial Ambiguity Resistant Precoder (PARP)

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Polynomial Ambiguity Resistant Precoder (PARP)

Xiang-Gen Xia

University of Delaware

 

In [1,2,3], we introduced the concept of polynomial resistant precoder (PARP) that can be applied to an intersymbol interference (ISI) channel, either single input single output (SISO) or multi-input multi-output (MIMO) channel.  With a PARP, in theory neither transmitter nor receiver needs to know the ISI channel, and the receiver can blindly identify an ISI channel and the transmitted signal up to a constant scaling difference. Below let me briefly introduce PARP.

 

A polynomial matrix H(z) of order p and size N × K is an N by K matrix whose all entries are polynomials of z-1 of order at most p, where there is at least one nonzero coefficient of the highest order z-p among all the polynomial entries.  A polynomial matrix H(z) is called irreducible if it has full rank for all nonzero z including = infinity.  A function matrix V(z) is a matrix where all entries are functions of z-1.

 

Definition 1: An N × K irreducible polynomial matrix G(z) is th order polynomial ambiguity resistant (PAR) if the following equation for a K × K function matrix V(z) has only trivial solutions of the form V(z)=a(z)IK for some nonzero polynomial a(z) of order at most :

E(z)G(z) = G(z)V(z)

where E(z) is an N × N nonzero polynomial matrix of order at most r, and IK is the K by K identity matrix. An th order PAR polynomial matrix is called an th order polynomial ambiguity resistant precoder (PARP).


The above polynomial ambiguity resistant property only requires the uniqueness of the right hand side matrix V(z) up to a nonzero polynomial.


Definition 2: An N × K irreducible polynomial matrix G(z) is strong th order polynomial ambiguity resistant if the following equation for an N × N nonzero polynomial matrix E(z) of order at most r and a K × K function matrix V(z) have only trivial solutions of the forms E(z)=a(z)IN and V(z)=a(z)IK for some nonzero polynomial a(z) of order at most :

E(z)G(z) = G(z)V(z).

A strong th order PAR polynomial matrix is called a strong th order PARP.


The above strong polynomial ambiguity resistant property requires a uniqueness up to a nonzero polynomial not only for the right-hand side matrix V(z) but also for the left-hand side nonzero polynomial matrix E(z). Obviously, strong PARP are PARP, and a (strong) r th order PARP is also a (strong) (-1)th order (strong) PARP. 

 

Some simple properties for PARP are, for example, K has to be less than N, i.e., K<N, and any constant matrix G cannot be PARP. This means that some redundancy and memory have to be added in a PARP. PARP and strong PARP have been applied to blind channel identification and/or equalization for both SISO and MIMO channels, and systematically studied and constructed in [1,2,3,4]. It turns out that a (strong) PARP is necessary and sufficient for the blind identifiability from the output and the precoder. More details are referred to [1,2,3,5].  Moreover, some optimality about PARP has been studied in [5], where a precoder is called modulated code (MC) and a PARP is renamed as PARMC. 

 

References

[1] H. Liu and X.-G. Xia, “Precoding techniques for undersampled multi-receiver communication systems,” IEEE Trans. on Signal Processing, vol. 48, pp, 1853-1863, Jul. 2000.

[2] X.-G. Xia and H. Liu, “Polynomial ambiguity resistant precoders: theory and applications in ISI/multipath cancellation,” Circuits, Systems, and Signal Processing, vol.19, no.2, pp.71-98, 2000.

[3] X.-G. Xia, W. Su, and H. Liu, “Filterbank precoders for blind equalization: Polynomial ambiguity resistant precoders (PARP),” IEEE Trans. on Circuits and Systems I, vol. 48, no. 2, pp. 193-209, Feb. 2001.

[4] G. C. Zhou and X.-G. Xia, “Ambiguity resistant polynomial matrices,” Linear Algebra and Its Applications, vol. 286, pp. 19-35, 1999.

[5] X.-G. Xia, Modulated Coding for Intersymbol Interference Channels, New York, Marcel Dekker, Oct. 2000.




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