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 z = 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 r 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)I_K for some nonzero polynomial a(z) of order at most r :

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

where E(z) is an N × N nonzero polynomial matrix of order at most r, and I_K is the K by K identity matrix. An r th order PAR polynomial matrix is called an r 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 r 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)I_N and V(z)=a(z)I_K for some nonzero polynomial a(z) of order at most r :

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

A strong r th order PAR polynomial matrix is called a strong r 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) (r -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 [1,2,3,4,5]. It turns out that a (strong) PARP is necessary and sufficient for the blind identifiability from the output and the precoder,  i.e., using only Y(z) and G(z) at the receiver where Y(z)=H(z)G(z)X(z). More details  are referred to [1,2,3,4,5,6].  Moreover, some optimality about PARP has been studied in [5, 6]. Note that a precoder is called modulated code (MC) and a PARP is renamed as PARMC in [6]. 

In the evening of the day of obtaining the definition of ambiguity resistant precoders in [1], I immediately and accidently and at the first time tried and came up with the polynomial matrix shown in (1.3) in the following figure, and asked myself to check it. After a short while, I found that it indeed satisfies the condition in Definition 1 with E(z)=E, i.e., constant matrix, and z^-1 is replaced by z. This therefore becomes the first example of ambiguity resistant precoders. 

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 and G. Zhou, "On optimal ambiguity resistant precoders in ISI/multipath cancellation," IEEE Trans. on Circuits and Systems - II, vol. 47, no. 8, pp. 746-756, Aug. 2000.

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

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