# Variants of the Memory Polynomial Model

## Orthogonal Memory Polynomial Model

One of the major drawbacks of the memory polynomial model is the ill-conditioning of its data matrix that needs to be constructed when identifying the model coefficients. To better illustrate this aspect, let’s rewrite Equation 5.3 for *L* samples (where *L*

**Figure 5.2 **Block diagram of the memory polynomial model

refers to the length of the data set used to identify the memory polynomial model coefficients):

where the data matrix *& _{MP}(n, L)* is given by:

where ф_{М? }(п) is as defined in Equation 5.4.

The nature of the memory polynomial model introduces a significant amount of correlation in the data matrix, *& _{MP}(n, L),* especially between the elements of each row (ф

_{М? }). Moreover, a lesser degree of correlation is also present between the various rows of

*®*due to the inherent correlation of the signal’s consecutive samples. The correlation of the data matrix results in an ill-conditioning problem that makes the linear system of Equation 5.6 vulnerable to disturbances [2] and numerical instability [3, 4]. This problem gets even more pronounced as the nonlinearity order of the model is increased.

_{MP}(n, L),To alleviate the ill-conditioning problem described here, the orthogonal memory polynomial model was proposed [3]. In this model, a new set of orthogonal basis functions is used. The basis functions are derived for signals whose magnitudes are uniformly distributed in the of range [0,1]. When used with standard compliant communication signals that have a different distribution function such as Raleigh distribution, the advantage of the orthogonal memory polynomial model in terms of condition number reduction, is significantly decreased. However, it still compares favorably with the memory polynomial model, as it achieves moderate condition number reduction for comparable performance and implementation complexity.

The output *(y _{O}MP*) of the orthogonal memory polynomial model is related to its input

*(x)*according to:

where *a*_{mk}, K, and *M* are as defined for Equation 5.2, and *щ[x(n - m)]* represents the basis function of the orthogonal memory polynomial model and is defined as:

The orthogonal memory polynomial model can be re-written in the generic formulation of Equation 5.1 as:

where A is the model coefficients’ vector defined as in Equation 5.5. However, in this case, the data vector Ф* _{OMP}(n)* is expressed, using the orthogonal memory polynomial basis function defined in Equation 5.9, by:

Figure 5.3 presents the block diagram of the orthogonal memory polynomial model. This shows that the model structure is similar to that of the memory polynomial model, and that the only difference lies in the expression of the basis functions adopted. Alternate orthogonal basis functions, such as Zernike polynomials [5, 6] and Gegenbauer polynomials [7], have also been proposed for the modeling and predistortion of power amplifiers and transmitters exhibiting memory effects.