# Generic Memory Polynomial Model Formulation

Memory polynomial based models are commonly used as standalone single-box models. However, they can also be part of two-box structures or in general multi-box structures. In this chapter, the main focus is memory polynomial based single-box models. Their implementation in two-box structures will be discussed in Chapter 6.

At this point, it is important to distinguish between the concept of a single-box model and a single-basis function model. A single-box model refers to a structure that is identified as a single function, in contrasts with a multi-box model where the sub-functions are determined successively. Thus, a single-box model can have more than one single basis function as, for example, is the case in the generalized memory polynomial model. This concept is further discussed in the subsequent sections of this chapter where single-box multi-basis function models are described and their mathematical formulations are derived.

All memory polynomial models can be formulated using the same generic linear system given by:

where *y(n)* is the model’s baseband complex output sample at instant n, ф(п) is a vector built using the baseband complex input signal samples according to the model’s basis functions set, and A is the vector containing the model coefficients.

The formulation of Equation 5.1 is valid for all memory polynomial models, independent of their type and the number of basis functions. The only difference is that vector ф(п) is defined depending on the model. It is important to notice here that all memory polynomial models are linear with respect to their coefficients, which enables the use of simple identification techniques.