Generalized Memory Polynomials

Two kinds of memory polynomials f are implemented:

• without cross-terms (cross_terms = False)

$$x_{\text{out}}(n) = f(x_\text{in}(n)) = \sum_{m=0}^{M-1} \sum_{k=0}^{K-1} a_{mk} \cdot x_{\text{in}}(n - m) \cdot | x_{\text{in}}(n - m) |^k$$

• with cross-terms (cross_terms = True)

$$x_{\text{out}}(n) = f(x_\text{in}(n)) = \sum_{m=0}^{M-1} c_m \cdot x_{\text{in}}(n - m) + \sum_{m=0}^{M-1} \sum_{j=0}^{M-1} \sum_{k=1}^{K-1} a_{mjk} \cdot x_{\text{in}}(n - m) \cdot | x_{\text{in}}(n - j) |^k$$

The parameter $M$ is called memory depth and the parameter $K$ is called degree.

To determine the coefficients ($a_{mk}$ or $c_m$ and $a_{mjk}$), a data matrix $X$ is constructed which collects all terms of the form $x_{\text{in}}(n - m)$ and $x_{\text{in}}(n - m) \cdot |x_{\text{in}}(n - j)|^k$. Then, if also the samples $x_{\text{out}}(n)$ are collected in a vector $x_{\text{out}}$ and if all coefficients are collected in a vector $c$, the solution is determined as

$$c = (X^{\mathrm{H}} X + \alpha I)^{-1} X^{\mathrm{H}} x_{\text{out}}$$

with a regularization parameter $\alpha \geq 0$ (alpha).

Example

To instantiate a memory polynomial without cross-terms and with $M=3$, $K=2$, $\alpha = 0.1$, write

mem_pol = gmp.MemoryPolynomial(degree=2, mem_depth=3, alpha=0.1, cross_terms=False)

and determine the coefficients via

c = mem_pol.fit(x_in, x_out)

To predict/equalize/postdistort some data y_in, use

y_out = mem_pol(y_in)

Reference

Morgan, Ma, Kim, Zierdt, Pastalan, "A Generalized Memory Polynomial Model for Digital Predistortion of Power Amplifiers," IEEE Trans. Signal Process., 2006.