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I want to build a memory polynomial model given that I know the corresponding output signals for known input signals. What is the easiest way to find the coefficients of it? And how do we take the memory and order into consideration while we build it.

This is what I know. if A and B are input and output signals of length n, then A_inverse x B will give me n by n matrix. But how can I introduce memory and order?, if i add any extra columns to A_inverse x B, it will change the dimensions to n by m where m > n. and i can no more operate it on A. 'x' means matrix multiplication.

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Memory polynomial system identification is a linear regression problem. Suppose you sent a $N$-element sequence $x(n)$ through a system and obtained an $N$-element output sequence $y(n)$. If the system were described by a memory polynomial, then your model would be...

$y(n) = \sum_{q=0}^{Q}[x(n-q)a(q)+x(n-q)|x(n-q)|^2 b(q)] + \epsilon(n)$

Here $a(q)$ and $b(q)$ are coefficient of the models and $\epsilon(n)$ is an error term. This is a linear model in $a$ and $b$, and can be expressed in matrix form as...

$\begin{bmatrix}y(Q)\\y(Q+1)\\...\\y(N-1)\end{bmatrix} = \begin{bmatrix}x(Q)&...&x(0)&x(Q)|x(Q)|^2&...&x(0)|x(0)|^2\\x(Q+1)&...&x(1)&x(Q+1)|x(Q+1)|^2&\ldots&x(1)|x(1)|^2\\\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\x(N-1)&\ldots&x(N-1-Q)&x(N-1)|x(N-1)|^2&\ldots&x(N-1-Q)|x(N-1-Q)|^2\\\end{bmatrix}\begin{bmatrix}a(0)\\...\\a(Q)\\b(0)\\...\\b(Q)\end{bmatrix}+\begin{bmatrix}\epsilon(0)\\\epsilon(1)\\...\\\epsilon(N-1)\end{bmatrix}$

You can solve for $a$ and $b$ using any of the linear least squares functions in MATLAB, Python, etc.

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