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For the following memory polynomial equation,

$y(n)=\displaystyle\sum_{m=0}^{M-1}\displaystyle\sum_{p =1\\p\,\textrm{odd}}^{P}a_{m_p} \, x(n-m) \, |x(n-m)|^{(p-1)}$

The total number of coefficients is $ M\frac{(P+1)}{2}$

I need to calculate:

1- the total number of complex multiplications

2- the total number of real multiplications

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  • $\begingroup$ The polynomial function does not seem fully generic to me (because of odd terms). Is that homework? What were your progress so far? $\endgroup$ Jan 3, 2020 at 10:09

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The direct computation should be straightforward. Since I suspect the exercice has other purposes (more advanced algorithmic complexity), there additionally are a couple of tricks to reduce the complexity of power evaluations.

First, if you have to evaluate all powers $y^0,y^1,\,\cdots,y^Q$, you can store the power-of-two powers, and use the binary expansion of a power $q$. It can be called exponentiation by squaring. For instance, to compute $y^7$, you need one multiply for $y^2 = y\times y$, one for $y^4=y^2\times y^2$ and two more for $y \times y^2 \times y^4$, so four multiplies instead of six by direct computation. And the $y^2$ and $y^4$ can be stored for the next powers.

Second, a classical polynomial evaluation is "Horner's scheme":

$$a_0 x^n + a_1 x^{n-1} + \cdots + a_n = \left(\left(\left( a_0x+a_1\right) x \right)\ldots a_{n-1}\right)x+ a_n$$ which can be adapted to your case.

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