New vector

We have a new vector defined as $x(0)=0$, $x(1)=0$, $x(2)=\beta_0$, $x(3)=\beta_1$, $x(4)=\beta_2$, $x(5)=\beta_3$, $x(6)=0$, and $x(7)=0$.

We have added four new elements, which are all zero. The benefit is that we can rewrite the equation for $\boldsymbol{y}$, with $i=0,1,\dots,5$,

$$y(i) = \sum_{k=0}^{k=m-1}w(k)x(i+(m-1)-k).$$

As an example, we have

$$y(4)=x(6)w(0)+x(5)w(1)+x(4)w(2)=0\times \alpha_0+\beta_3\alpha_1+\beta_2\alpha_2,$$

as before except that we have an additional term $x(6)w(0)$, which is zero.

Similarly, for the fifth-order term we have

$$y(5)=x(7)w(0)+x(6)w(1)+x(5)w(2)=0\times \alpha_0+0\times\alpha_1+\beta_3\alpha_2.$$

The zeroth-order term is

$$y(0)=x(2)w(0)+x(1)w(1)+x(0)w(2)=\beta_0 \alpha_0+0\times\alpha_1+0\times\alpha_2=\alpha_0\beta_0.$$

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