Closed-form expressions for FIR least squares inverse filters

[Linder88]

Homework Statement

The assignment is to find a closed-form expression for the FIR least squares inverse filter of length N for each of the following systens

Homework Equations

$$
1.G( z ) = \frac{1}{1 - \alpha z^{-1}}; | \alpha | < 1 \\
2. G(z) = 1 - z^{-1} \\
3. G(z) = \frac{\alpha - z^{-1}}{1 - \alpha z^{-1}}; |\alpha| < 1
$$

The Attempt at a Solution

Anybody have any ideas, I can't really understand what is meant by close-form either from the book or from wikipedia. My guess is:
$$
1. G( z ) = \frac{z}{z - \alpha} \\
2. G( z ) = z-1 \\
3. G( z ) = \frac{z - 1 }{z - \alpha}
$$
Every helping hand is welcome

 

[haruspex]

FIR least squares inverse filter of length N

Your attempted solution does not seem to bear any relationship to that clause. Not an area I know anything about, but it is clear that you are not merely being asked to present G(z) in a closed form (those expressions already are).

what is meant by close-form

Closed form means an equation of the form function = (some combination of standard functions).
That is, the right hand side cannot contain any references back to the function being expressed, nor integrals, nor sums, nor any special functions defined for the purpose. There are some grey areas.

 

[Linder88]

Yes, you are right. I realized that I have misunderstood the quetion, I'm supposed to first tale the inverse of $G(z)$
$$
1. G^{-1}(z)=\frac{1}{G(z)}=1-\alpha z^{-1} \\
2. G^{-1}(z)=\frac{1}{1-z^{-1}} \\
3. G^{-1}(z)=\frac{1-\alpha z^{-1}}{\alpha-z^{-1}}
$$
Now, I only need to make the inverse z-transform
$$
1. g(n) = -\alpha \delta(n-1) \\
2. g(n) = -u(n-1) \\
3.
$$
I'm not sure about the third

 

[haruspex]

I'm not sure about the third

Expand as constant+constant/(α-z^-1)?