Z-transform of an affine function

What is the transfer function of the system described by the following affine input ($$x$$)-output ($$y$$) relationship: $$y[n] = \alpha x[n] + \beta.$$

Using the Z-transform we find: $$Y[z] = \alpha X[z]+ \beta$$ and $$H[z] = \frac{Y[z]}{X[z]} = \alpha + \frac{\beta}{X[z]}$$

But I don't know how to take it from here to compute the transfer function! I'm not sure if this can be done. Any help would be appreciated.

• You have a static system. No memory. No state. Why do you need its Z-transform? Mar 27 '21 at 9:45
• If a system is LTI, then $Y (z) = H (z) X (z)$. Thus, if $X (z) = 0$, then $Y (z) = 0$. Is that the case for your static affine system? Mar 27 '21 at 9:50
• @RodrigodeAzevedo, I have a system with an output that I know can be described by this affine relationship and for which I do not know the input. It is when I was thinking of estimating the parameters $\alpha$, $\beta$ using a blind system identification approach and I started thinking about identifiability questions that I thought about getting its transfer function, then got stuck! Mar 27 '21 at 15:00

That system cannot be described by a transfer function because it is not linear. It neither satisfies the condition of additivity nor does it satisfy the condition of homogeneity.

If $$y_1[n]$$ is the response to an input $$x_1[n]$$, and $$y_2[n]$$ is the response to an input $$x_2[n]$$, then the response to $$x_1[n]+x_2[n]$$ doesn't equal $$y_1[n]+y_2[n]$$ (additivity is violated). Furthermore, the response to $$\alpha x_1[n]$$ doesn't equal $$\alpha y_1[n]$$ (homogeneity is violated).

This is unrelated to existence questions concerning the $$\mathcal{Z}$$-transform. The $$\mathcal{Z}$$-transform of a constant $$\beta$$ doesn't equal $$\beta$$, but it simply doesn't exist.

It does not have a Z-transform.

The constant sequence $$f[n] = \beta$$, $$\forall n, \beta \neq 0$$ does not have a Z-transform, as the sum $$F(z) = \sum_{n=-\infty}^{\infty} \beta z^{-n}$$ diverges for any $$z$$; i.e., there's no set of $$z$$ values (region of convergence - ROC) for which the sum is finite (convergent).

Hence your affine function $$y[n] = \alpha x[n] + \beta$$ does not have a Z-transform.

May be you want to restrict the domain of input to $$n\geq 0$$ in which case you will have a mapping of $$y[n] = (\alpha x[n] + \beta) u[n]$$ which may have a Z-transform, provided taht $$x[n]u[n]$$ does have a Z-transform.