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.

  • 1
    $\begingroup$ You have a static system. No memory. No state. Why do you need its Z-transform? $\endgroup$ Mar 27 '21 at 9:45
  • $\begingroup$ 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? $\endgroup$ Mar 27 '21 at 9:50
  • $\begingroup$ @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! $\endgroup$
    – Likely
    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.


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