# Find transfer function given impulse

I am given the following discrete time transfer function : $$G_d(z^{-1})=z^{-d}\frac{b_0+b_1z^{-1}}{1+a_1z^{-1}}$$ which has the following impulse response $$g_d[n]=\{0,1,-0.1,-0.05,...\}$$

How can I find the transfer function?($b_0$, $b_1$ etc)

• are you asking, how do I find $b_0$ $b_1$, and $a_1$? from the given impulse response? – user28715 Jun 23 '17 at 21:56

\begin{align} G_d(z^{-1})&=z^{-d}\frac{b_0+b_1z^{-1}}{1+a_1z^{-1}}\\ &=z^{-d}\frac{b_0}{1+a_1z^{-1}}+z^{-(d+1)}\frac{b_1}{1+a_1z^{-1}} \end{align} Hence, considering
$$\mathcal{Z}^{-1}\{\frac{1}{1-az^{-1}}\}=a^nu[n]$$
$$\mathcal{Z}^{-1}\{z^{-n_0}X(z)\}=x[n-n_0]$$
we have $$g_d[n]=b_0a_1^{(n-d)}u[n-d]+b_1a_1^{n-(d+1)}u[n-(d+1)]$$ Now the unknowns can be found by equating the terms apropriately.