Find transfer function given impulse
$$\begin{align}
G_d(z^{-1})&amp;=z^{-d}\frac{b_0+b_1z^{-1}}{1+a_1z^{-1}}\\
&amp;=z^{-d}\frac{b_0}{1+a_1z^{-1}}+z^{-(d+1)}\frac{b_1}{1+a_1z^{-1}}
\end{align}$$
Hence, considering 


  $$\mathcal{Z}^{-1}\{\frac{1}{1-az^{-1}}\}=a^nu[n]$$


and


  $$\mathcal{Z}^{-1}\{z^{-n_0}X(z)\}=x[n-n_0]$$


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.