Amplification factor of a two-sided one-step stencil
= Amplification factor of a two-sided one-step stencil
For
$$
\sum_{k=r}^sa_ku^{n+1}_{m+k}
=\sum_{k=r}^sb_ku^n_{m+k},
$$
the Fourier convention $\widehat u(\theta)=\sum_me^{-im\theta}u_m$ gives
$$
H(\theta)=
\frac{\sum_{k=r}^sb_ke^{ik\theta}}
{\sum_{k=r}^sa_ke^{ik\theta}},
$$
provided the denominator does not vanish.