Non zero mean ARMA

Consider the ARMA process $y(t)=\frac{C(z)}{A(z)}e(t)$ where is a non zero mean white noise$e(t) \sim WN(\mu,\lambda^2)$.

Gain theorem

If $W(z)=\frac{C(z)}{A(z)}$ is assintotically stable, then: $$m_y=\mathbb{E}[y(t)]=\frac{C(1)}{A(1)}\mu$$

Unbiased processes

Considering $y(t)$ we can defined the unbiased processes as: $$\tilde{y}(t)=y(t)-m_y$$ $$\tilde{e}(t)=e(t)-\mu$$ which are both zero mean processes: $$\mathbb{E}[\tilde{y}]=\mathbb{E}[y(t)-m_y]=m_y-m_y=0$$ $$\mathbb{E}[\tilde{e}]=\mathbb{E}[e(t)-\mu]=\mu-\mu=0$$

We can then sobstitute these processes in the recursive equation: $$\tilde{y}(t)=\frac{C(z)}{A(z)}\tilde{e}(t)$$ Which is a zero mean ARMA.

If we compute the covariance function of $\tilde{y}(t)$: $$ \gamma_{\tilde{y}}(\tau)=\mathbb{E}[\tilde{y}(t)\tilde{y}(t-\tau)]=\newline \mathbb{E}[(y(t)-m_y)(y(t-\tau)-m_y)]=\gamma_y(\tau) $$

So the unbiased process $\tilde{y}(t)$ has the same covariance of $y(t)$.