let us consider following two series
$$y[t]=a_1\sin(\omega_1 t + \phi_1) + a_2\sin(\omega_2 t + \phi_2)+ \cdots + a_p\sin(\omega_p t+\phi_p) + z_1(t)$$
and
$$y_1 [t] = A_1(\sin(\omega_1 t+\phi_1) + A_2 \sin(\omega_2 t+\phi_2) + \cdots + A_p \sin(\omega_p t+\phi_p) + z(t)$$
suppose that all parameters are fixed,phases even we can consider as $0$,frequencies are same,just amplitudes and random term(white noise) are different,i want to know how strong statistical relationship will be between these two series?i think that two two series will be related to each other,as if we consider this as linear equation,then we will get two linear equation with same constant values instead of sinusoidal components and therefore relationship between them will be just linear form right?or correlation coefficient between each variable should be strong,level of strongest depend on how amplitudes are related to each other right?thanks in advance
Let, $$y(t)=\sum_{k=1}^p a_i\sin (w_kt+\phi_k)+z(t)\\ y_1(t)=\sum_{k=1}^p A_i\sin (w_kt+\phi_k)+z_1(t)$$ where $z(t),z_1(t)$are i.i.d.$\sim \mathcal{N}(\mu,\sigma^2)\forall t\in \mathbb{R}$. The cross correlation between the two processes is $$E(y(t)y_1(t+\tau))=\sum_{i,j}a_iA_j\sin (w_it+\phi_i)\sin (w_j (t+\tau)+\phi_j)$$ The processes are clearly non-stationary. The processes, however become WSS processes if $\phi\sim\mathcal{U}[0,2\pi)$ and are i.i.d.; in that case the cross correlation becomes $$\frac{1}{2}\sum_{i}a_iA_i\cos w_i\tau$$