# correlation between two series

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

• Are the amplitudes random? – Samrat Mukhopadhyay May 7 '14 at 17:15
• no fixed,they are all fixed – dato datuashvili May 7 '14 at 17:17
• I think I understand what you are saying. If the amplitudes are all fixed, but different for the two signals, then the cross correlation between the two signals will be just the product of the non random terms of the two signals and will depend on the relative levels of the amplitudes. – Samrat Mukhopadhyay May 7 '14 at 17:20
• that means that if amplitudes are close to each other correlation coefficient will be high right? – dato datuashvili May 7 '14 at 17:24
• Yes, if their signs are same. – Samrat Mukhopadhyay May 7 '14 at 17:28

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 $$\frac{1}{2}\sum_{i}a_iA_i\cos w_i\tau$$