Multidimensional signal synchronization

Question

I am trying to find ways of synchronizing multidimensional waves gotten from the brain. For one dimensional signals, I used Cross correlation and was able to synchronise, but for multi dimensional signal, After searching online, I couldn't find somewhere to start from, can someone help with a list of different methods I can use. Materials that can help will also be appreciated.

Answer 1

Abdulmueez—that's an interesting problem statement. Here is one idea (maybe you already tried it)—Could you just generalize the one-dimensional case to $p$-dimensional?

Assuming stationary sequences, the cross-correlation can be defined as $R_{xy}(m){\triangleq}E[x(m)y^\ast(0)]$ (where $E$ is the expectation operator). If $x(m)$ and/or $y(m)$ are random sequences, then $R_{xy}(m)$ is also a random sequence$\ldots$ but of course it can be estimated. One of the most common estimators $\hat{R}_{xy}(m)$ of the random sequence $R_{xy}(m)$ is the humble average such that $$\hat{R}_{xy}(m)\triangleq\frac{1}{N}\sum_{n=1}^N x(n+m)y^{\ast}(n)$$ If the dimension is $p=2$, then at this point you could try just letting $x(m)$ and $y(m)$ be complex valued (sequences of ordered pairs) and using the real part for one dimension and the imaginary part as the other.

If $p\ge3$, then for the $p$-dimensional $x(m_1,m_2,\ldots,m_p)$ and $y(m_1,m_2,\ldots,m_p)$ you could define a $p$-dimensional cross-correlation function as $$R_{xy}(m_1,m_2,\ldots,m_p){\triangleq}E[x(m_1,m_2,\ldots,m_p)y^\ast(0,0,\ldots,0)]$$ and compute a $p$-dimensional estimate $\hat{R}_{xy}(m_1,m_2,\ldots,m_p)$ of this $p$-dimensional random sequence $R_{xy}(m_1,m_2,\ldots,m_p)$ by again averaging: $$\hat{R}_{xy}(m_1,m_2,\ldots,m_p)\triangleq\frac{1}{N^{p}}\sum_{n_1=1}^N \sum_{n_2=1}^N\cdots\sum_{n_p=1}^N x(n_1+m_1,n_2+m_2,\ldots,n_p+m_p)y^{\ast}(n_1,n_2,\ldots n_p)$$