Periodic Sequences

$\diamond$

A sequence is rectangularly periodic if

$$\tilde{x}(n_1, n_2+N_2)=\tilde{x}(n_1, n_2)$$

$$\tilde{x}(n_1+N_1, n_2)=\tilde{x}(n_1, n_2)$$

$N_1$: Horizontal Period
$N_2$: Vertical Period

$\diamond$

More generally, $\tilde{x}(n_1,n_2)$ is periodic with periodicity matrix $\bf N$ if

$$\tilde{x}({\bf n}) = \tilde{x}({\bf n} + {\bf N} {\bf r}), \forall{\bf n} \in {\bf\cal I}, \forall{\bf r} \in {\bf\cal I}$$

where ${\bf\cal I}$ is the set of all integer vectors of same dimension as ${\bf n}$.

1. $\vert \det {\bf N} \vert \ne 0$ is the number of samples in one period of $\tilde{x}$.
2. ${\bf N}$ is an integer matrix and $\vert \det {\bf N} \vert$ is a positive integer.
3. The columns of ${\bf N}$ represent periodicity vectors.
4. ${\bf N} \mbox{ diagonal } \Longrightarrow \mbox{ rectangular periodicity}$.