In this post, I call $\hat{v}\in\mathbb{C}^N$ the Discrete Fourier Transform of $v\in\mathbb{C}^N$ the vector such that: $$ \hat{v}_j = \sum_{k=0}^{N-1} v_k \exp\left(-2i\pi \frac{kj}{N}\right) $$
For instance, if a signal $v\in \mathbb{C}^{N}$ of size $N = P\times Q$ can be written as $v_{i\times Q + j} = u_j$ for a given $u\in \mathbb{C}^Q$, then its discrete Fourier Transform is sparse in the sense that it reads $\hat{v}_{j\times P} = \hat{u}_j$ and other values are zero. Intuitively, the signal is periodic wrt a sub-period of N, which explains only some harmonics are present in its Discrete Fourier Transform.
Similarly, what can be said about $\hat{v}$ if $v_{i\times Q + j} = u_i$ for a given $u\in \mathbb{C}^P$ ? Intuitively, the signal only takes $P$ values, and is constant on every interval of the form $[iQ,\ (i+1)Q[$.
Thank you for your help,