Dimension of channel matrix elements in MIMO systems In a MIMO system we have:
$y= \mathbf{H} \star \mathbf{x} +n$ with $x \in \mathbb{C}^{N_t \times 1}$ ($N_t$ the number of transmission antennas) and $\star$ the convolution product.

For the matrix, $\mathbf{H}$ we have: $\mathbf {H}=(h_{i,j}(t,\tau))$ with $i \in \{1,...,N_r \}$, $j \in \{1,...,N_t \}$, $t$ the time and $\tau$ the delay for the impulse response

Here is my question:
What is the dimension of $h_{i,j}(t,\tau)$?It's a number or a vector because in many book, there is $size(\mathbf{H})=N_r \times N_t$. But if $h_{i,j} (t,\tau)$ is a vector, it's false.

An example is article Survey of channel and radio propagation models for wireless MIMO systems.
Sorry, I'm starting again.
$\mathbf{H(t,\tau)}=(h_{ij}(t,\tau))$. 

If I have $\mathbf{H}(t,\tau)=\mathbf{H(\tau)}$ and $N_t=1$ then $\mathbf{H}(\tau) \in \mathbb{C}^{N_r \times 1}$