Dimension of Matrix in MIMO with multi-path

Question

In a MIMO system, the received signal is supposed to be $y = Hx + n$, where $H$ is the channel of dimension $N_t \times N_r$, $x$ is the transmitted signal with dimension of $N_t\times 1$ , and $n$ is the noise, finally the resulted signal $y$ represents the received signal of dimension of $N_r\times1$.

Assuming that I need to use channel $H$ as multi-path channel whose number of taps is $e.g.$ 5 .. The question is:

• Will the dimension of channel $H$ will be $5N_t\times N_r$?
• If so, how can we perform the matrix multiplication of $Hx$ in $y = Hx + n$, so the received signal $y$ will be of dimension $5N_r\times1$?

Answer 1

The short answer is: if you transmit one block of $N$ symbols (no data before it for at least $L$ symbol times) over a frequency-selective channel of length $L$, then the channel matrix will be of dimension $NN_TN_R\times N$.

The long answer: start with $N_T=N_R=1$. Then the $n^{\text{th}}$ received sample can be written as

$$y_n=\sum_{l=0}^Lh_lx_{n-l}+z_n$$

If you arrange these samples in a vector-matrix form you get

$$\mathbf{y}_{N\times 1}=\mathbf{H}_{N\times N}\mathbf{x}_{N\times 1} + \mathbf{z}_{N\times 1}$$

Now let $N_T=1$ while the number of receive antennas $N_R$, then the $n^{\text{th}}$ received sample over $m^{\text{th}}$ receive antenna is

$$y_n^{(m)}=\sum_{l=0}^Lh_l^{(m)}x_{n-l}+z^{(m)}_n$$

for $m=1,\,2,\,\ldots,\,N_R$. If you arrange these sample in a vector-Matrix form, you get

$$\mathbf{y}_{NN_R\times 1}=\mathbf{H}_{NN_R\times N}\mathbf{x}_{N\times 1} + \mathbf{z}_{NN_R\times 1}$$

Similarly, you can do the general case for $N_T$ transmit and $N_R$ receive antennas.

As you can see, to make this work on block-by-block basis, you need to padding $L$ zeros before each block to eliminate the inter-block interference. Alternatively, you can use cyclic-prefixed (CP)-OFDM, which is more efficient at the receiver side.

Answer 2

Will the dimension of channel $H$ will be $5N_t$x$N_r$ ?? If so, How can we perform the matrix multiplication of $Hx$ in $y = Hx + n$ ? ... So the received signal $y$ will be of dimension $5N_r$x1, right?

No!

$\mathbf H$ is defined to work as following: With $\mathbf x$ being the transmit vector, and $\mathbf y$ being the receive vector, and in the absence of noise, it is

$$\mathbf y = \mathbf H\mathbf x\text.$$

Thus, since the dimensions of $\mathbf x$ and $\mathbf y$ are fixed (at $N_r$ and $N_t$, respectively), the only choice for the dimension of $\mathbf H$ is $N_t\times N_r$.

A multipath channel can't be represented with a simple construct such as 2D Matrix. So, channel Matrix and multipath will not work together unless you redefine $\mathbf x$ and probably also $\mathbf y$.

I like your idea of "inflating" the transmit vector i.e. you could argue that individual delay slots are active orthogonally to each other and could restrict $\mathbf H$ to be causal. However, I'm personally not aware of anyone who's done that – so, finding a good provable notation that works out like the actual multipath MIMO channel would be worth giving it a shot. I'm pretty sure that the result won't be a simple $5N_t \times N_r$ Matrix, but probably be a tensor of higher dimensionality ($N_t \times N_r \times 5 \times 5$) to describe the convolution that happens between every path, but it'll be certainly interesting.

In short: You will need a more complicated system model.

In fact, describing and correcting multipath MIMO channels is so complicated that in practice we don't. Instead, modern communication systems that use MIMO (that being WiFi and LTE) use techniques like OFDM to split the wideband multipath MIMO channel in many narrowband single-path MIMO channels and then deal with these individually, because these single-path MIMO channels are the MIMO channels we know how do deal with using the well-established "boring" math.