What is Channel Vector?

Question

When I read research paper, there is a word named "Channel Vector". I think it is too abstract, so it is hard to understand for me.

What does it mean? For example:

Consider the downlink communication between a BS equipped with \$M\$ antenna elements and \$K\$ single-antenna mobile users. We assume that this communication takes place via a RIS with \$N\$ reflecting elements deployed on the facade of a building existing in the vicinity of both communication ends, as illustrated in Fig. 1. The direct signal path between the BS and the mobile users is neglected due to unfavorable propagation conditions. Then, the discrete-time signal received at mobile user \$k\$, with \$k = 1, 2,\ldots,K\$, is written as $$ y_k=h_{2,k}\mathbf{\Phi}\mathbf{H}_1\mathbf{x}+w_k,$$ where \$\mathbf h_2,k \in \mathbb C^{1\times N}\$ denotes the channel vector between the RIS and user \$k\$, \$\mathbf H_1 \in \mathbb C^{N\times M}\$ denotes the channel matrix between the BS and the RIS

Answer 1

Preliminary / What you'll need to understand

OK, if you're reading about RISes, then you will need a solid understanding about the math behind MIMO. If a vector is too abstract for you, then you won't get far, I'm afraid!

You'll want to be very familiar with the channels models used. As a book, I'd recommend Tse / Viswanath: Fundamentals of Wireless Communications, which is available freely. You'll really want to read this from the beginning to the end of at least Chapter 7, to understand the channel model and why you want to do anything like this, before going deeper into research papers.

Brief answer

A channel vector in this case describes the flat MISO channel, i.e. the channel coefficients for the \$N\$ paths between the \$N\$ antennas of the RIS and the respective user.

End thoughts

Again, this is really the extreme basics, and reading a research paper might be very frustrating if this is hard to understand for you!
Improve your life, and make progress much, much faster, by investing the time to read the textbook I linked to above (or any other MIMO-introducing comms engineering textbook for that matter, but Tse/Viswanath is the standard as far as I can tell, and well-written). Afterwards, you'll be a lot faster understanding the papers you're reading. This is really an investment of time - it comes back with having more time later on. And also, you'll actually understand what you're doing!