# Frequency-selective channel: Basic in theory

In a lecture, it was covered a topic "frequency selective and flat fading channel models".

For case, the frequency selective channel we have a channel impulse response H for each frequency, is varying in frequency.

Last week I have a lab task where I have implemented the different precoding algorithms. Now I am thinking about how you calculate a precoding matrix for the frequency selective channel. You have more than one channel response, so you have to calculate for each of them. A calculation takes long. Of course, Matlab can do that very fast but you can have more than thousands response and it takes a long for Matlab too. How do you calculate in that case a precoding matrix? Do you take an average?

Best regards

Noel

In theory you need to recompute your precoding for every subcarrier.

In practice you might not want to do this as the computational complexity is pretty high. What one then typically does is grouping adjacent subcarriers and time indices together into "chunks" (I believe they are called resource blocks in LTE?) and just compute one precoding strategy for the entire chunk. The size of the chunk is a trade-off: if we make it too large, the quality of the precoding degrades because the channels becomes less and less similar over frequency and time. If we make it too small, computational complexity becomes an issue.

To determine the effective channel for the entire chunk that is used to compute the precoding, various strategies exist. The simplest might be just averaging the channels, for the MIMO case we often prefer to average the second-order statistics.

• "grouping adjacent subcarriers", did you mean a grouping channel matrices? – Noel Miller Sep 23 '19 at 12:01
• Sure, in the MIMO case, the channels would be matrices. The point is to say that the channels in adjacent subcarriers are still somewhat similar so we can group them together. – Florian Sep 23 '19 at 12:04
• As I remember the lecture about precoding techniques, If we add a precoding step, a performance of the channel will be bad. So if you group subcarriers, then the performance will be even worse, right? – Noel Miller Sep 23 '19 at 12:04
• Huh? I don't get your sentence: "if we add precoding, the performance will be bad"? Why would we do precoding? On the contrary, precoding should help you to improve the rate you can transfer over a channel... – Florian Sep 23 '19 at 12:27
• one more question, I don't get one thing. You have written you prefer to average the second-order statistics. let assume 4 channel matrices and I group them: 1. with 2., 3. with 4. The next step is to average. If I take the average (mean), then I get a new 2 matrices. For the precoding strategy, I will use now these two new matrices. As I understood, I have 2 channel matrices, which I will multiply with Tx signal, decode, right? – Noel Miller Sep 23 '19 at 12:29