Capacity of channels with pilots?

Question

[Note] I've asked this question at https://dsp.stackexchange.com/questions/1434/capacity-of-channels-with-pilots with NO answers.

Question is about "channel capacity" http://en.wikipedia.org/wiki/Channel_capacity I guess it is pretty elementary for experts, however I have not seen it discussed in e.g. Proakis book or Tse "Fundamentals of Wireless Com".

Question (roughly speaking): is channel capacity known for the fading channel with "pilots" ? (discrete, memoryless channel). "Pilots" - means we transmit on certain positions symbols known to receiver and transmitter both. It is common for all modern telecommunication standards GSM, UMTS, LTE to have "pilots".

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Simplest example:

Consider the following signal model ("block" fading), block length $N$, pilot number $k$

$r_{i,l} = h_l s_{i,l} + n_{i,l}$

here $i=1...N$ - index of symbol inside block, $N$ - block length $l=1...infinity$ - index of block

$h_l$ - are random i.i.d, but constant on block i.e. does not depend on "i",

$s_{i,l}$ sent symbols, assume that for $i=1...k$ and all $l$ they are pilots i.e. they are equal to $1$. For other $i$ these sent information signals - assume also i.i.d.

$n_{i,l}$ - noise - random i.i.d.

What is the capacity of this channel ? Assume that distributions of all random variables are given - for simplicity can assume they are Gaussian.

i.i.d. - "independent identically distributed"

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Let me explain how pilots are used in the standard algorithms. We do not know $h$ - so first we proceed "channel estimation", we make some estimate of $h$ from known pilots and than estimate data symbols with "known" h.

Answer 1

This is a classical problem which goes under the name "non-coherent detection". Here are some facts:

1. Capacity of the block fading channel is not achieved using pilots.

2. Capacity of the channel is not achieved using Gaussian data.

3. Capacity of the channel is generally unknown.

4. Only capacity expansions in the low and high SNR regime are known to date.

5. With Gaussian data, the situation is analyzed in e.g. "forgot the title" by Rusek, Lozano, and Jindal" TIT 2012.

6. Input block should optimally consist of a Haar unitary matrix plus a certain, to date unknown, pdf of the singular values.

7. Grassmanian codebooks are usually used in practical situations.

8. With pilots: estimating the channel and using the estimate as it is correct is analyzed in e.g. "forgot the title" by Lapidoth and Shamai, TIT 2002