Is this a ergodic channel? please help!

[vincemarat]

ergodic channel capacity+ fast fading

A two-input (x1, x2) and one-output channel is given by the figure where the
switch toggle between r1 and r2 randomly with probability p (to be in r1) and 1-p to be in r2. The switch toggles its position per symbol.
Is the above channel an ergodic channel? Why?

Please help. Thank u!

 

[rramya]

stationary ergodic channel

Hi,
it depends on the channel gain i.e., H matrix.
Let me give you a brief idea:
Definitions of Ergodic & Non-Ergodic channel:
Ergodic channel
The channel gain process is ergodic, i.e., the time average is equal to the ensemble average.
In other words, the randomness of the channel gain can be averaged out (removed) over time.
So long-term constant bit rates can be supported (like AWGN channels).

Non-ergodic channel
The channel gain is a random variable and does not change with time.
The channel gain process is stationary but not ergodic, i.e., the time average
is not equal to the ensemble average. In other words, the randomness of the
channel gain can not be averaged out (removed) over time.
So long-term constant bit rates can not be supported.


Usually, We do convert ISI channel (to remove ISI) to the parallel AWGN channel (Power constriant)
esp in MIMO-OFDM sytems......
total capacity of a MIMO channel is made up
by the sum of parallel AWGN SISO subchannels.

(It is often seen in time invariant frequency selective channel, fast fading, fast fading freq selctive channels)
A collection of non-interfering AWGN channels is called a parallel AWGN
channel. The individual AWGN channels within the collection are known as sub-channels.
Here, each of the AWGN channels has a different operating SNR

The matrix notation to depict the input-output relation
y = Hx + w

x , y and w are the input, output and noise vectors (all of dimension 2x1, in your case)

The channel matrix H is a 2x2 dimensional matrix. (in your case)
The matrix H has an important feature: the diagonal elements are all
identical. Such a matrix is said to be Toeplitz.

Note:
If the linear transformation H was diagonal, then we would have a parallel
AWGN channel.


w is an i.i.d. Gaussian random vector.
We know that w1 and w2 are i.i.d. Gaussian random variables with mean 0 and same variance sigma^2 and should be uncorrelated.

Common (idealized) assumptions:
• The channel matrix H is random and the receiver has perfect channel
knowledge.
• The channel is memoryless, i.e., for each use of the channel an independent
realization of H is drawn.
• The entries of the channel matrix hij are complex-valued normal distributed
random variables of the channel gains |hij| are Rayleigh distributed.

that is.....................

We will consider several scenarios for the matrix H:
1. H is deterministic.
2. H is a random matrix , chosen according
to a probability distribution, and each use of the channel corresponds to an
independent realization of H.
3. H is a random matrix, but is fixed once it is chosen.

ergodic Channel:
when the channel is memoryless, i.e., when
each use of the channel employs an independent realization of H. In
particular, if the process that generates H is ergodic.

actually, Decompose channel into parallel, AWGN broadcast channels
Intuitively, more data is transmitted when the channel is strong.
Ergodic (Shannon) Capacity: maximum long-term rates averaged over the fading process.
Shannon capacity applied directly to fading channels.
Delay depends on channel variations.
Variable transmission rate fluctuating with fading conditions.

Ergodic capacity takes advantage of channel fluctuations by allocating
additional power to stronger channel states.

Non-ergodic channelchannel gain H process is stationary but not ergodic)
In contrast, for the case in which H is chosen randomly at the beginning of all time
and is held fixed for all the uses of the channel.

• Diversity needed : coding, interleaving, …
• Outage probability


hope U get a starting point to learn further...

Happy learning.

 

[vincemarat]

You gave so much valuable information!
Thank u so much!!!