How to generate a time-varying channel

[David83]

Hi all;

I would like to generate a discrete-time time-varying multipath (doubly selective) channel. How can I do that? Also, I need the power delay profile of the multipath channel to decay exponentially. Any help will be highly appreciated.

Thanks

 

[Shug]

You can find a pretty realistic channel model for DVB-T2 in the common simulation platform here:

https://sourceforge.net/projects/dvb-t2-csp/

The standard will probably take you a little time to weed through, but you can find their Rayleigh channel model in the file "t2_ch_dvbtpscen_getch_dvbtp.m," located in ".\model\ch\ch_dvbtpscen" (all their channel models can be found there). The parameters are generated there; the channel response is used in "t2_ch_dvbtpscen_conv.m". These are really the main two files you need to generate a multipath channel; I haven't checked, but I am assuming the power delay profile decays exponentially.

Note that this is not a time-varying channel. In order to vary the channel, you would have to modify the channel parameters (in "t2_ch_dvbtpscen_getch_dvbtp.m") through time (probably over several symbols). You may want to try Clarke's model for this purpose, but there are probably better models to use; the main parameter that should vary over time is the phases, so you can assign random phases and keep the delays and gains constant.

 

[David83]

Thanks for the information and the link. When the only source of time variation is the relative motion between the transceiver, it is clear how to generate the channel. However, in my case, the channel itself is inherently time-varying. I have a paper where the authors generated the time-varying channel by adding a random Gaussian walk process to the original channel. I do not understand why this walk process would add time variation to the original channel.

 

[Shug]

What do you mean the channel itself is inherently time-varying? What is the source of the time variance?

Many theoretical papers use the model where the channel is i.i.d. in time, which means that the channel has no correlation from one "timestep" to the next (timestep here can be defined in many ways; for very fast fading it is from modulated symbol to modulated symbol, but otherwise can be from one OFDM symbol to the next, etc.) In reality such a model can occur when the receiver is moving very fast compared to the transmitter, or when the surroundings are changing quickly, so the paths to the receiver are changing quickly. However the primary reason this model is used is for ease of theoretical analysis and is not necessarily reflective of real-life situations.

 

[David83]

What do you mean the channel itself is inherently time-varying? What is the source of the time variance?

Many theoretical papers use the model where the channel is i.i.d. in time, which means that the channel has no correlation from one "timestep" to the next (timestep here can be defined in many ways; for very fast fading it is from modulated symbol to modulated symbol, but otherwise can be from one OFDM symbol to the next, etc.) In reality such a model can occur when the receiver is moving very fast compared to the transmitter, or when the surroundings are changing quickly, so the paths to the receiver are changing quickly. However the primary reason this model is used is for ease of theoretical analysis and is not necessarily reflective of real-life situations.

Inherently time-varying means that the reflection points of the multi-paths change with time even though the transmitter and receiver are fixed, like in underwater acoustic channels, over which I am working. This is different when the source of time variation is Doppler scaling, where usually the paths' gains assumed to be constant over one block, and just the phase changes.

 

[Shug]

That certainly sounds like a challenging problem! Acoustic underwater environments are tough.

It seems you're certainly more familiar with the literature on acoustic underwater than I am, but I think the code in the files I mentioned above can be modified in a straightforward fashion to generate a reasonable model. You would need the number of paths to consider for each symbol you are transmitting - this could be a random variable; you could, for example, use a poisson process, but for simplicity right now let us assume it is fixed at N. You would also want to consider the time interval over which you are considering these paths; let's call that T. You would then need to generate the following:

path arrival times (Tau): Generate N uniform random variables from 0 to T, and sort them in ascending order.

phases (Phi): Generate N uniform random variables from 0 to 2*pi.

channel gains (Ro): Generate N i.i.d. Rayleigh random variables (or just complex Gaussian random variables and take the magnitude) a_1...a_n. Then, to get exponential decay, generate Ro as follows:
\[\rho_n = a_ne^{-\lambda\tau_n}\]

The parameter lambda would depend on the channel characteristics. Once you generate Ro, Phi, and Tau, you can use the DVB-T2 code provided (starting line 67 on in "t2_ch_dvbtpscen_getch_dvbtp.m", just replace Ro, Phi, and Tau with the above) to generate the channel for the current symbol. Then for each successive symbol, you would repeat the above to generate a new channel.

 

[David83]

I am working in underwater as my PhD dissertation framework. It is so tough, and I am facing a progress issue:-(

Anyway, thanks for the details. I want to know in more details about the exponential delay profile. See attached paper View attachment Mitigation of Intercarrier Interference for OFDM Over Time-Varying Underwater Acoustic Channels.pdf. In section 4.A the author details how to generate the time varying channel, by first generating L+1 independent taps with exponential delay profile where the power in the last tap is half the power of the first tap on average. Then a tunable random Gaussian walk process is added to add variation. He does not mention how the decaying function looks like. In the reference he mention, it is written that the onde-sided exponential decaying looks like:
\[A_c(t)=\frac{1}{T_m}e^{-\tau/T_m}\]
Based on these information, how to generate the attached channel?

Thanks

 

[Shug]

I don't quite understand the "random walk" part of his argument; I am not sure if he is changing the arrival path times or just the channel gains themselves. Also, I have not seen the reference he is talking about.

However, the power in the last tap being half the power in the first tap is similar to the half-life argument from chemistry; for a parameter lambda, the last multipath component (since we are only considering multipath components that are up to 1/2 of the largest multipath) is given as (log(2)/t), or in your notation T_m*log(2). Goldsmith probably discusses how the paths are distributed between the first and the last multipath components, but reasonable assumptions are either randomly or uniformly distributed.

I think the model that I provided above will yield a similar model to the one in the paper you discuss; it should at least give reasonable results. But as far as I can tell without looking at Goldsmith's book, the decaying function is just a simple exponential with the last multipath component located at the point where it would be half the amplitude of the first multipath component.

 

[David83]

I don't quite understand the "random walk" part of his argument; I am not sure if he is changing the arrival path times or just the channel gains themselves. Also, I have not seen the reference he is talking about.

However, the power in the last tap being half the power in the first tap is similar to the half-life argument from chemistry; for a parameter lambda, the last multipath component (since we are only considering multipath components that are up to 1/2 of the largest multipath) is given as (log(2)/t), or in your notation T_m*log(2). Goldsmith probably discusses how the paths are distributed between the first and the last multipath components, but reasonable assumptions are either randomly or uniformly distributed.

I think the model that I provided above will yield a similar model to the one in the paper you discuss; it should at least give reasonable results. But as far as I can tell without looking at Goldsmith's book, the decaying function is just a simple exponential with the last multipath component located at the point where it would be half the amplitude of the first multipath component.

He considers a channel model such that the received signal is written as: \[y[n]=\sum_{l=0}^Lh[n;l]x[n-l]+z[n]\], and he does not assume any thing about the channel except that it is time-varying to make it general.

Based on your modeling we can write the channel vector as \[\mathbf{h}=[h_0e^{-\lambda\tau_0} ... h_Le^{-\lambda\tau_L}]^T\], where \[\{h_l\}_{l=0}^L\] are i.i.d. complex Gaussian random variables, right? The author says that the last tap has half the power of that of the first tap, on average. I would translate that as the average power of the last tap is half of that of the first tap. This implies that \[\lambda=\frac{\ln 2}{2(\tau_L-\tau_0)}\]. Is this right?

 

[Shug]

I think the implication is that
\[\lambda = \frac{ln 2}{\tau_L-\tau_0},\]
without the additional factor of 2 you have included. the log(2) term takes care of the 1/2 power term. I think everything else is correct.

 

[David83]

I think the implication is that
\[\lambda = \frac{ln 2}{\tau_L-\tau_0},\]
without the additional factor of 2 you have included. the log(2) term takes care of the 1/2 power term. I think everything else is correct.

OK, thanks for the discussion. That was fruitful