What is the suitable mathematical form of the multipath channel impulse response?

Question

I am trying to understand the nature of a wireless multipath channel impulse response but I am a bit confused while interpreting the time domain multipath impulse response. There are different mathematical models of the wireless multipath channels in some of the papers.

As far as I know the mathematical form of the time domain multipath impulse response is modelled in the following equation as:

\begin{align} h(\tau, t) &= \sum_i a_i(t)\delta(t - \tau_i (t))\tag{Eq. 1.1}\\ H(f; t) &= \sum_i a_i(t)\displaystyle e^{-j2\pi f\tau_i (t)}\tag{Eq. 1.2} \end{align}

Here is the link for this channel model.

So in this model; $N$ is the number of paths, $\tau$ is the delay corresponding to each path,and there is no doppler introduced in the channel. Only the path amplitudes and the path delays are changing for each symbol transmission. Amplitudes and delays are assumed to be constant during one symbol transmission.

If we want to get the frequency represantation of this channel then it takes the form in frequency domain in ($\rm Eq.\ 1.2$). Then we obtain the corresponding frequency response of this channel by giving values within the frequency range which we are dealing with.

But in the equation ($\rm Eq.\ 2.1$) below the same multipath channel is modelled in the time domain as:

\begin{align} h(t)&=\sum_{k=1}^K a_k\cdot e^{j\phi_k}\cdot e^{-j2\pi f_c\tau_k}\delta(t - \tau_k)\tag{Eq. 2.1}\\ H(f)&=\sum_{k=1}^K a_k\cdot e^{j\phi_k}\cdot e^{-j2\pi f_c\tau_k}\cdot e^{-j2\pi f\tau_k}\tag{Eq. 2.2} \end{align}

Here is the link for this model.

This makes me confused because this second model has the phase shift $2\pi f_c \tau$ and I don't understand why this term has came into place here. Because this term, $2\pi f_c\tau$, appears in the form of the frequency response of the channel shown in the first equation,($\rm Eq.\ 1.2$). But here it is a parameter of the time domain represantation.

Additionally the second model has the phase shift $\phi_k$ which I think this is a phase shift caused by some of the scatterers in the channel environment.

Could you please help me answering the following questions:

1. What is the difference between the time domain impulse responses of the channel models shown in ($Eq.\ 1.1$) and ($Eq.\ 2.1$)?

2. In order to get the time domain represantation of the channel paths (as $a+jb$), do I first get the frequency response of the channel within a specific frequency range which I want to transmit my data over those frequencies?

   For example: If I want to transmit an OFDM data over 64 subcarrier frequencies, $900\ \rm{MHz}, 901\ \rm{MHz},\ldots, 963\ \rm{MHz}$) through my multipath channel, with the command filter in MATLAB, do I first need to find the channel frequency response for all 64 frequencies then its time domain values by performing IFFT, then applying the data to the filter (channel impulse response)?

   timeDomainRxsignal=filter(channelImpulseResponse,1,txData)
   

   I know there is a command in MATLAB for Rayleigh or Rician channel models but here I would like to make this channel manually for my practice.

Answer 1

Let's say we have a function that we transmit which is: $$x(t)$$ Transmitting the function is kinda easy part. We assume there is just the function we are trying to send.

However things are a little bit ugly in the air. Let's look at the receiver part.

What we generally consider first is the AWGN. Which is: $$r(t) = y(t) + n(t)$$

$r(t)$ is our received function. $y(t)$ is our function that reaches our hand and we assume we don't know what it is. In this case, for now, please consider $y(t) = x(t)$. Lastly we have gaussian distributed noise which is: $n(t)$.

Well, we have some fading types should be added. Let's combine it together and call them Channel Response. Channel responses should vary over time but for this situation I will assume it is constant.

Let's call this $h_k$.

The subscript of $h$ basically means we have different channel response for each channels. And hence we have: $$h*y(t) + n(t)$$

But, hey! I don't see that at the receiver! There is some odd peaks at the signal! What is that thing?

Sadly, this is multipath and, at the receiver, you are staring at this signal:

$$(\sum\limits_{k=1}^K = h_k*y(t)) + n(t)$$

Aaaand no! They are not coming at the same time. Let's add a delay $\tau_k$

$$(\sum\limits_{k=1}^K = h_k*y(t - \tau_k)) + n(t)$$

$K$ is the number of multipath. Please notice that as the multipath occurs, each path has it's own channel. As you can see, there are different channel responses and delay times for each.

I hope it will be useful.

As always, have a nice day.