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In the Question Design of equalizer for wireless communication the diagram of a basic communication system is presented with informative points and discussions. My Questions are:

  1. Considering the channel modeled as a FIR filter having a single input and single output. If the impulse response (channel coefficients) take values in the complex domain then am I looking at a Rayleigh Fading channel?

  2. What constitutes a multi-path channel? Would multi-path channel be typically not a single input and single output?

  3. In the diagram, the input to the channel is Tx. Is Tx digital? I mean can it take bits or multiple level bits? Suppose, we want to send an image or a real-valued time series through a communication system, then is it transmitted in real values or digital in wireless communication?

  4. At which stage is digital modulation performed? Then the output of the modulation should be digital. So, is this digital output the Tx that is sent to a channel?

  5. What is the difference between modulation and encoding? When we are doing encoding, we are basically transforming the information from one representation to another. Isn't this what modulation also does?

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Well, these are very fundamental questions, and I think they are written in any good book. But maybe it's hard to extract the exact information from them, which you ask for. So, let me briefly answer:

  1. In wireless, the channel is usually modelled as a FIR filter with impulse response $h(\tau)$. But, this channel might change over time (due to e.g. mobility), to make it a time-variant impulse response $h(t,\tau)$. Moreover, the channel is often modeled as a tapped delay line, $h(t,\tau)=\sum_ia_i(t)\delta(\tau-\tau_i(t))$ where $a_i(t)$ are time-varying gains of the taps and $\tau_i(t)$ are the time-varying delays of the taps. The fact that $a_i(t)$ has complex values does not make it Rayleigh fading. Rayleigh fading describes the temporal distribution of $a_i(t)$. If over a long time range the distribution of each path gain is a Gaussian, it becomes Rayleigh fading. The doppler spread of the Rayleigh fading describes how quickly the value $a_i(t)$ changes over time. In the special case of block-fading $a_i(t)$ is a constant (the channel remains static) for the whole transmission. Then, when a new transmission starts, a new channel is generated. Then, the realizations $a_i$ follow a Gaussian, to make it block Rayleigh fading.

  2. A multipath channel corresponds to the tapped-delay line model. Have a look at one of my articles about multipath propagation, maybe it helps. The number of inputs and outputs depends on the number of TX and RX antennas of your system (talking about MIMO systems here) or the number of users. For single-antenna systems there is just one input and one output.

  3. The input to the physical channel is analog, if you model it as a tapped delay lines. However, you can assume transmission in passband, then all signals are real. Or, you can do an equivalent analysis in baseband, where all signals are complex but the math is easier to treat. For the description of transition from baseband to passband, you can look at another article of mine.

    Though, there are other channel models such as BSC or BEC, though these are not used to model the transmission of an analog signal over the wireless physical channel, but abstract away from that.

  4. Modulation is performed after encoding ;-). Modulation means to generate a signal which is adapted to the channel, a a function of the transmit bits. It can e.g. include mapping the bits to complex QAM constellation points.

  5. Encoding takes your bits you want to transmit (the payload) and adds redundancy via a channel code. This way, if at the receiver side, one bit is lost, it can be recovered by appropriate decoding.

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  • $\begingroup$ Thank you very much for your point wise answer, indeed it is very hard to extract these from a book. I have one last question, can you please clarify? Assume that $z(t)$ is the output of the channel : $z(t)=\sum_{l=1}^L h(l)u(t−l)$ where $h(l)$ is the channel coefficients (impulse response), u(t) is the input and L is the order or delay of the model. This model is FIR. I did not understand your first point where you said that the channel is modeled as a delay line. Can I say that $L$ is the number of paths? Here, I have included the delay l. Is this Rayleigh fading channel? $\endgroup$ – Ria George May 11 '17 at 15:38
  • $\begingroup$ what you model is a time-invariant multipath (i.e. frequency-selective) channel, where $h(l)$ is the coefficient of the $l$th path. In your model, the $l$th path has (discrete) delay of $l$. This can be considered as a tapped-delay-line. This channel is not necessarily a Rayleigh fading channel. As I said Rayleigh fading has something to do with change over time. You can make it a Rayleigh block-fading (some people call this quasi-static) channel, if for each channel realization, each $h(l)$ is drawn from a circular gaussian complex distribution. $\endgroup$ – Maximilian Matthé May 11 '17 at 19:04

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