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I would like to model a transmitter and received pair, both sampling at the same bandwidth (not clock synchronized), where there are a number of reflectors. The time delay due to the reflectors can be a non-integer multiple of the receiver resolution (1 / sampling bandwidth), and I would like to accurately model that.

This means that if I take the transmitted signal at the nominal sample rate, and sum a bunch of time-delayed versions of the signal (with the time delay equal to the difference in time it takes light to travel the direct path and the reflected path), this will be inaccurate because the time delays will be quantized to the sampling period, which is too coarse.

Obviously I can upsample and then do the sum of the upsampled discrete-time signals, which will decrease the quantization error of the time delay. But I'm wondering if there's a way to model the received digital signal without any quantization error. In other words, I'd like to model the continuous-time reflections.

I'm modeling this for a RADAR application, so I'd like the model to be phase-accurate. I'm assuming that each reflector can be modeled by the distance to the transmitter and the receiver, and some real-valued attenuation constant. Is this assumption accurate for RADAR applications? I'm also assuming that the received signal is a sum of time-delayed signals with time-delay and attenuation from the reflector's distances and attenuation constant. Is this accurate for RF, or do I need to account for a discrete phase jump at each reflection?

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So, to give you something to read up on first, the channel you describe is a Rician or Rayleigh channel, depending on whether you have a dominant line-of-sight path or not.

So, as a first approach, to delay something in time, you don't have to shift it by a whole sample – you can also do it in frequency domain, by DFT'ing your signal, multiplying it with a $e^{j\frac{\Delta t}{f_s} f}$, and then IDFT it back. That's how I've built all my radar simulations.

I'm assuming that each reflector can be modeled by the distance to the transmitter and the receiver, and some real-valued attenuation constant.

A radar target typically also phase-shifts the electromagnetic wave – on perpendicularly hitting a metal surface, by 180°, but differently for more complex surfaces. In reality, we often model that stochastically. What is appropriate for you will very much depend on what you simulate. A model for scattering of reflections e.g. on leaves will not be very useful for modeling reflections e.g. inside an oil tank, and vice versa. Both are use cases for radar.

For many scenarios, there's "standard" models, such as the COST-Hata urban model, the ITU indoor models and many, many more. It really depends on which environment you're considering, and also, which group of researchers you ask.

Many of these models are defined for what I call the "easy RF" ranges below ~5.8 GHz. Since radar, for Doppler and wavelength reasons, often operates above, things often aren't too well represented by these models (i.e. they simply don't apply to common radar frequencies > 8GHz). This has two interesting aspects:

  • The higher your frequency, the more quasi-optical your models get for most useful spatial scales, and
  • Since millimeter waves are very en vogue right now (with all the 5G research going on), you might find very interesting RF propagation models when looking at very recent papers. As an example for a very practical, hands-on-testing based propagation/reflection description, try Simić, Perpinias and Petrova: 60 GHz Outdoor Urban Measurement Study of the Feasibility of Multi-Gbps mm-Wave Cellular Networks. Downside of these considerations is always that, albeit multipath does play an important role for these researchers, the individual phase really doesn't matter (only the interference with other paths does).
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