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I am trying to understand why exponential decay power delay profiles are used to model fading. In other words, why power of taps of channel impulse response decays exponentially and not, for instance, linearly.

The only explanation I could come up was that since taps are modeled as circularly symmetric Gaussian random variables, their magnitude is Rayleigh distributed and their power is Chi-squared distributed with two degrees of freedom, or in other words, exponentially distributed. But this does not sound correct. I suppose this has something to do with channel acting as low pass filter, but I was not able to figure out exact relation.

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While it is correct that discrete-time channel tap is usually modeled as circularly symmetric Gaussian, having power being exponentially distributed (see this answer and the cited book therein for the philosophy of modelization), the classic exponential PDP comes from the log distance path loss model where the power in decibel decrease approximately linearly with time, a phenomenon which, to the best of my knowledge, arises from real-life measurement.

Differently put, the power values of discrete-time channel taps are exponentially distributed around their mean values; whereas the exponential decay PDP governs the relationships among these mean values (to be precise, this is the result of averaging scattering function that represents the correlation among the taps, with delay becomes the sole parameter). Same function, different meaning.

For example, we can have taps modelized as Rayleigh, Rician, Nagakami, (see this for the difference), etc. random variables with the aforementioned exponential decay PDP. Also, it is totally legitimate to model a channel of Rayleigh taps with PDP being not exponential decay (see the industrial 3GPP defined PDP for LTE).

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  • $\begingroup$ I don't quite see the connection between the path loss model, which does decay exponentially as being the outer area of a sphere, and the coefficients of the channel response which is the delay spread of the channel? $\endgroup$ Commented Apr 15, 2022 at 15:09
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    $\begingroup$ @DanBoschen misleading words due to laziness (bad habit) sorry. I meant "the result of averaging scattering function that represents the correlation among the taps, with delay becomes the sole parameter". $\endgroup$
    – AlexTP
    Commented Apr 15, 2022 at 16:01
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    $\begingroup$ ok nice - that makes more sense $\endgroup$ Commented Apr 15, 2022 at 16:03
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The suspicion is correct. Although there are different fading models for different conditions, this one is applicable to terrestrial channels with lots of obstructions and reflections and no strong direct line of site path. The multiple paths with sufficient delay relative to the symbol rate can ensure independence such that the summed signal as received appears Gaussian distributed, consistent with the Central Limit Theorem. As a complex Gaussian signal, it will be well modelled by a uniform phase distribution and a Rayleigh amplitude distribution. This is referred to as "Rayleigh fading". Further consideration is the temporal change in that fading relative to the symbol rate to determine if it is "fast fading" or "slow fading".

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