Deciding for the correct way to calculate coherence time

Question

I have to calculate the coherence time of an indoor environment. The signal I am sending is modulated with GFSK, the carrier frequency is about 2.4 GHz and my symbol time is 1 microsecond.

During my research, I had encountered some formulas like

$T_c = \frac1{f_d}$

$T_c = \frac1{B_d} $(doppler spread)

$T_c = \frac1{2f_d} $(doppler frequency)

$T_c =\frac{9}{16f_d}$

$T_c = \frac{0.423}{f_d}$

$T_c = \frac1{2\pi f_d}$

So I am a little bit confused here. Which formula suits best for my case and how to decide the right formula in different cases?

P.S.: The Doppler shift shouldn't be that much since it is an indoor experiment and the movement with the highest velocity should be walking quickly.

Answer 1

This is, as always, a question of what you define something to be. So, context of the different formulas to be key.

For example, the formulas that only differ by a factor of 2 will be taken from two different publications, one understanding bandwidths as double-sided, the other as only single-sided. The definitions of $f_d$ will definitely also differ between these different publications etc.

So, you're not doing your homework 100% honestly: You can't just compare formulas and assume that the same symbols in all of them mean the same thing.

After all, coherence is a term with different meanings for different people: On one hand, a channel is said to be "coherent" over a given time (for a limited bandwidth!) when it doesn't change "too much". Whatever "too much" is for your application. The time within that is the case is called "coherence time".

On the other hand, we often just assign a quantity that we call "coherence time" to a channel, abstracted from whatever you want to do over that channel, based purely on the dynamics of that channel, and those dynamics are captured in the change of the channel over time, which, mathematically, is also a change of phase over time – a freuquency – and hence the term "Doppler Spread" describes that. From that understanding (describing the dynamics of a channel rather than the ability of any specific transceiver to use it) you'll find all the formulas that derive or approximate coherence time from frequency shifts or, rather, bandwidths.

So, the second case applies to your formulas, here. I can't tell you whether that is appropriate for your research – that's up to you as a researcher to figure out (or just: define).

It's (IMHO) very important to realize that this all isn't a deterministic model of what happens to your signal – there's always an underlying, stochastic channel model. For example, you'd model the channel at any instant in time as Rayleigh channel in a tapped delay line model, and then you model the changes to that channel's coefficients with some probability density function. Based on that model, you make a statement about what is "likely" to change (for example, you say "this and that changes with the change following $\mathcal N(0,\sigma^2)$", so let's consider what's within $2\sigma$ as likely) within a given time (which is the inverse of the bandwidth of the channel). And based on that, you calculate a Doppler spread. So, there's quite a lot of things your channel model defines that contributes to your definition of $T_c$:

1. A definition of "coherent", typically based on a limit for a (relative) change in some aspect of the channel impulse response
2. The instantaneous channel model's bandwidth
3. The stochastic model of how the model changes over time, typically a PDF
4. An (often arbitrary) definition of what is considered likely to describe any observation over finite time
5. A way to convert that to a coherence time, typically:
   1. a conversion from some stochastic moment of the PDF to a change-over-time, and an interpretation of that as a frequency
   2. a conversion of that frequency to a time, typically its inverse with some factor depending on normalizations used elsewhere

So, you'll need to pick a channel model first; then things can be derived from, or sensibly defined in accordance to, that model.

There's a lot of good literature out there, but seriously, stochastic, time-variant channels can be a bit daunting at first – take it slowly. Assuming you understand the basics of (deterministic, static) channels: Learn about stochastic, time-invariant channels first, then move on to time-variant channels [1]. The references below are just suggestions. I really can't tell you what works best for you as literature – but if you happen to have a researcher advising you: Researchers have favorite literature that use a notation that they are familiar with, and that is (unlike the wildly collected formulas you cited) coherent, and they often don't mind discussing things in that literature, so take the chance and ask them for a book on channels that they'd recommend, and read that, and discuss with them.

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[1] Proakis: Digital Communications has a chapter "Communication through Fading Multipath Channels" that is the culmination of the channel models it introduces.
[2] Sklar: Digital Communications, as Sklar is traditionally very concerned with a good description of channels; hence, both books introduce channels in the order I recommend above.