Could anyone please provide a simple explanation of what Coherence Bandwidth means, and how is that related to Delay Spread.
I read some articles but they don't help me to form the relation between the two concepts. I know that the Delay Spread is something which occurs when we have multipaths. Whereas, the Coherence Bandwidth describes the "stability" of the channel in terms of frequency.
Please correct me if there are errors in my understanding and question.
The coherence bandwidth measures how much a channel is statisticallt flat given a fixed size window through which we watch it.
Imagine a transmitter and a receiver, the transmitted signal being \$x(t)\rightleftharpoons X(f)\$. The receiver ideally gets the same \$x(t)\$ at the antenna, but unfortunately there's the mighty channel \$c(t)\rightleftharpoons C(f)\$. The receiver will then get \$Y(f) =X(f)\ast C(f)\$.
That \$C(f)\$ is ideally 1, i.e. it's perfectly flat for each frequency (please note that \$C(f)\$ is a complex function, i.e. \$C(f) : \mathbb{R} \rightarrow \mathbb{C}\$). That's not what happens in real life. If we measure just \$|C(f)|\$ completaly disregarding the phase, that is fundamental in almost all modulation techniques, we find out that it's everything but flat.
Now just close the math-ish door and let all the engineering folks in. There's already one shouting "Hey, if you look close enough that \$C(f)\$ is flat indeed.". Well, he's right and that's where the coherence bandwidth kicks in: you have the channel transfer function, you look at it close enough, and that becomes flat. The coherence bandwidth tells you how much is close enough: if that's 1kHz, well you've got to magnify your \$x\$ axis to see only a 1kHz portion at a time. If that's 1MHz... Well, you guessed it.
So why is it a statistical parameter? Well, you can't certainly measure all the channels you want to transmit on. Some guys at IEEE one day decided "ok, if you are in a city with tall buildings you should expect a \$B_C\$ of this much, if you are on a flat desert \$B_C\$ would be that much", and so on, and various models were born.
When does the delay spread kick in? Well I see you quite grasp what it is, and as you say \$B_C\$ and \$D\$ are close friend. As you know \$D\$ measures how much delay we should expect between the direct (the most direct) path and the others. That number tells us how long before our signal gets compromised by itself. Well it appears that a nice rule of thumb (read: there's little to no physical meaning associated to the following formula) is \$B_C=\frac{1}{D}\$. And we love to know the \$B_C\$ of a channel because it tells us how much bandwidth our signal can use without using advanced techniques such as an equalizer.
Added after OP request
What is delay spread? Here is what I have in my notes:
$$\Delta\overline{\tau}=\sum_{l=1}^Np_l\tau_l$$
Where:
The delay spread hence express something like the delay after what I expect most of the power is arrived at the receiver. Consider the situation with: $$N=3\\ \mathbf{p}=[0.7, 0.2, 0.1] \\ \mathbf{\tau} = [1,2,5]ms$$ You get \$\Delta\overline{\tau}=1.6ms\$
While if: $$N=4\\ \mathbf{p}=[0.3, 0.4, 0.2, 0.1] \\ \mathbf{\tau} = [1,2,4,7]ms$$ You get \$\Delta\overline{\tau}=2.6ms\$
As you can see the delay spread measures when most of the power will arrive, and it's quite useful to appropriately tune the receiving equalizer to get the most out of your signal.
Coherence bandwidth refers to the situation in which all the frequencies in that band are subject to the same amount of attenuation. Given that, then they should be well behaved and predictable with frequency/wavelength.
What you are probably not understanding is what a multi-path signal looks like in the frequency domain. When you have multi-path, which is subject to variable delay from the environment, you can have both attenuation or peaking, both of which makes the frequency that is being subjected to it, different than it's neighbours and therefore it no longer fits the "coherence bandwidth" definition. i.e. it needs to be dealt with differently.
Another way of looking at this is that you want to control your bandwidth sufficiently so that you get all the related signals into the receiver before multi-path can affect the reception and corrupt the signal. It makes the design of the receiver circuits easier.
