Here is the open-loop bandwidth of a certain op-amp shown in red: -
The blue line is when certain closed-loop components are applied to the op-amp.
Bandwidth is normally measured at the 3dB point of the frequency response and in the case of an op-amp (open-loop) this will be at 24Hz in the diagram.
If closed loop components were present, the gain would be reduced to (say) 20dB (blue line) but the bandwidth would increase to 1MHz.
The above example is for simple resistors "closing" the loop with negative feedback and the resulting bandwidth (3dB point) is always greater.
However, if an op-amp filter circuit was required that cut-off frequencies above 10Hz, the filter would have a bandwidth of 10Hz. In this example the closed-loop bandwidth is less than the open-loop bandwidth.
Does this help?
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:
- N is the number of paths
- \$\tau_l\$ is the delay associated with the l-th path
- \$p_l\$ is the normalised power of the l-th path so that \$\sum p_l=1\$
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.
Best Answer
It is always undesirable to have a fading channel, whether fast, slow, frequency selective, or whatever.
But we don't get to choose how the channel behaves, so have to design coding schemes to overcome the fading. The better the channel, the less forward error correction is needed, either reduction in payload, complexity, latency, it's all bad and is avoided if possible.
Multipath fading causes frequency selectivity. If the channel is very narrow, the receiver will only see a small width of the fade, and it will appear flat. If the channel is wide, then the receiver will see a significant variation in the signal strength across the channel.
Schemes with narrow channels suffer from frequency tracking for Doppler which limits mobility, and potentially long drop-outs due to slow fades. Hence most systems designed in the last decade or two are relatively wide.
If portions of a channel are known to work well, with other portions being poor, but it's not known a priori which portions, then a good coding scheme will spread redundancy across the channel, so that whatever gets lost can be reconstructed.
In the case of a CD recording channel, which is very good until there is a scratch, the coding scheme seeks to spread the redundant data out in time, so there is enough good data to reconstruct the 100s of bits lost during a scratch.
OFDM will seek to spread the redundant data apart in frequency, across the channel, and in time, to combat both impulse noise and frequency selective fading. Multiple receive antennae are used to combat slow fading with spatial diversity.
Once the system is coded to cope with multipath, Single Frequency Networks (SFNs) can be deployed, to make much more efficient use of the spectrum for broadcast (DAB for instance). 4G radio can also use downlinks from multiple base stations on the same frequency to exploit the same.
Having said that lot, don't get too hung-up on the technology. Many (too many) of the decisions about what gets deployed are made for commercial politics.
Organisations like ETSI etc are made up of manufacturers who have lots of patents for bits of the technology. The rather bizarre word for how they work is coopertition. That is, they cooperate to create the global specification (they have to, you can't make a market from 10 different non-interoperable systems), and then compete with each other to make money from it. It's a wonder the process works at all.
During the standards setting process, company A wants to use their hyperbligual multiplex patent. Company B says OK, but only if it's modified with our left-handed frequency channeliser. And both get written into the standard, even if just one of them would deliver the benefit. So the basics, channel width, OFDM, are reasonable. The details, OMG, nightmare, ice-pack on the forehead needed to read the standards. But free-market commerce hasn't come up with a better way to do it.