Filters can be coupled both inductively and capacitively:
The top circuit is an inductively coupled filter. The two at the bottom are capacitively coupled filters.
In inductively coupled filters, the windings of the inductors are wrapped around the same core. The ratio VL2:VL1 (omitting C1 and C2) is called coefficient of coupling k. This coefficient has a major influence of the frequency response:
From left to right, k increases. This will result in an increasing output voltage. When k reaches a certain value, the frequency response starts having two top values. A filters are critically coupled when this just doen't happen. This appears to be the case when k = 1/Q, in which Q is the quality factor of both inductors. Of course, both inductors need to have the same Q. If k < 1/Q; the filter is undercoupled and will be narrow banded. If k > 1/Q; the filter is overcoupled, and will be wide banded.
In capacitively coupled filters, k is deternined by capacitors.
The cirtuit on the left shows top coupling. Here, k = CT/√(C3∙C4).
The filter on the right shows foot coupling, where k = √(C6∙C7)/CF.
When we compare coupled filters with LC filters, we'll see that coupled filters are much steeper. In the narrow banded LC-filter the difference in output voltage at 150kHz and 159kHz is about 15dB. In the critically couple filter, this difference is about 35dB! Note however that the output voltage of a coupled filter is much less than that of an LC filter.
The picture above is a part of the circuit diagram of a radio receiver.
The transistor amplifies the input voltage and the coupled filter picks out the wanted signal. You notice that the manufacturer used a tapped inductor. The voltage at pin 9 will be less than at pin 8, which reduces the current flow drawn by the next stage. This is important, because this current will change the resonance frequency if it's too high.