The relation between fractional bandwidth and Q factor

The functional relation between the fractional bandwidth and the quality factor of a radiating system is studied from simple first-order-like RLC-like systems to higher-order radiating systems. It is demonstrated that, considering a first-order system, only the quality factor based on differentiation of the input impedance has unique proportionality to the fractional bandwidth. In the general case of higher-order systems, it is however shown that this property is lost.