where is a constant, related to the high cutoff frequency by the following relation: Even if this kind frequency response is not realizable by a causal filter, its usefulness lies in the fact that behaviour of a cascade connection of first order low pass filters approaches the behaviour of this system more closely as the number of cascaded stages asymptotically rises to infinity. The corresponding impulse response can be calculated using the inverse Fourier transform of the shown frequency response Applying directly the definition of step response, To determine the 10% to 90% rise time of the system it is necessary to solve for time the two following equations: By using known properties of the error function, the value is found: since , and finally One-stage low-pass RC network For a simple one-stage low-pass RC network, the 10% to 90% rise time is proportional to the network time constant : The proportionality constant can be derived from the knowledge of the step response of the network to a unit step function input signal of amplitude: Solving for time and finally, Since and are such that solving these equations we find the analytical expression for and : The rise time is therefore proportional to the time constant: Now, noting that then and since the high frequency cutoff is equal to the bandwidth, Finally note that, if the 20% to 80% rise time is considered instead, becomes: One-stage low-pass LR network Even for a simple one-stage low-pass RL network, the 10% to 90% rise time is proportional to the network time constant . The formal proof of this assertion proceed exactly as shown in the previous section: the only difference between the final expressions for the rise time is due to the difference in the expressions for the time constant of the two different circuits, leading in the present case to the following result Rise time of damped second order systems According to , for underdamped systems used in control theory rise time is commonly defined as the time for a waveform to go from 0% to 100% of its final value: accordingly, the rise time from 0 to 100% of an underdamped 2nd-order system has the following form: The quadratic approximation for normalized rise time for a 2nd-order system, step response, no zeros is: where is the damping ratio and is the natural frequency of the network. Rise time of cascaded