Bounds Existing on the Time and Frequency Responses of Various Types of Networks

The transient response of fixed, lumped, linear, and stable networks is investigated and many bounds are shown to exist on the.impulse and step responses of various classes of system functions. Conversely, if the impulse response is restricted in certain ways, bounds must then exist on the frequency response. These bounds, which are obtained by manipulating the Fourier transforms, have many practical implications. For instance, the response due to a unit impulse of current on a passive driving point impedance which has a shunting capacity, C, across its input terminals is bounded by ±(1/C) and the rise time for a low pass system of this form must be greater than rC where r equals the value of the impedance under dc conditions. Similar statements may be made for transfer functions satisfying certain mathematical restrictions. As other examples, more severe lower bounds have been found on the settling time for such systems. Also the overshoot or undershoot of the response to a step input of current for an RC driving point impedance cannot be greater than one hundred per cent.