Bounds on the envelopes of the response of systems to bandlimited inputs

The response of a linear system to an input may be considered in terms of the analytic signals associated with input, system impulse response function and output. Following from a convolution relationship linking the analytic signals it is shown that the envelope of the response signal is contained between bounds given in terms of a convolution of the envelopes of the system and input, and also a measure of the instantaneous phase of system and input. Consideration of the envelope of the response is sometimes more appropriate than obtaining the detailed time history, and methods of bounding the behaviour of the envelope are therefore useful. Analytic results for modulated harmonic (and swept sine) inputs to simple systems are derived and computational results for more complex inputs into multi-modal systems are presented.