One type of hysteresis is a lag between input and output. An example is a sinusoidal input  that results in a sinusoidal output , but with a phase lag :
Such behavior can occur in linear systems, and a more general form of response is
where  is the instantaneous response and  is the impulse response to an impulse that occurred  time units in the past.  In the frequency domain, input and output are related by a complex generalized susceptibility that can be computed from ; it is mathematically equivalent to a transfer function in linear filter theory and analogue signal processing.
This kind of hysteresis is often referred to as rate-dependent hysteresis. If the input is reduced to zero, the output continues to respond for a finite time. This constitutes a memory of the past, but a limited one because it disappears as the output decays to zero. The phase lag depends on the frequency of the input, and goes to zero as the frequency decreases.
When rate-dependent hysteresis is due to dissipative effects like friction, it is associated with power loss.