N roots of the characteristic polynomial and will therefore be the poles of the transfer function. Consider the case of a transfer function with a single pole where . The Laplace transform of a general sinusoid of unit amplitude will be . The Laplace transform of the output will be and the temporal output will be the inverse Laplace transform of that function: The second term in the numerator is the transient response, and in the limit of infinite time it will diverge to infinity if σP is positive. In order for a system to be stable, its transfer function must have no poles whose real parts are positive. If the transfer function is strictly stable, the real parts of all poles will be negative, and the transient behavior will tend to zero in the limit of infinite time. The steady-state output will be: The frequency response (or "gain") G of the system is defined as the absolute value of the ratio of the output amplitude to the steady-state input amplitude: which is just the absolute value of the transfer function evaluated at . This result can be shown to be valid for any number of transfer function poles. Signal processing Let be the input to a general linear time-invariant system, and be the output, and the bilateral Laplace transform of and be Then the output is related to the input by the transfer function as and the transfer function itself is therefore In particular, if a complex harmonic signal with a sinusoidal component with amplitude , angular frequency and phase , where arg is the argument where is input to a linear time-invariant system, then the corresponding component in the output is: Note that, in a linear time-invariant system, the input frequency has not changed, only the amplitude and the phase angle of the sinusoid has been changed by the system. The frequency response describes this change for every frequency in terms of gain: and phase shift: The phase delay (i.e., the frequency-dependent amount of delay introduced to the sinusoid by the transfer function) is: The group delay (i.e., the frequency-dependent amount of delay introduced to the envelope of the sinusoid by the transfer function) is found by computing the derivative of the phase shift with respect to angular frequency , The transfer function can also be shown using the Fourier transform which is only a special case of