I'm a bit confused about the frequency response and the state of a system. Is it simply the ratio of the output of the zero state response to the input of the zero state response? Or does it include the zero input response? Any clarifications would be greatly welcomed.

  • $\begingroup$ there seems to be some confused semantic. i dunno if, by "frequency response", you mean LTI system (or filter) or something else. Matt may have guessed correctly. frequency response is either (A) the ratio of the output Fourier Transform to the input Fourier Transform evaluated as a complex function of frequency, $f$ or $\omega$.or (B) the log of (A) usually expressed in dB and degrees (but the mathematical natural units would be nepers and radians.) $\endgroup$ – robert bristow-johnson Jan 31 '17 at 8:38

A frequency response describes a linear time-invariant (LTI) system. It is the Fourier transform of the system's impulse response, and only LTI systems are fully characterized by their impulse response.1

A system with non-zero initial conditions is not LTI because its output has a component which only depends on the initial conditions and which is independent of the input signal. Consequently, a system with non-zero initial conditions cannot fully be described by a frequency response. If you assume zero initial conditions, then such a system (if otherwise LTI) can be described by a frequency response.

So to answer your question, a system can indeed only be fully described by a frequency response if its zero-input response (ZIR) is zero, because if its ZIR were not zero it wouldn't be an LTI system.

Also take a look at these two answers to related questions: answer 1, answer 2.

1Also linear time-varying (LTV) systems can be characterized by an impulse response, but it is a two-dimensional function, and LTV systems don't have a frequency response in the conventional sense.

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