# Does the frequency response assume the zero input response to be 0?

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.

• 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.) Jan 31, 2017 at 8:38

## 1 Answer

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.

• A system is not LTI if it is not linear and/or time-invariant. I don't see how a system with non-zero initial conditions makes the system non-linear and/or time-invariant. That would make a simple RC-circuit with an initial capacitor voltage a non-LTI system which cannot be true. I think you are wrong here, Matt.
– Carl
Jan 25, 2022 at 15:47
• @Carl: A system with non-zero initial condition doesn't obey the rule of homogeneity: if you scale the input by some constant, the output doesn't scale accordingly, because the contribution to the output from the initial conditions doesn't scale with the input signal. So, you're right, a simple RC-circuit with non-zero initial conditions is non-linear, at least according to this very common definition of system linearity. Jan 25, 2022 at 16:08
• However, it is possible to define system linearity is another way, which interpretes the initial conditions as a separate input, leading to the concepts of zero-input linearity and zero-state linearity. Jan 25, 2022 at 16:08