I wonder if the following acoustic system is i) considered to be linear time-invariant, and ii) if it is fully described by it's frequency response?

A speaker produces waves at the input location of a duct system with a branch and the output is measured after the branch point. The anechoic terminations completely absorb the sound waves, and it is assumed that only 1D low frequency waves exist.

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So the system, when subject to the input $$ x(t) = e^{j \omega t}, \quad t \ge 0 $$

experiences the output $$ y(t) = \left\{ \begin{array}{l l} 0 & \quad 0 \le t < t_0 \\ A e^{j \omega t} & \quad \text{$t_0 \le t<t_1$}\\ A e^{j \omega t} + B e^{j \omega t}& \quad \text{$t \ge t_1$} \end{array} \right., $$

where A and B are complex constants depending on the duct geometry, and $t_1$is the delay time due to the side branch geometry.

Yes the system itself is not changing with time, however due to the finite speed of sound there is a transient period before the reflection from a side branch has occurred.

The frequency response at $\omega$ (a complex number capturing only magnitude and phase relationship between input and output) would be related to the final part of the piecewise defined output (steady state part) and offer no information about the time $t_1$. So, for example, a second system which had a longer side branch may have the same frequency response yet a different value of $t_1$. Therefore my argument would be that the FR is inadequate to characterise the transient behaviour of the system. Is this correct?


No, if your system can (with sufficient accuracy) be modeled as a linear time-invariant system, then it is fully characterized by its impulse response. Since the frequency response is the Fourier transform of the impulse response, the system is also fully characterized by its frequency response. The frequency response does not only describe steady-state behavior but also transients (because it simply is a full description of the system). See also this answer to a related question.

  • $\begingroup$ "No" meaning no it is not an LTI system? Or that my reasoning is not correct? I am aware that the FR describes transients, but not these specific transients. The previous question/answer does not deal with this $\endgroup$ – xyz Apr 7 '14 at 7:20
  • $\begingroup$ "No" to your last question ("is this correct?"). Why would the FR describe transients in general but "not these specific transients"? What's special about them? $\endgroup$ – Matt L. Apr 7 '14 at 7:22
  • $\begingroup$ Piece wise defined output, two separate systems have the same input/output phase and magnitude relation at steady state $\endgroup$ – xyz Apr 7 '14 at 7:23
  • $\begingroup$ That's just a kind of echo, isn't it? $\endgroup$ – Matt L. Apr 7 '14 at 7:24
  • $\begingroup$ Yes. My apologies that this is probably a very basic question that I am having trouble understanding. But for the said example then how could the FR capture an echo? $\endgroup$ – xyz Apr 7 '14 at 7:28

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