In general, How do we design a system that takes input x(t) and outputs x(t/2)? I know that if system outputs x(t/2) then the frequency domain of output signal has the form X(2f). But what should be the impulse response for such systems?

  • $\begingroup$ hmmm stupid question, if this is continuous-signal continuous time, how are you physically dealing with the fact that with 1 hour of input, you produce 2 hours of output? You don't have a time machine… So you can't actually design a system like that. You can do theoretical considerations, but you can't build it. Or is this not actually time-continuous, and you're in the (imho) easier case of discrete-time signals? $\endgroup$ Commented Mar 13 at 13:30
  • $\begingroup$ In both cases, you need unbounded memory, don't you? $\endgroup$
    – MBaz
    Commented Mar 13 at 13:38
  • $\begingroup$ The red-shift (doppler) of stars that for the foreseeable future moving away from us would seem to satisfy this question (with the exception that the scaling probably is 1.000001 rather than 2)? $\endgroup$
    – Knut Inge
    Commented Mar 13 at 13:48
  • $\begingroup$ @MarcusMüller if not practically, I wonder if it is possible to make such a system in theory? And yes the input is continuous time. $\endgroup$
    – user133933
    Commented Mar 13 at 13:56
  • $\begingroup$ @user133933 not at all. you can't say "make" and "in theory"; that's just a paradox, unless you state which axioms you are willing to ignore. $\endgroup$ Commented Mar 13 at 13:56

1 Answer 1


But what should be the impulse response for such systems?

There is non. It's not an LTI system and hence it can't be described by it's transfer function or impulse response.

Of course there are time-stretchers or compressors in (for example) audio. They typically look for snippets of the signal that are more or less periodic and then cut out or replicate entire periods. The details there depend a lot on the type of signal and what your specific requirements are.


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