The following Convolve expression doesn't produce a result.

a = 0.1;
ImpulseResponse[t_] := 
  (1/(2.*Pi*10.*Sqrt[1 - a*a]))*Exp[-a*2*Pi*10*t]*Sin[2.*Pi*10.*Sqrt[1 - a*a]*t]
pulse[x_] := Cos[2*Pi*x]
Convolve[ImpulseResponse[t], pulse[t], t, x]

I am trying to figure out why and make it work, but I am not really successful at that.

  • 3
    $\begingroup$ My guess is because your impulse response is an exponential that goes to infinity as t goes to negative infinity. Notice in the docs that Convolve executes an integral from -infinity to infinity. You probably want to multiply the impulse response you've given by HeavisideTheta[t]. $\endgroup$ – N.J.Evans Dec 15 '16 at 20:31
  • $\begingroup$ @N.J.Evans You are absolutely right. `HeavisideTheta[t]´ does the job. $\endgroup$ – skrat Dec 15 '16 at 22:20

Your impulse response function isn't defined over {-∞, ∞}. But if you extend it as required with UnitStep, you will get a result.

With[{a = 1/10},
  impulseResponse[t_] := 
    (1/(2*Pi*10*Sqrt[1 - a*a]))*Exp[-a*2*Pi*10*t]*Sin[2*Pi*10*Sqrt[1 - a*a]*t] *
pulse[t_] := Cos[2*Pi*t]
Convolve[impulseResponse[t], pulse[t], t, x]

(3*Sqrt[11]*(99*Cos[2*Pi*x] + 2*Sin[2*Pi*x]))/(19610*Pi)


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