(* ::Package:: *) (* ::Input:: *) (*$Assumptions={\[Tau] \[Element] Reals, \[Tau]>0, A \[Element] Reals, A>0, \[Alpha] \[Element] Reals, \[Alpha]>0, t \[Epsilon] Reals, t>-\[Infinity], f0 \[Element] Reals, f0>0, f \[Element] Reals}*) (* ::Text:: *) (*Sine-Gaussian waveform*) (* ::Input:: *) (*hp=1/2 A(1+\[Alpha]^2) Exp[-t^2/\[Tau]^2]Sin[2 \[Pi] f0 t+\[CurlyPhi]]*) (*hc=A \[Alpha] Exp[-t^2/\[Tau]^2]Cos[2 \[Pi] f0 t+\[CurlyPhi]]*) (*h=hp -I hc;*) (* ::Text:: *) (*Angular mean of h^2*) (* ::Input:: *) (*h2mean=Integrate[hp^2+hc^2,{\[Alpha],-1,1}]/2//Simplify (* -d\[Alpha] =-d cos\[Theta] = sin\[Theta] d\[Theta] *)*) (* ::Text:: *) (*rss value*) (* ::Input:: *) (*hrss=Sqrt[Integrate[h2mean,{t,-\[Infinity],\[Infinity]}]]//FullSimplify*) (* ::Text:: *) (*rss value in one particular case for the phase*) (* ::Input:: *) (*hrss/. \[CurlyPhi]->\[Pi]/4//N*) (* ::Text:: *) (*rss value considering only the duration of the signal*) (* ::Input:: *) (*hrss4=Sqrt[Integrate[h2mean,{t,-2\[Tau],2\[Tau]}]]//FullSimplify*) (* ::Text:: *) (*rss value for this particular phase that agrees with the previous result*) (* ::Input:: *) (*hrss4/. \[CurlyPhi]->\[Pi]/4//N*)