\[Omega]0 = 20;
amplitude[\[Omega]_, \[Sigma]_] := E^(-((\[Omega] - \[Omega]0)^2/(2 \[Sigma]^2)))/(Sqrt[2 \[Pi]] \[Sigma]);
rpgenerator = Total[Table[ RandomReal[{-1, 1}]*Sin[j/5 \[Omega] + 2 \[Pi] RandomReal[{0, 1}]], {j, 1, 80}]];
randomphase[\[Omega]_, flag_] := flag*I rpgenerator;
phase[\[Omega]_, flag_] := Exp[randomphase[\[Omega], flag]];
argphase = randomphase[\[Omega] - \[Omega]0, 1];
spectrum[\[Omega]_, \[Sigma]_, flag_] := amplitude[\[Omega], \[Sigma]]*phase[\[Omega], flag]

plot[\[Sigma]_, flag_, string_] := (
  \[Delta]\[Omega] = \[Sigma]/150;
  hsamples = Total@Table[ spectrum[\[Omega], \[Sigma], flag]*E^(I \[Omega] t), {\[Omega], \[Omega]0 - 10 \[Sigma], \[Omega]0 + 10 \[Sigma], \[Delta]\[Omega]}];
  tplot = Re@hsamples/FindMaximum[Abs@hsamples, {t, 0}][[1]];
  Grid[{{Text[Style[string, Purple, Bold, FontSize -> 18]], 
     SpanFromLeft}, {},
    {
     Show[
      Plot[ Abs[amplitude[\[Omega], \[Sigma]]/amplitude[\[Omega]0, \[Sigma]]], {\[Omega], 10, 30}, PlotRange -> {-1.1, 1.2}, Ticks -> None, Axes -> {True, False}, AxesStyle -> Thick, PlotStyle -> Directive[Thick, Black], PlotLegends -> Placed[LineLegend[{Black, Orange}, {"Amplitude", "Phase"},  LegendFunction -> (Framed[#, RoundingRadius -> 5, Background -> White] &), LegendMargins -> 5], {Right, Top}], PlotLabel -> "Frequency domain", LabelStyle -> Directive[Bold, Black, FontSize -> 14], ImageSize -> Large, PlotPoints -> 200]
      ,
      Plot[ Evaluate@(Im[randomphase[\[Omega] - \[Omega]0, flag]]/(4 \[Pi])), {\[Omega], 10, 30}, PlotStyle -> {Thick, Orange}, Axes -> False]
      ]
     ,
     Show[
      Plot[Cos[\[Omega]0 t], {t, -5, 5}, PlotRange -> {-1.1, 1.2}, Ticks -> None, Axes -> {True, False}, AxesStyle -> Thick, PlotStyle -> Directive[LightGray], PlotLegends -> Placed[LineLegend[{Black, LightGray}, {"Signal", "Carrier"}, LegendFunction -> (Framed[#, RoundingRadius -> 5, Background -> White] &), LegendMargins -> 5], {Right, Top}], LabelStyle -> Directive[Bold, Black, FontSize -> 14]]
      ,
      Plot[tplot, {t, -5, 5}, PlotRange -> {-1.1, 1.2}, PlotStyle -> Directive[Thick, Black]], ImageSize -> Large, PlotLabel -> "Time domain", LabelStyle -> Directive[Bold, Black, FontSize -> 14]]
     }}]
  )

frame1 = plot[0.01, 0, "Time coherence tells us how much our signal looks like a perfect sinusoidal"];
frame2 = plot[0.01, 0, "A signal with a very narrow bandwidth is almost exactly a sinusoidal"];
sinstep[t_] := Sin[\[Pi]/2 t]^2;
frame3 = Table[plot[N[0.01 + sinstep[k]*(3 - 0.01)], 0, "A signal with a large bandwidth stops looking like a simple sinusoidal"], {k, 0, 1, 0.01}];
frame4 = plot[3, 0, "If the phase of the spectrum is flat, all sinusoidal components add constructively at t=0, and we get a pulse"];
frame5 = Table[plot[3, sinstep[k], "But if the phase is random, the result doesn't look like a sinusoidal at all (short time coherence)"], {k, 0, 1, 0.01}];
frame6 = Table[plot[3 - (3 - 0.01)*sinstep[k], 1, "If the bandwidth is small, the randomness of the phase doesn't matter, and we recover a nice sinusoidal"], {k, 0, 1, 0.01}];
frame7 = Table[plot[0.01, 1 - sinstep[k], "If the bandwidth is small, the randomness of the phase doesn't matter, and we recover a nice sinusoidal"], {k, 0, 1, 0.01}];

ListAnimate[Join[Table[frame1, {100}],  Table[frame2, {100}], frame3, Table[frame3[[-1]], {30}], Table[frame4, {100}], frame5, able[frame5[[-1]], {30}], frame6, Table[frame6[[-1]], {30}], frame7, Table[frame7[[-1]], {30}]  ]]