波のフーリエ変換(2)

フーリエ変換 (Subscript[X, (Subscript[T, 0])]) (f) のグラフ（周期\(\(T\_0\) を持つ\)波）

\(\(\( (*\ \(X\_\(T\_0\)\) \((f)\) の定義\ *) \)\(
\)\(lx[f_] := 
        1\/π\ \(I\ \((2  f)\) \((1 - E\^\(\(-I\)\ 2\ π \((\ 2  f)\)\))\)\)\/\(1 - \((2  f)\)\^2\);\)\)\)

\(\(\( (*\ 
      Re\ \(X\_\(T\_0\)\) \((f)\) のグラフ\ *) \)\(
\)\(Plot[Re[lx[f]], {f, \(-4.5\), 4.5}, 
      Axes -> Automatic, ImageSize -> {440, Automatic}, 
      PlotRange -> {{\(-4.5\), 4.5}, {\(-0.5\)*1.2, 0.5*1.2}}, 
      Ticks -> {{\(-4\), \(-3\), \(-2\), \(-1\), 0, 1, 2, 3, 
            4}, {\(-0.5\), 0.5}}]\)\)\)

\(\(\( (*\ 
      Im\ \(X\_\(T\_0\)\) \((f)\) のグラフ\ *) \)\(
\)\(Plot[Im[lx[f]], {f, \(-4.5\), 4.5}, 
      Axes -> Automatic, ImageSize -> {440, Automatic}, 
      PlotRange -> {{\(-4.5\), 4.5}, {\(-0.5\)*1.2, 0.5*1.2}}, 
      Ticks -> {{\(-4\), \(-3\), \(-2\), \(-1\), 0, 1, 2, 3, 
            4}, {\(-0.5\), 0.5}}]\)\)\)

Plot[Abs[lx[f]], {f, -4.5, 4.5}, 
      Axes -> Automatic, ImageSize -> {440, Automatic}, 
      PlotRange -> {{-4.5, 4.5}, {-0.2, 1.2}}, 
      Ticks -> {{-4, -3, -2, -1, 0, 1, 2, 3, 4}, {0.5, 
            1}}]

\(\(\( (*\ 偏角ϕのグラフ\ *) \)\(
\)\(Plot[Arg[lx[f]], {f, \(-4.5\), 4.5}, 
      Axes -> Automatic, ImageSize -> {440, Automatic}, 
      PlotRange -> {{\(-4.5\), 4.5}, Automatic}, 
      Ticks -> {{\(-4\), \(-3\), \(-2\), \(-1\), 0, 1, 2, 3, 
            4}, {\(-π\), π}}]\)\)\)

フーリエ変換 (Subscript[X, (Subscript[T, 0])]) (f) のグラフ（合成された波）

\(\(\( (*\ \(X\_\(T\_0\)\) \((f)\) の定義\ *) \)\(
\)\(lx[
          f_] := \(1\/2\) \((1\/π\ \(I\ \((2  f)\) \((1 - E\^\(\(-I\)\ 2\ π \((\ 2\ f)\)\))\)\)\/\(1 - \((2  f)\)\^2\) + 
              2\/\(2\ π\)\ \(I\ f \((1 - E\^\(\(-I\)\ 2\ π\ f*2\))\)\)\/\(1 - f\^2\))\);\)\)\)

\(\(\( (*\ 
      Re\ \(X\_\(T\_0\)\) \((f)\) のグラフ\ *) \)\(
\)\(Plot[Re[lx[f]], {f, \(-4.5\), 4.5}, 
      Axes -> Automatic, ImageSize -> {440, Automatic}, 
      PlotRange -> {{\(-4.5\), 4.5}, {\(-1\)*1.2, 1*1.2}}, 
      Ticks -> {{\(-4\), \(-3\), \(-2\), \(-1\), 0, 1, 2, 3, 
            4}, {\(-1\), \(-0.5\), 0.5, 1}}]\)\)\)

\(\(\( (*\ 
      Im\ \(X\_\(T\_0\)\) \((f)\) のグラフ\ *) \)\(
\)\(Plot[Im[lx[f]], {f, \(-4.5\), 4.5}, 
      Axes -> Automatic, ImageSize -> {440, Automatic}, 
      PlotRange -> {{\(-4.5\), 4.5}, {\(-1\)*1.2, 1*1.2}}, 
      Ticks -> {{\(-4\), \(-3\), \(-2\), \(-1\), 0, 1, 2, 3, 
            4}, {\(-1\), \(-0.5\), 0.5, 1}}]\)\)\)

Plot[Abs[lx[f]], {f, -4.5, 4.5}, 
      Axes -> Automatic, ImageSize -> {440, Automatic}, 
      PlotRange -> {{-4.5, 4.5}, {-0.2, 1.2}}, 
      Ticks -> {{-4, -3, -2, -1, 0, 1, 2, 3, 4}, {0.5, 
            1}}]

\(\(\( (*\ 偏角ϕのグラフ\ *) \)\(
\)\(Plot[Arg[lx[f]], {f, \(-4.5\), 4.5}, 
      Axes -> Automatic, ImageSize -> {440, Automatic}, 
      PlotRange -> {{\(-4.5\), 4.5}, Automatic}, 
      Ticks -> {{\(-4\), \(-3\), \(-2\), \(-1\), 0, 1, 2, 3, 
            4}, {\(-π\), π}}]\)\)\)

フーリエ変換 (Subscript[X, (Subscript[T, 0])]) (f) のグラフ（初期位相を持つ波）

\(\(\( (*\ \(X\_\(T\_0\)\) \((f)\) の定義\ *) \)\(
\)\(\(lc = 1;\)

    \(lx[f_] := \((\(\@2\/2\) lc)\) 
            1\/\(2  π\)\ \(I\ f \((1 - E\^\(\(-I\)\ 2\ π\ f\))\)\)\/\(1 - f\^2\) - \((\(\@2\/2\) lc)\) 
            1\/\(2\ π\)\ \(1 - E\^\(\(-I\)\ 2\ π\ f\)\)\/\(1 - f\^2\);\)\)\)\)

\(\(\( (*\ 
      Re\ \(X\_\(T\_0\)\) \((f)\) のグラフ\ *) \)\(
\)\(Plot[Re[lx[f]], {f, \(-4.5\), 4.5}, 
      Axes -> Automatic, ImageSize -> {440, Automatic}, 
      PlotRange -> {{\(-4.5\), 4.5}, {\(-lc\)/2*1.2, lc/2*1.2}}, 
      Ticks -> {{\(-4\), \(-3\), \(-2\), \(-1\), 0, 1, 2, 3, 
            4}, {{\(-1\)/2, "-C/2"}, {1/2, "C/2"}}}]\)\)\)

\(\(\( (*\ 
      Im\ \(X\_\(T\_0\)\) \((f)\) のグラフ\ *) \)\(
\)\(Plot[Im[lx[f]], {f, \(-4.5\), 4.5}, 
      Axes -> Automatic, ImageSize -> {440, Automatic}, 
      PlotRange -> {{\(-4.5\), 4.5}, {\(-lc\)/2*1.2, lc/2*1.2}}, 
      Ticks -> {{\(-4\), \(-3\), \(-2\), \(-1\), 0, 1, 2, 3, 
            4}, {{\(-1\)/2, "-C/2"}, {1/2, "C/2"}}}]\)\)\)

Plot[Abs[lx[f]], {f, -4.5, 4.5}, 
      Axes -> Automatic, ImageSize -> {440, Automatic}, 
      PlotRange -> {{-4.5, 4.5}, {-lc/2*0.2, lc/2*1.2}}, 
      Ticks -> {{-4, -3, -2, -1, 0, 1, 2, 3, 
            4}, {{-1/2, "-C/2"}, {1/2, "C/2"}}}]

\(\(\( (*\ 偏角ϕのグラフ\ *) \)\(
\)\(Plot[Arg[lx[f]], {f, \(-4.5\), 4.5}, 
      Axes -> Automatic, ImageSize -> {440, Automatic}, 
      PlotRange -> {{\(-4.5\), 4.5}, Automatic}, 
      Ticks -> {{\(-4\), \(-3\), \(-2\), \(-1\), 0, 1, 2, 3, 
            4}, {\(-π\), π}}]\)\)\)