FourierSeries_RealvsComplex_sixstep.ipynb

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    {
      "cell_type": "markdown",
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      "source": [
        "<a href=\"https://colab.research.google.com/github/bingsen-wang/PowerElectronics/blob/main/FourierSeries_RealvsComplex_sixstep.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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      "cell_type": "markdown",
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      "source": [
        "<center><h1><b>Fourier Series: Complex vs Real Coefficients for Six-Step Waveform\n",
        "</h1></b>\n",
        "\n",
        "Dr. Bingsen Wang\n",
        "\n",
        "12/16/2023\n",
        "</center>"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "#Complex Fourier Coefficiencts"
      ],
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      "source": [
        "A function $f(x)$ can be decomposed into Fourier series with complex Fourier coefficients\n",
        "$$f(x) = \\sum_{n=-\\infty}^{\\infty}c_ne^{jn\\omega x}\\tag{1}$$\n",
        "where the complex coefficient is determined by\n",
        "$$c_n = \\frac{1}{2\\pi}\\int_0^{2\\pi} f(x)e^{-jn\\omega x}dx\\tag{2}$$\n",
        "\n",
        "It is worth noting that $f(x)$ can be a complex function although $x$ is real. If f(x) is real, then $c_n = c_{-n}^*$ with $*$ deonoting the conjugate."
      ],
      "metadata": {
        "id": "9J72oziV-6ot"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#Real Fourier Coefficients"
      ],
      "metadata": {
        "id": "ORyDoTjFYE7_"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Let $f(x)$ in (1) be real function an rewrite (1) as\n",
        "$$f(x) = c_0 + \\sum_{n=1}^{\\infty}(c_ne^{jn\\omega x} +c_{-n}e^{-jn\\omega x})\\tag{3}$$\n",
        "\n",
        "Based on Euler's identify\n",
        "$$\\begin{align}\n",
        "e^{jn\\omega x} &= \\cos(n\\omega x) + j\\sin(n\\omega x)\\\\\n",
        "e^{-jn\\omega x} &= \\cos(n\\omega x) - j\\sin(n\\omega x)\n",
        "\\end{align}\\tag{4}$$\n",
        "\n",
        "Substituting (4) into (3) yields\n",
        "$$\\begin{align}\n",
        "f(x) &= c_0 + \\sum_{n=1}^{\\infty}\\left\\{c_n\\left[\\cos(n\\omega x) + j\\sin(n\\omega x)\\right] +c_{-n}\\left[\\cos(n\\omega x) - j\\sin(n\\omega x)\\right]\\right\\}\\\\\n",
        " &= c_0 + \\sum_{n=1}^{\\infty}\\left[(c_n + c_{-n})\\cos(n\\omega x) + j(c_n - c_{-n})\\sin(n\\omega x) \\right]\n",
        "\\end{align}\\tag{5}$$\n",
        "\n",
        "The Fourier series with real coefficients is commonly written as\n",
        "$$\\begin{align}\n",
        "f(x) &= a_0 + \\sum_{n=1}^{\\infty}\\left[a_n\\cos(n\\omega x) + b_n\\sin(n\\omega x) \\right]\n",
        "\\end{align}\\tag{6}$$\n",
        "By comparison of (5) and (6), the real and complex coeffiencts are related by\n",
        "$$\\begin{align}\n",
        "a_0 &= c_0\\\\\n",
        "a_n &= c_n + c_{-n}\\\\\n",
        "b_n &= j(c_n + c_{-n})\n",
        "\\end{align}\\tag{7}$$\n",
        "\n"
      ],
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    },
    {
      "cell_type": "markdown",
      "source": [
        "Substituting (2) into (7) yields\n",
        "$$\\begin{align}\n",
        "a_0 &= \\frac{1}{2\\pi}\\int_0^{2\\pi} f(x)dx\\\\\n",
        "a_n &= \\frac{1}{\\pi}\\int_0^{2\\pi} f(x)\\cos(n\\omega x)dx\\\\\n",
        "b_n &= \\frac{1}{\\pi}\\int_0^{2\\pi} f(x)\\sin(n\\omega x)dx\n",
        "\\end{align}\\tag{8}$$"
      ],
      "metadata": {
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    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Hv0YxDrWU4h8"
      },
      "source": [
        "#Python Code\n",
        "The Python code illustrates the complex Fourier series using six-step waveform as an example."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 17,
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        "outputId": "84593c5a-5d8a-4d0f-c530-3beef3f3678b"
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      "outputs": [
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          "text": [
            "1\n",
            "1\n",
            "5\n",
            "7\n",
            "11\n",
            "13\n",
            "1\n"
          ]
        },
        {
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          "data": {
            "text/plain": [
              "<Figure size 900x900 with 4 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ],
      "source": [
        "import matplotlib\n",
        "from os import linesep\n",
        "from math import ceil\n",
        "from numpy import linspace,cos,pi,sin,random,exp,real, imag,heaviside,fft,array\n",
        "import matplotlib.pyplot as plt\n",
        "from scipy.integrate import quad\n",
        "from matplotlib import animation,rc\n",
        "rc('animation', html='jshtml')\n",
        "plt.rcParams.update({\"font.family\" : \"serif\",\"mathtext.fontset\" : \"cm\"})\n",
        "\n",
        "def sixstep(t): # six step on complex plane\n",
        "  ha = heaviside(cos(2*pi*t),0)\n",
        "  hb = heaviside(cos(2*pi*t-2*pi/3),0)\n",
        "  hc = heaviside(cos(2*pi*t+2*pi/3),0)\n",
        "  return ha+hb*exp(1j*2*pi/3)+hc*exp(1j*4*pi/3)\n",
        "\n",
        "def fftcplx(zt):\n",
        "  rt = zt.real\n",
        "  it = zt.imag\n",
        "  return (fft.fft(rt) + 1j*fft.fft(it))/len(zt)\n",
        "\n",
        "def recon(zfft,t,n):\n",
        "  zrec = [] #len 2n+1\n",
        "  radius = [] #radius\n",
        "  for i in range(n+1):\n",
        "    if i==0:\n",
        "      zrec.append(zfft[i]+t*0)\n",
        "      radius.append(0*t)\n",
        "    else:\n",
        "      radius.append(zfft[i]*exp(1j*2*pi*i*t))\n",
        "      radius.append(zfft[-i]*exp(-1j*2*pi*i*t))\n",
        "\n",
        "      zrec.append(zrec[2*i-2] + radius[2*i-1])\n",
        "      zrec.append(zrec[2*i-1] + radius[2*i])\n",
        "  return [zrec,radius]\n",
        "\n",
        "# draw dotted line between three points in three different plots\n",
        "def l2Ax(fig,ax1,ax2,ax3,p1,p2,p3,colors):\n",
        "  transFigure = fig.transFigure.inverted()\n",
        "  p1_g = transFigure.transform(ax1.transData.transform(p1))\n",
        "  p2_g = transFigure.transform(ax2.transData.transform(p2))\n",
        "  p3_g = transFigure.transform(ax3.transData.transform(p3))\n",
        "  line1 = matplotlib.lines.Line2D((p1_g[0],p2_g[0]),(p1_g[1],p2_g[1]),\n",
        "                      transform=fig.transFigure,ls='--',color=colors[1], lw=1)\n",
        "  line2 = matplotlib.lines.Line2D((p2_g[0],p3_g[0]),(p2_g[1],p3_g[1]),\n",
        "                      transform=fig.transFigure,ls='--',color=colors[2], lw=1)\n",
        "  fig.lines= [line1,line2]\n",
        "  return\n",
        "\n",
        "def vectorDraw(ax,vecs,n,nn,colors): #2*nn+1 vectors at nth instant\n",
        "  ax.clear()\n",
        "  ax.set_xlim(-1.5,1.5)\n",
        "  ax.set_ylim(-1.5,1.5)\n",
        "  ax.axis('off')\n",
        "  for k in range(2*nn+1):\n",
        "    if k==0:\n",
        "      center = vecs[0][0][n]\n",
        "      ax.arrow(0,0,center.real,center.imag,\n",
        "             width = 0.002,head_width=0.01,length_includes_head=True,color=colors[4])\n",
        "    else:\n",
        "      center = vecs[0][k-1][n]\n",
        "      ccl = vecs[1][k]\n",
        "      radius = ccl[n]\n",
        "      ax.arrow(center.real,center.imag,radius.real,radius.imag,\n",
        "             width = 0.002,head_width=0.01,length_includes_head=True,color=colors[4])\n",
        "      ax.plot(center.real+ccl.real,center.imag+ccl.imag,lw=1,color=colors[5],alpha=0.5)\n",
        "  ax.plot(vecs[0][2*nn][:n].real,vecs[0][2*nn][:n].imag,lw=3,color=colors[1])\n",
        "\n",
        "  return\n",
        "#parameters\n",
        "clst=['lightskyblue','violet','aqua','cornflowerblue','yellow','w',(.1,.1,.1)] #colors dark bg\n",
        "# clst=['mediumblue','m','brown','navy','brown',(.1,.1,.1),'w'] #colors light bg\n",
        "Nf = 256\n",
        "t=linspace(0,1,1024)\n",
        "traj = sixstep(t) #trajector on complex plane\n",
        "trajfft = fftcplx(traj) #complex fft\n",
        "trajrec = recon(trajfft,t,100) #reconstructed vectors, circles\n",
        "fig = plt.figure(figsize=(9,9),facecolor=clst[-1])\n",
        "fig.tight_layout()\n",
        "ax_frame = [[[0, 0, .5, .5], [-1.5,1.5], [-1.5,1.5]], #complex plane\n",
        "            [[0, 0, .5, .5], [-1.5,1.5], [-1.5,1.5]], #animated vectors\n",
        "            [[.5, .5, .5, .5], [-.1,1.1], [-1.5,1.5]], #square wave in time\n",
        "            [[0, .5, .5, .5], [-1.5,1.5], [-1.5,1.5]], #mirror\n",
        "            ]# [pos-boundary, xlim, ylim] for subplots\n",
        "ax_lbl=[['$Im$', '$Re$'],\n",
        "        ['', ''],\n",
        "        ['$f(t)$', '$t$'],\n",
        "        ['', ''],\n",
        "        ] #variables for yx axes of each subplot\n",
        "\n",
        "ax_lst=[] #axis list or the list of subplots\n",
        "ytxt_lst=[] #list of y texts\n",
        "line_lst = [] #array containing all the line segments\n",
        "for k in range(len(ax_frame)):\n",
        "  xn,xm,yn,ym = ax_frame[k][1]+ax_frame[k][2]\n",
        "  ax=fig.add_axes(ax_frame[k][0],xlim=[xn,xm], ylim=[yn,ym],fc='none') #no fc\n",
        "  ax.axis('off') #turn off axis frames\n",
        "  ax_lst.append(ax)\n",
        "  if k==3:\n",
        "    ax.plot([-1.5,1.5], [-1.5,1.5], color=clst[-2], lw=3)\n",
        "    for i in range(100):\n",
        "      ax.plot([-1.6+i*3/100, -1.53+i*3/100],[-1.5+i*3/100,-1.5+i*3/100], color='lightgray', lw=1)\n",
        "  else:\n",
        "    ax.annotate(\"\", (xm, 0), (xn, 0),arrowprops={'arrowstyle':\"->\",'color':clst[-2]}) #x-axis\n",
        "    ax.annotate(\"\", (0,ym), (0,yn),arrowprops={'arrowstyle':\"->\",'color':clst[-2]}) #y-axis\n",
        "  line_lst.append(ax.plot([], [],color=clst[k], lw=2)[0])\n",
        "  ax.text(xm-.02,0,ax_lbl[k][-1],size=20,va='top',ha='right',color=clst[-2]) #x-axis label\n",
        "  if k==0:\n",
        "    ytxt_lst.append(ax.text(0.01,ym,ax_lbl[k][0],color=clst[-2],size=20,va='center')) #y label\n",
        "  else:\n",
        "    ytxt_lst.append(ax.text(0.01,ym-0.02,ax_lbl[k][0],color=clst[k],size=20,va='top')) #y label\n",
        "ax_lst[0].plot(traj.real,traj.imag,color=clst[0],marker='o',markersize=6, lw=1)\n",
        "ax_lst[2].plot(t,traj.real,color=clst[0],lw=2,alpha=0.4)\n",
        "# animation function. This is called sequentially\n",
        "def animate(i):\n",
        "  ii = (i%Nf)*4\n",
        "  k=ceil(int(i/Nf)/2)*6 + (-1)**int(i/Nf)\n",
        "  if ii==0:\n",
        "    print(k)\n",
        "  line_lst[2].set_data(t[:ii+1],trajrec[0][2*k][:ii+1].real)\n",
        "  # line_lst[2].set_data(t,trajrec[0][2*k].real)\n",
        "  vectorDraw(ax_lst[1],trajrec,ii,k,clst)\n",
        "  p1 = [trajrec[0][2*k][ii].real,min(0,trajrec[0][2*k][ii].imag)]\n",
        "  p2 = [trajrec[0][2*k][ii].real,trajrec[0][2*k][ii].real]\n",
        "  p3 = [t[ii].real,trajrec[0][2*k][ii].real]\n",
        "  l2Ax(fig,ax_lst[0],ax_lst[3],ax_lst[2],p1,p2,p3,clst)\n",
        "  return\n",
        "\n",
        "anim = animation.FuncAnimation(fig, animate, frames=Nf*5, interval=200)\n",
        "# anim #uncomment to generate animation in the output area\n",
        "# to save the animation, uncomment the following three lines\n",
        "fn = r\"FourierSeries_RealVsComplex_sixstep.mp4\"\n",
        "writervideo = animation.FFMpegWriter(fps=64)\n",
        "anim.save(fn, writer=writervideo,dpi = 120)\n",
        "\n",
        "#debug\n",
        "# print(len(trajrec[0]))"
      ]
    }
  ]
}