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# Answer to Question #114898 in Electrical Engineering for drfg effe

Question #114898
Find the complex exponential Fourier series coefficients for:
a. x(t)=3cos(4w0t)
b. x(t)=sin^2(t)
2. Find the DTFT of:
x(n)={1 0<=n<=3
0 otherwise}
1
2020-05-11T14:26:46-0400

a. Find the complex and exponential form:

"x(t)=3\\text{cos}(4w_0t)\\iff0+j\\frac{3}{\\sqrt{2}}\\iff\\frac{3}{\\sqrt{2}}e^{j90^\\circ}."

Fourier series coefficients from -π to π:

"a_0=\\frac{\\int^\\pi_{-\\pi}\\text{cos}(4\\omega_0t)dx}{\\pi},\\\\\na_n=\\frac{\\int^\\pi_{-\\pi}\\text{cos}(4\\omega_0t)\\text{cos}(tk)dx}{\\pi},\\\\\nb_n=\\frac{\\int^\\pi_{-\\pi}\\text{sin}(tk)\\text{cos}(4\\omega_0t)dx}{\\pi},\\\\\nk=1,2,3,..."

b. Find the complex and exponential form:

"x(t)=\\text{sin}^2(t)=\\frac{1}{2}-\\frac{\\text{cos(2t)}}{2}\\iff\\frac{1}{2}-\\frac{j}{2\\sqrt2}\\iff\\\\\\iff\\frac{1}{2}-\\frac{e^{j90^\\circ}}{2\\sqrt2}."

Fourier series coefficients from -π to π:

"a_0=\\frac{\\int^\\pi_{-\\pi}\\text{sin}^2(t)dx}{\\pi},\\\\\na_n=\\frac{\\int^\\pi_{-\\pi}\\text{sin}^2(t)\\text{cos}(tk)dx}{\\pi},\\\\\nb_n=\\frac{\\int^\\pi_{-\\pi}\\text{sin}^2(t)\\text{sin}(tk)dx}{\\pi},\\\\\nk=1,2,3,..."

2. Find the DTFT of

"x(n)=\\begin{Bmatrix}\n 1,& \\text{if }1\\leq n\\leq3 \\\\\n 0&\\text{otherwise}\n\\end{Bmatrix}"

"X(\\Omega)=\\sum^3_1e^{-j\\Omega n}=e^{-3j\\Omega}(e^{j\\Omega}+e^{2j\\Omega}+1)."

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