Question

Find the discrete Fourier approximation g_{2}(x) for f(x)f(x) based on the table information.x | 0 | \pif(x) | 0 | 2

Discrete math
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asked 2021-05-16

Find the discrete Fourier approximation \(g_{2}(x)\) for \(f(x)f(x)\) based on the table information.
\(x | 0 | \pi\)
\(f(x) | 0 | 2\)

Expert Answers (1)

2021-05-17
Find coefficients
\(\displaystyle{a}_{{0}}={\frac{{{1}}}{{{2}}}}{\left({f{{\left({0}\right)}}}+{f{{\left(\pi\right)}}}\right)}={\frac{{{1}}}{{{2}}}}{\left({1}+{2}\right)}={\frac{{{3}}}{{{2}}}}\)
\(\displaystyle{a}_{{1}}={\left({f{{\left({0}{\cos{{\left({0}\right)}}}+{f{{\left(\pi\right)}}}{\cos{{\left(\pi\right)}}}\right)}}}={\left({1}{\cos{{\left({0}\right)}}}+{2}{\cos{{\left(\pi\right)}}}\right)}=-{1}\right.}\)
\(\displaystyle{b}_{{1}}={\left({f{{\left({0}\right)}}}{\sin{{\left({0}\right)}}}+{f{{\left(\pi\right)}}}{\sin{{\left(\pi\right)}}}\right)}={\left({1}{\sin{{\left({0}\right)}}}+{2}{\sin{{\left(\pi\right)}}}\right)}={0}\)
Fourier appromation is
\(\displaystyle{g}_{{1}}={\frac{{{3}}}{{{2}}}}-{\cos{{\left({x}\right)}}}\)
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