How to calculate the Fourier coefficients, of a given input : $$u(t)=\sum_{k=-\infty}^{+\infty}\overline u_k e^{ik\omega_0t}$$ of this output: $$v(t)=u(t)(1-cos^2(\omega_0t))$$ So how is the mathematical definition of this, maybe started from the definition of Fourier series. So how can I start this ?
How to calculate the Fourier coefficients
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fourier-series
signal-processing
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1What's your formula for the fourier coefficient? Does $c_n = \frac{1}{2\pi}\int_0^{2\pi} f(t) e^{2\pi int}dt$ look familiar? – 2017-02-20
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0Hmmmm I think I have to found $a_0, a_n \ and \ b_n$ or not? – 2017-02-20
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1Try expanding $1-\cos^2(\omega_0 t)$ into complex exponentials. – 2017-02-20
1 Answers
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$$v(\Omega)=\frac{1}{2\pi}\int_{-\pi}^{\pi}u(t)e^{{i}\Omega{t}}dt-\frac{1}{2\pi}\int_{-\pi}^{\pi}u(t)\cos^{2}(\omega_{0}t)e^{{i}\Omega{t}}dt=$$ $$=\frac{1}{2}u(\Omega)-\frac{1}{2\pi}\int_{-\pi}^{\pi}u(t)\frac{e^{2i\omega_{0}t}+e^{-2i\omega_{0}t}}{4}e^{{i}\Omega{t}}dt=$$ $$=\frac{1}{2}u(\Omega)-\frac{1}{4}u(\Omega+2\omega_{0})-\frac{1}{4}u(\Omega-2\omega_{0})$$
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0Sorry, but $\Omega$ is instant of what? – 2017-02-20
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0$\Omega$ is a summation index. You name it $k$ – 2017-02-20