fourier

Funciones ortogonales. Fourier.

Un conjunto de funciones \phi_k (t) es ortogonal en un intervalo a<t<b si para dos funciones cualesquiera \phi_m (t) y \phi_n (t) pertenecientes al conjunto \phi_k (t), cumple:

\displaystyle \int_{a}^{b}{\phi_m (t) \phi_n (t) dt} = \left\{ \begin{matrix} 0, \quad para \quad m \ne n \\ r_n, \quad para \quad m=n \end{matrix} \right.

Considerando como ejemplo, un conjunto de funciones senoidales, mediante el cálculo elemental se puede demostrar que

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t}dt}=0      para m \ne 0

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t}dt}=0      para todo valor de m

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t} \cos{n\omega_o t} dt} = \left\{ \begin{matrix} 0, \quad \quad \quad para \quad m \ne n \\ \frac{T}{2}, \quad para \quad m=n\ne 0 \end{matrix} \right.

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \sin{n\omega_o t} dt} = \left\{ \begin{matrix} 0, \quad \quad \quad para \quad m \ne n \\ \frac{T}{2}, \quad para \quad m=n\ne 0 \end{matrix} \right.

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t}\cos{n\omega_o t}dt}=0      para todo valor de m y n

donde \displaystyle \omega_o=\frac{2\pi}{T}.

Estas relaciones demuestran que las funciones 1, \cos{\omega_o t}\cos{2\omega_o t} ,\cos{3\omega_o t}, …, \cos{n\omega_o t}, …, \sin{\omega_o t}\sin{2\omega_o t}\sin{3\omega_o t}, …, \sin{n\omega_o t}, …, forman un conjunto de funciones ortogonales en el intervalo -\frac{T}{2}<t<\frac{T}{2}.

Demostraciones de las integrales mencionadas

Demostración de la primera integral

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t}dt} = \left[\frac{1}{m\omega_o} \sin{m\omega_o t}\right]_{-\frac{T}{2}}^{\frac{T}{2}}

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t}dt} = \left[\frac{1}{m\omega_o} \sin{m\omega_o \left(-\frac{T}{2}\right)}\right] -  \left[\frac{1}{m\omega_o} \sin{m\omega_o \left(\frac{T}{2}\right)}\right]

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t}dt} = \left[\frac{1}{m\omega_o} \sin{m \left(\frac{2\pi}{T} \right) \left(-\frac{T}{2}\right)}\right] -  \left[\frac{1}{m\omega_o} \sin{m \left(\frac{2\pi}{T} \right) \left(\frac{T}{2}\right)}\right]

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t}dt} = \left[\frac{1}{m\omega_o} \sin{(-m\pi)}\right] -  \left[\frac{1}{m\omega_o} \sin{(m\pi)} \right] = 0

Finalmente

\displaystyle \therefore \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t}dt}=0

Demostración de la segunda integral

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t}dt} = \left[-\frac{1}{m\omega_o} \cos{m\omega_o t}\right]_{-\frac{T}{2}}^{\frac{T}{2}}

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t}dt} = \left[-\frac{1}{m\omega_o} \cos{m\omega_o \left(-\frac{T}{2}\right)}\right] -  \left[-\frac{1}{m\omega_o} \cos{m\omega_o \left(\frac{T}{2}\right)}\right]

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t}dt} = \left[-\frac{1}{m\omega_o} \cos{m \left(\frac{2\pi}{T} \right) \left(-\frac{T}{2}\right)}\right] -  \left[-\frac{1}{m\omega_o} \cos{m \left(\frac{2\pi}{T} \right) \left(\frac{T}{2}\right)}\right]

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t}dt} = \left[-\frac{1}{m\omega_o} \cos{(-m\pi)}\right] -  \left[-\frac{1}{m\omega_o} \cos{(m\pi)} \right]

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t}dt} = - \frac{1}{m\omega_o} \cos{(m\pi)} + \frac{1}{m\omega_o} \cos{(m\pi)} = 0

Finalmente

\displaystyle \therefore \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t}dt}=0

Demostración de la tercera integral

Si m=n \ne 0

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t} \cos{n\omega_o t} dt} = \int_{-\frac{T}{2}}^{\frac{T}{2}}{\frac{1}{2} \left[\cos{(m+n)\omega_o t} + \cos{(m-n)\omega_o t}\right] dt}

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t} \cos{n\omega_o t} dt}  = \frac{1}{2} \left[\frac{1}{(m+n)\omega_o} \sin{(m+n)\omega_o t} + \frac{1}{(m-n)\omega_o} \sin{(m-n)\omega_o t} \right]_{-\frac{T}{2}}^{\frac{T}{2}}

\displaystyle = \frac{1}{2} \left[\frac{1}{(m+n)\omega_o} \sin{(m+n)\left(\frac{2\pi}{T}\right) \left(\frac{T}{2}\right)} + \frac{1}{(m-n)\omega_o} \sin{(m-n)\left(\frac{2\pi}{T}\right)\left(\frac{T}{2}\right)} \right] - \frac{1}{2} \left[\frac{1}{(m+n)\omega_o} \sin{(m+n)\left(\frac{2\pi}{T}\right) \left(-\frac{T}{2}\right)} + \frac{1}{(m-n)\omega_o} \sin{(m-n)\left(\frac{2\pi}{T}\right)\left(-\frac{T}{2}\right)} \right]

\displaystyle = \frac{1}{2} \left[\frac{1}{(m+n)\omega_o} \sin{(m+n)\left(\pi \right)} + \frac{1}{(m-n)\omega_o} \sin{(m-n)\left(\pi \right)} \right] - \frac{1}{2} \left[\frac{1}{(m+n)\omega_o} \sin{(m+n)\left(-\pi \right)} + \frac{1}{(m-n)\omega_o} \sin{(m-n)\left(-\pi \right)} \right]

\displaystyle = \frac{1}{2(m+n)\omega_o} \sin{(m+n)\left(\pi \right)} + \frac{1}{2(m-n)\omega_o} \sin{(m-n)\left(\pi \right)} + \frac{1}{2(m+n)\omega_o} \sin{(m+n)\left(\pi \right)} + \frac{1}{2(m-n)\omega_o} \sin{(m-n)\left(\pi \right)}

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t} \cos{n\omega_o t} dt} = \frac{1}{(m+n)\omega_o} \sin{(m+n)\left(\pi \right)} + \frac{1}{(m-n)\omega_o} \sin{(m-n)\left(\pi \right)}

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t} \cos{n\omega_o t} dt}= \frac{1}{(m+n)\omega_o} \sin{2m\left(\pi \right)}  =0

Si m = n \ne 0

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t} \cos{n\omega_o t} dt} = \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos^2{m\omega_o t} \, dt}

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t} \cos{n\omega_o t} dt} = \int_{-\frac{T}{2}}^{\frac{T}{2}}{\left(\frac{1}{2} + \frac{1}{2}\cos{2mt}\right) \, dt} = \left[\frac{1}{2}t + \frac{1}{4m}\sin{2mt}+C \right]_{-\frac{T}{2}}^{\frac{T}{2}}

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t} \cos{n\omega_o t} dt} = \left[\frac{1}{2} \left( \frac{T}{2} \right) + \frac{1}{4m}\sin{2m\left( \frac{T}{2} \right)} \right] - \left[\frac{1}{2}\left( -\frac{T}{2} \right) + \frac{1}{4m}\sin{2m\left( -\frac{T}{2} \right)} \right]

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t} \cos{n\omega_o t} dt} = \left[\frac{T}{4} + \frac{1}{4m}\sin{(mT)} \right] - \left[-\frac{T}{4} - \frac{1}{4m}\sin{(mT)} \right]

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t} \cos{n\omega_o t} dt} = \frac{T}{4} + \frac{1}{4m}\sin{(mT)} + \frac{T}{4} + \frac{1}{4m}\sin{(mT)}

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t} \cos{n\omega_o t} dt} = \frac{T}{2} + \frac{1}{2m}\sin{(mT)} = \frac{T}{2}

donde m es un número entero.

Finalmente

\displaystyle \therefore \int_{-\frac{T}{2}}^{\frac{T}{2}}{\cos{m\omega_o t} \cos{n\omega_o t} dt} = \left\{ \begin{matrix} 0, \quad \quad \quad \text{para} \quad m \ne n \\ \frac{T}{2}, \quad \text{para} \quad m=n\ne 0 \end{matrix} \right.

Demostración de la cuarta integral

Si m=n \ne 0

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \sin{n\omega_o t} dt} = \int_{-\frac{T}{2}}^{\frac{T}{2}}{\frac{1}{2} \left[-\cos{(m+n)\omega_o t} + \cos{(m-n)\omega_o t}\right] dt}

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \sin{n\omega_o t} dt} = \frac{1}{2} \left[-\frac{1}{(m+n)\omega_o} \sin{(m+n)\omega_o t} + \frac{1}{(m-n)\omega_o} \sin{(m-n)\omega_o t} \right]_{-\frac{T}{2}}^{\frac{T}{2}}

\displaystyle = \frac{1}{2} \left[-\frac{1}{(m+n)\omega_o} \sin{(m+n)\left(\frac{2\pi}{T}\right) \left(\frac{T}{2}\right)} + \frac{1}{(m-n)\omega_o} \sin{(m-n)\left(\frac{2\pi}{T}\right)\left(\frac{T}{2}\right)} \right] - \frac{1}{2} \left[-\frac{1}{(m+n)\omega_o} \sin{(m+n)\left(\frac{2\pi}{T}\right) \left(-\frac{T}{2}\right)} + \frac{1}{(m-n)\omega_o} \sin{(m-n)\left(\frac{2\pi}{T}\right)\left(-\frac{T}{2}\right)} \right]

\displaystyle = \frac{1}{2} \left[-\frac{1}{(m+n)\omega_o} \sin{(m+n)\left(\pi \right)} + \frac{1}{(m-n)\omega_o} \sin{(m-n)\left(\pi \right)} \right] - \frac{1}{2} \left[-\frac{1}{(m+n)\omega_o} \sin{(m+n)\left(-\pi \right)} + \frac{1}{(m-n)\omega_o} \sin{(m-n)\left(-\pi \right)} \right]

\displaystyle = - \frac{1}{2(m+n)\omega_o} \sin{(m+n)\left(\pi \right)} + \frac{1}{2(m-n)\omega_o} \sin{(m-n)\left(\pi \right)} - \frac{1}{2(m+n)\omega_o} \sin{(m+n)\left(\pi \right)} + \frac{1}{2(m-n)\omega_o} \sin{(m-n)\left(\pi \right)}

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \sin{n\omega_o t} dt} = - \frac{1}{(m+n)\omega_o} \sin{(m+n)\left(\pi \right)} + \frac{1}{(m-n)\omega_o} \sin{(m-n)\left(\pi \right)}

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \sin{n\omega_o t} dt} = - \frac{1}{(m+n)\omega_o} (0) + \frac{1}{(m-n)\omega_o} (0) =0

Si m = n \ne 0

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \sin{n\omega_o t} dt} = \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin^2{m\omega_o t} \, dt}

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \sin{n\omega_o t} dt} = \int_{-\frac{T}{2}}^{\frac{T}{2}}{\left(\frac{1}{2} - \frac{1}{2}\cos{2mt}\right) \, dt} = \left[\frac{1}{2}t - \frac{1}{4m}\sin{2mt}+C \right]_{-\frac{T}{2}}^{\frac{T}{2}}

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \sin{n\omega_o t} dt} = \left[\frac{1}{2} \left( \frac{T}{2} \right) - \frac{1}{4m}\sin{2m\left( \frac{T}{2} \right)} \right] - \left[\frac{1}{2}\left( -\frac{T}{2} \right) - \frac{1}{4m}\sin{2m\left( -\frac{T}{2} \right)} \right]

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \sin{n\omega_o t} dt} = \left[\frac{T}{4} - \frac{1}{4m}\sin{(mT)} \right] - \left[-\frac{T}{4} + \frac{1}{4m}\sin{(mT)} \right]

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \sin{n\omega_o t} dt} = \frac{T}{4} - \frac{1}{4m}\sin{(mT)} + \frac{T}{4} - \frac{1}{4m}\sin{(mT)}

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \sin{n\omega_o t} dt} = \frac{T}{2} - \frac{1}{2m}\sin{(mT)} = \frac{T}{2}

donde m es un número entero.

Finalmente

\displaystyle \therefore \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \sin{n\omega_o t} dt} = \left\{ \begin{matrix} 0, \quad \quad \quad \text{para} \quad m \ne n \\ \frac{T}{2}, \quad \text{para} \quad m=n\ne 0 \end{matrix} \right.

Demostración de la quinta integral

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \cos{n\omega_o t} dt} = \int_{-\frac{T}{2}}^{\frac{T}{2}}{\frac{1}{2} \left[\sin{(m+n)\omega_o t} + \sin{(m-n)\omega_o t}\right] dt}

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \cos{n\omega_o t} dt} = \frac{1}{2} \left[-\frac{1}{(m+n)\omega_o} \cos{(m+n)\omega_o t} - \frac{1}{(m-n)\omega_o} \cos{(m-n)\omega_o t} \right]_{-\frac{T}{2}}^{\frac{T}{2}}

\displaystyle = \frac{1}{2} \left[-\frac{1}{(m+n)\omega_o} \cos{(m+n)\left(\frac{2\pi}{T}\right) \left(\frac{T}{2}\right)} - \frac{1}{(m-n)\omega_o} \cos{(m-n)\left(\frac{2\pi}{T}\right)\left(\frac{T}{2}\right)} \right] - \frac{1}{2} \left[-\frac{1}{(m+n)\omega_o} \cos{(m+n)\left(\frac{2\pi}{T}\right) \left(-\frac{T}{2}\right)} - \frac{1}{(m-n)\omega_o} \cos{(m-n)\left(\frac{2\pi}{T}\right)\left(-\frac{T}{2}\right)} \right]

\displaystyle = \frac{1}{2} \left[-\frac{1}{(m+n)\omega_o} \cos{(m+n)\left(\pi \right)} - \frac{1}{(m-n)\omega_o} \cos{(m-n)\left(\pi \right)} \right] - \frac{1}{2} \left[-\frac{1}{(m+n)\omega_o} \cos{(m+n)\left(-\pi \right)} - \frac{1}{(m-n)\omega_o} \cos{(m-n)\left(-\pi \right)} \right]

\displaystyle = - \frac{1}{2(m+n)\omega_o} \cos{(m+n)\left(\pi \right)} - \frac{1}{2(m-n)\omega_o} \cos{(m-n)\left(\pi \right)} + \frac{1}{2(m+n)\omega_o} \cos{(m+n)\left(\pi \right)} + \frac{1}{2(m-n)\omega_o} \cos{(m-n)\left(\pi \right)}

\displaystyle \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t} \cos{n\omega_o t} dt} = 0

Finalmente

\displaystyle \therefore \int_{-\frac{T}{2}}^{\frac{T}{2}}{\sin{m\omega_o t}\cos{n\omega_o t}dt}=0


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