ASN3 - Class Notes October 5

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$1/T\longrightarrow df$

$n/T\longrightarrow f$

$\sum _{n=-\infty }^{\infty }{\frac {1}{T}}\longrightarrow \int _{-\infty }^{\infty }()df\!$

Can we make an unperiodic signal and make it periodic by taking the limit?

$x(t)=\lim _{T\to \infty }{\frac {1}{T}}(\int _{-{\frac {T}{2}}}^{\frac {T}{2}}x(t')e^{\frac {-j2\pi nt'}{T}}dt')e^{\frac {j2\pi nt}{T}}\!$

note that $X(F)={\mathcal {F}}[x(t)]\int _{-{\frac {T}{2}}}^{\frac {T}{2}}x(t')e^{\frac {-j2\pi nt'}{T}}dt'\!$

becomes as the limit is taken n/t becomes f $x(t)=\int _{-\infty }^{\infty }[\int _{-\infty }^{\infty }x(t')e^{-j2\pi ft'}dt']e^{j2\pi ft}df\!$

$x(t)=\int _{-\infty }^{\infty }x(t')[\int _{-\infty }^{\infty }e^{j2\pi f(t'-t)}df]dt'\!$

note that the defination of the delta function is $\int _{-\infty }^{\infty }e^{j2\pi f(t'-t)}df\!$

$x(t)=\int _{-\infty }^{\infty }x(t')\delta _{(}t'-t)dt'\!$ THE GAME

LTI (Linear Time Invariant System)

Input LTI Output Reason

$x2(t)\!$ . $e^{\frac {-j2\pi mt}{T}}=\int _{-{\frac {T}{2}}}^{\frac {T}{2}}\sum _{n=0}^{\infty }a_{n}e^{\frac {j2\pi nt}{T}}e^{\frac {-j2\pi mt}{T}}dt=\sum _{n=0}^{\infty }a_{n}\int _{-{\frac {T}{2}}}^{\frac {T}{2}}e^{\frac {j2\pi (n-m)t}{T}}dt=\sum _{n=0}^{\infty }a_{n}T\delta _{mn}\!$