ASN3 - Class Notes October 5

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$1/T⟶df$

$n/T⟶f$

$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{T}⟶\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}()df$

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

$x(t)=\underset{T\to \mathrm{\infty }}{\lim }\frac{1}{T}(\underset{-\frac{T}{2}}{\overset{\frac{T}{2}}{\int }}x(t^{\prime })e^{\frac{-j2\pi n{t}^{\prime }}{T}}d{t}^{\prime })e^{\frac{j2\pi nt}{T}}$

note that $X(F)={ℱ}[x(t)]\underset{-\frac{T}{2}}{\overset{\frac{T}{2}}{\int }}x(t^{\prime })e^{\frac{-j2\pi n{t}^{\prime }}{T}}d{t}^{\prime }$

becomes as the limit is taken n/t becomes f $x(t)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}[\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t^{\prime })e^{-j2\pi f{t}^{\prime }}d{t}^{\prime }]e^{j2\pi ft}df$

$x(t)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t^{\prime })[\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{j2\pi f(t^{\prime }-t)}df]d{t}^{\prime }$

note that the defination of the delta function is $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{j2\pi f(t^{\prime }-t)}df$

$x(t)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t^{\prime }){\delta }_{(}{t}^{\prime }-t)d{t}^{\prime }$ THE GAME

LTI (Linear Time Invariant System)

Input LTI Output Reason

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