ASN3 - Class Notes October 5

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$1/T⟶df$

$n/T⟶f$

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

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

$X(F)={ℱ}[x(t)]<math{>}^{‴}THEGAM{E}^{‴}LTI(LinearTimeInvariantSystem)InputLTIOutputReason<math>x2(t)$ . $e^{\frac{-j2\pi mt}{T}}=\underset{-\frac{T}{2}}{\overset{\frac{T}{2}}{\int }}\sum _{n=0}^{\mathrm{\infty }}{a}_n{e}^{\frac{j2\pi nt}{T}}{e}^{\frac{-j2\pi mt}{T}}dt=\sum _{n=0}^{\mathrm{\infty }}{a}_n\underset{-\frac{T}{2}}{\overset{\frac{T}{2}}{\int }}{e}^{\frac{j2\pi (n-m)t}{T}}dt=\sum _{n=0}^{\mathrm{\infty }}{a}_{n}T{\delta }_{mn}$