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1 / T ⟶ d f {\displaystyle 1/T\longrightarrow df}
n / T ⟶ f {\displaystyle n/T\longrightarrow f}
∑ n = − ∞ ∞ 1 T ⟶ ∫ − ∞ ∞ ( ) d f {\displaystyle \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 → ∞ 1 T ( ∫ − T 2 T 2 x ( t ′ ) e − j 2 π n t ′ T d t ′ ) e j 2 π n t T {\displaystyle 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 ) = F [ x ( t ) ] ∫ − T 2 T 2 x ( t ′ ) e − j 2 π n t ′ T d t ′ {\displaystyle 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 ) = ∫ − ∞ ∞ [ ∫ − ∞ ∞ x ( t ′ ) e − j 2 π f t ′ d t ′ ] e j 2 π f t d f {\displaystyle x(t)=\int _{-\infty }^{\infty }[\int _{-\infty }^{\infty }x(t')e^{-j2\pi ft'}dt']e^{j2\pi ft}df\!}
x ( t ) = ∫ − ∞ ∞ x ( t ′ ) [ ∫ − ∞ ∞ e j 2 π f ( t ′ − t ) d f ] d t ′ {\displaystyle x(t)=\int _{-\infty }^{\infty }x(t')[\int _{-\infty }^{\infty }e^{j2\pi f(t'-t)}df]dt'\!}
THE GAME
LTI (Linear Time Invariant System)
Input LTI Output Reason
x 2 ( t ) {\displaystyle x2(t)\!} . e − j 2 π m t T = ∫ − T 2 T 2 ∑ n = 0 ∞ a n e j 2 π n t T e − j 2 π m t T d t = ∑ n = 0 ∞ a n ∫ − T 2 T 2 e j 2 π ( n − m ) t T d t = ∑ n = 0 ∞ a n T δ m n {\displaystyle 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}\!}
![{\displaystyle X(F)={\mathcal {F}}[x(t)]\int _{-{\frac {T}{2}}}^{\frac {T}{2}}x(t')e^{\frac {-j2\pi nt'}{T}}dt'\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1fcaa22a06769a7b820ac28954e5b5269784f04)
![{\displaystyle x(t)=\int _{-\infty }^{\infty }[\int _{-\infty }^{\infty }x(t')e^{-j2\pi ft'}dt']e^{j2\pi ft}df\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4809b434871f43b380b8822632711c5dc65c2b4a)
![{\displaystyle x(t)=\int _{-\infty }^{\infty }x(t')[\int _{-\infty }^{\infty }e^{j2\pi f(t'-t)}df]dt'\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1de446d6534ca544dad93116a3b89752b1c5e31)
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