Back to my Home Page
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\!}
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}\!}
. 