Continuous-time Fourier transform
However, the periodic signal using a linear combination of complex exponential signal, for analyzing linear time invariant system is a very important property. One of the most important contributions is to periodic signals Fourier extended to non-periodic signals, Fourier had thought: a non-periodic signal can be seen as an infinitely long period periodic signal.
Now consider a signal \ (X (T) \) , which has a limited duration, namely a \ (T_l \) , when the \ (| t |> T_1 \ ) when, \ (X (T) = 0 \) As shown in FIG. a:
From this non-periodic signal, it may constitute a periodic signal \ (\ tilde {x} ( t) \) , so \ (x (t) \) is \ (\ tilde {x} ( t) \) of a period , b as shown in FIG. When \ (\ T) when large, \ (\ tilde {x} (T) \) and in a longer period of time \ (x (t) \) consistent with \ (\ tilde {x} (T) \ rightarrow \ infty \) , for any finite time \ (T \) value terms, \ (\ tilde are {X} (T) = X (T) \) . There Fourier series can be drawn:
\ [\ tilde are {X} (T) = \ sum_ K = {- \ infty} ^ {+ \ infty} {a_ke JK ^ {{\ omega O}}} T \ ]
\ [a_k = \ frac1T \ int _ {- T / 2} ^ {+ T / 2} {\ tilde are {X} (T) E ^ {- JK {\ omega O} T}} \]
where \ (\ omega O 2 = \ PI / T \) , while we \ (\ tilde {x} ( t) \) replaced by \ (X (T) \) , to give:
\ [a_k = \ frac1T \ int _ {- T / 2} ^ {+ T / 2} {\ tilde {x} (t) e ^ {- jk {\ omega_0} t}} = \ frac1T \ int _ {- \ infty} ^ {+ \ infty}
{x (t) e ^ {- jk {\ omega_0} t}} \] Thus, we define \ (Ta_k \) envelope \ (X (jw) \) to
\ [ X (jw) = \ int _
{- \ infty} ^ {+ \ infty} {x (t) e ^ {-jk {\ omega_0} t}} \, {\ rm d} t \] In this case, the coefficient \ (a_k \) can be written as
\ [a_k = \ frac1TX (jw_0) \]
\ (\ tilde are {X} (T) \) to
\ [\ tilde {x} ( t) = \ sum_ {k = - \ infty } ^ {+ \ infty} {
\ frac1TX (jk \ omega_0) e ^ {jk \ omega_0t}} \] Alternatively, since the \ (2 \ PI / T = \ omega O \) , \ (\ tilde are {X} (T ) \) and can be expressed as
\ [\ tilde {x} ( t) = \ frac1 {2 \ pi} \ sum_ {k = - \ infty} ^ {+ \ infty} {X (jk \ omega_0) e ^ { jk \ omega_0t} \ omega_0} \
] with \ (T \ rightarrow \ infty \) , \ (\ tilde are {X} (T) \ rightarrow X (T) \) , \ (\ omega O \ rightarrow 0 \) , denoted \ (\ omega_0 = \ omega \ ) then:
\ [X (T) = \ frac1 {2 \ PI} \ int _ {- \ infty} ^ {+ \ infty} {X-(JK \ Omega) E ^ {JK \ Omega T}} \, {\ RM D} \ Omega \]
\ [X-(JW ) = \ int _ {- \
infty} ^ {+ \ infty} {x (t) e ^ {-jk {\ omega} t}} \, {\ rm d} \] t on both Fourier transform formula Discipline for (fourier transform pair). Function \ (X (j \ omega) \) is called \ (x (t) \) Fourier transform or Fourier integral (fourier integral). Two equations are called synthetic formula and analytical formulas