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Fourier Transform Pairs. The Fourier transform transforms a function of time, f(t), into a function of frequency, F(s):. F {f(t)}(s) = F(s) = /.
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Fourier Transform Pairs
The Fourier transform transforms a function of time, f (t), into a function of frequency, F(s):
F { f (t)}(s) = F(s) =
Z (^) ∞
−∞
f (t)e−^ j^2 πstdt.
The inverse Fourier transform transforms a func- tion of frequency, F(s), into a function of time, f (t):
F −^1 {F(s)}(t) = f (t) =
Z (^) ∞
−∞
F(s)e j^2 πstds.
The inverse Fourier transform of the Fourier trans- form is the identity transform:
f (t) =
Z (^) ∞
−∞
−∞
f (τ)e−^ j^2 πsτdτ
e j^2 πstds.
Fourier Transform Pairs (contd).
Because the Fourier transform and the inverse Fourier transform differ only in the sign of the exponential’s argument, the following recipro- cal relation holds between f (t) and F(s):
f (t) −→F F(s)
is equivalent to
F(t) −→F f (−s).
This relationship is often written more econom- ically as follows:
f (t) ←→F F(s)
where f (t) and F(s) are said to be a Fourier transform pair.
Fourier Transform of Gaussian (contd.)
F(s) = e−πs
2 Z (^) ∞
−∞
e−π(t+^ js)
2 dt
After substituting u for t + js and du for dt we see that:
F(s) = e−πs
2 Z (^) ∞
−∞
e−πu
2 du ︸ ︷︷ ︸ 1
It follows that the Gaussian is its own Fourier transform:
e−πt
(^2) F ←→ e−πs
2 .
Fourier Transform of Dirac Delta Function
To compute the Fourier transform of an impulse we apply the definition of Fourier transform:
F {δ(t − t 0 )}(s) = F(s) =
Z (^) ∞
−∞
δ(t − t 0 )e−^ j^2 πstdt
which, by the sifting property of the impulse, is just: e−^ j^2 π^ s t^0.
It follows that:
δ(t − t 0 ) F −→ e−^ j^2 π^ s t^0.
Fourier Transform of Sine and Cosine
We can compute the Fourier transforms of the sine and cosine by exploiting the sifting prop- erty of the impulse: Z (^) ∞
−∞
f (x)δ(x − x 0 )dx = f (x 0 ).
Z (^) ∞
−∞
[δ(s + s 0 ) + δ(s − s 0 )] e j^2 πstds.
Fourier Transform of Sine and Cosine (contd.)
Expanding the above yields the following ex- pression for f (t): Z (^) ∞
−∞
δ(s + s 0 )e j^2 πstds +
Z (^) ∞
−∞
δ(s − s 0 )e j^2 πstds
Which by the sifting property is just:
f (t) = e j^2 π^ s^0 t^ + e−^ j^2 π^ s^0 t = 2 cos( 2 π s 0 t).
Fourier Transform of the Pulse
To compute the Fourier transform of a pulse we apply the definition of Fourier transform:
F(s) =
Z (^) ∞
−∞
Π(t)e−^ j^2 πstdt
Z (^12)
−^12
e−^ j^2 πstdt
− j 2 πs
e−^ j^2 πst
(^12)
− j 2 πs
e−^ jπs^ − e jπs
πs
e jπs^ − e−^ jπs
2 j
Using the fact that sin(x) = (
e jx−e−^ jx) 2 j we see that:
F(s) =
sin(πs) πs
Fourier Transform of the Shah Function
Recall the Fourier series for the Shah function:
1 2 π
t 2 π
2 π
∞
ω=−∞
e jω^ t.
By the sifting property,
III
t 2 π
∞
ω=−∞
Z (^) ∞
−∞
δ(s − ω)e jstds.
Changing the order of the summation and the integral yields
III
t 2 π
Z (^) ∞
−∞
∞
ω=−∞
δ(s − ω)e jstds.
Factoring out e jst^ from the summation
III
t 2 π
Z (^) ∞
−∞
e jst^
∞
ω=−∞
δ(s − ω)ds
Z (^) ∞
−∞
e jstIII(s)ds.
