5. Review of Fourier Series
Deal
with continuous-time periodic signals.
Discrete frequency spectra.
A Periodic Signal
A Periodic Signal
f(t)
t
T
2T
3T
6. Two Forms for Fourier Series
Sinusoidal
a0 ∞
2πnt ∞
2πnt
f (t ) = + ∑ an cos
+ ∑ bn sin
Form
2 n =1
T
T
n =1
2 T /2
a0 = ∫
f (t )dt
−T / 2
T
Complex
Form:
f (t ) =
∞
∑ cn e
n = −∞
jnω0t
2 T /2
an = ∫
f (t ) cos nω0tdt
T −T / 2
2 T /2
bn = ∫
f (t ) sin nω0tdt
T −T / 2
1
cn =
T
∫
T /2
−T / 2
f (t )e − jnω0t dt
7. How to Deal with Aperiodic Signal?
A Periodic Signal
A Periodic Signal
f(t)
t
T
If T→∞, what happens?
9. Fourier Integral
fT (t ) =
∞
∑c e
n = −∞
n
jnω0t
1
cn =
T
∫
T /2
−T / 2
fT (t )e − jnω0t dt
∞
1 T /2
=∑ ∫
fT (τ)e − jnω0 τ dτ e jnω0t
−T / 2
n = −∞ T
1 ∞ T /2
=
fT (τ)e − jnω0 τ dτ ω0 e jnω0t
∑
2π n = −∞ ∫−T / 2
1 ∞ T /2
=
fT (τ)e − jnω0 τ dτ e jnω0t ∆ω
∑
2π n = −∞ ∫−T / 2
1 ∞ ∞
=
fT (τ)e − jωτ dτ e jωt dω
2π ∫−∞ ∫−∞
ω0 =
2π
T
1 ω0
=
T 2π
Let ∆ω = ω0 =
2π
T
T → ∞ ⇒ dω = ∆ω ≈ 0
10. Fourier Integral
1 ∞ ∞
− jωτ
e jωt dω
f (t ) =
∫−∞ ∫−∞ f (τ)e dτ
2π
F(jω )
1 ∞
jω t
f (t ) =
∫−∞ F ( jω)e dω
2π
∞
F ( jω) = ∫ f (t )e
−∞
− jω t
dt
Synthesis
Analysis
11. Fourier Series vs. Fourier Integral
Fourier
Series:
f (t ) =
cn e jnω0t
∑
Period Function
n = −∞
1
cn =
T
Fourier
Integral:
∞
∫
T /2
−T / 2
Discrete Spectra
fT (t )e − jnω0t dt
1 ∞
f (t ) =
F ( jω)e jωt dω
2π ∫−∞
∞
F ( jω) = ∫ f (t )e − jωt dt
−∞
Non-Period
Function
Continuous Spectra
21. Notation
F [ f (t )] = F ( jω)
F [ F ( jω)] = f (t )
-1
f (t ) ←
→ F ( jω)
F
22. Linearity
a1 f1 (t ) + a2 f 2 (t ) ←
→ a1 F1 ( jω) + a2 F2 ( jω)
F
orrk !!
Wo k
H om e W
!!Home
23. Time Scaling
1 ω
f (at ) ←
→
F j
|a| a
F
orrk !!
Wo k
H om e W
!!Home
24. Time Reversal
f ( −t ) ←
→ F ( − jω)
F
Pf) F [ f (−t )] = ∞ f (−t )e − jωt dt = t =∞ f (−t )e − jωt dt
∫−∞
∫t =−∞
=∫
− t =∞
−t = −∞
= −∫
=∫
f (t )e jωt d ( −t )
f (t )e d ( −t )
−t = −∞
t = −∞
t =∞
∞
− t =∞
j ωt
f (t )e dt = ∫
j ωt
t =∞
t = −∞
f (t )e jωt dt
= ∫ f (t )e jωt dt = F (− jω)
−∞
25. Time Shifting
f (t − t0 ) ←
→ F ( jω) e
F
− jωt 0
Pf) F [ f (t − t )] = ∞ f (t − t )e − jωt dt = t =∞ f (t − t )e − jωt dt
0
0
0
∫−∞
∫t =−∞
=∫
t +t0 =∞
=e
− j ωt 0
=e
− j ωt 0
t + t 0 = −∞
f (t )e − jω(t +t0 ) d (t + t0 )
∫
t =∞
∫
∞
t = −∞
−∞
f (t )e − jωt dt
− jω t
f (t )e − jωt dt = F ( jω)e 0
26. Frequency Shifting (Modulation)
f (t )e
Pf)
jω0t
¬ F [ j (ω − ω0 ) ]
→
F [ f (t )e
F
jω 0 t
∞
] = ∫ f (t )e jω0t e − jωt dt
−∞
∞
= ∫ f (t )e − j ( ω−ω0 )t dt
−∞
= F [ j (ω − ω0 )]
27. Symmetry Property
F [ F ( jt )] = 2πf (−ω)
Proof
∞
2πf (t ) = ∫ F ( jω)e jωt dω
−∞
∞
2πf (−t ) = ∫ F ( jω)e − jωt dω
−∞
Interchange symbols ω and t
∞
2πf (−ω) = ∫ F ( jt )e − jωt dt = F [ F ( jt )]
−∞
28. Fourier Transform for
Real Functions
If f(t) is a real function, and F(jω) = FR(jω) + jFI(jω)
F(−jω) = F*(jω)
∞
F ( jω) = ∫ f (t )e
− jωt
−∞
∞
dt
F * ( jω) = ∫ f (t )e dt = F (− jω)
−∞
jωt
29. Fourier Transform for
Real Functions
If f(t) is a real function, and F(jω) = FR(jω) + jFI(jω)
F(−jω) = F*(jω)
FR(jω) is even, and FI(jω) is odd.
F R jω ) = F R F (− jω ) = − F (jω )
(−
(jω ) I
I
Magnitude spectrum |F(jω)| is even, and
phase spectrum φ(ω) is odd.
30. Fourier Transform for
Real Functions
If f(t) is real and even
F(jω) is real
Pf)
Even
If f(t) is real and odd
√
f (t ) = f (−t )
F(jω) is pure imaginary
Pf)
Odd
F ( jω) = F (− jω)
Real
F (− jω) = F * ( jω)
F ( jω) = F * ( jω)
√
f (t ) = − f (−t )
F ( jω) = − F (− jω)
Real
F (− jω) = F * ( jω)
F ( jω) = − F * ( jω)
31. Example:
F [ f (t )] = F ( jω)
Sol)
F [ f (t ) cos ω0t ] = ?
1
f (t ) cos ω0t = f (t )(e jω0t + e − jω0t )
2
1
1
jω 0 t
F [ f (t ) cos ω0t ] = F [ f (t )e ] + F [ f (t )e − jω0t ]
2
2
1
1
= F [ j (ω − ω0 )] + F [ j (ω + ω0 )]
2
2
34. 1
Example:
sin at
f (t ) =
πt
Sol)
wd(t)
t
−d/2
d/2
F ( jω) = ?
2 ωd
Answer is
Wd ( jω) = sin
just
ω 2
opposite to
as expected
2 td
F [Wd ( jt )] = F sin = 2πwd (−ω)
2
t
0 ω <| a |
sin at
F [ f (t )] = F
= w2 a (−ω) = 1 ω >| a |
πt
35. Fourier Transform of f’(t)
f (t ) ←
→ F ( jω) and lim f (t ) = 0
F
t → ±∞
f ' (t ) ←F jωF ( jω)
→
Pf) F [ f ' (t )] = ∞ f ' (t )e − jωt dt
∫−∞
= f (t )e
− j ωt ∞
−∞
= jωF ( jω)
∞
+ jω∫ f (t )e − jωt dt
−∞
36. Fourier Transform of f (t)
(n)
f (t ) ←
→ F ( jω) and lim f (t ) = 0
F
t → ±∞
f ( n ) (t ) ←F ( jω) n F ( jω)
→
orrk !!
Wo k
H om e W
!!Home
37. Fourier Transform of f (t)
(n)
f (t ) ←
→ F ( jω) and lim f (t ) = 0
F
t → ±∞
f ( n ) (t ) ←F ( jω) n F ( jω)
→
orrk !!
Wo k
H om e W
!!Home
38. Fourier Transform of Integral
f (t ) ←
→ F ( jω) and
F
∫
∞
−∞
f (t )dt = F ( 0 ) = 0
t f ( x)dx = 1 F ( jω)
F ∫
−∞
jω
Let φ(t ) =
∫
t
−∞
f ( x)dx
lim φ(t ) = 0
t →∞
F [φ' (t )] = F [ f (t )] = F ( jω) = jωΦ ( jω)
1
Φ ( jω) =
F ( jω)
jω
39. The Derivative of Fourier Transform
dF ( jω)
F [− jtf (t )] ←
→
dω
F
Pf)
∞
F ( jω) = ∫ f (t )e − jωt dt
−∞
∞
dF ( jω) d ∞
∂ − j ωt
− j ωt
=
∫−∞ f (t )e dt = ∫−∞ f (t ) ∂ω e dt
dω
dω
∞
= ∫ [− jtf (t )]e − jωt dt = F [− jtf (t )]
−∞