Lecture IX: Fourier transform
Maxim Raginsky
BME 171: Signals and Systems
Duke University
October 8, 2008
Maxim Raginsky
Lecture IX: Fourier transform
This lecture
Plan for the lecture:
1
Recap: Fourier series representation of periodic signals
2
Frequency content of aperiodic signals: the Fourier transform
3
The inverse Fourier transform
4
Properties of the Fourier transform
5
Generalized Fourier transform
6
Bandlimited and timelimited signals
7
Frequency response of LTI systems
Maxim Raginsky
Lecture IX: Fourier transform
Recap: Fourier series
Recall from the last lecture that any sufficiently regular T -periodic
continuous-time signal x(t) can be expanded, e.g., in a complex
exponential Fourier series:
x(t) =
∞
X
ck ejkω0 t ,
k=−∞
where ω0 = 2π/T is the fundamental frequency, and the Fourier
coefficients {ck } are given by
ck =
1
T
Z
T /2
x(t)e−jkω0 t dt,
k = . . . , −2, −1, 0, 1, 2, . . .
−T /2
The Fourier coefficients {ck } tell us about the frequency content (or
spectral content) of x(t).
Maxim Raginsky
Lecture IX: Fourier transform
Spectral content of aperiodic signals: the Fourier transform
What about aperiodic signals?
Any continuous-time signal x(t) that has finite “energy”, i.e.,
Z ∞
x2 (t)dt < +∞,
−∞
can be represented in the frequency domain via the Fourier transform:
Z ∞
x(t)e−jωt dt
X(ω) =
−∞
We will also write
X(ω) = F [x(t)]
to denote the fact that X(ω) is the Fourier transform of x(t).
Maxim Raginsky
Lecture IX: Fourier transform
Example: rectangular pulse
Consider the rectangular pulse
pτ (t) =
F [pτ (t)] =
=
1, |t| ≤ τ /2
0, |t| > τ /2
Z
∞
pτ (t)e−jωt dt
−∞
Z τ /2
e−jωt dt
−τ /2
=
=
=
=
Maxim Raginsky
−
1 −jωt
e
jω
τ /2
−τ /2
ejωτ /2 − e−jωτ /2
jω
2 sin(ωτ /2)
ω τω
τ sinc
.
2π
Lecture IX: Fourier transform
Inverse Fourier transform
The signal x(t) can be recovered from its Fourier transform
X(ω) = F [x(t)] using the inverse Fourier transform formula
Z ∞
1
x(t) = F −1 [X(ω)] =
X(ω)ejωt dω
2π −∞
Note:
There is a factor of 1/2π in front of the integral.
The integration is with respect to ω, for a fixed value of t.
We will also write
x(t) ↔ X(ω)
and say that x(t) [time domain] and X(ω) [freq. domain] are a Fourier
transform pair.
Maxim Raginsky
Lecture IX: Fourier transform
Proof:
1
2π
Z ∞ Z ∞
′
1
x(t′ )e−jωt dt′ ejωt dω
X(ω)ejωt dω =
2π −∞
−∞
−∞
Z ∞
Z ∞
′
1
=
x(t′ )
ej(t−t )ω dω dt′ .
2π −∞
−∞
Z
∞
1
2π
Z
Z πΩ
′
1
lim
ej(t−t )ω dω
Ω→∞
2π
−∞
−πΩ
iπΩ
h
1
j(t−t′ )ω
e
= lim
′
Ω→∞ 2πj(t − t )Ω
−πΩ
′
sin(πΩ(t − t ))
= lim
Ω→∞
πΩ(t − t′ )
= lim sinc(Ω(t − t′ ))
∞
′
ej(t−t )ω dω =
Ω→∞
= δ(t − t′ )
Hence,
1
2π
Q.E.D.
Z
∞
X(ω)ejωt dω =
−∞
Z
∞
x(t′ )δ(t − t′ )dt′ = x(t)
−∞
Maxim Raginsky
Lecture IX: Fourier transform
Properties of the Fourier transform
The Fourier transform has many useful properties that make calculations
easier and also help thinking about the structure of signals and the action
of systems on signals.
The properties are listed in any textbook on signals and systems. We will
look at and prove a few of them.
Maxim Raginsky
Lecture IX: Fourier transform
Linearity
The Fourier transform is linear: if
x1 (t) ↔ X1 (ω)
and
x2 (t) ↔ X2 (ω),
then
c1 x1 (t) + c2 x2 (t) ↔ c1 X1 (ω) + c2 X2 (ω)
for any two numbers c1 and c2 .
Proof: obvious –
F [c1 x1 (t) + c2 x2 (t)] =
Z
∞
[c1 x1 (t) + c2 x2 (t)] e−jωt dt
−∞
= c1
Z
∞
x1 (t)e
−jωt
dt + c2
−∞
Z
∞
x2 (t)e−jωt dt
−∞
= c1 X1 (ω) + c2 X2 (ω)
Q.E.D.
Maxim Raginsky
Lecture IX: Fourier transform
Time shift
If x(t) ↔ X(ω), then x(t − c) ↔ X(ω)e−jωc for any constant c.
Proof:
F [x(t − c)]
=
Z
∞
x(t − c)e−jωt dt
−∞
Z ∞
x(t)e−jω(t+c) dt
Z ∞
−jωc
= e
x(t)e−jωt dt
=
−∞
−∞
= X(ω)e−jωc .
Q.E.D.
Maxim Raginsky
Lecture IX: Fourier transform
Multiplication by a complex exponential
If x(t) ↔ X(ω), then x(t)ejω0 t ↔ X(ω − ω0 ) for any real ω0 .
Proof:
F x(t)e
jω0 t
=
=
Z
∞
x(t)ejω0 t e−jωt dt
−∞
Z ∞
x(t)e−j(ω−ω0 )t dt
−∞
= X(ω − ω0 ).
Q.E.D.
Maxim Raginsky
Lecture IX: Fourier transform
Multiplication by a cosine
If x(t) ↔ X(ω), then x(t) cos(ω0 t) ↔ 12 [X(ω + ω0 ) + X(ω − ω0 )].
Proof: use linearity and the last property to get
1
jω0 t
−jω0 t
x(t) e
+e
F [x(t) cos(ω0 t)] = F
2
1 1 =
F x(t)ejω0 t + F x(t)e−jω0 t
2
2
1
=
[X(ω − ω0 ) + X(ω + ω0 )] .
2
Q.E.D.
Maxim Raginsky
Lecture IX: Fourier transform
Convolution in time domain
If x(t) ↔ X(ω) and v(t) ↔ V (ω), then
x(t) ⋆ v(t) ↔ X(ω)V (ω)
Proof:
F [x(t) ⋆ v(t)]
Z
∞
Z
∞
[x(t) ⋆ v(t)]e−jωt dt
Z ∞
=
x(λ)v(t − λ)dλ e−jωt dt
−∞
−∞
Z ∞
Z ∞
=
x(λ)
v(t − λ)e−jωt dt dλ
−∞
−∞
|
{z
}
=
−∞
Z ∞
F [v(t−λ)]
x(λ)V (ω)e−jωλ dλ
Z ∞
x(λ)e−jωλ dλ
= V (ω)
=
−∞
−∞
= X(ω)V (ω).
Q.E.D.
Maxim Raginsky
Lecture IX: Fourier transform
Parseval’s theorem
Let x(t) and v(t) be real-valued signals. Then
Z ∞
Z ∞
1
X(ω)V (ω)dω
x(t)v(t)dt =
2π −∞
−∞
Proof:
Z
∞
x(t)v(t)dt
=
−∞
Z
∞
−∞
=
=
=
V (ω)ejωt dω) dt
−∞
Z ∞
V (ω)
x(t)ejωt dt dω
x(t)
1
2π
Z
∞
Z ∞
1
2π −∞
−∞
Z ∞
1
V (ω)X(−ω)dω
2π −∞
Z ∞
1
X(ω)V (ω)dω,
2π −∞
where we used the fact that, since x(t) is real,
Z ∞
X(ω) =
x(t)ejωt dt = X(−ω).
−∞
Q.E.D.
Maxim Raginsky
Lecture IX: Fourier transform
Parseval’s theorem: cont’d
An important consequence of Parseval’s theorem is that
Z ∞
Z ∞
1
2
x (t)dt =
|X(ω)|2 dω.
2π
−∞
−∞
In other words, signal energy can be computed both in time domain and
in frequency domain (up to a factor of 1/2π).
Maxim Raginsky
Lecture IX: Fourier transform
Duality
If x(t) ↔ X(ω), then X(t) ↔ 2πx(−ω).
Proof:
F [X(t)] =
=
=
Z
∞
X(t)e−jωt dt
−∞
Z ∞
1
X(t)e−jωt dt
2π ·
2π −∞
Z ∞
′
1
2π ·
X(ω ′ )e−jωω dω ′
2π −∞
|
{z
}
=F −1 [X(ω)](−ω)
=
2π ·
1
x(−ω)
2π
Q.E.D.
Maxim Raginsky
Lecture IX: Fourier transform
Duality: an example
Let
x(t) = τ sinc
τt
2π
.
Then by duality we have
X(ω) = 2πpτ (ω).
In more detail:
pτ (t) ↔ τ sinc
Thus, by duality,
τ sinc
τt
2π
Maxim Raginsky
τω 2π
↔ 2πτ (ω).
Lecture IX: Fourier transform
Generalized Fourier transform
The Fourier transform is defined only for signals with finite energy.
However, we can extend its scope by allowing singularity functions.
We begin by computing the Fourier transform of the unit impulse δ(t).
Z ∞
F [δ(t)] =
δ(t)e−jωt dt
−∞
Z ∞
=
δ(t)dt
−∞
= 1,
where we used the sifting property of the unit impulse.
By duality, we have
1 ↔ 2πδ(−ω) = 2πδ(ω).
Maxim Raginsky
Lecture IX: Fourier transform
Fourier transform of the cosine
The cosine signal x(t) = cos(ω0 t) does not have the Fourier transform in
the ordinary sense. It does, however, have a generalized Fourier
transform:
1 jω0 t
−jω0 t
F [cos(ω0 t)] = F
(e
+e
)
2
1 1 F 1 · ejω0 t + F 1 · e−jω0 t
=
2
2
1
=
[2πδ(ω − ω0 ) + 2πδ(ω + ω0 )]
2
= πδ(ω − ω0 ) + πδ(ω + ω0 ).
Q.E.D.
Maxim Raginsky
Lecture IX: Fourier transform
Fourier transform of a periodic signal
Using the generalized Fourier transform, we can analyze periodic signals
that do not have a Fourier transform in the ordinary sense. Thus, if x(t)
is a T -periodic signal, we can expand it in a complex exponential Fourier
series as
∞
X
ck ejkω0 t .
x(t) =
k=−∞
X(ω) =
=
=
F
"
∞
X
k=−∞
∞
X
k=−∞
∞
X
ck e
jkω0 t
#
ck F ejkω0 t
2πck δ(ω − kω0 ).
k=−∞
Thus, the (generalized) Fourier transform of a periodic signal is a train of
impulses located at integer multiples of the fundamental frequency ω0 .
Maxim Raginsky
Lecture IX: Fourier transform
Bandlimited and timelimited signals
A signal x(t) is called:
bandlimited if there exists a number B > 0 (called the
bandwidth), such that
X(ω) = 0,
for all |ω| ≥ B.
timelimited if there exists a number T > 0, such that
x(t) = 0,
for all |t| ≥ T.
It can be proved that a bandlimited signal cannot be timelimited, and
vice versa. We’ve seen an example of this with the transform pairs
τω τt
and
τ sinc
↔ 2πpτ (ω)
pτ (t) ↔ τ sinc
2π
2π
However, a signal can be approximately timelimited and bandlimited —
that is, there exist numbers B > 0 and T > 0, such that |x(t)| is small
for |t| ≥ T and |X(ω)| is small for |ω| ≥ B.
Maxim Raginsky
Lecture IX: Fourier transform
Frequency response of LTI systems
Consider an LTI system with the impulse response h(t). Then the output
of the system due to input x(t) is given by the convolution integral,
Z ∞
x(λ)h(t − λ)dλ.
y(t) = x(t) ⋆ h(t) =
−∞
In frequency domain, the action of the system can be described as
follows:
Y (ω) = H(ω)X(ω).
This is a consequence of the fact that convolution in time domain
corresponds to multiplication in frequency domain.
The Fourier transform H(ω) of the impulse response h(t) is called the
frequency response of the system.
Maxim Raginsky
Lecture IX: Fourier transform