Post on 05-May-2018
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 -periodiccontinuous-time signal x(t) can be expanded, e.g., in a complexexponential Fourier series:
x(t) =∞∑
k=−∞
ckejkω0t,
where ω0 = 2π/T is the fundamental frequency, and the Fouriercoefficients ck are given by
ck =1
T
∫ T/2
−T/2
x(t)e−jkω0tdt, k = . . . ,−2,−1, 0, 1, 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.,
∫ ∞
−∞
x2(t)dt < +∞,
can be represented in the frequency domain via the Fourier transform:
X(ω) =
∫∞
−∞
x(t)e−jωtdt
We will also writeX(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) =
1, |t| ≤ τ/20, |t| > τ/2
F [pτ (t)] =
∫ ∞
−∞
pτ (t)e−jωtdt
=
∫ τ/2
−τ/2
e−jωtdt
=
[
−1
jωe−jωt
]τ/2
−τ/2
=ejωτ/2 − e−jωτ/2
jω
=2 sin(ωτ/2)
ω
= τsinc(τω
2π
)
.
Maxim Raginsky Lecture IX: Fourier transform
Inverse Fourier transform
The signal x(t) can be recovered from its Fourier transformX(ω) = F [x(t)] using the inverse Fourier transform formula
x(t) = F−1[X(ω)] =1
2π
∫ ∞
−∞
X(ω)ejωtdω
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 writex(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
∫∞
−∞
X(ω)ejωtdω =1
2π
∫∞
−∞
(∫∞
−∞
x(t′)e−jωt′dt′)
ejωtdω
=
∫∞
−∞
x(t′)
(1
2π
∫∞
−∞
ej(t−t′)ωdω
)
dt′.
1
2π
∫∞
−∞
ej(t−t′)ωdω =1
2πlim
Ω→∞
∫ πΩ
−πΩ
ej(t−t′)ωdω
= limΩ→∞
1
2πj(t − t′)Ω
[
ej(t−t′)ω]πΩ
−πΩ
= limΩ→∞
sin(πΩ(t − t′))
πΩ(t − t′)
= limΩ→∞
sinc(Ω(t − t′))
= δ(t − t′)
Hence,
1
2π
∫ ∞
−∞
X(ω)ejωtdω =
∫ ∞
−∞
x(t′)δ(t − t′)dt′ = x(t)
Q.E.D.
Maxim Raginsky Lecture IX: Fourier transform
Properties of the Fourier transform
The Fourier transform has many useful properties that make calculationseasier and also help thinking about the structure of signals and the actionof systems on signals.
The properties are listed in any textbook on signals and systems. We willlook 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),
thenc1x1(t) + c2x2(t) ↔ c1X1(ω) + c2X2(ω)
for any two numbers c1 and c2.
Proof: obvious –
F [c1x1(t) + c2x2(t)] =
∫ ∞
−∞
[c1x1(t) + c2x2(t)] e−jωtdt
= c1
∫∞
−∞
x1(t)e−jωtdt + c2
∫∞
−∞
x2(t)e−jωtdt
= c1X1(ω) + c2X2(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)] =
∫ ∞
−∞
x(t − c)e−jωtdt
=
∫ ∞
−∞
x(t)e−jω(t+c)dt
= e−jωc
∫∞
−∞
x(t)e−jωtdt
= 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)ejω0t ↔ X(ω − ω0) for any real ω0.
Proof:
F[x(t)ejω0t
]=
∫∞
−∞
x(t)ejω0te−jωtdt
=
∫ ∞
−∞
x(t)e−j(ω−ω0)tdt
= 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(ω0t) ↔12 [X(ω + ω0) + X(ω − ω0)].
Proof: use linearity and the last property to get
F [x(t) cos(ω0t)] = F
[1
2x(t)
(ejω0t + e−jω0t
)]
=1
2F
[x(t)ejω0t
]+
1
2F
[x(t)e−jω0t
]
=1
2[X(ω − ω0) + X(ω + ω0)] .
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)] =
∫∞
−∞
[x(t) ⋆ v(t)]e−jωtdt
=
∫∞
−∞
(∫∞
−∞
x(λ)v(t − λ)dλ
)
e−jωtdt
=
∫ ∞
−∞
x(λ)
(∫ ∞
−∞
v(t − λ)e−jωtdt
)
︸ ︷︷ ︸
F [v(t−λ)]
dλ
=
∫ ∞
−∞
x(λ)V (ω)e−jωλdλ
= V (ω)
∫∞
−∞
x(λ)e−jωλdλ
= 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∫ ∞
−∞
x(t)v(t)dt =1
2π
∫ ∞
−∞
X(ω)V (ω)dω
Proof:∫
∞
−∞
x(t)v(t)dt =
∫∞
−∞
x(t)
(1
2π
∫∞
−∞
V (ω)ejωtdω)
)
dt
=1
2π
∫∞
−∞
V (ω)
(∫∞
−∞
x(t)ejωtdt
)
dω
=1
2π
∫ ∞
−∞
V (ω)X(−ω)dω
=1
2π
∫ ∞
−∞
X(ω)V (ω)dω,
where we used the fact that, since x(t) is real,
X(ω) =
∫ ∞
−∞
x(t)ejωtdt = 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
∫ ∞
−∞
x2(t)dt =1
2π
∫ ∞
−∞
|X(ω)|2dω.
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)] =
∫∞
−∞
X(t)e−jωtdt
= 2π ·1
2π
∫ ∞
−∞
X(t)e−jωtdt
= 2π ·1
2π
∫ ∞
−∞
X(ω′)e−jωω′
dω′
︸ ︷︷ ︸
=F−1[X(ω)](−ω)
= 2π ·1
2πx(−ω)
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
(τω
2π
)
Thus, by duality,
τ sinc
(τt
2π
)
↔ 2πτ (ω).
Maxim Raginsky 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).
F [δ(t)] =
∫ ∞
−∞
δ(t)e−jωtdt
=
∫∞
−∞
δ(t)dt
= 1,
where we used the sifting property of the unit impulse.
By duality, we have1 ↔ 2πδ(−ω) = 2πδ(ω).
Maxim Raginsky Lecture IX: Fourier transform
Fourier transform of the cosine
The cosine signal x(t) = cos(ω0t) does not have the Fourier transform inthe ordinary sense. It does, however, have a generalized Fouriertransform:
F [cos(ω0t)] = F
[1
2(ejω0t + e−jω0t)
]
=1
2F
[1 · ejω0t
]+
1
2F
[1 · e−jω0t
]
=1
2[2πδ(ω − ω0) + 2πδ(ω + ω0)]
= 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 signalsthat 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 Fourierseries as
x(t) =
∞∑
k=−∞
ckejkω0t.
X(ω) = F
[∞∑
k=−∞
ckejkω0t
]
=
∞∑
k=−∞
ckF[ejkω0t
]
=
∞∑
k=−∞
2πckδ(ω − kω0).
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 thebandwidth), 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, andvice versa. We’ve seen an example of this with the transform pairs
pτ (t) ↔ τ sinc(τω
2π
)
and τ sinc
(τt
2π
)
↔ 2πpτ (ω)
However, a signal can be approximately timelimited and bandlimited —that is, there exist numbers B > 0 and T > 0, such that |x(t)| is smallfor |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 outputof the system due to input x(t) is given by the convolution integral,
y(t) = x(t) ⋆ h(t) =
∫∞
−∞
x(λ)h(t − λ)dλ.
In frequency domain, the action of the system can be described asfollows:
Y (ω) = H(ω)X(ω).
This is a consequence of the fact that convolution in time domaincorresponds 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