Lecture X: Discrete-time Fourier transform
Maxim Raginsky
BME 171: Signals and Systems
Duke University
October 15, 2008
Maxim Raginsky
Lecture X: Discrete-time Fourier transform
This lecture
Plan for the lecture:
1
Recap: Fourier transform for continuous-time signals
2
Frequency content of discrete-time signals: the DTFT
3
Examples of DTFT
4
Inverse DTFT
5
Properties of the DTFT
Maxim Raginsky
Lecture X: Discrete-time Fourier transform
Recap: Fourier transform
Recall from the last lecture that any sufficiently regular (e.g.,
finite-energy) continuous-time signal x(t) can be represented in frequency
domain via its Fourier transform
Z ∞
x(t)e−jωt dt.
X(ω) =
−∞
We can recover x(t) from X(ω) via the inverse Fourier transform formula:
Z ∞
1
X(ω)ejωt dω.
x(t) =
2π −∞
Maxim Raginsky
Lecture X: Discrete-time Fourier transform
Spectral content of discrete-time signals
In this lecture, we will look at one way of describing discrete-time signals
through their frequency content: the discrete-time Fourier transform
(DTFT).
Any discrete-time signal x[n] that is absolutely summable, i.e.,
∞
X
|x[n]| < +∞,
n=−∞
has a DTFT X(Ω), −∞ < Ω < ∞, given by
X(Ω) =
∞
X
x[n]e−jnΩ
n=−∞
Note that, even though the underlying signal x[n] is discrete-time, the
DTFT is a function of a continuous frequency Ω.
Maxim Raginsky
Lecture X: Discrete-time Fourier transform
Periodicity of the DTFT
The first thing to note is that the DTFT X(Ω) of x[n] is 2π-periodic:
∞
X
X(Ω + 2π) =
=
x[n]e−jn(Ω+2π)
n=−∞
∞
X
n=−∞
=
∞
X
x[n]e−jnΩ |e−j2πn
{z }
=1
x[n]e−jnΩ
n=−∞
=
X(Ω).
This periodicity is due to the discrete-time nature of the signal. Thus,
when working with DTFT’s, we only need to look at the range
0 ≤ Ω ≤ 2π (or −π ≤ Ω ≤ π).
Maxim Raginsky
Lecture X: Discrete-time Fourier transform
Computing DTFT’s: an example
Consider
x[n] =
Then
X(Ω) =
=
an , q1 ≤ n ≤ q2
0,
otherwise
q2
X
n=q1
q2
X
an e−jnΩ
(ae−jΩ )n
n=q1
=
(ae−jΩ )q1 − (ae−jΩ )q2 +1
1 − ae−jΩ
In the last step, we used the formula
q2
X
rn =
n=q1
rq1 − rq2 +1
,
1−r
valid whenever q1 and q2 are integers with q2 > q1 and r is any real or
complex number.
Maxim Raginsky
Lecture X: Discrete-time Fourier transform
Computing DTFT’s: another example
Consider the signal
x[n] = an u[n],
where |a| < 1. Then
X(Ω) =
=
∞
X
an e−jnΩ
n=0
∞
X
(ae−jΩ )n
n=0
=
1
,
1 − ae−jΩ
where we used the formula
∞
X
rn =
n=0
1
,
1−r
valid for any real or complex number r satisfying |r| < 1.
Maxim Raginsky
Lecture X: Discrete-time Fourier transform
Computing DTFT’s: another example
Consider the rectangular pulse
1, n = −q, −q + 1, . . . , q − 1, q
x[n] =
0, otherwise
Then
X(Ω) =
q
X
e−jnΩ
n=−q
=
=
=
=
(e−jΩ )−q − (e−jΩ )q+1
1 − e−jΩ
jqΩ
e
− e−jqΩ e−jΩ ejΩ/2
· jΩ/2
1 − e−jΩ
e
j(q+1/2)Ω
−j(q+1/2)Ω
e
−e
ejΩ/2 − e−jΩ/2
sin[(q + 1/2)Ω]
sin(Ω/2)
Maxim Raginsky
Lecture X: Discrete-time Fourier transform
Inverse DTFT
We can recover the original signal x[n] from its DTFT X(Ω) via the
inverse DTFT formula
Z 2π
1
X(Ω)ejnΩ dΩ.
x[n] =
2π 0
Proof: use orthonormality of complex exponentials –
1
2π
Z
2π
X(Ω)e
jnΩ
dΩ
=
0
1
2π
Z
2π
0
∞
X
∞
X
m=−∞
1
=
x[m] ·
2π
m=−∞
|
=
∞
X
x[m]e
−jmΩ
Z
2π
0
!
ejnΩ dΩ
ej(n−m)Ω dΩ
{z
}
=δ[n−m]
x[m]δ[n − m],
m=−∞
= x[n]
Maxim Raginsky
Lecture X: Discrete-time Fourier transform
Properties of the DTFT
Like its continuous-time counterpart, the DTFT has several very useful
properties. These are listed in any text on signals and systems. We will
take a look at a couple of them.
First of all, the DTFT is linear: if
x1 [n] ↔ X1 (Ω)
and
x2 [n] ↔ X2 (Ω),
then
c1 x1 [n] + c2 x2 [n] ↔ c1 X1 (Ω) + c2 X2 (Ω)
for any two constants c1 , c2 .
The proof is obvious from definitions.
Maxim Raginsky
Lecture X: Discrete-time Fourier transform
Convolution in time domain
If x[n] ↔ X(Ω) and v[n] ↔ V (Ω), then
x[n] ⋆ v[n] ↔ X(Ω)V (Ω).
Proof: let y[n] = x[n] ⋆ v[n]. Then
Y (Ω) =
∞
X
(x[n] ⋆ v[n])e
−jnΩ
=
n=−∞
∞
X
=
n=−∞
∞
X
x[k]
v[n − k]e−jnΩ
k=−∞
n=−∞
∞
X
∞
X
=
x[k]
=
x[k]e−jkΩ
k=−∞
|
′
v[n ]e
−j(n′ +k)Ω
{z
=X(Ω)
!
∞
X
Maxim Raginsky
{z
=V (Ω)
!
x[k]v[n − k] e−jnΩ
!
v[n]e−jnΩ
n=−∞
}|
∞
X
k=−∞
!
n′ =−∞
k=−∞
∞
X
∞
X
!
}
Lecture X: Discrete-time Fourier transform
Parseval’s theorem
If x[n] and v[n] are real-valued signals, then
Z 2π
∞
X
1
x[n]v[n] =
X(Ω)V (Ω)dΩ.
2π 0
n=−∞
Proof:
∞
X
x[n]v[n]
=
n=−∞
∞
X
x[n]
n=−∞
=
=
=
1
2π
Z
2π
1
2π
Z
2π
1
2π
1
2π
2π
V (Ω)
dΩ
x[n]e
jΩn
!
x[n]e
−j(−Ω)n
V (Ω)e
n=−∞
V (Ω)
0
∞
X
n=−∞
2π
|
{z
=X(−Ω)
V (Ω)X(Ω)dΩ
jΩn
0
∞
X
0
Z
Z
dΩ
!
dΩ
}
0
where we used the fact that x[n] is real-valued.
Maxim Raginsky
Lecture X: Discrete-time Fourier transform
Parseval’s theorem: cont’d
An important consequence of Parseval’s theorem is that the signal energy
∞
X
x2 [n]
n=−∞
can be computed also in the frequency domain:
∞
X
n=−∞
x2 [n] =
1
2π
Maxim Raginsky
Z
2π
|X(Ω)|2 dΩ
0
Lecture X: Discrete-time Fourier transform
Summary of the DTFT
The discrete-time Fourier transform (DTFT) gives us a way of
representing frequency content of discrete-time signals.
The DTFT X(Ω) of a discrete-time signal x[n] is a function of a
continuous frequency Ω. One way to think about the DTFT is to view
x[n] as a sampled version of a continuous-time signal x(t):
x[n] = x(nT ),
n = . . . , −2, −1, 0, 1, 2, . . . ,
where T is a sufficiently small sampling step. Then X(Ω) can be thought
of as a discretization of X(ω).
Due to discrete-time nature of the original signal, the DTFT is
2π-periodic. Hence, Ω = 2π is the highest frequency component a
discrete-time signal can have.
The DTFT possesses several important properties, which can be
exploited both in calculations and in conceptual reasoning about
discrete-time signals and systems.
Maxim Raginsky
Lecture X: Discrete-time Fourier transform