EE351–Spectrum Analysis and Discrete Time Systems
University of Saskatchewan
Definition of Discrete-Time Fourier Transform (DTFT)
F{x[n]} = X(ejω ) ,
+∞
X
x[n]e−jωn
n=−∞
1
F −1 {X(ejω )} = x[n] ,
2π
Z
X(ejω )ejωn dω
2π
• Why use the above awkward notation for the transform?
• Answer: It is consistent with the z-transform
X(z) ,
∞
X
x[n]z −n ⇒ X(ejω ) = X(z) |z=ejω
n=−∞
• This also enables us to distinguish between the discrete-time (DT) and
continuous-time (CT) Fourier transforms.
Dr. H. Nguyen
Page 203
EE351–Spectrum Analysis and Discrete Time Systems
University of Saskatchewan
Some Observations
X(ejω ) =
∞
X
x[n]e−jωn ;
n=−∞
|
{z
Analysis equation
}
1
x[n] =
2π
|
Z
X(ejω )ejωn dω
2π
{z
}
Synthesis equation
• Since ejωn = ej(ω+`2π)n , for any pair of integers ` and n ⇒ X(ejω ) is a periodic
function of ω with a fundamental period of 2π.
• Unlike the DT Fourier series, the frequency ω is continuous (Recall that the CT
Fourier transform X(jω) is not periodic in general).
• Thus the DT synthesis integral can be taken over any continuous interval of
length 2π
• This is similar to the CT Fourier series analysis equation
Dr. H. Nguyen
Page 204
EE351–Spectrum Analysis and Discrete Time Systems
University of Saskatchewan
Frequency Concept for Discrete-Time Signals
• Recall that ej(ω+`2π)n = ejωn for any integer `
• If seemingly very high-frequency discrete-time signals, cos[(ω + `2π)n], are equal
to low-frequency discrete-time signals, cos(ωn), what do low and high
frequencies mean for discrete-time signals?
• Note that the unit of ω is “radians per sample”
• A sinusoid with a frequency of 0.1 radians per sample is the same as one with a
frequency of 0.1 + 2π radian per sample
• No DT signal can oscillate “faster” between two consecutive samples of opposite
magnitudes
• No DT signal can oscillate “slower” than 0 radians per sample
• Thus
(i) ω = π, `(π + 2π) is highest perceivable frequency for DT signals
(ii) ω = 0, `(2π) is lowest perceivable frequency
Dr. H. Nguyen
Page 205
EE351–Spectrum Analysis and Discrete Time Systems
5 Portland
State University
ECE 223
University of Saskatchewan
Discrete-Time Fourier Transform
Ver. 1.14
6
Frequency Concept for Discrete-Time Signals (Cont’d)
Discrete-Time Frequency Concepts Continued
X(ejω )
1
−4π −3π −2π −π
0
π
ω
2π
3π
4π
X(ejω )
1
−4π −3π −2π −π
0
π
ω
2π
3π
4π
•• Low
frequencies
are those
that
Low frequencies
are those
that are
nearare
0 near 0
•• High
frequencies
are those
near ±π
High frequencies
are those
near ±π
•• Intermediate
frequencies
are inthose
in between
Intermediate frequencies
are those
between
•• Note
thatthethe
highest
frequency,
π per
radians
sample
equalperto
Note that
highest
frequency,
π radians
sampleper
is equal
to 0.5iscycles
0.5
cycles per sample
sample
• We will encounter this concept again when we discuss sampling
Dr. H. Nguyen
7 Portland State University
Page 206
ECE 223
Discrete-Time Fourier Transform
Ver. 1.14
8
EE351–Spectrum Analysis and Discrete Time Systems
University of Saskatchewan
Examples
Example 1: Find the Fourier transform of h[n] = an u[n] where a < 1. Sketch the
transform over range of −3π to 3π for a = 0.5 and a = −0.5.
Example 2: Find the Fourier transform of the following pulse signal. Sketch the
transform over range of −3π to 3π for N = 5, 10.

 1 |n| ≤ N
pN [n] =
 0 |n| ≤ N
Dr. H. Nguyen
Page 207
14
EE351–Spectrum Analysis and Discrete Time Systems
9 Portland State University
ECE 223
Discrete-Time Fourier Transform
University of Saskatchewan
Ver. 1.14
10
Example 1: First-Order Filter a = 0.5
Example 3: First-Order Filter a = 0.5
n
Fourier Transform of (0.5) u[n]
1
x[n]
0.5
0
−6
−4
−2
0
2
4
6
8
−6.2832
−3.1416
0
3.1416
6.2832
9.4248
−6.2832
−3.1416
0
3.1416
Frequency (rad/sample)
6.2832
9.4248
jω
|X(e )|
−0.5
−8
2
∠ X(ejω) (o)
.14
1
0
−9.4248
50
0
−50
−9.4248
11 Portland State University
Dr. H. Nguyen
ECE 223
Discrete-Time Fourier Transform
Ver. 1.14
12
Page 208
EE351–Spectrum Analysis and Discrete Time Systems
University of Saskatchewan
Example
First-Order Filter
Filteraa==−0.5
−0.5
Example 1:
4: First-Order
Find
tran
n
Fourier Transform of (−0.5) u[n]
1
x[n]
0.5
0
−0.5
−8
−6
−4
−2
0
2
4
6
8
∠ X(ejω) (o)
jω
|X(e )|
2
1
0
−9.4248
50
−6.2832
−3.1416
0
3.1416
6.2832
9.4248
−6.2832
−3.1416
0
3.1416
Frequency (rad/sample)
6.2832
9.4248
0
−50
−9.4248
Portland State University
ECE 223
Discrete-Time Fourier Transform
Dr. H. Nguyen
Ver. 1.14
13 Portland
Page 209
Example 5: Workspace
14
EE351–Spectrum Analysis and Discrete Time Systems
13 Portland State University
ECE 223
Discrete-Time Fourier Transform
University of Saskatchewan
Ver. 1.14
14
Example 2: Fourier Transform of A Pulse N = 5
Example 6: Pulse Transform for N = 5
Fourier Transform of p5[n]
10
8
6
jω
P(e )
14
4
2
0
−2
−9.4248
−6.2832
15 Portland State University
Dr. H. Nguyen
−3.1416
ECE 223
0
3.1416
6.2832
Discrete-Time Fourier Transform
9.4248
Ver. 1.14
16
Page 210
EE351–Spectrum Analysis and Discrete Time Systems
University of Saskatchewan
Example
2: Fourier
Transform
of for
A Pulse
N = 10
Example
7: Pulse
Transform
N = 10
Fourier Transform of p [n]
10
20
10
P(ejω)
P(ejω)
15
5
0
−9.4248
−6.2832
Portland State University
−3.1416
ECE 223
0
3.1416
6.2832
Discrete-Time Fourier Transform
Dr. H. Nguyen
9.4248
Ver. 1.14
17 Portland
Page 211
Example 9: Inverse of Impulse Train
EE351–Spectrum Analysis and Discrete Time Systems
University of Saskatchewan
Convergence
X(ejω ) =
+∞
X
x[n]e−jωn ;
n=−∞
1
x[n] =
2π
Z
X(ejω )ejωn dω
2π
• Sufficient conditions of the convergence of the discrete time Fourier transform of
a bounded discrete-time signal:
– Finite duration: There exits an N such that x[n] = 0 for |n| > N .
P∞
– Absolutely summable:
n=−∞ |x[n]| < ∞.
P∞
2
– Finite energy:
n=−∞ |x[n]| < ∞
• The synthesis equation always converges
• There is no Gibb’s phenomenon
Dr. H. Nguyen
Page 212
EE351–Spectrum Analysis and Discrete Time Systems
University of Saskatchewan
Properties
Periodicity
X(ej(ω+`2π) ) = X(ejω )
Linearity
F
a1 x1 [n] + a2 x2 [n] ⇐⇒ a1 X1 (ejω ) + a2 X2 (ejω )
Time Shifting
F
x[n − n0 ] ⇐⇒ e−jωn0 X(ejω )
Frequency Shifting
F
ejω0 n x[n] ⇐⇒ X(ej(ω−ω0 ) )
Conjugation
F
x∗ [n] ⇐⇒ X ∗ (e−jω )
Conjugate Symmetry
If x[n] is real, then
X(ejω )
=
X ∗ (e−jω )
Differencing
F
x[n] − x[n − 1] ⇐⇒ (1 − e−jω )X(ejω )
This is similar to a continuous-time derivative.
Dr. H. Nguyen
Page 213
EE351–Spectrum Analysis and Discrete Time Systems
University of Saskatchewan
Properties (Cont’d)
Accumulation
n
X
∞
X
1
jω
j0
x[n] ⇐⇒
X(e ) + πX(e )
δ(ω − 2π`)
−jω
1
−
e
m=−∞
F
`=−∞
This is similar to a continuous-time integration.
Time Reversal
F
x[−n] ⇐⇒ X(e−jω )
Frequency Differentiation
dX(ejω )
n · x[n] ⇐⇒ j
dω
F
Parseval’s Relation
+∞
X
1
|x[n]| =
2π
n=−∞
2
Z
|X(ejω |2 dω
2π
Convolution
F
y[n] = x[n] ∗ h[n] ⇐⇒ Y (ejω ) = X(ejω )H(ejω )
Multiplication
1
x[n] · w[n] ⇐⇒
2π
F
Dr. H. Nguyen
Z
X(eju )W (ej(ω−u) )du
2π
Page 214
EE351–Spectrum Analysis and Discrete Time Systems
University of Saskatchewan
DT Fourier Transform: Summary
• X(ejω ) is a periodic function of ω with a fundamental period of 2π
• Two discrete-time complex exponentials with frequencies that differ by a
multiple of 2π are equal: ejωn = ej(ω+2`π)n
• The highest perceivable discrete-time frequency is π radians per sample
• The lowest perceivable discrete-time frequency is 0 radians per sample
• The analysis equation may or may not converge, the synthesis equation always
converges
• The energy of the signal in the time-domain is proportional to energy of the
DTFT in an interval of 2π
• The DTFT shares many of the properties of CTFT and Fourier series
Dr. H. Nguyen
Page 215