Digital Signal Processing Reference
In-Depth Information
20
15
l
g
r
,
y
i
d
.
,
©
,
L
s
10
5
0
-5
-
π
-
π/
2
0
π/
2
π
Figure 4.10
The DTFT of the signal in (4.19).
The DTFT of this particular signal turns out to be real (we will see later
that this is a consequence of the signal's symmetry) and it is plotted in Fig-
ure 4.10. When, as is very often the case, the DTFT is complex-valued, the
usual way to represent it graphically takes themagnitude and the phase sep-
arately into account. The DTFT is always a 2
-periodic function and the
standard convention is to plot the interval from
π
−
π
π
.Largerintervals
can be considered if the periodicity needs to be made explicit; Figure 4.11,
for instance, shows five full periods of the same function.
to
20
15
10
5
0
-5
-5
π
-4
π
-3
π
-2
π
-
π
0
π
2
π
3
π
4
π
5
π
Figure 4.11
The DTFT of the signal in (4.19), with explicit periodicity.
4.4.1 The DTFT as the Limit of a DFS
A way to gain some intuition about the structure of the DTFT formulas is
to consider the DFS of periodic sequences with larger and larger periods.
Intuitively, as we look at the structure of the Fourier basis for the DFS, we
can see that the number of basis vectors in (4.9) grows with the length
N
of the period and, consequently, the frequencies of the underlying complex
