Digital Signal Processing Reference
In-Depth Information
1.
cos(Æ
0
n)
is periodic only if, from (9.39),
Æ
0
2p
=
k
N
,
where
k
and
N
are integers.
2.
cos
(Æ
0
n)
is periodic in
Æ
with period
2p;
that is, with
k
any integer,
cos(Æ
0
n) = cos(Æ
0
+ 2pk)n.
e
jÆ
0
n
.
Of course, the same properties apply to and
In this section, the properties of even and odd were defined with respect to
discrete-time signals. These properties are useful in the applications of the discrete-
time Fourier transform, as shown in Chapter 12. Next, the properties of periodic dis-
crete-time signal were investigated. The property that a discrete-time sinusoid is
periodic in frequency has great implications with respect to the sampling of continu-
ous-time signals, as shown in Chapters 5 and 6.
cos(Æ
0
n + u)
9.4
COMMON DISCRETE-TIME SIGNALS
In Section 2.3, we defined some common continuous-time signals that occur in the
transient response of certain systems. In this section, equivalent discrete-time signals
are introduced; these signals can appear in the transient response of certain discrete-
time systems.
One such signal, the sinusoid, was mentioned in Section 9.3. For example, a
digital computer can be programmed to output a discrete-time sinusoid to generate
an audible tone of variable frequency. The discrete-time sinusoidal signal is trans-
mitted from the computer to a digital-to-analog converter (D/A), which is an elec-
tronic circuit that converts binary numbers into a continuous-time voltage signal.
(See Section 1.3.) This voltage is then applied through a power amplifier to a speaker.
The operation is depicted in Figure 9.16. A timing chip in the computer is used to de-
termine the sample period
T
and, hence, the frequency of the tone.
We now use an example of a system to introduce a common discrete-time sig-
nal. The block shown in Figure 9.17(a) represents a memory device that stores a
number. Examples of this device are shift registers or memory locations in a digital
computer. Every
T
seconds, we shift out the number stored in the device. Then a dif-
ferent number is shifted into the device and stored. If we denote the number shifted
into the device as
x
[
n
], the number just shifted out must be A device used
in this manner is called an
ideal time delay
. The term
ideal
indicates that the num-
bers are not altered in any way, but are only delayed.
Suppose that we connect the ideal time delay in the system shown in Figure
9.17(b). The number shifted out of the delay is multiplied by the constant
a
to form
the next number to be stored, resulting in the system equation
x[n - 1].
x[n] = ax[n - 1].
(9.44)
