Introduction to signals - Continuous and Discrete Time Signals and Systems - page 8

+

time instant t = 0 . 5 τ is defined as t = 0 . 5 τ + ε . The value of the rectangular

−

pulse at the discontinuity t = 0 . 5 τ is, therefore, specified by x (0 . 5 τ ) = 1

+

and x (0 . 5 τ ) = 0. Likewise, the value of the rectangular pulse at its other

discontinuity t

− +

) = 1.
A CT signal that is continuous for all t except for a finite number of instants
is referred to as a piecewise CT signal. The value of a piecewise CT signal at the
point of discontinuity t 1 can either be specified by our earlier notation, described
in the previous paragraph, or, alternatively, using the following relationship:

=− 0 . 5 τ is specified by x ( − 0 . 5 τ ) = 0 and x ( − 0 . 5 τ

+
1 ) + x ( t

−
1 )

x ( t 1 ) = 0 . 5 x ( t . (1.1)

Equation (1.1) shows that x ( 0 . 5 τ ) = 0 . 5 at the points of discontinuity t =
0 . 5 τ . The second approach is useful in certain applications. For instance,
when a piecewise CT signal is reconstructed from an infinite series (such as the
Fourier series defined later in the text), the reconstructed value at the point of
discontinuity satisfies Eq. (1.1). Discussion of piecewise CT signals is continued
in Chapter 4, where we define the CT Fourier series.

1.1.2 Analog and digital signals

A second classification of signals is based on their amplitudes. The amplitudes
of many real-world signals, such as voltage, current, temperature, and pressure,
change continuously, and these signals are called analog signals. For example,
the ambient temperature of a house is an analog number that requires an infinite
number of digits (e.g., 24.763 578. . . ) to record the readings precisely. Digital
signals, on the other hand, can only have a finite number of amplitude values.
For example, if a digital thermometer, with a resolution of 1

◦

C and a range

◦ ◦

of [10 C], is used to measure the room temperature at discrete time

instants, t = kT , then the recordings constitute a digital signal. An example of
a digital signal was shown in Fig. 1.1(h), which plots the temperature readings
taken once a day for one week. This digital signal has an amplitude resolution
of 0.1

C, 30

◦

C, and a sampling interval of one day.
Figure 1.5 shows an analog signal with its digital approximation. The analog
signal has a limited dynamic range between [ − 1, 1] but can assume any real
value (rational or irrational) within this dynamic range. If the analog signal is
sampled at time instants t = kT and the magnitude of the resulting samples are
quantized to a set of finite number of known values within the range [ − 1, 1],
the resulting signal becomes a digital signal. Using the following set of eight
uniformly distributed values,

[ − 0.875, − 0.625, − 0.375, − 0.125, 0.125, 0.375, 0.625, 0.875],

within the range [ − 1, 1], the best approximation of the analog signal is the
digital signal shown with the stem plot in Fig. 1.5.
Another example of a digital signal is the music recorded on an audio com-
pact disc (CD). On a CD, the music signal is first sampled at a rate of 44 100