Matemáticas Aplicadas
Unidad 1
Señales y Sistemas.
Dra. Ericka Reyes Sánchez
1.-Señales y Sistemas.
Objetivo Especifico: Que el alumno clasifique las señales y los sistemas, e identifique sus
aplicaciones en la ingeniería. (7 hrs.)
1.1 Clasificación de Señales.
1.1.1 Señales Continuas y Discretas.
1.1.2 Funciones como Señales.
1.1.3 Energía y Potencia de Señales.
1.1.4 Señales Periódicas.
1.1.5 Señales Pares e Impares.
1.2 Tipos Especiales de Funciones.
1.2.1 Funciones Generalizadas.
1.2.2 Funciones Exponenciales Complejas.
1.3 Sistemas y sus Propiedades.
1.3.1 Sistemas Continuos y Discretos.
1.3.2 Sistemas con Memoria.
1.3.3 Invertibilidad y Sistemas Inversos.
1.3.4 Sistemas Invariantes en el Tiempo.
1.3.5 Sistemas Lineales.
SOME USEFUL SIGNAL OPERATIONS
We discuss here three useful signal operations: shifting, scaling, and
inversion. Since the independent variable in our signal description is
time, these operations are discussed as time shifting, time scaling, and
time reversal (inversion). However, this discussion is valid for functions
having independent variables other than time (e.g., frequency or
distance).
Time Shifting
Consider a signal 𝑥(𝑡) (Fig. 1.4a) and the same signal delayed by 𝑇
seconds (Fig. 1.4b), which we shall denote by 𝜑(𝑡). Whatever happens
in 𝑥(𝑡) (Fig. 1.4a) at some instant 𝑡 also happens in 𝜑(𝑡) (Fig. 1.4b) 𝑇
seconds later at the instant 𝑡 + 𝑇. Therefore
𝜑 𝑡 + 𝑇 = 𝑥(𝑡) and
𝜑 𝑡 = 𝑥(𝑡 − 𝑇)
Therefore, to time-shift a signal by T, we
replace t with 𝑡 − 𝑇 . Thus 𝑥(𝑡 −
𝑇) represents 𝑥(𝑡) time-shifted by 𝑇 seconds.
If 𝑇 is positive, the shift is to the right (delay),
as in Fig. 1.4b. If 𝑇 is negative, the shift is to
the left (advance), as in Fig. 1.4c. Clearly,
𝑥(𝑡 − 2) is 𝑥(𝑡) delayed (right-shifted) by 2
seconds, and 𝑥(𝑡 + 2) is 𝑥(𝑡) advanced (leftshifted) by 2 seconds.
Time Scaling
The compression or expansion of a signal in time is known as
𝑡𝑖𝑚𝑒 𝑠𝑐𝑎𝑙𝑖𝑛𝑔. Consider the signal x(t) of Fig. 1.6a. The signal 𝜑(𝑡) in
Fig. 1.6b is 𝑥(𝑡) compressed in time by a factor of 2. Therefore,
whatever happens in 𝑥(𝑡) at some instant 𝑡 also happens to 𝜑(𝑡) at the
instant 𝑡/2 so that
𝜑
𝑡
2
= 𝑥(𝑡)
and
𝜑 𝑡 = 𝑥(2𝑡)
Observe that because 𝑥(𝑡) = 0 at 𝑡 = 𝑇1 and 𝑇2 , we must have
𝜑(𝑡) = 0 at 𝑡 = 𝑇1 2 and 𝑇2 2, as shown in Fig. 1.6b. If 𝑥(𝑡) were
recorded on a tape and played back at twice the normal recording
speed, we would obtain 𝑥(2𝑡). In general, if x(t) is compressed in time
by a factor 𝑎(𝑎 > 1), the resulting signal 𝜑(𝑡) is given by
𝜑 𝑡 = 𝑥(𝑎𝑡)
Using a similar argument, we can show that
𝑥(𝑡) expanded (slowed down) in time by a factor
𝑎(𝑎 > 1) is given by
𝑡
𝜑 𝑡 =𝑥
𝑎
Figure 1.6c shows 𝑥(𝑡/2), which is
𝑥(𝑡) expanded in time by a factor of 2. Observe
that in a time-scaling operation, the origin 𝑡 =
0 is the anchor point, which remains unchanged
under the scaling operation because at 𝑡 = 0,
𝑥(𝑡) = 𝑥(𝑎𝑡) = 𝑥(0).
In summary, to time-scale a signal by a factor a,
we replace t with at. If 𝑎 > 1, the scaling results
in compression, and if 𝑎 < 1, the scaling results
in expansion.
Time Reversal
Consider the signal 𝑥(𝑡) in Fig. 1.8a. We can view 𝑥(𝑡) as a rigid wire frame hinged at
the vertical axis. To time-reverse 𝑥(𝑡), we rotate this frame 180◦ about the vertical
axis. This time reversal [the reflection of 𝑥(𝑡) about the vertical axis] gives us the
signal 𝜑(𝑡) (Fig. 1.8b). Observe that whatever happens in Fig. 1.8a at some instant t
also happens in Fig. 1.8b at the instant −t, and vice versa. Therefore,
𝜑 𝑡 = 𝑥(−𝑡)
Thus, to time-reverse a signal we replace t with −𝑡, and the time reversal of signal
𝑥(𝑡) results in a signal 𝑥(−𝑡). We must remember that the reversal is performed
about the vertical axis, which acts as an anchor or a hinge. Recall also that the reversal
of 𝑥(𝑡) about the horizontal axis results in −𝑥(𝑡).
Tarea 3