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 π₯(π‘) (Fig. 1.4a) and the same signal delayed by π seconds (Fig. 1.4b), which we shall denote by π(π‘). Whatever happens in π₯(π‘) (Fig. 1.4a) at some instant π‘ also happens in π(π‘) (Fig. 1.4b) π seconds later at the instant π‘ + π. Therefore π π‘ + π = π₯(π‘) and π π‘ = π₯(π‘ − π) Therefore, to time-shift a signal by T, we replace t with π‘ − π . Thus π₯(π‘ − π) represents π₯(π‘) time-shifted by π seconds. If π is positive, the shift is to the right (delay), as in Fig. 1.4b. If π is negative, the shift is to the left (advance), as in Fig. 1.4c. Clearly, π₯(π‘ − 2) is π₯(π‘) delayed (right-shifted) by 2 seconds, and π₯(π‘ + 2) is π₯(π‘) advanced (leftshifted) by 2 seconds. Time Scaling The compression or expansion of a signal in time is known as π‘πππ π ππππππ. Consider the signal x(t) of Fig. 1.6a. The signal π(π‘) in Fig. 1.6b is π₯(π‘) compressed in time by a factor of 2. Therefore, whatever happens in π₯(π‘) at some instant π‘ also happens to π(π‘) at the instant π‘/2 so that π π‘ 2 = π₯(π‘) and π π‘ = π₯(2π‘) Observe that because π₯(π‘) = 0 at π‘ = π1 and π2 , we must have π(π‘) = 0 at π‘ = π1 2 and π2 2, as shown in Fig. 1.6b. If π₯(π‘) were recorded on a tape and played back at twice the normal recording speed, we would obtain π₯(2π‘). In general, if x(t) is compressed in time by a factor π(π > 1), the resulting signal π(π‘) is given by π π‘ = π₯(ππ‘) Using a similar argument, we can show that π₯(π‘) expanded (slowed down) in time by a factor π(π > 1) is given by π‘ π π‘ =π₯ π Figure 1.6c shows π₯(π‘/2), which is π₯(π‘) expanded in time by a factor of 2. Observe that in a time-scaling operation, the origin π‘ = 0 is the anchor point, which remains unchanged under the scaling operation because at π‘ = 0, π₯(π‘) = π₯(ππ‘) = π₯(0). In summary, to time-scale a signal by a factor a, we replace t with at. If π > 1, the scaling results in compression, and if π < 1, the scaling results in expansion. Time Reversal Consider the signal π₯(π‘) in Fig. 1.8a. We can view π₯(π‘) as a rigid wire frame hinged at the vertical axis. To time-reverse π₯(π‘), we rotate this frame 180β¦ about the vertical axis. This time reversal [the reflection of π₯(π‘) about the vertical axis] gives us the signal π(π‘) (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, π π‘ = π₯(−π‘) Thus, to time-reverse a signal we replace t with −π‘, and the time reversal of signal π₯(π‘) results in a signal π₯(−π‘). 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 π₯(π‘) about the horizontal axis results in −π₯(π‘). Tarea 3