(a) Sketch the Bode plots for the following frequency responses: (i) 1 + (jw/10) (ii) 1

QUESTION:

(a) Sketch the Bode plots for the following frequency responses: (i) 1 + (jw/10) (ii) 1 - (jw/10) (iii) 16 (iv) 1-(jw/10) (Jw+2)4 l+jw (v) (jw/10)-1 (vi) I +(jw/10) I+Jw I+Jw (vii) 1-(jw/10) (viii) IO+Sjw+ IO(jw)2 (jw)2+(jw)+l I +(jw/10) (ix) 1 + jw + (jw)2 (x) 1- jw + (jw)2 ( ") (jw+ 10)(10jw+ I) XI [(jw/100+ I )][((jw )2 + jw +I)] (b) Determine and sketch the impulse response and the step response for the system with frequency response (iv). Do the same for the system with frequency response (vi). The system given in (iv) is often referred to as a non-minimum-phase system, while the system specified in (vi) is referred to as being a minimum phase. The corresponding impulse responses of (iv) and (vi) are referred to as a non -minimum-phase signal and a minimum-phase signal, respectively. By comparing the Bode plots of these two frequency responses, we can see that they have identical magnitudes; however, the magnitude of the phase of the system of (iv) is larger than for the system of (vi). We can also note differences in the time-domain behavior of the two systems. For example, the impulse response of the minimum-phase system has more of its energy concentrated near t = 0 than does the impulse response of the non-minimum-phase system. In addition, the step response of (iv) initially has the opposite sign from its asymptotic value as t ~ oo, while this is not the case for the system of (vi). The important concept of minimum- and non -minimum-phase systems can be extended to more general LTI systems than the simple first-order systems we have treated here, and the distinguishing characteristics of these systems can be described far more thoroughly than we have done.