6.28. (a) Sketch the Bode plots for the following frequency responses: (ii) 1 - (jw/10) (iv) (vi) (i) (iii) (v) 1 + (jw/10) 16 (jw+2)* (jw/10)-1 1+jw (vii) (ix) (xi) 1-(jos/10)

1+ jw 1+{jw/10) 1+ jw 1-(jw/10) (jw)²+(jw)+1 1+ jw + (jw)² (jw+10)(10jw+1) [Cjw/100+1)][(jar)² + jw+1)] (b) Determine and sketch the impulse response and the step response for the sys- tem with frequency response (iv). Do the same for the system with frequency response (vi). (viii) (x) 10+5jw+10(jw)² 1+(jw/10) 1- jw + (jw)² 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 sys- tems. 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 → ∞, 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.