(2+ja)(1+ja) .6.28. (a) Sketch the Bode plots for the following frequency responses: (1) 1 + (joo/10) (ii) 1-(jw/10) (iii) 16 (iv) (10) 1+ ja 1+0/10) w/10-1 (v) (vi) 1-(10) (ja)²+(jo)+1 1

+ jw + (jo)² j+1010+1) C100+1+ jau + 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). (vii) (ix) (xi) (viii) 10+5+10) 1+(ja/10) (x) 1-jw + (jw)? 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 →, 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.