Question

PROBLEM 7.2: x[n] 0.4 0.81 A system is shown in signal flow-graph form above; it represents the cascade of a 2nd order system and a 1st order system. (a) Give the overall system response, H(). (b) Is the overall system minimum phase? Explain briefly, (c) Is the overall system stable? Explain briefly. (d) Draw the signal flow graph of this system in direct-form-Il-transposed form.

Answer 1

a).

> > > > >oyo) . Y.o.8 0.4 0.81 A system is shown in signal flow graph from above.it represents the cascade of a 2nd order system and a 1st orde system. @ Give the overall system response HCZ) Solution By Marson's gain formula we have foor forward paths and 3 individual loops HCZ) = 1 + 2 Z+0.812241.6223 1-(oge'+0.422 0 87') +(032'xo8€!)+10:42 082) = 1 +92 +0-812+1.6273 1- 0.64 2² to 571 -0.3223 HCZ) 73+22? +0817 +1.62 z²+0.52² - 0.64 2 - 0:32

b).

6 is the overall system minimum phase ? Explain briefiy ? Sorotan A system is called minimum phase of the all poles and zero's lies inside the unit circle. H(2) = 2² +22² +0.812 +1.62 23+0.582-0642-0.32 poles at located at Z-0-8 Zg = -0.8 Zz = -05 zeros are located at 2 = -2 Zz = 0.91 Zz = -ogi unit As we can see one zero is outside the circle ie 2,= -2. So the given system not minimum phase system cole Grieta

c).

© is the overall system stable ? Explain brighgl The system is stable if all the poles les inside the unit circe. we can see all poles ic 2 = 0.8, Z 2-0.8, 2q = -0.5 les inside the unit circe. So the given system is stable.

d).

Draw the signal flow graph of this system direct form - II transposed form, in Solution HCZ) = 23+az’+0.812 +1.62 52²_0.642 -0.32 xon) o gon) A &-057 10.817 1.62 0-32