(A). According to Eq.(9.3), We Will Get

Chapter 9

9.2Solution:

(a). According to eq.(9.3), we will get

ROC:Re{s}>-5

(b)., Re{s}<-5

then it’s obviously that A=-1, , Re{s}<-5.

9.5 Solution:

(a). 1, 1

it has a zero in the finite s-plane, that is

And because the order of the denominator exceeds the order of the numerator by 1

X(s) has 1 zero at infinity.

(b). 0, 1

it has no zero in the finite s-plane.

And because the order of the denominator exceeds the order of the numerator by 1

X(s) has 1 zero at infinity.

(c). 1, 0

it has a zero in the finite s-plane, that is

And because the order of the denominator equals to the order of the numerator

X(s) has no zero at infinity.

9.7 Solution:

There are 4 poles in the expression, but only 3 of them have different real part.

The s-plane will be divided into 4 strips which parallel to the jw-axis and have no cut-across.

There are 4 signals having the same Laplace transform expression.

9.8 Solution:

ROC: R(x)+Re{2}

And x(t) have three possible ROC strips:

g(t) have three possible ROC strips:

Then the ROC of is (-1,1)

is two sides.

9.9 Solution:

It is obtained from the partial-fractional expansion:

We can get the inverse Laplace transform from given formula and linear property.

9.10 Solution:

(a).

It’s lowpass.

(b).

It’s bandpass.

(c).

It’s highpass.

9.13 Solution:

,and

The Laplace transform : and ,

From the scale property of Laplace transform, ,

So ,

From given ,

We can determine :

9.21 Solution:

(a).

(b).

(i).

(f).

9.22 Solution:

(a). ,

(b). ,

So ,

From the property of shifting in the s-domain,we can get

and

9.28. Solution:

(a). All possible ROCs:

(b). It’s obviouse to see:

unstable &uncausal

stable &uncausal

unstable &causal

9.31. Solution:

(a).

(b).1. The system is stable.

ROC:(-1,2)

2. The system is causal.

ROC:

3. The system is neither stable nor causal

ROC:

9.32.Solution:

from (1)

, for all tand x(t) is a eigen function

from (2)

when ,

9.33. Solution:

9.35. Solution:

According to the block-diagram, we will know

(a), Re{s}>-1

(b) It’s obviouse that this system is stable.

9.37. Solution:

(a)

9.45. Solution:

,-1<Re{s}<2

,Re{s}<2

, Re{s}>-1

9.60. Solution:

(a)

Re{s}:

(b)

the zeros of

If , then . But couldn’t reach infinite in finite s-plane. So there are no poles.

(c). (d) are as follow:

Extra problems:

X(s)?

Solution:

,Re[s]>0