VE216 Homework 6 Solved

1.       [5] Is this system stable? Explain. (Note: the system is causal.)

 

2.       [5] How many signals have a Laplace transform that may be experessed as

 

in its region of convergence?

3.       [5] Use geometric evaluation from the pole-zero plot to determine the magnitude of the Fourier transform of the signal whose Laplace transform is specified as

 

4.       [5] Consider two right-sided signals x(t) and y(t) related through the differential equations

 

and

 

Determine Y(S) and X(S), along with their regions of convergence.

5.       [10] A causal LTI system S with impluse response h(t) has its input x(t) and output y(t) related through a linear constant-coefficient differential equation of the form

 

(a)                    If

 

,

how many poles does G(s) have?

(b)                    For what real values of the parameter α is S guaranteed to be stable?

6.       [10] Draw a direct-form representation for the causal LTI systems with the following system fuctions:

(a)                     (b)

 

(c)

 

7.       [10] A causal LTI system with impulse response h(t) has the following properties:1.When the input to the system is x(t) = e2t for all t, the output is  for all t. 2.The impulse respose h(t) satisfies the differential equation  ), where b is an unkhown constant.

Determine the system function H(s) of the system, consistent with information above. There should be no unkhowm constants in your answer, that is, the constant b should not appear in the answer.

8.       [10] A unit step signal is applied to a system consisting of two LTI system connected in parallel. The pole-zero plots of each of the system are shown below. Determine the output signal. Assume that each of the system has unit gain at DC.

 

Hint: first find the Laplace transform Y (s) of the output signal using the convolution and linearity properties of the Laplace transform, Then take the inverse Laplace transform to get y(t) using PFE. The “unit gain at DC” specifies H1(0) and H2(0), which you can use to determine the scaling factor.

9.       [10] Consider an LTI system with input x(t) = e−tu(t) and impulse response h(t) = e−2tu(t).

(a)                    Determine the Laplace transform of x(t) and h(t).

(b)                    Using the convolution property, determine the Laplace transform Y (s) of the output y(t).

(c)                     From the Laplace transform of y(t) as obtained in part(b), determine y(t).

(d)                    Verify your result in part (c) by explicitly convolving x(t) and h(t).

10.   [10] The system function of a causal LTI system is

 

Determine and sketch the response y(t) when the input is

 

11.   [20] In this problem, we consider the construction of various type of block diagram representations for a causal LTI system S with input x(t), output y(t), and system function

 

To derive the direct-diagram representation of S, we first consider a causal LTI system S1 that has the same input x(t) as S, but whose system function is

 

With the output of S1 denoted by y1(t), the direct-form diagram representation of S1 is shown in Figure

1. The signals e(t) and f(t) indicates in the figure represent respective inputs into the two integrators.

(a)    Express y(t) (the output of S) as a linear combination of y1(t), dy1(t)/dt, and d2y1(t)/dt2.

(b)    How is dy1(t)/dt related to f(t).

(c)    How is d2y1(t)/dt2 related to e(t).

(d)   Express y(t) as a linear combination of e(t), f(t), y1(t).

(e)    Use the result from the previous part to extend the direct-form block diagram representation of S1 and create a block diagram representation of S.

(f)     Observing that

 

draw a block diagram representation for S as a cascade combination of two subsystems.

(g)    Observing that

 

draw a block-diagram representation for S as parallel combination of three subsystems.