Linear Signals and Systems “ Laplace Transform Analysis of Continuous-Time Systems”
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ECE303: Linear Signals and Systems HW3: Chapter Four
July 6, 2021 Due: July 20, 2021
(180 points)
1. [25] Find the inverse Laplace transform of
a. 2
2 4 3 ( ) 1
1 9 s
X s s s s
= + + − − +
b. 4
( ) ( 1)( 3)
X s s s
= + +
c. 2
3
2 4 1 ( )
( 1)( 2) s s
X s s s
+ + =
+ +
d. 2
1 ( )
( 3)( 2 5) s
X s s s s
+ =
+ + +
e. 2 3
2 ( )
( 5 4)
s sse e X s
s s s
− −+ =
+ +
Check your results using MATLAB … useful functions are ilaplace() and partfrac()
2. [10] If the output of a system is ���� = � �� cos�4�� ��� when an input, ���� = 10� �� ���, what is the transfer function of the system at hand?
3. [10] Suppose the transfer function of a system is given by 2
( ) 6
s H s
s =
+
Calculate its impulse response and the output yZS(t) if the input is ���� = � ��� ���.
4. [15] Use Laplace transform to solve the following differential equations (i.e., solve for y(t))
a. 2
2 3 4 ( )t
d y dy y e u t
dt dt −+ − = with y(0) = 1 and
(0) 0
dy
dt =
b. 2
2 6 25 2
d y dy dx y x
dt dt dt + + = + where x(t) = 25u(t) and ICs are y(0) = 1 and
(0) 1
dy
dt =
2
c. D2y(t)+Dy(t)+2y (t)= x(t) where x(t) = 2u(t) and ICs are zero
5. [15] Obtain the impulse response of the systems characterized by the following differential equations
a. 2
2
( ) ( ) 2 ( ) ( )
d y t dy t y t x t
dt dt + + =
b. ( ) ( )
5 ( ) ( ) dy t dx t
y t x t dt dt
+ = +
c. 2 2
2 2
( ) ( ) ( ) 2 ( )
d y t dy t d x t y t
dt dt dt + + =
6. [35] Investigate the stability of the following systems (it is recommended
you check your results using Matlab’s pzmap.m)
a. 2
2 ( )
( 5 6) s
H s s s
+ =
− −
b. 2
2 ( )
( 9) s
H s s
+ =
+
c. 2
2
6 ( )
( 5 4) s
H s s s
+ =
+ +
d. ( ) sin(4 )th t e t−= for the case where t ≥0
e. 2
2 3 2
d y dy dx y x
dt dt dt − + = −
f. 2
2 3 2
d y dy dx y x
dt dt dt + + = +
g. 2
2 2
d y dy dx y x
dt dt dt + − = −
7. [30] A linear system is represented by the following transfer function
2 5 ( )
( 2)( 3) s
H s s s
+ =
+ +
Assume zero initial conditions a. Derive its impulse response and plot it in MATLAB. Use tf() and
impulse() to compute the impulse response using MATLAB. Compare the two waveforms by plotting them in the same figure window.
b. Repeat part (a) if input is unit step function. Use tf() and step() to get the step response in MATLAB.
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c. Derive and plot output y(t) if input is ���� = � ��� ���. Here, use conv() in the time domain to compare your results
d. Repeat part (c) if input is a sinusoid of amplitude 1Volts and frequency 3Hz. Here, use gensig() to generate the input signal and lsim() to excite the system.
e. Provide a pole-zero plot in the s-plane. Use pzmap() f. Provide a bode plot. Use bode(). Let the frequency axis be in Hz and
the magnitude be absolute (Hint look into the properties of the obtained figure).
8. [10] For the circuit shown, find y(t) using Laplace transform techniques. Assume the initial conditions to be:
( ) ( ) ( )1 2 20 1V, 0 3V, 0 1mAc c Lv v i− = − = − − =
9. [10] For the circuit below use Laplace to find its transfer function, H(s).
Let C4 = 0.3333 mF and C3 = 2C4, R1 = R2 = 2kΩ, R5 = 1kΩ.
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10. [20] Circuit Analysis using Laplace Transform a. Refer to the circuit shown below. Use Laplace to solve for y(t) if
2
1 ( )
( 1) F s
s =
+
And the output of interest is across the 4 Ohms resistor (labeled R3 below). Assume zero initial conditions.
b. For the circuit shown below with f(t) = 12u(t) Volts and ICs are ����0� = 4 ����, and ����0� = 2 ����, Use Laplace transforms to solve for {y1(t), y2(t)} and the voltage across resistor R3.
11. [10] For the circuit shown below, find
a. The steady state voltage across the resistor, R2 b. The power dissipated by the resistor, R2