The document describes MATLAB exercises for a circuits, signals and systems workshop. It includes 9 exercises that students will carry out over 10 weeks exploring topics like Fourier series analysis and synthesis of waveforms, filter circuits, and more. Exercise 2 focuses on using Fourier series to progressively synthesize a square wave by adding weighted harmonics. Exercise 3 similarly synthesizes a triangular wave. Exercise 4 analyzes the Fourier series of a half-wave rectified sine waveform.

1. AMIR BAGHDADI
Student Number 2802166
Second Year
CIRCUITS, SIGNALS AND SYSTEMS WORKSHOP
MATLAB and Simulink Exercises during a Computer Workshop – 1 hour per week for
the first 10 weeks. You will carry out the following nine exercises:
Exercise 1: Simulink Model to display a sinusoidal signal on an oscilloscope. Investigate
the effect of changing its amplitude, frequency, dc bias, and phase. Also displays the
RMS
value of the signal.
Exercise 2: Progressive synthesis of a bipolar square wave from sinusoidal signals in
the
Trigonometric Fourier Series of this waveform. Shows plots of the fundamental
frequency
alone, progressively adds the third, fifth, seventh harmonics weighted with coefficients
up
until the 19
th
harmonic. Finishes with a 3-D surface representing the gradual
transformation of a sine wave into a square wave.
Exercise 3: Progressive synthesis of a unipolar triangular wave from sinusoidal signals
in
the Trigonometric Fourier Series of this waveform. Shows a plot of the DC offset
followed
by the fundamental frequency added to it, progressively adds the third, fifth, seventh
harmonics weighted with coefficients up until the 19
th
harmonic. Finishes with a 3-D
surface representing the gradual transformation of a sine wave into a triangular wave.
Exercise 4: Progressive synthesis of a half-wave rectified sine waveform from sinusoidal
signals in the Trigonometric Fourier Series of this waveform. Shows a plot of the DC
offset followed by the fundamental frequency added to it, progressively adds the
second,
fourth, sixth harmonics weighted with coefficients up until the 8
th
harmonic.
Exercise 5: Find the Trigonometric Fourier series of a sawtooth waveform and
synthesize
it by writing a MATLAB program to verify the solution.
Exercise 6: Find the Discrete Fourier Transform (DFT) of a signal f(t) by using the Fast
Fourier Transform (FFT) algorithm in MATLAB. With exercises 6.1: Find the frequencies
in a noiseless (deterministic) signal y(t); and 6.2 Find the frequencies in a noisy signal
y(t)
Exercise 7: Find the power frequency spectrum of three signals with exercises 7.1, 7.2
and
7.3.
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2. AMIR BAGHDADI
Student Number 2802166
Second Year
Exercise 8: Simulink prediction of the unit step response of some first and second order
circuits. Model the circuit with a transfer function and analytically predict its time
response. Verify with a MATLAB/SIMULINK simulation.
Exercise 9: MATLAB prediction of the frequency response of filter circuits. Sketch Bode
plots of given circuit transfer functions. Compare you sketch with a MATLAB Bode plot.
Predict the frequency response of Low pass, High pass, Band pass, Band stop, Phase
Advance and Phase Lag circuits.
EXERCISE 2: FOURIER SERIES OF SQUARE WAVE
Aim: To show that the Fourier series expansion for a square-wave is made up of a
sum of odd harmonics.
You will show this graphically using MATLAB. Type in step one only to see what it does
(there is no need for the pause at his stage. A sine wave will be displayed.)
1. Start by forming a time vector running from 0 to 10 in steps of 0.1, and take the
sine of all the points. Let's plot this fundamental frequency.
t = 0:.1:10; (%creates a vector of values from zero to ten in steps of 0.1%)
y = sin(t); (%creates a vector y with values that are the sine of the values in the
vector t%)
plot(t,y); (% plots the vector t against the vector y%)
pause; (% the pause is overcome by pressing any button%)
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3. AMIR BAGHDADI
Student Number 2802166
Second Year
3

4. AMIR BAGHDADI
Student Number 2802166
Second Year
2. Now add the third harmonic weighted by a third to the fundamental, and plot it.
y = sin(t) + sin(3*t)/3; (%creates a vector y%)
plot(t,y); (% plot the time vector t against the vector y %)
pause;
4

5. AMIR BAGHDADI
Student Number 2802166
Second Year
5

6. AMIR BAGHDADI
Student Number 2802166
Second Year
3. Now add the first, third, fifth, seventh, and ninth harmonics each weighted
respectively by a third, a fifth, a seventh, and a ninth)
y = sin(t) + sin(3*t)/3 + sin(5*t)/5 + sin(7*t)/7 + sin(9*t)/9;
plot(t,y);
pause;
6

7. AMIR BAGHDADI
Student Number 2802166
Second Year
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8. AMIR BAGHDADI
Student Number 2802166
Second Year
4. For a finale, we will go from the fundamental to the 19th harmonic, creating
vectors of successively more harmonics, and saving all intermediate steps as the
rows of a matrix. These vectors are plotted on the same figure to show the
evolution of the square wave.
t = 0:.02:3.14; (% create a vector with values from zero Pi %)
y = zeros(10, length(t));
x = zeros(size(t));
for k=1:2:19
x = x + sin(k*t)/k;
y((k+1)/2,:) = x;
end
plot(y(1:2:9,:)')
title('The building of a square wave')
pause;
5. Here is a 3-D surface representing the gradual transformation of a sine wave into a
square wave.
surf(y);
8

9. AMIR BAGHDADI
Student Number 2802166
Second Year
shading interp
axis off ij
9

10. AMIR BAGHDADI
Student Number 2802166
Second Year
EXERCISE 3: FOURIER SERIES OF TRIANGULAR WAVE
Aim: To show that the Fourier series expansion for a triangular-wave shown in
figure 2 is made up of a sum of odd harmonics and a constant term.
You will show this graphically using MATLAB.
Then start by forming a time vector running from 0 to 10 in steps of 0.1.
1. t = [0:0.1:10];
2. Lets plot the DC term V/2
t =[0:0.1:10];
y =5;
figure;
plot(t,y);
title('The constant term = 5 V')
10

11. AMIR BAGHDADI
Student Number 2802166
Second Year
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12. AMIR BAGHDADI
Student Number 2802166
Second Year
12

13. AMIR BAGHDADI
Student Number 2802166
Second Year
The constant term=5V
6
5.8
5.6
5.4
5.2
5
4.8
4.6
4.4
4.2
4
0 1 2 3 4 5 6 7 8 9 10
3. Next plot the DC term plus the fundamental
y = 5 + (40/(pi*pi))*cos(t);
figure;
plot(t,y);
title('The constant term of 5 V plus fundamental frequency')
13

14. AMIR BAGHDADI
Student Number 2802166
Second Year
14

15. AMIR BAGHDADI
Student Number 2802166
Second Year
15

16. AMIR BAGHDADI
Student Number 2802166
Second Year
The constant term of 5V plus fundamental frequency
10
9
8
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 8 9 10
4. Now add the constant 5, the fundamental and third harmonic, and plot it.
y = 5+ (40/(pi*pi))*cos(t) + (40/((3*pi)*(3*pi)))*cos(3*t);
figure;
plot(t,y);
title('The constant term plus fundamental frequency plus thirdharmonic')
16

17. AMIR BAGHDADI
Student Number 2802166
Second Year
17

18. AMIR BAGHDADI
Student Number 2802166
Second Year
18

19. AMIR BAGHDADI
Student Number 2802166
Second Year
The constant term fundamental frequency plus third harmonic
10
9
8
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 8 9 10
5. Plot the sum of constant , fundamental, 3
rd
and 5
th
harmonics
y = 5+ (40/(pi*pi))*cos(t) + (40/((3*pi)*(3*pi)))*cos(3*t) +
(40/((5*pi)*(5*pi)))*cos(5*t);
figure;
plot(t,y);
title('Constant term, fundamental, 3rd and 5th harmonics')
19

20. AMIR BAGHDADI
Student Number 2802166
Second Year
20

21. AMIR BAGHDADI
Student Number 2802166
Second Year
21

22. AMIR BAGHDADI
Student Number 2802166
Second Year
Constant term, fundamental, 3rd and 5th harmonics
10
9
8
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 8 9 10
6. Finally, plot individual terms in the series on the same graph up to the 19
th
harmonic
x = 5;
for k = 1:2:19
x = x + (40/((k*pi)*(k*pi)))*cos(k*t);
y((k+1)/2,:) = x;
end
figure;
plot(y(1:2:9,:)');
title('Constant term, fundamental, 3rd, 5th, 7th upto 19th harmonics');
22

23. AMIR BAGHDADI
Student Number 2802166
Second Year
23

24. AMIR BAGHDADI
Student Number 2802166
Second Year
24

25. AMIR BAGHDADI
Student Number 2802166
Second Year
Constant term, fundamental, 3rd, 5th, 7th upto 19th harmonics
10
9
8
7
6
5
4
3
2
1
0
0 20 40 60 80 100 120
7. Show the above as a 3-D surface representing the gradual transformation of cosine
waves into a triangular wave. Note that the coefficients decrease as 1/n
surf(y);
shading interp
axis off ij
25

26. AMIR BAGHDADI
Student Number 2802166
Second Year
26

27. AMIR BAGHDADI
Student Number 2802166
Second Year
27

28. AMIR BAGHDADI
Student Number 2802166
Second Year
3D Surface of a triangular wave
EXERCISE 4: Fourier Series Of Half Rectified Sine Wave
Aim: To show that the Fourier series expansion for a half-wave rectified sine
waveform is made up of a sum a constant term and even harmonics
of sine and cosine waves.
You will show this graphically using MATLAB.
t = [0:0.1:10];
y =10/pi;
figure;
plot(t,y);
title('The constant term ')
28

29. AMIR BAGHDADI
Student Number 2802166
Second Year
29

30. AMIR BAGHDADI
Student Number 2802166
Second Year
y = (10/pi)*(1 + (pi/2))*sin(t);
figure;
plot(t,y);
title('The constant term plus fundamental frequency')
30

31. AMIR BAGHDADI
Student Number 2802166
Second Year
31

32. AMIR BAGHDADI
Student Number 2802166
Second Year
y = (10/pi)*(1 + (pi/2)*sin(t)-(2/3)*cos(2*t));
figure;
plot(t,y);
title('The constant term plus fundamental frequency plus thirdharmonic')
32

33. AMIR BAGHDADI
Student Number 2802166
Second Year
33

34. AMIR BAGHDADI
Student Number 2802166
Second Year
y = (10/pi)*(1 + (pi/2)*sin(t)-(2/3)*cos(2*t)-(2/15)*cos(4*t));
figure;
plot(t,y);
title('Constant term, fundamental, 2nd and 4th harmonics')
34

35. AMIR BAGHDADI
Student Number 2802166
Second Year
35

36. AMIR BAGHDADI
Student Number 2802166
Second Year
y = (10/pi)*(1 + (pi/2)*sin(t)-(2/3)*cos(2*t)-(2/15)*cos(4*t)-
(2/35)*cos(6*t));
figure;
plot(t,y);
title('Constant term, fundamental, 2nd and 4th harmonics')
36

37. AMIR BAGHDADI
Student Number 2802166
Second Year
37

38. AMIR BAGHDADI
Student Number 2802166
Second Year
EXERCISE 5: Exercise in solving Trigonometric Fourier Series
Show that the Trigonometric Fourier series of the sawtooth wave
38

39. AMIR BAGHDADI
Student Number 2802166
Second Year
The constant term
7.5
7
6.5
6
5.5
5
0 1 2 3 4 5 6 7 8 9 10
39

40. AMIR BAGHDADI
Student Number 2802166
Second Year
The constant term plus fundamental frequency
8
6
4
2
0
-2
-4
-6
-8
0 1 2 3 4 5 6 7 8 9 10
40

41. AMIR BAGHDADI
Student Number 2802166
Second Year
The constant term, fundamental frequency and third harmonic
10
8
6
4
2
0
-2
-4
-6
-8
-10
0 1 2 3 4 5 6 7 8 9 10
41

42. AMIR BAGHDADI
Student Number 2802166
Second Year
The constant term, fundamental frequency, 2nd harmonic and 3rd harmonic
10
8
6
4
2
0
-2
-4
-6
-8
-10
0 1 2 3 4 5 6 7 8 9 10
42

43. AMIR BAGHDADI
Student Number 2802166
Second Year
The constant term, fundamental frequency, 2nd, 3rd and 4th harmonics
10
8
6
4
2
0
-2
-4
-6
-8
-10
0 1 2 3 4 5 6 7 8 9 10
EXERCISE 6: Find the Discrete Fourier Transform (DFT) of a signal f(t) by using
the Fast Fourier Transform (FFT) algorithm in MATLAB
Fourier analysis is extremely useful for data analysis, as it breaks down a signal into
constituent sinusoids of different frequencies.
1. For analogue (continuous-time) signals, the Fourier Transform of a signal f(t) is
dt e t f G
t jw -
+8
8-
∫
= ) ( ) (ω
2. For sampled vector data, Fourier analysis is performed using the Discrete Fourier
transform (DFT).
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Continuous-time Signal
Sampled signal
(Sampling period = 1 mS)
Figure 1: Sample and Hold circuit (ADC)
The fast Fourier transform (FFT) is an efficient algorithm for computing the DFT of a
43

44. AMIR BAGHDADI
Student Number 2802166
Second Year
sequence; it is not a separate transform. It is particularly useful in areas such as signal
and
image processing, where its uses range from filtering, convolution, and frequency
analysis
to power spectrum estimation.
The following exercises (6.1, 6.2, 7.1, 7.2 and 7.3) will investigate the Discrete Fourier
Transform and show you that a real continuous-time signal y(t) can be sampled (e.g.
with
an analogue to digital converter ADC as shown in figure 1) and the vector of discrete-
time
data ‘y’ fed to the MATLAB algorithm fft(y) which will compute the Fourier Transform
and plot the frequency spectrum.
•The signal y(t) is sampled with a constant sampling period.
•The sampled data is input to MATLAB by putting the data in a vector ‘y’.
•The FFT algorithm operates on the data vector y and returns the frequency
spectrum in a vector Y where Y = fft(y, N). The elements of Y are complex
numbers to accommodate the fact that the spectrum has both a modulus and phase
at each frequency. N is the number of points over which the FFT will be computed
•To present the results, each element of Y is multiplied by its complex conjugate to
give a real value and this value is normalized by dividing it by the number of
points N in the FFT algorithm.
Yy = Y.* conj(Y)/N
•The final result is plotted to show the power at each contributing frequency
44

45. AMIR BAGHDADI
Student Number 2802166
Second Year
Deterministic signal with two frequency components at 50Hz and 120 Hz
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
0 5 10 15 20 25 30 35 40 45 50
time (milliseconds)
45

46. AMIR BAGHDADI
Student Number 2802166
Second Year
Frequency content of y
80
70
60
50
40
30
20
10
0
0 50 100 150 200 250 300 350 400 450 500
frequency (Hz)
46

47. AMIR BAGHDADI
Student Number 2802166
Second Year
Noisy signal with two frequency components
8
6
4
2
0
-2
-4
-6
-8
0 5 10 15 20 25 30 35 40 45 50
time (milliseconds)
47

48. AMIR BAGHDADI
Student Number 2802166
Second Year
Frequency content of y
120
100
80
60
40
20
0
0 50 100 150 200 250 300 350 400 450 500
frequency (Hz)
A. Exercise 6.1 Find the frequencies in a noiseless (deterministic) signal y(t).
Create the M-file FFT_deterministic.m, or cut and paste from Blackboard.
clear all;
t = 0:0.001:0.6;
y = sin(2*pi*50*t) + sin(2*pi*120*t) %+ sin(2*pi*200*t)
a = 1000*t(1:50);
b = y(1:50);
figure; plot(a,b);
48

49. AMIR BAGHDADI
Student Number 2802166
Second Year
49

50. AMIR BAGHDADI
Student Number 2802166
Second Year
clear all;
t = 0:0.001:0.6;
y = sin(2*pi*50*t) + sin(2*pi*120*t) %+ sin(2*pi*200*t)
a = 1000*t(1:50);
b = y(1:50);
figure; plot(a,b);
%figure; plot(t,y)
Y = fft(y,512);
Pyy = Y.*conj(Y)/512;
50

51. AMIR BAGHDADI
Student Number 2802166
Second Year
f = 1000*(0:256)/512;
figure; plot(f,Pyy(1:257))
51

52. AMIR BAGHDADI
Student Number 2802166
Second Year
B. Exercise 6.2 Find the frequencies in a noisy signal y(t). Use the M-file
FFT_noisy.m
clear all;
t = 0:0.001:0.6;
x = sin(2*pi*50*t) + sin(2*pi*120*t); % + sin(2*pi*200*t)
n = 1.2*randn(size(t));
y = x + n;
a = 1000*t(1:50);
b = y(1:50);
figure; plot(a,b);
52

53. AMIR BAGHDADI
Student Number 2802166
Second Year
53

54. AMIR BAGHDADI
Student Number 2802166
Second Year
clear all;
t = 0:0.001:0.6;
x = sin(2*pi*50*t) + sin(2*pi*120*t); % + sin(2*pi*200*t)
n = 1.2*randn(size(t));
y = x + n;
a = 1000*t(1:50);
b = y(1:50);
figure; plot(a,b);
%figure; plot(t,y)
Y = fft(y,512);
54

55. AMIR BAGHDADI
Student Number 2802166
Second Year
Pyy = Y.*conj(Y)/512;
f = 1000*(0:256)/512;
figure; plot(f,Pyy(1:257))
55

56. AMIR BAGHDADI
Student Number 2802166
Second Year
56

57. AMIR BAGHDADI
Student Number 2802166
Second Year
57

58. AMIR BAGHDADI
Student Number 2802166
Second Year
EXERCISE 7.1 Find the frequency spectrum of a periodic rectangular signal whose
amplitude is ± 1 and the fundamental frequency is 50 Hz.
You will find the spectrum by first creating a rectangular signal from its Fourier series
and
then applying the Fast Fourier transform (FFT) algorithm to this signal to find the
frequencies contained in it. Recall that the Fourier series of this rectangular pulse was
used
to synthesize the signal in Exercise 2.
clear all,
= 0:0.001:6;
= zeros(size(t));
for k=1:2:19
x = x + sin(2*pi*50*k*t)/k;
end
y = x;
figure; plot(1000*t(1:50),y(1:50))
58

59. AMIR BAGHDADI
Student Number 2802166
Second Year
59

60. AMIR BAGHDADI
Student Number 2802166
Second Year
clear all,
t = 0:0.001:6;
x = zeros(size(t));
for k=1:2:19
x = x + sin(2*pi*50*k*t)/k;
end
y = x;
figure;
plot(1000*t(1:50),y(1:50))
title('Square Wave Signal')
xlabel('time (milliseconds)')
pause
60

61. AMIR BAGHDADI
Student Number 2802166
Second Year
Squarewave Signal
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
0 5 10 15 20 25 30 35 40 45 50
time (milliseconds)
Y = fft(y,512);
Pyy = Y.* conj(Y) / 512;
f = 1000*(0:356)/512;
figure;
plot(f,Pyy(1:357)) % this is a plot of the frequency versus signal power
61

62. AMIR BAGHDADI
Student Number 2802166
Second Year
62

63. AMIR BAGHDADI
Student Number 2802166
Second Year
Y = fft(y,512);
Pyy = Y.* conj(Y) / 512;
f = 1000*(0:356)/512;
figure;
plot(f,Pyy(1:357)) % this is a plot of the frequency versus signal power
title('Frequency content of y')
xlabel('Frequency (HZ)')
63

64. AMIR BAGHDADI
Student Number 2802166
Second Year
Frequency content of y
70
60
50
40
30
20
10
0
0 100 200 300 400 500 600 700
Frequency (Hz)
EXERCISE 7.2 Find the power frequency spectrum of a single shot (non-repeating)
rectangular pulse. Let amplitude A = 1 and pulse width = 1 second. Use the M-file
FFT_Pulse.m
t = 0:0.1:2;
y = [1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0];
figure;
plot(t(1:20),y(1:20))
64

65. AMIR BAGHDADI
Student Number 2802166
Second Year
65

66. AMIR BAGHDADI
Student Number 2802166
Second Year
title('Signal: Single shot rectangular pulse')
xlabel('time (seconds)')
66

67. AMIR BAGHDADI
Student Number 2802166
Second Year
Signal: Single shot rectangular pulse
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time (seconds)
Y = fft(y,512);
Pyy = Y.* conj(Y)/512;
f = 10*(0:256)/512;
figure;
plot(f,Pyy(1:257))
67

68. AMIR BAGHDADI
Student Number 2802166
Second Year
68

69. AMIR BAGHDADI
Student Number 2802166
Second Year
title('Frequency content of y')
xlabel('frequency (Hz)')
69

70. AMIR BAGHDADI
Student Number 2802166
Second Year
Frequency content of y
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
frequency (Hz)
EXERCISE 7.3 Find the power frequency spectrum of a single shot (non-repeating)
unit pulse (area 1). Let amplitude A = 10 to represent infinity and pulse width = 0
second. Use the M-file FFT_UnitPulse.m
t = 0:0.1:2;
y = [10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ];
figure;
plot(t(1:20),y(1:20))
70

71. AMIR BAGHDADI
Student Number 2802166
Second Year
71

72. AMIR BAGHDADI
Student Number 2802166
Second Year
title('Signal: Unit impulse')
xlabel('time (seconds)')
72

73. AMIR BAGHDADI
Student Number 2802166
Second Year
Signal: Unit Impulse
10
9
8
7
6
5
4
3
2
1
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time (seconds)
Y = fft(y,512);
Pyy = Y.* conj(Y)/512;
f = 10*(0:256)/512;
figure;
plot(f,Pyy(1:257))
73

74. AMIR BAGHDADI
Student Number 2802166
Second Year
74

75. AMIR BAGHDADI
Student Number 2802166
Second Year
title('Frequency content of y')
xlabel('frequency (Hz)')
75

76. AMIR BAGHDADI
Student Number 2802166
Second Year
Frequency content of y
1.5
1
0.5
0
-0.5
-1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
frequency (Hz)
EXERCISE 8: Find the Transfer Function of a circuit, find its step response and
frequency spectrum (using MATLAB) by carrying out steps 1 to 5:
1. Show that equation 1 is the differential equation describing the dynamics of the
following series RLC circuit
76

77. AMIR BAGHDADI
Student Number 2802166
Second Year
Bode Diagram
0
-20
Magnitude (dB)
-40
-60
-80
0
-45
Phase (deg)
-90
-135
-180
0 1 2 3 4
10 10 10 10 10
Frequency (rad/sec)
Step Response
1
0.9
0.8
0.7
0.6
Amplitude
0.5
0.4
0.3
0.2
0.1
0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Time (sec)
77

78. AMIR BAGHDADI
Student Number 2802166
Second Year
Bode Diagram
0
-20
Magnitude (dB)
-40
-60
-80
-100
0
-45
Phase (deg)
-90
-135
-180
0 1 2 3 4
10 10 10 10 10
Frequency (rad/sec)
78

79. AMIR BAGHDADI
Student Number 2802166
Second Year
Step Response
1
0.9
0.8
0.7
0.6
Amplitude
0.5
0.4
0.3
0.2
0.1
0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Time (sec)
79