1. Continuous Time Signals
Basic Signals – Singularity Functions
Transformations of Continuous Time Signals
Signal Characteristics
Common Signals

2. Continuous-Time Signals
Assumptions:
Functions, x(t), are of the one independent variable that typically represents time, t.
Time t can assume all real values: -∞ < t < ∞,
Function x(t) is typically a real function.
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3. Singularity Functions
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4. Unit Step Function Unit step function definition: 20 April 2017
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5. Unit Step Function Properties
Scaling:
Unit step function can be scaled by a real constant K (positive or negative)
Multiplication:
Multiplication of any function, say x(t), by a unit step function u(t) is equivalent to defining the signal x(t) for t≥0.
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6. Unit Ramp Function Unit Ramp Function is defined as: 20 April 2017
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7. Unit Ramp Function Properties
Scaling:
Unit step function can be scaled by a real constant K (positive or negative)
Integral of the unit step function is equal to the ramp function:
Derivative of the unit ramp function is the unit step function.
Slope of the straight line
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8. Unit Impulse Function
Unit Impulse Function, also know as Dirac delta function, is defined as:
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9. Unit Impulse Function Properties
Scaling:
Unit impulse function can be scaled by a real constant K (positive or negative)
Delta function can be approximated by a pulse centered at the origin
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10. Unit Impulse Function Properties
Unit impulse function is related to unit step function:
Conversely:
Proof:
t<0
t>0
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11. Time Transformation of Signals
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12. Time Reversal:
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13. Time Scaling |a| > 1 – Speed Up |a| < 1 – Slow Down
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14. Time Shifting
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15. Example 1
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16. Independent Variable Transformations
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17. Example 2
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18. Example 3
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19. Independent Variable Transformations
Replace t with , on the original plot of the signal.
Given the time transformation:
Solve for
Draw the transformed t-axis directly below the -axis.
Plot y(t) on the t-axis.
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20. Amplitude Transformations
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21. Example 4
Consider signal in the figure. Suppose the signal is applied to an amplifier with the gain of 3 and introduces a bias (a DC value) of -1. That is:
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22. Example 5
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23. Transformations of Signals
Name
y(t)
Time reversal
x(-t)
Time scaling
x(at)
Time shifting
x(t-t0)
Amplitude reversal
-x(t)
Amplitude scaling
Ax(t)
Amplitude shifting
x(t)+B
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24. Signal Characteristics

25. Even and Odd Signals xe(t)=xe(-t) xo(t)=-xo(-t) Even Functions
Odd Functions
xo(t)=-xo(-t)
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26. Even and Odd Signals
Any signal can be expressed as the sum of even part and on odd part:
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27. Average Value
Average Value of the signal x(t) over a period of time [-T, T] is defined as:
The average value of a signal is contained in its even function (why?).
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28. Properties of even and odd functions
The sum of two even functions is even.
The sum of two odd functions is odd.
The sum of an even function and an odd function is neither even nor odd.
The product of two even functions is even.
The product of two odd functions is even.
The product of an even function and an odd function is odd.
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29. Periodic Signals
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30. Periodic Signals Continuous-time signal x(t) is periodic if:
T is period of the signal.
A signal that is not periodic is said to be aperiodic.
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31. Periodic Signals
If constant T is a period of of a function x(t) than nT is also its period, where T>0 and n is any positive integer.
The minimal value of the constant T >0 is a that satisfies the definition x(t)= x(t+ T) is called a fundamental period of a signal and it is denoted by T0.
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32. Examples of Periodic Signals
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33. Sinusoidal Signal Properties
A – Amplitude of the signal
 - is the frequency in rad/sec
 - is phase in radians
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34. Sinusoidal Function Properties
Note:
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35. Periodicity of Sinusoidal Signal
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36. Example: Sawtooth Periodic Waveform
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37. Period and Frequency Fundamental Period T0 – Measured in seconds.
Fundamental Frequency f0 – Measured in Hz – number of periods (cycles) per second or equivalently in radian frequency rad/s.
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38. Testing for Periodicity
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39. Testing for Periodicity
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40. Composite Signals
Each signal can be decomposed into a sum of series of pure periodic signals (Taylor Series Expansion/Fourier Series Expansion)
The sum of continuous-time periodic signals is periodic if and only if the ratios of the periods of the individual signals are ratios of integers.
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41. Composite Signals
If a sum of N periodic signals is periodic, the fundamental period can be found as follows:
Convert each period ratio, To1/Toi≤ i ≤ N , to a ratio of integers, where To1 is the period of the first signal considered and Toi is the period of one of the other N-1 signals. If one or more of these ratios is not rational, the sum of signals is not periodic.
Eliminate common factors from the numerator and denominator of each ratio of integers.
The fundamental period of the sum of signals is To=koTo1 ; kois the least common multiple of the denominators of the individual ratios of integers.
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42. Composite Signals If x1(t) is periodic with period T1, and
Then
x1(t)+x2(t) is periodic with period equal to the least common multiple (T1, T2) if the ratio of the two periods is a rational number, where k1 and k2 are integers:
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43. Composite Signals Let T’= k1T1 = k2T2 y(t) = x1(t)+x2(t) Then
y(t+T’) = x1(t+T’)+x2(t+T’)= x1(t+ k1T1)+x2(t+ k2T2)= x1(t)+x2(t) = y(t)
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44. Example 2.7 a)
Assume that v(t) is a sum of periodic signals given below. Determine if the signal is periodic and what its periodicity?
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45. Solution
Determine whether v(t) constituent signals have periods with ratios that are integers (rational numbers):
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46. Solution
Ratios of periods are rational numbers thus the composite signal v(t) is periodic.
Elimination of common factors:
T01/T02 = 4/7
T01/T03 = 7/21=1/3
Least common multiple of the denominator ratios:
n1= 3*7=21
Fundamental period of v(t) is:
T0= n1 T01 = 21*2/3.5=12
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47. Example 2.7 b)
Assume that to v(t) is added a periodic signal x4(t) given below. Determine if the signal is periodic and what its periodicity?
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48. Solution
Since ratio of the x1(t) and x4(t) periods is not a rational number the v(t) is not periodic.
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49. Homework #1: For x(t)=Acos(t+) find
What are its maximum and minimum values? What are corresponding times when they occur?
What is the value of the function when it crosses vertical y- axis (ordinate) and horizontal x-axis (abscissa)?
At what time instances the function becomes zero?
Indicate all the above point values in a plot.
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50. Homework #1
Use the following MATLAB script to test your calculations and plot the function:
function pfunc(A, f, th1, th2) %
% Periodic Sine Function
% A - gain (1)
% f - frequency (1)
% th1 - phase of the first signal (0)
% th2 - phase of the second signal (pi/6)
w = 2.*pi.*f; % radial frequency
fs = *f;
mint = -pi*f/2;
maxt = pi*f/2;
miny = -1.2*A;
maxy = 1.2*A;
t = mint:fs:maxt; % time axis
y = A*cos(w*t+th1);
plot(t, y, 'b', 'LineWidth',2); title('Periodic Signal');
grid on; hold; axis([mint maxt miny maxy]);
y = A*cos(w*t+th2);
plot(t, y, 'r', 'LineWidth',2);
ylabel('cos(\omegat+\theta)'); xlabel('Angle x\pi [rad]');
x=-0.8;
text(x,A*cos(w*x+th1),sprintf('%s+%3.2f)','\leftarrow cos(-\pit',th1),...
'HorizontalAlignment','left',...
'BackgroundColor','b');
x=-0.6;
text(x,A*cos(w*x+th2),sprintf('%s+%3.2f)','\leftarrow cos(-\pit',th2),...
'BackgroundColor','r');
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51. Homework #1
Problems 2.1, 2.2, 2.9, 2.10, 2.13, 2.14, 2.20.
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52. Example
Consider the signal to the left and its time reversed version. The signal is decomposed into its even and odd functions:
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53. Common Signals in Engineering
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54. Common Signals
Continuous-time physical systems are typically modeled with ordinary linear differential equations with constant coefficients.
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55. Exponential Signals Useful Complex Exponential Relations 20 April 2017
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56. Euler’s Formula
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57. Example of Exponential Functions
C and a real, x(t)=Ceat
a=s, s>0 - Increasing Exponential:
Chemical Reactions, Uninhibited growth of bacteria, human population?
a=s, s<0 - Decaying Exponential:
Radioactive decay, response of an RC circuit, damped mechanical system.
a=s, s=0 - Constant (DC) signal.
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58. Time Constant of the Exponential Function
The constant parameter t is called the time constant in of the exponential function presented below.
To relate to the time constant the following is necessary:

59. Example of Exponential Functions
C complex, a imaginary, x(t)=Ceat
a=jw, s=0; C=Aejf – A and f are real:
For C – real (f=0)
x(t) is periodic:
Why x(t) is periodic?
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60. Periodicity of Complex Exponential
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61. Example of Complex Exponentials
C complex, a complex, x(t)=Ceat
C=Aejf , A and f are real;
a=s+jw, s, w are also real.
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62. Time-Shifted Signals Time-Shifted Impulse Function Shift d(t) by t0:
“Sifting” Property of the Impulse Function:
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63. Examples of Impulse Functions
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64. Properties of the Unit Impulse Function
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65. Shifted Unit Step Function
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66. Continuous & Piece-wise Continuous Functions
Rectangular Function
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67. Rectangular Function
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68. Triangular Pulse Function
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69. Straight Line Equation
[x0,y0] - a point on the line
[x1,y1] - a point on the line
[x2,y2] - a point on the line
m = (y2-y1)/(x2-x1) - slope
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70. Composite Signal from Straight Lines
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71. Example of Composite Signal
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