1. Signals and Systems Analysis
NET 351
Instructor: Dr. Amer El-Khairy
د. عامر الخيري

2. Brief description of basic learning outcomes
Understand basic knowledge of signal analysis and processing
Deal with different domains and systems
Understand basic knowledge of sampling theory
Solve signal problems using correlation
Solve signal problems using Fourier, and Laplace Transforms
Apply signal analysis and processing on some applications
Use some analytical tools (e.g. MATLAB)
Signals and systems analysis د. عامر الخيري

3. Signals and systems analysis د. عامر الخيري
Course description
Introduction to the course content, text book(s), reference(s) and course plane.
Introduction to Signal Processing and its Applications.
Types of Signals & its Properties.
Singularity Functions.
Signals in the Time & Frequency Domains.
Continuous-Time Linear Time System.
Correlation & Convolution Theory.
Signals and systems analysis د. عامر الخيري

4. Course description (continued)
Fourier series.
Fourier Transform.
Fourier Applications.
Laplace Transform.
Inverse Laplace Transform.
Sampling Theory.
Signal Reconstruction.
Introduction to some Analytical Tools (e.g. MATLAB).
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5. Signals and systems analysis د. عامر الخيري
Assessment
Assessment weight (%)
Due week
The nature of the evaluation function (e.g.: article, quiz, group project, etc.)
index
20%
Week 7
First exam 1
Week 12
Second exam 2
10%
Weeks 5 & 10
Theoretical assignments 3
50%
After Week 15
Final exam 4
100%
Total:
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6. Textbook and References
Textbook: Oppenheim Willsky, and Nawab, "Signals and Systems", Printice-Hall, The Latest Edition.
Reference: Won Y. Yang, Tae G. Chang, Ik H. Song, "Signals and Systems with MATLAB", Springer-Verlag Berlin Heidelberg 2009.
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7. Recommended Reading Material
Signals and Systems, Oppenheim & Willsky
Signals and Systems, Haykin & Van Veen
Mastering Matlab 6
Mastering Simulink 4
Many other introductory sources available. Some background reading at the start of the course will pay dividends when things get more difficult.
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8. Signals and systems analysis د. عامر الخيري
What is a Signal?
A signal is a pattern of variation of some form
Signals are variables that carry information
Examples of signal include:
Electrical signals
Voltages and currents in a circuit
Acoustic signals
Acoustic pressure (sound) over time
Mechanical signals
Velocity of a car over time
Video signals
Intensity level of a pixel (camera, video) over time
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9. How is a Signal Represented?
Mathematically, signals are represented as a function of one or more independent variables.
For instance a black & white video signal intensity is dependent on x, y coordinates and time t f(x,y,t)
On this course, we shall be exclusively concerned with signals that are a function of a single variable: time t
f(t)
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10. Signals and systems analysis د. عامر الخيري
What a Signal is?
A signal is a mathematical representation that describes a physical phenomenon.
Examples
Speech signal  one-dimensional signal that descries the acoustic pressure variation as a function of time, t
Picture signal  two-dimensional signal that describes the gray level as a function of spatial coordinates, x and y.
Only one-dimensional signals are considered in this course. The independent variable is referred to as the time.
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11. Continuous Time (CT) and Discrete Time (DT) Signals
A signal is a continuous-time (CT) signal if it is defined for a continuum of values of the independent variable, t.
A signal is discrete-time (DT) if it is defined only at discrete times and the independent variable takes on only a discrete set of values
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12. Signals and systems analysis د. عامر الخيري
Analog vs. Digital
If a continuous-time signal x(l) can take on any value in the continuous interval (a, b), where a may be -  and b may be +, then the continuous-time signal x(t) is called an analog signal.
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13. Analog vs. Digital (continued)
Digital is a discrete or non-continuous waveform with examples such as computer 1s and 0s
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14. Analog vs. Digital (continued)
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15. Real and complex signals
A signal x(t) is a real signal if its value is a real number, and a signal x(t) is a complex signal if its value is a complex number. A general complex signal x(t) is a function of the form
x( t ) = x1( t ) + j x2( t )
where x1( t ) and x2( t ) are real signals and j=
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16. Deterministic and Random Signals
Deterministic signals are those signals whose values are completely specified for any given time. Thus, a deterministic signal can be modeled by a known function of time t.
Random signals are those signals that take random values at any given time and must be characterized statistically. Random signals will not be discussed in this text
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17. Signals and systems analysis د. عامر الخيري
Even and Odd Signals
A signal x(t) or x[n] is referred to as an even signal if
x(-t) = x(t)
x[-n] = x[n]
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18. Even and Odd Signals (continued)
A signal x(t) or x[n] is referred to as an odd signal if
x(-t) = -x(t)
x[-n] = -x[n]
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19. Even and Odd Signals (continued)
Any signal x(t) or x[n] can be expressed as a sum of two signals, one of which is even and one of which is odd. That is,
x(t) = xe(t) + xo(t)
x[n] = xe[n] + xo[n]
where xe(t) = ½{ x(t) + x(-t) }
xo(t) = ½{ x(t) - x(-t) }
xe[n] = ½{ x[n] + x[-n] }
xo[n] = ½{ x[n] - x[-n] }
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20. Periodic and Non-periodic Signals
A continuous-time signal x(t) is said to be periodic with period T if there is a positive nonzero value of T for which
x(t + T) = x(t) for all t
An example of such a signal is shown below
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21. Periodic and Non-periodic Signals (continued)
From this equation
x(t + T) = x(t)
it follows that
x(t + mT) = x(t)
for all t and any integer m.
Note that this definition does not work for a constant signal x(t).
Any continuous-time signal which is not periodic is called a non-periodic (or aperiodic ) signal.
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22. Periodic and Non-periodic Signals (continued)
Periodic discrete-time signals are defined analogously. A sequence (discrete-time signal) x[n] is periodic with period N if there is a positive integer N for which
x[n + N] = x[n] for all n
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23. Periodic and Non-periodic Signals (continued)
From the following equation
x[n + N] = x[n]
It follows that
x[n + mN] = x[n]
for all n and any integer m.
Any sequence which is not periodic is called a non-periodic (or aperiodic) sequence.
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24. Causal vs. Anticausal vs. Noncausal Signals
Causal signals are signals that are zero for all negative time.
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25. Causal vs. Anticausal vs. Noncausal Signals
Anti-causal are signals that are zero for all positive time.
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26. Causal vs. Anticausal vs. Noncausal Signals
Non-causal signals are signals that have nonzero values in both positive and negative time.
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27. Signal Energy and Power
The total energy over an interval t1<=t<=t2 in a continuous time signal x(t) is defined as 
For discrete-time sequence x[n] 
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28. Signal Energy and Power
A signal is called “energy signal” if
A signal is called “Power signal” if
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29. Transformation of Independent Variables
Reflection
Shifting
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30. Signals and systems analysis د. عامر الخيري
Scaling
Example
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31. Basic continuous-time signals
The Unit Step Function:
The unit step function u(t) is defined as
Note that it is discontinuous at t = 0 and that the value at t = 0 is undefined.
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32. Basic continuous-time signals (continued)
Similarly, the shifted unit step function u(t - to) is defined as
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33. Basic continuous-time signals (continued)
The Unit Impulse Function:
The unit impulse function d(t), also known as the Dirac delta function, plays a central role in system analysis. Traditionally, d(t) is often defined as the limit of a suitably chosen
conventional function having unity area over an infinitesimal time interval
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34. Basic continuous-time signals (continued)
The unit impulse function d(t) properties:
But an ordinary function which is everywhere 0 except at a single point must have the integral 0. Thus, d(t) cannot be an ordinary function and
mathematically it is defined by
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35. Basic continuous-time signals (continued)
Similarly, the delayed delta function
d(t – t0) is defined by
where f (t) is any regular function continuous at t = t0. For convenience, d(t) and d(t-t0) are depicted graphically
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36. Basic continuous-time signals (continued)
Some additional properties of d(t) are:
Sifting property
if x(t) is continuous at t=0
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37. Basic continuous-time signals (continued)
using the two equations
and
We can conclude that any signal x(t) can be
expressed as follows:
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38. Basic continuous-time signals (continued)
Complex Exponential Signals:
Using Euler's formula, this signal can be defined as
Thus, x(t) is a complex signal whose real part is cos w0t and imaginary part is sin w0t.
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39. Basic continuous-time signals (continued)
Complex Exponential Signals:
x(t) is a periodic signal whose fundamental period is
w0 is called the fundamental frequency.
General Complex Exponential Signals:
Let s = s + jw be a complex number. We define x(t) as
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40. Basic continuous-time signals (continued)
Signal x(t) known as a general complex exponential signal whose real part est coswt and imaginary part est sinwt are exponentially increasing (s > 0) or decreasing (s < 0) sinusoidal signals
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41. Basic continuous-time signals (continued)
Real Exponential Signals:
if s = s (a real number), then
reduces to a real
exponential signal.
If s > 0 then x(t) is a
growing exponential.
If s < 0 then x(t) is a
decaying exponential.
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42. Basic continuous-time signals (continued)
Sinusoidal Signals:
A continuous-time sinusoidal signal can be expressed as
where A is the amplitude (real), w0 is the radian frequency in radians per second, and f is the phase angle in radians.
The sinusoidal signal x(t) is periodic with fundamental period
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43. Basic continuous-time signals (continued)
Sinusoidal Signals:
The reciprocal of the fundamental period T0 is called the fundamental frequency f0:
From the previous two equations we can conclude the following relation:
w0 is the fundamental angular frequency.
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44. Basic continuous-time signals (continued)
Sinusoidal Signals:
Using Euler's formula, the sinusoidal signal can be expressed as
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45. Basic continuous-time signals (continued)
f - Frequency
The number of times a signal makes a complete cycle within a given time frame; frequency is measured in Hertz (Hz), or cycles per second e.g. S=5cos(25t) here f=5Hz
Spectrum – Range of frequencies that a signal spans from minimum to maximum e.g. S=S1+S2 where S1=cos(25t) and S2=cos(27t). Here spectrum SP={5Hz,7Hz}.
Bandwidth – Absolute value of the difference between the lowest and highest frequencies of a signal. In the above example bandwidth is BW=2Hz.
Consider an average voice
The average voice has a frequency range of roughly 300 Hz to 3100 Hz
The spectrum would be 300 – 3100 Hz
The bandwidth would be 2800 Hz
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46. Basic continuous-time signals (continued)
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47. Basic continuous-time signals (continued)
 - Phase
The position of the waveform relative to a given moment of time or relative to time zero, e.g. S1=cos(25t) and S2=cos(25t + /2). Here S1 has phase =0 and S2 has phase = /2.
A change in phase can be any number of angles between 0 and 360 degrees
Phase changes often occur on common angles, such as /4=45, /2=90, 3/4=135, etc
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48. Basic continuous-time signals (continued)
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49. Basic continuous-time signals (continued)
Ramp Function:
The ramp function is closely related to the unit-step discussed above. Where the unit-step goes from zero to one instantaneously, the ramp function better resembles a real-world signal, where there is some time needed for the signal to increase from zero to its set value, one in this case. We define a ramp function as follows:
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50. Basic continuous-time signals (continued)
Ramp Function:
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51. Basic Discrete-time signals
The Unit Step Sequence:
The unit step sequence u[n] is defined as
Note that the value of u[n] at n = 0 is defined [unlike the continuous-time step function u(t) at t = 0] and equals unity.
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52. Basic Discrete-time signals
The Unit Step Sequence:
The shifted unit step sequence u[n-k] is defined as
Note that the value of u[n] at n = 0 is defined [unlike the continuous-time step function u(t) at t = 0] and equals unity.
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53. Basic Discrete-time signals
The Unit impulse Sequence:
The unit impulse or unit sample sequence d[n] is defined as
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54. Basic Discrete-time signals
The Unit impulse Sequence:
The shifted unit impulse or shifted unit sample sequence d[n-k] is defined as
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55. Basic Discrete-time signals
The Unit impulse Sequence:
Unlike the continuous-time unit impulse function d(t), d[n] is defined without mathematical complication or difficulty. From above definitions it is readily seen that
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56. Basic Discrete-time signals
The Unit impulse Sequence:
Unlike the continuous-time unit impulse function d(t), d[n] is defined without mathematical complication or difficulty. From above definitions it is readily seen that
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57. Basic Discrete-time signals
The Sinusoidal Sequence:
A sinusoidal sequence can be expressed as
If n is dimensionless, then both Wo, and q have units of radians. In order for the sequence to be periodic with period N > 0, Wo must satisfy the following condition: (m is a positive integer)
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58. Basic Discrete-time signals
The Sinusoidal Sequence (continued):
Thus the sequence is not periodic for any value of Wo. It is periodic only if Wo / 2p is a rational number.
Thus, if Wo satisfies the periodicity condition, Wo ≠ 0, N and m have no factors in common, then the fundamental period of the sequence x[n] is No given by:
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59. Basic Discrete-time signals
The following sequence is periodic:
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60. Basic Discrete-time signals
The following sequence is non-periodic:
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61. Basic Discrete-time signals
Complex Exponential Sequences:
The complex exponential sequence is of the form
Again, using Euler's formula, x[n] can be expressed as
In order for x[n] to be periodic with period N > 0, Wo must satisfy the following condition: (m is a positive integer)
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62. Basic Discrete-time signals
Complex Exponential Sequences (continued):
Thus the sequence is not periodic for any value of Wo. It is periodic only if Wo / 2p is a rational number.
Thus, if Wo satisfies the periodicity condition, Wo ≠ 0, N and m have no factors in common, then the fundamental period of the sequence x[n] is No given by:
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63. Basic Discrete-time signals
Complex Exponential Sequences (continued):
in dealing with discrete-time exponentials, we need only consider an interval of length 2p in which to choose Wo.
Usually, we will use the interval
0 ≤ Wo< 2p
or the interval
-p ≤ Wo< p.
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64. Basic Discrete-time signals
General Complex Exponential Sequences:
The most general complex exponential sequence is often defined as
x[n] = Can
where C and a are in general complex numbers. Note that Equation is a special case of
the above equation with
C = 1 and a =
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65. Basic Discrete-time signals
Real Exponential Sequences:
If C and a are both real, then x[n] is a real exponential sequence.
Four distinct cases can be identified:
a > 1,
0 < a < 1,
-1 < a < 0, and
a < - 1.
Note that if a = 1, x[n] is a constant sequence, whereas if a = - 1, x[n] alternates in value between +C and -C.
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66. Basic Discrete-time signals
Real Exponential Sequences (continued):
a > 1
0 < a < 1
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67. Basic Discrete-time signals
Real Exponential Sequences (continued):
0 > a > -1
a < -1
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68. Signals and systems analysis د. عامر الخيري
What is a System?
Systems process input signals to produce output signals
Examples:
A CD player takes the signal on the CD and transforms it into a signal sent to the loud speaker
A system takes a signal as an input and transforms it into another signal
System
Input signal
x(t)
Output signal
y(t)
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69. What is a System(continued)?
Let x and y be the input and output signals, respectively, of a system. Then the system is viewed as a transformation (or mapping) of x into y. This transformation is represented by the mathematical notation
y = Tx
where T is the operator representing some well-defined rule by which x is transformed into y.
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70. Classifications of System
Continuous-Time and Discrete-Time Systems:
If the input and output signals x and y are continuous-time signals, then the system is called a continuous-time system. If the input and output signals are discrete-time signals or sequences, then the system is called a discrete-time system.
continuous-time system
discrete-time system
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71. Classifications of System
Systems with Memory and without Memory:
A system is said to be memoryless if the output at any time depends on only the input at that same time. Otherwise, the system is said to have memory.
An example of a memoryless system:
An example of a system with memory:
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72. Classifications of System
Causal and Noncausal Systems:
A system is called causal if its output y(t) at an arbitrary time t = to, depends on only the input x(t) for t ≤ to. That is, the output of a causal system at the present time depends on only the present and/or past values of the input, not on its future values. Thus, in a causal system, it is not possible to obtain an output before an input is applied to the
system. A system is called noncausal if it is not causal.
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73. Classifications of System
Causal and Noncausal Systems:
All ”realtime” systems must be causal, since they
can not have future inputs available to them.
Examples of noncausal systems are:
Examples of causal systems are
Note that all memoryless systems are causal, but not vice versa.
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74. Classifications of System(cntd.)
Linear Systems and Nonlinear Systems:
If the operator T in Equation y = Tx satisfies the following two conditions, then T is called a linear operator and the system represented by a linear operator T is called a linear system:
1- Additivity:
Given that Tx1 = y1 and Tx2 = y2, then
T{x1 + x2} = y1 + y2
for any signals x1 and x2.
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75. Classifications of System(cntd.)
Linear Systems and Nonlinear Systems:
2- Homogeneity or (scaling):
Tax = ay
for any signal x and any scalar a.
Any system that does not satisfy both conditions is classified as a nonlinear system. Both conditions can be combined into a single condition as
T {a1x1 + a2x2} = a1y1 + a2y2
where a, and a, are arbitrary scalars. This final Equation is known as the superposition property.
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76. Classifications of System(cntd.)
Linear Systems and Nonlinear Systems: (illustration using diagrams)
1- Additivity:
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77. Classifications of System(cntd.)
Linear Systems and Nonlinear Systems: (illustration using diagrams)
2- Homogeneity or (scaling):
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78. Classifications of System(cntd.)
Time-Invariant and Time-Varying Systems:
A system is called time-invariant if a time shift (delay or advance) in the input signal causes the same time shift in the output signal. Thus, for a continuous-time system, the system is time-invariant if for any real value of t.
For a discrete-time system, the system is time-invariant (or shift-invariant ) if for any integer k
A system that doesn’t satisfy this condition is called time-varying.
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79. Classifications of System(cntd.)
Linear Time-Invariant System:
A system is called Linear time-invariant (LTI) if this system is both linear and time-invariant.
Stable Systems:
A system is bounded-input/bounded-output (BIBO) stable if for any bounded input x defined by the corresponding output y is also bounded defined by
where k1 and k2 are finite real constants.
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80. Classifications of System(cntd.)
Stable Systems: (Graphical illustration)
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81. Classifications of System(cntd.)
Invertible Systems: a system S is invertible if the input signal can always be uniquely recovered
from the output signal. The inverse system, formally written as S–1 (this is not the arithmetic inverse), is such that the cascade interconnection in Figure below is equivalent to the identity system, which leaves the input unchanged.
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82. System block diagrams (Interconnections)
Systems may be interconnections of other systems. For example, the discrete-time system.
For example, the discrete-time system shown as a block diagram in the Figure below can be described by the following system equations:
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83. System block diagrams (Interconnections)
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84. System block interconnections
Cascade Interconnection:
The cascade interconnection shown in the Figure below is a successive application of two (or more) systems on an input signal:
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85. System block interconnections
Parallel Interconnection:
The parallel interconnection shown in the Figure below is an application of two (or more) systems to the same input signal, and the output is taken as the sum of the outputs of the individual systems.
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86. System block interconnections
Feedback Interconnection
The feedback interconnection of two systems as shown in Figure below is a feedback of the output of system G1 to its input, through system G2.
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