Signals and Systems Analysis NET 351 Instructor Dr

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

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 ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 2

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 ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 3

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). Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 4

Assessment index The nature of the evaluation function (e. g. : article, quiz, group project, etc. ) Due week Assessment weight (%) 1 First exam Week 7 20% 2 Second exam Week 12 20% 3 Theoretical assignments Weeks 5 & 10 10% 4 Final exam After Week 15 50% Total: 100% Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 5

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. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 6

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. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 7

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 Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 8

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 f(t) t Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 9

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. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 10

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 Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 11

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. Signals and systems 12 analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ

Analog vs. Digital (continued) Digital is a discrete or non-continuous waveform with examples such as computer 1 s and 0 s Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 13

Analog vs. Digital (continued) Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 14

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 ) = x 1( t ) + j x 2( t ) where x 1( t ) and x 2( t ) are real signals and j=. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 15

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 Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 16

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] Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 17

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] Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 18

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] } Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 19

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 Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 20

Periodic and Non-periodic Signals (continued) • From this equation x(t + T) = x(t) it follows that x(t + m. T) = 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. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 21

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 Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 22

Periodic and Non-periodic Signals (continued) • From the following equation x[n + N] = x[n] It follows that x[n + m. N] = x[n] for all n and any integer m. • Any sequence which is not periodic is called a non-periodic (or aperiodic) sequence. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 23

Causal vs. Anticausal vs. Noncausal Signals • Causal signals are signals that are zero for all negative time. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 24

Causal vs. Anticausal vs. Noncausal Signals • Anti-causal are signals that are zero for all positive time. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 25

Causal vs. Anticausal vs. Noncausal Signals • Non-causal signals are signals that have nonzero values in both positive and negative time. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 26

Signal Energy and Power • The total energy over an interval t 1<=t<=t 2 in a continuous time signal x(t) is defined as • For discrete-time sequence x[n] Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 27

Signal Energy and Power • A signal is called “energy signal” if • A signal is called “Power signal” if Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 28

Transformation of Independent Variables • Reflection • Shifting Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 29

• Scaling • Example Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 30

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. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 31

Basic continuous-time signals (continued) • Similarly, the shifted unit step function u(t - to) is defined as Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 32

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 Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 33

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 Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 34

Basic continuous-time signals (continued) • Similarly, the delayed delta function d(t – t 0) is defined by where f (t) is any regular function continuous at t = t 0. For convenience, d(t) and d(t-t 0) are depicted graphically Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 35

Basic continuous-time signals (continued) • Some additional properties of d(t) are: Sifting property if x(t) is continuous at t=0 Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 36

Basic continuous-time signals (continued) • using the two equations and We can conclude that any signal x(t) can be expressed as follows: Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 37

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 w 0 t and imaginary part is sin w 0 t. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 38

Basic continuous-time signals (continued) • Complex Exponential Signals: x(t) is a periodic signal whose fundamental period is w 0 is called the fundamental frequency. General Complex Exponential Signals: Let s = s + jw be a complex number. We define x(t) as Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 39

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 Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 40

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. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 41

Basic continuous-time signals (continued) Sinusoidal Signals: A continuous-time sinusoidal signal can be expressed as where A is the amplitude (real), w 0 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 Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 42

Basic continuous-time signals (continued) Sinusoidal Signals: The reciprocal of the fundamental period T 0 is called the fundamental frequency f 0: From the previous two equations we can conclude the following relation: w 0 is the fundamental angular frequency. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 43

Basic continuous-time signals (continued) Sinusoidal Signals: Using Euler's formula, the sinusoidal signal can be expressed as Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 44

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=5 cos(2 5 t) here f=5 Hz – Spectrum – Range of frequencies that a signal spans from minimum to maximum e. g. S=S 1+S 2 where S 1=cos(2 5 t) and S 2=cos(2 7 t). Here spectrum SP={5 Hz, 7 Hz}. – Bandwidth – Absolute value of the difference between the lowest and highest frequencies of a signal. In the above example bandwidth is BW=2 Hz. – 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 Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 45

Basic continuous-time signals (continued) Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 46

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. S 1=cos(2 5 t) and S 2=cos(2 5 t + /2). Here S 1 has phase =0 and S 2 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 Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 47

Basic continuous-time signals (continued) Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 48

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: Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 49

Basic continuous-time signals (continued) Ramp Function: Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 50

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. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 51

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. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 52

Basic Discrete-time signals The Unit impulse Sequence: The unit impulse or unit sample sequence d[n] is defined as Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 53

Basic Discrete-time signals The Unit impulse Sequence: The shifted unit impulse or shifted unit sample sequence d[n-k] is defined as Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 54

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 Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 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 Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 56

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) Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 57

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 / 2 p 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: Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 58

Basic Discrete-time signals The following sequence is periodic: Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 59

Basic Discrete-time signals The following sequence is non-periodic: Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 60

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) Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 61

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 / 2 p 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: Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 62

Basic Discrete-time signals Complex Exponential Sequences (continued): in dealing with discrete-time exponentials, we need only consider an interval of length 2 p in which to choose Wo. Usually, we will use the interval 0 ≤ Wo < 2 p or the interval -p ≤ W o < p. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 63

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 = Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 64

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: Ø a > 1, Ø 0 < a < 1, Ø -1 < a < 0, and Ø 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. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 65

Basic Discrete-time signals Real Exponential Sequences (continued): a>1 0<a<1 Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 66

Basic Discrete-time signals Real Exponential Sequences (continued): 0 > a > -1 a < -1 Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 67

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 Input signal x(t) System Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ Output signal y(t) 68

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. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 69

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 Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 70

Classifications of 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: Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 71

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. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 72

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. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 73

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 Tx 1 = y 1 and Tx 2 = y 2, then T{x 1 + x 2} = y 1 + y 2 for any signals x 1 and x 2. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 74

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 {a 1 x 1 + a 2 x 2} = a 1 y 1 + a 2 y 2 where a, and a, are arbitrary scalars. This final Equation is known as the superposition property. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 75

Classifications of System(cntd. ) Linear Systems and Nonlinear Systems: (illustration using diagrams) 1 - Additivity: Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 76

Classifications of System(cntd. ) Linear Systems and Nonlinear Systems: (illustration using diagrams) 2 - Homogeneity or (scaling): Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 77

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 timeinvariant (or shift-invariant ) if for any integer k A system that doesn’t satisfy this condition is called time-varying. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 78

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 k 1 and k 2 are finite real constants. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 79

Classifications of System(cntd. ) Stable Systems: (Graphical illustration) Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 80

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. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 81

System block diagrams (Interconnections) Systems may be interconnections of other systems. For example, the discrete-time system shown as a block diagram in the Figure below can be described by the following system equations: Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 82

System block diagrams (Interconnections) Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 83

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: Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 84

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. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 85

System block interconnections Feedback Interconnection The feedback interconnection of two systems as shown in Figure below is a feedback of the output of system G 1 to its input, through system G 2. Signals and systems analysis ﻋﺎﻣﺮ ﺍﻟﺨﻴﺮﻱ. ﺩ 86