Chapter (1)

1. Signals and Systems Chapter (1) Networks and Communication Department

2. 1.1 What does a signal mean ? • A signal is a function representing a physical quantity or variable, typically is contains information about the behavior or nature of the phenomenon. • A system is viewed as transformation (mapping) of into , to process input signals to produce output signals. • Types of Signals in our lives and some examples: • Electrical Signals: (e.g. voltage and current) • Electromagnetic Signals: (e.g. radio, waves and light) • Sound Signals: (e.g. human sounds and music) System Output signal Input signal Signals and Systems Analysis

3. 1.2 Classifications of Signals • Continuous-Time and Discrete-Time Signals (Basic types of Signals) • Continuous-Time Signals: A signal is continuous-time signal if the independent variable t is continuous. • Discrete-Time Signals: • A signal is defined at discrete times, a discrete-time signal is often identified as a sequence of numbers, denoted by • A very important class of discrete-time signals is obtained by sampling a continuous-time signal Signals and Systems Analysis

4. 1.2 Classifications of Signals (Cont.) • Analog and Digital Signals • Analog signals: If a continuous-time signal can take on any value in the continuous interval (, b), where may be -and b may be +, then the continuous-time signal is called an analog signal. • Digital signals: Itis a discrete or non-continuous waveform, that could take only a finite number of values; with examples such as computer 1s and 0s. Signals and Systems Analysis

5. 1.2 Classifications of Signals (Cont.) Signals and Systems Analysis

6. 1.2 Classifications of Signals (Cont.) E. Even and Odd Signals A signal or is referred to as an even signal if = = Signals and Systems Analysis

7. 1.2 Classifications of Signals (Cont.) E. Even and Odd Signals (Cont.) A signal or is referred to as an oddsignal if = = Signals and Systems Analysis

8. 1.2 Classifications of Signals (Cont.) E. Even and Odd Signals (Cont.) • 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, = e+ o = e[] + o[] wheree(t) = ½{ + } even part of o(t) = ½{ - } odd part of e[] = ½{ +} even part of o[] = ½{ - } even part of Signals and Systems Analysis

9. 1.2 Classifications of Signals (Cont.) • Periodic and Non-periodic signals: Periodic signal: Pattern repeated over time. Non-periodic signal(Aperiodic): Pattern not repeated over time. Signals and Systems Analysis

10. 1.2 Classifications of Signals (Cont.) • Periodic and Non-periodic Signals: • A continuous-time signalis said to be periodicwith period Tif there is a positive nonzero value of T for which (t + T) = for all t • In other words, a periodic signal has the property that it is unchanged by a time shift of T (called period T). An example of such a signal is shown below Signals and Systems Analysis

11. 1.2 Classifications of Signals (Cont.) • Periodic and Non-periodic Signals (Cont.): • From this equation (t + T) = it follows that (t + mT) = for all t and any integer m. Note that this definition does not work for a constant signal . • Any continuous-time signal which is not periodic is called a non-periodic (or aperiodic ) signal. Signals and Systems Analysis

12. 1.2 Classifications of Signals (Cont.) • Periodic and Non-periodic Signals (Cont.): Signals and Systems Analysis

13. 1.2 Classifications of Signals (Cont.) • Periodic and Non-periodic Signals (Cont.): Signals and Systems Analysis

14. 1.2 Classifications of Signals (Cont.) • Exponential and Sinusoidal Signals: Signals and Systems Analysis

15. 1.2 Classifications of Signals (Cont.) 1.2.1 Continuous-Time Complex Exponential and Sinusoidal Signals(Cont.): • Real Exponential Signals (Cont.): Signals and Systems Analysis

16. 1.2 Classifications of Signals (Cont.) 1.2.1 Continuous-Time Complex Exponential and Sinusoidal Signals (Cont.): • Periodic Complex Exponential and Sinusoidal Signals: Euler's formula, • An important property of this signal is that it is periodic. • The fundamental period ( ) is the smallest positive value of T, Real Part Imaginary Part Signals and Systems Analysis

17. 1.2 Classifications of Signals (Cont.) 1.2.1 Continuous-Time Complex Exponential and Sinusoidal Signals (Cont.): • Sinusoidal Signals: • A continuous-time sinusoidal signal can be expressed as WhereA is the amplitude (real), 0 is the radian frequency in radians per second, and is the phase angle in radians. • Sinusoidal signal is periodic with fundamental period given by Signals and Systems Analysis

18. 1.2 Classifications of Signals (Cont.) 1.2.1 Continuous-Time Complex Exponential and Sinusoidal Signals (Cont.): • Sinusoidal Signals (Cont.) Signals and Systems Analysis

19. 1.2 Classifications of Signals (Cont.) 1.2.1 Continuous-Time Complex Exponential and Sinusoidal Signals (Cont.): • Sinusoidal Signals (Cont.) • The reciprocal of the fundamental period T0is called the fundamental frequency 0: • The frequency is the number of times a signal makes a complete cycle within a given time frame. • From the previous two equations we can conclude the following relation: • 0 is the fundamental angular frequency. Signals and Systems Analysis

20. 1.2 Classifications of Signals (Cont.) • Frequency, Spectrum and Bandwidth: • Frequency is measured in Hertz (Hz), or cycles per second = A cos(t) e.g. S = 5cos (25t) here =5Hz. • Spectrum: Range of frequencies that a signal spans from minimum to maximum, e.g. =+ where =cos(25t) and =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. Signals and Systems Analysis

21. 1.2 Classifications of Signals (Cont.) •  - Phase • The position of the waveform relative to a given moment of time or relative to time zero, e.g. =cos(25t) and=cos(25t + /2). Here has phase =0 and 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

22. 1.2 Classifications of Signals (Cont.) 1.2.1 Continuous-Time Complex Exponential and Sinusoidal Signals (Cont.): The relationship between fundamental frequency and period for continuous-time sinusoidal signals. Here Which implies that Signals and Systems Analysis

23. 1.2 Classifications of Signals (Cont.) 1.2.2 Discrete-Time Complex Exponential and Sinusoidal Signals: Where and are complex (real) numbers. • Sinusoidal Sequence Signals: • In order for the sequence to be periodic only if N > 0 must satisfy the following condition: (m is a positive integer) N is the fundamental period Signals and Systems Analysis

24. 1.2 Classifications of Signals (Cont.) 1.2.2 Discrete-Time Complex Exponential and Sinusoidal Signals (Cont.): • Real Exponential Sequence Signals 0 > > -1 > 1 < -1 0 < < 1 Signals and Systems Analysis

25. 1.3 Transformations of the independent variable: • Reflection • Shifting Signals and Systems Analysis

26. 1.3 Transformations of the independent variable (Cont.): • Scaling • Exercise :Sketch the signal x[n+2] Signals and Systems Analysis

27. 1.4 The Unit Impulse and Unit Step Function 1.4.1 The Discrete-Time Unit Impulse The shifted unit impulse (or sample) sequence is defined as: Signals and Systems Analysis

28. 1.4 The Unit Impulse and Unit Step Function (Cont.) 1.4.1 The Discrete-Time Unit Step (Cont.) Note:that the value of u[n] at n = 0 is defined. The shifted unit impulse (or sample) sequence is defined as: Note:that the value of at n = 0 is defined. step Signals and Systems Analysis

29. 1.4 The Unit Impulse and Unit Step Function (Cont.) 1.4.2 The Continuous-Time Unit Step: Note: The unit step is discontinuous at and that value is undefined. Similarly, the shifted unit step function is defined as Signals and Systems Analysis

30. 1.4 The Unit Impulse and Unit Step Function (Cont.) Signals and Systems Analysis

31. 1.5 Continuous-Time and Discrete-Time Systems System Representation: Continuous-Time and Discrete-Time Systems: Signals and Systems Analysis

32. 1.5 Continuous-Time and Discrete-Time Systems Cascade(series) Interconnection: Parallel Interconnection: Signals and Systems Analysis

33. 1.5 Continuous-Time and Discrete-Time Systems Series-parallel Interconnection: Feedback Interconnection: Signals and Systems Analysis

34. 1.6 Basic Systems Properties System 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 memorylesssystem: • An example of a system with memory: Signals and Systems Analysis

35. 1.6 Basic Systems Properties (Cont.) • Inevitability and inverse systems: • Invertible system: a system S is invertible if the the input signal can always be uniquely recovered from the output signal Y(t)=2x(t) • The inverse system: formally written as ,such that the cascade interconnection in the figure below is equivalent to the identity system, which leaves the input unchanged Signals and Systems Analysis

36. 1.6 Basic Systems Properties (Cont.) Causality Systems: • Causal Systems: if the output at any time depends on values of the input at only the present and past time. • Thus, in a causal system, it is not possible to obtain an output before an input is applied to the system. • All ”real time” systems must be causal, since they can not have future inputs available to them. • A system is called non-causal if it is not causal. Examples of causal systems are: Examples of non-causalsystems are: Note:that all memoryless systems are causal, but not vice versa. Signals and Systems Analysis

37. 1.6 Basic Systems Properties (Cont.) Time-Invariant and Time-Varying Systems: • A system is called time-invariantif 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. Signals and Systems Analysis

38. 1.6 Basic Systems Properties (Cont.) • 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. Signals and Systems Analysis

39. 1.6 Basic Systems Properties (Cont.) • Linear Systems and Nonlinear Systems (Cont.): • 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. Signals and Systems Analysis

40. 1.6 Basic Systems Properties (Cont.) Linear Time-Invariant System: • A system is called Linear time-invariant (LTI)if this system is both linear and time-invariant. Signals and Systems Analysis