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# Signals and systems( chapter 1)

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### Signals and systems( chapter 1)

1. 1. Signals and Systems: A computer without signals - without networking, audio and video.
2. 2. Terms • Signal • voltage over time • State • the variables of a differential equation • System • linear time invariant transfer function
3. 3. Chapter 1 : Signals And Systems Signals & Systems • Signal – physical form of a waveform – e.g. sound, electrical current, radio wave • System – a channel that changes a signal that passes through it – e.g. a telephone connection, a room, a vocal tract Input Signal System Output Signal Input signal system Output signal 1.0 Introduction Signals and Systems subject is focusing on a signal involving dependent variable (i.e : time even though it can be others such as a distance , position , temperature , pressure and others)
4. 4. Chapter 1 : Signals And Systems 1.1 Signals and Systems Definition a) Signal • A function of one/more variable which convey information on the natural of a physical phenomenon. • Examples : human speech, sound, light, temperature, current etc b) Systems • An entity that processes of manipulates one or more signals to accomplish a function, thereby yielding new signal. • Example: telephone connection
5. 5. Chapter 1 : Signals And Systems 1.2 Classification of Signals There are several classes of signals a) Continuous time and discrete time signals b) Analog and digital signals c) Real and Complex signals d) Even and Odd Signals e) Energy and power signals f) Periodic and aperiodic signals
6. 6. Chapter 1 : Signals And Systems a) Continuous time and discrete time signals • Continuous signals : signal that is specified for a continuum (ALL) values time t : can be described mathematically by continuous function of time as : x(t) = A sin (ω0 t + ɸ) where • A ω ɸ : Amplitude : Radian freq in rad / sec : phase angle in rad / degree Discrete time signals : signal that is specified only at discrete values of t
7. 7. Chapter 1 : Signals And Systems b) Analog and digital signals • Analog signals : signal whose amplitude can take on any value in a continuous range • Digital signals : signal whose amplitude can take only a finite number of values (signal which associated with computer since involve binary 1 / 0 )
8. 8. Chapter 1 : Signals And Systems Examples of signals
9. 9. • Example 1:
10. 10. • Example 2:
11. 11. Chapter 1 : Signals And Systems c) Deterministic and probabilistic signals • Deterministic signals : a signal whose physical description is known completely either in a mathematical form or a graphical form and its future value can be determined. • Probabilistic signals : a signal whose values cannot be predicted precisely but are known only in terms of probabilistic value such as mean value / mean-squared value and therefore the signal cannot be expressed in mathematical form.
12. 12. Chapter 1 : Signals And Systems d) Energy and power signals • Energy signals : a signal with finite energy signal • Power signals : a signal with finite and nonzero power Finite energy signal Infinite energy signal
13. 13. Chapter 1 : Signals And Systems 1.9 Energy and Power Signals For an arbitrary signal x(t) , the total energy , E is defined as The average power , P is defined as
14. 14. Chapter 1 : Signals And Systems 1.9 Energy and Power Signals Based on the definition , the following classes of signals are defined : a) x(t) is energy signal if and only if 0 < E < ∞ so that P = 0. b) x(t) is a power signal if and only if 0 < P < ∞ thus implying that E = ∞. c) Signals that satisfy neither property are therefore neither energy nor power signals.
15. 15. Exercise 1) Calculate the total energy of the rectangular pulse shown in figure. 2) Given a signal as listed below, determine whether x(t) is energy, power or neither signal. Justify the answer. a. x(t) = cos t b. x(t) = 3 e-4t u(t)
16. 16. Chapter 1 : Signals And Systems e) Periodic and aperiodic signal • Periodic signals : signal that repeats itself within a specific time or in other words, any function that satisfies : f (t ) = f (t + T ) where T is a constant and is called the fundamental period of the function. T= • Aperiodic signals 2π ω0 : signal that does not repeats itself and therefore does not have the fundamental period.
17. 17. Chapter 1 : Signals And Systems Examples of signals Periodic signal Aperiodic signal
18. 18. • Example: Find the period for t f (t ) = cos 3
19. 19. Chapter 1 : Signals And Systems e) Periodic and aperiodic signal (continue) • Any continuous time signal x(t) is classified as periodic if the signal satisfies the condition : x(t) = x(t + nT) • where n = 1 , 2 , 3 .... The sum of two or more signals is periodic if the ratio (evaluation of two values) of their periods can be expressed as rational number. The new fundamental period and frequency can be obtained from a periodic signal. A rational number is a number that can be written as a simple fraction (i.e. as a ratio). • The sum of two or more signals is aperiodic if the ratio (evaluation of two values) of their periods is expressed as irrational number and no new fundamental period can be obtained.
20. 20. • Periodic and aperiodic signal (continue) x(t) = x(t + nT) Tο = 2π / ωο π Ωο = m 2π N π
21. 21. Chapter 1 : Signals And Systems Exercise 1 • Determine whether listed x(t) below is periodic or aperiodic signal. If a signal is periodic, determine its fundamental period. T= a) x(t) = sin 3t b) x(t) = 2 cos 8πt c) x(t) = 3 cos (5πt + π/2) d) x(t) = cos t + sin √2 t e) x(t) = sin2t j[(π/2)t-1] f) x(t) = e 2π ω0
22. 22. EXERCISE 2 a) b) c) d) j(π/4)n x[n]=e x[n]=cos1/4n x[n]= cos π/3 n + sin π/4n x[n]=cos2π/8n
23. 23. Chapter 1 : Signals And Systems Exercise 2 • Determine whether the following signals are periodic or aperiodic. Find the new fundamental period if necessary. a) x3(t) = 6 x1(t) + 2 x2(t) b) x5(t) = 6 x1(t) + 2 x2(t) + x4(t) where : x1(t) = sin 13t x2(t) = 5 sin (3000t + π/4) x4(t) = 2 cos (600πt – π/3)
24. 24. Chapter 1 : Signals And Systems Exercise 3 • Given x1(t) = 2 sin (5t) , x2(t) = 5 sin (3t) and x3(t) = 5 sin (2t + 25o) . • If x(t) = x1(t) – 3x2(t) + 2x3(t) , determine whether x(t) is periodic or aperiodic signal . • If it is periodic, determine it’s period and frequency
25. 25. Chapter 1 : Signals And Systems • Even and Odd A signal x ( t ) or x[n] is referred to as an even signal if x(-t)=x(r) x[-n]=x[n] A signal x ( t ) or x[n] is referred to as an odd signal if x(-t)=-x(t) x[-n]=-x[n] Examples of even and odd signals are shown in Fig. 1-2.
26. 26. Chapter 1 : Signals And Systems 1.4) Singularity Functions is an important and unique sub-class of aperiodic signals they are either discontinuous or continuous derivatives they are basic signals to represent other signals a) Unit Step, u(t) u(t) u(t) = 1 t 0 ;t<0 1 ;t≥0
27. 27. Chapter 1 : Signals And Systems b) Unit Ramp , r(t) r(t) 1 1 r(t) = t 0 ;t<0 t ;t≥0
28. 28. Chapter 1 : Signals And Systems c) Unit impulse, δ(t) δ(t) δ(t) = 1 t 1 ;t=0 0 ;t≠0
29. 29. 1.5) Representation of Signals A deterministic signal can be represented in terms of: 1. sum of singularity functions 2. sum of steps functions and 3. piece-wise continuous functions
30. 30. Chapter 1 : Signals And Systems Sum Of Singularity Function Express signal in term of sum of singularity function Note : the ramp function r(t) can be described by step function as : r(t) = t u(t) r(t±a) = (t±a) u(t±a) Example Express the following signal in term of sum of singularity function. x(t) Answer x(t) = u(t+1) – r(t+1) + r(t-1) + u(t-1) 1 = u(t+1) – (t+1)u(t+1) + (t-1)u(t-1) + u(t-1) 1 -1 -1 t
31. 31. Chapter 1 : Signals And Systems Exercise Express the following signals in term of sum of singularity function. x(t) Answer 1 1 2 3 4 t x(t)= r(t) – 2r(t-1) + 2r(t+3) +r(t-4) = r(t) – 2(t-1)u(t-1) +2(t+3)u(t+3) +(t-4)u(t-4) -1 x(t) 2 Answer 1 -2 -1 1 2 t x(t) = 2δ(t+2) - u(t+2) + r(t+1) – r(t-1) – u(t-1) + δ(t+2)
32. 32. Chapter 1 : Signals And Systems Exercise Sketch the following signal if the sum of singularity function of the signal is given as : a) x(t) = r(t) + r (t-1) - u (t-1) b) y(t) = u(t+1)-r(t+1)+r(t-1)+u(t-1) c) x(t) = r(t) + r(t+1) + 2u(t+1) – r(t+1) + 2r(t) – r(t-1) + u(t-2) – 2u(t-3)
33. 33. Chapter 1 : Signals And Systems Piece – wise continous function Description of signal from a general form of y = mx + c Example Given the signal x(t) as shown below , express the signals x(t) in terms of piece wise continuous function x(t) Solution 1 -1 0 1 2 t x(t) = -t-1 t 1 0 ; -1 < t < 0 ;0<t<1 ;1<t<2 ; elsewhere
34. 34. Chapter 1 : Signals And Systems Exercise Example Given the signal x(t) as shown below , express the signals x(t) in terms of piece wise continuous function . x(t) 1 1 -1 0 2 t
35. 35. Chapter 1 : Signals And Systems Exercise Example Given the signal x(t) as shown below , express the signals x(t) in terms of piece wise continuous function . x(t) Solution 1 1 -1 0 2 t x(t) = t+1 -1 0 ; -1 < t < 0 ;1<t<2 ; elsewhere
36. 36. Chapter 1 : Signals And Systems 1.6 Properties of Signals There are 4 properties of signals a) Magnitude scaling b) Time reflection c) Time scaling d) Time shifting
37. 37. Chapter 1 : Signals And Systems a) Magnitude scaling : Any arbitrary real constant is multiplied to a signal and the result is, for a unit step the amplitude changes, for a unit ramp, the slope changes and for a unit impulse, the area changes A -A A 3 slope = 2 t t 3u(t) -2 t 1 -2u(t) A 2r(t) A t 2 0 3 A -0.5 slope = -2 -2 -2r(t) 0 3δ(t) t -0.5δ(t) t
38. 38. Chapter 1 : Signals And Systems b) Time reflection : The mirror image of the signal with respect to the y-axis u(t) δ(t) r(t) 1 1 0 u(-t) t slope = -1 t r(-t) 0 δ(-t) t
39. 39. Chapter 1 : Signals And Systems c) Time scaling : The expansion or compression of the signal with respect to time t axis x(0.5t) a x(kt) a 2b b t t x(2t) x(kt) a k > 1 compression k < 1 expansion 0.5b x(2t) t
40. 40. Chapter 1 : Signals And Systems f) Time shifting : The shifting of the signal with respect to the x - axis u(t) u(t) r(t) 1 1 slope = 1 t 1 u(t-1) t -1 u(t+1) r(t) r(t-1) δ(t) -2 3 t r(t+1) 2 0 3δ(t-2) t -0.5 slope = 1 -1 t 1 t -0.5δ(t+2)
41. 41. Chapter 1 : Signals And Systems Example 1 Given the signal x(t) as shown below , sketch y(t) = 3x (1- t/2) . x(t) 1 t -1 0 1 2
42. 42. Chapter 1 : Signals And Systems Example 2 Given the signal x(t) as shown below , sketch y(t) = 2x (-0.5t+1) using both graphical and analytical method. x(t) 1 -1 t 1 -1 2
43. 43. Chapter 1 : Signals And Systems Exercise Given the signal x(t) as shown below , sketch y(t) = -2x (2-0.5t) + 1 using both graphical and analytical method. x(t) 1 t -2 1 -1 -1 2 3
44. 44. Chapter 1 : Signals And Systems 1.9 Energy and Power Signals For an arbitrary signal x(t) , the total energy , E is defined as The average power , P is defined as
45. 45. Chapter 1 : Signals And Systems 1.9 Energy and Power Signals Based on the definition , the following classes of signals are defined : a) x(t) is energy signal if and only if 0 < E < ∞ so that P = 0. b) x(t) is a power signal if and only if 0 < P < ∞ thus implying that E = ∞. c) Signals that satisfy neither property are therefore neither energy nor power signals.
46. 46. Chapter 1 : Signals And Systems 2.0 Classification Of System i) a) With memory(dynamic) b) Without memory(static) ii) a) Causal : the present output depends on past and/or future input. : the present output depends only on present input. : the output does not depends on future but can depends on the past or the present input.. b) Non-causal : the output depends on future input
47. 47. Chapter 1 : Signals And Systems 2.0 Classification Of System iii) a) Time variant : y2 (t) ≠ y1 (t- to) : same input produces different output at different time b) Time invariant : y2 (t) = y1 (t- to) : same input produces same output at different time where : y1 (t- to) is the output corresponding to the time shifting , (t- to) at y1 (t) y2(t) is the output corresponding to the input x2(t) where x2(t) = x1 (t- to)
48. 48. Chapter 1 : Signals And Systems 2.0 Classification Of System 4) a) Linear : y(t) = ay1(t) + by2(t) (superposition applied) b) Non linear : y(t) ≠ ay1(t) + by2(t) (superposition not applied ) where : If an excitation x1[t] causes a response y1[t] and an excitation x2 [t] causes a response y2[n] , then an excitation : x [t = ax1[t] + bx2[t] will cause the response y [t] = ay1[t] + by2[t] (to be presented as y(t) in solution)
49. 49. Chapter 1 : Signals And Systems Example The following system is defined by the input – output relationship where x(t) is the input and y(t) is the output. a) y(t) = 10x2 (t+1) e) y’(t) + y(t) = x(t) b) y(t) = 10x(t) + 5 f) y’(t) + 10y(t) + 5 = x(t) c) y(t) = cos (t) x(t) + 5 g) y’(t) + 3y(t) = x(t) + 2x2(t) Determine whether the system is : i. Static or dynamic ii. Causal or non-causal iii. Time - variant or time invariant iv. Linear or non-linear