Lecture123

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Lecture123

  1. 1. Signals and SystemsSignals and Systems 6552111 Signals and Systems6552111 Signals and Systems Sopapun Suwansawang Lecture #1 1 Lecture #1 Elementary Signals and Systems Week#1-2
  2. 2. SignalsSignals Signals are functions of independent variables that carry information. FUNCTIONS OF TIME AS SIGNALS 6552111 Signals and Systems6552111 Signals and Systems Sopapun Suwansawang 2 Figure : Domain, co-domain, and range of a real function of continuous time. )(tfv =
  3. 3. SignalsSignals For example: Electrical signals voltages and currents in a circuit Acoustic signals audio or speech 6552111 Signals and Systems6552111 Signals and Systems Acoustic signals audio or speech signals (analog or digital) Video signals intensity variations in an image Biological signals sequence of bases in a gene Sopapun Suwansawang 3
  4. 4. There are two types of signals: Continuous-time signals (CT) are functions of a continuous variable (time). Discrete-time signals (DT) are functions of 6552111 Signals and Systems6552111 Signals and Systems SignalsSignals Discrete-time signals (DT) are functions of a discrete variable; that is, they are defined only for integer values of the independent variable (time steps). 4Sopapun Suwansawang
  5. 5. CT and DT SignalsCT and DT Signals 6552111 Signals and Systems6552111 Signals and Systems CT DT 5Sopapun Suwansawang Signal such as :)(tx ),...(),...,(),( 10 ntxtxtx or in a shorter form as : ,...,...,, ],...[],...,1[],0[ 10 nxxx nxxx or
  6. 6. where we understand that 6552111 Signals and Systems6552111 Signals and Systems )(][ nn txnxx == and 's are called samples and the time interval between them is called the sampling interval. When nx CT and DT SignalsCT and DT Signals 6Sopapun Suwansawang between them is called the sampling interval. When the sampling intervals are equal (uniform sampling), then n )()(][ snTtn nTxtxnxx s === = where the constant is the sampling intervalsT
  7. 7. 6552111 Signals and Systems6552111 Signals and Systems    < ≥ = 0,0 0,8.0 )( t t tx t    < ≥ = 0,0 0,8.0 ][ n n nx n CT and DT SignalsCT and DT Signals 7Sopapun Suwansawang )(tx t 0 1 0 1 ][nx n 1 2 3 4 5
  8. 8. A discrete-time signal x[n] can be defined in two ways: 1. We can specify a rule for calculating the nth value of the sequence. (see Example 1) 6552111 Signals and Systems6552111 Signals and Systems CT and DT SignalsCT and DT Signals value of the sequence. (see Example 1) 2. We can also explicitly list the values of the sequence. (see Example 2) 8Sopapun Suwansawang
  9. 9. 6552111 Signals and Systems6552111 Signals and Systems DT SignalsDT Signals     ≥ ==       0 ][ 2 1 n xnx n n Example 1 9Sopapun Suwansawang    <  00 n ...}, 8 1 , 4 1 , 2 1 ,1{}{ =nx
  10. 10. 6552111 Signals and Systems6552111 Signals and Systems DT SignalsDT Signals Example 1: Continue 10Sopapun Suwansawang
  11. 11. DT SignalsDT Signals 6552111 Signals and Systems6552111 Signals and Systems Example 2 11Sopapun Suwansawang
  12. 12. DT SignalsDT Signals 6552111 Signals and Systems6552111 Signals and Systems The sequence can be written as Example 2 : continue ,...}0,0,2,0,1,0,1,2,2,1,0,0{...,}{ =nx 12Sopapun Suwansawang }2,0,1,0,1,2,2,1{}{ =nx We use the arrow to denote the n = 0 term. We shall use the convention that if no arrow is indicated, then the first term corresponds to n = 0 and all the values of the sequence are zero for n < 0.
  13. 13. Example 3 Given the continuous-time signal specified by DT SignalsDT Signals 6552111 Signals and Systems6552111 Signals and Systems    ≤≤−− = otherwise tt tx 0 111 )( Determine the resultant discrete-time sequence obtained by uniform sampling of x(t) with a sampling interval of 0.25 s 13Sopapun Suwansawang   otherwise0
  14. 14. Solve : Ts=0.25 s, Ts=1 s, DT SignalsDT Signals 6552111 Signals and Systems6552111 Signals and Systems      = ↑ ,...0,25.0,5.0,75.0,1,75.0,5.0,25.0,0...,][nx    = ,...0,1,0...,][nxTs=1 s, 14Sopapun Suwansawang      = ↑ ,...0,1,0...,][nx
  15. 15. Analog signals 6552111 Signals and Systems6552111 Signals and Systems Analog and Digital SignalsAnalog and Digital Signals If a continuous-time signal x(t) can take on any value in the continuous interval (-∞∞∞∞ , +∞∞∞∞), then the continuous-time signal x(t) is called an analog Digital signals A signal x[n] can take on only a finite number of distinct values, then we call this signal a digital signal. 15Sopapun Suwansawang the continuous-time signal x(t) is called an analog signal.
  16. 16. CT and DT 6552111 Signals and Systems6552111 Signals and Systems Digital SignalsDigital Signals 16Sopapun Suwansawang
  17. 17. CT Binary signal Multi-level signal 6552111 Signals and Systems6552111 Signals and Systems Digital SignalsDigital Signals 17Sopapun Suwansawang
  18. 18. Intuitively, a signal is periodic when it repeats itself. A continuous-time signal x(t) is periodic if there exists a positive real T for which 6552111 Signals and Systems6552111 Signals and Systems Periodic SignalsPeriodic Signals for all t and any integer m.The fundamental period T0 of x(t) is the smallest positive value of T 18Sopapun Suwansawang )()( mTtxtx += 0 0 2 ω π =T
  19. 19. Fundamental frequency 6552111 Signals and Systems6552111 Signals and Systems Periodic SignalsPeriodic Signals 0 0 1 T f = Hz Fundamental angular frequency 19Sopapun Suwansawang 0 00 2 2 T f π πω == rad/sec
  20. 20. A discrete-time signal x[n] is periodic if there exists a positive integer N for which 6552111 Signals and Systems6552111 Signals and Systems Periodic SignalsPeriodic Signals ][][ mNnxnx += for all n and any integer m.The fundamental period N0 of x[n] is the smallest positive integer N 20Sopapun Suwansawang 0 0 2 Ω = π N
  21. 21. Any sequence which is not periodic is called a non-periodic (or aperiodic) sequence. 6552111 Signals and Systems6552111 Signals and Systems NonperiodicNonperiodic SignalsSignals 21Sopapun Suwansawang
  22. 22. 6552111 Signals and Systems6552111 Signals and Systems Periodic SignalsPeriodic Signals CT 22Sopapun Suwansawang DT
  23. 23. Example 3 Find the fundamental frequency in figure below. 6552111 Signals and Systems6552111 Signals and Systems Periodic SignalsPeriodic Signals 23Sopapun Suwansawang Hz T f 4 11 0 0 ==.sec40 =T (sec.)
  24. 24. Exercise Determine whether or not each of the following signals is periodic. If a signal is periodic, determine its fundamental period. 6552111 Signals and Systems6552111 Signals and Systems Periodic SignalsPeriodic Signals ttx ) 4 cos()(.1 π += 24Sopapun Suwansawang nnnx nnx enx tttx tttx nj 4 sin 3 cos][.6 4 1 cos][.5 ][.4 2sincos)(.3 4 sin 3 cos)(.2 4 )4/( ππ ππ π += = = += +=
  25. 25. Solve EX.1 6552111 Signals and Systems6552111 Signals and Systems Periodic SignalsPeriodic Signals 1 4 cos) 4 cos()( 00 =→      +=+= ω π ω π tttx ππ 22 25Sopapun Suwansawang π π ω π 2 1 22 0 0 ===T x(t) is periodic with fundamental period T0 = 2π.
  26. 26. Solve EX.2 6552111 Signals and Systems6552111 Signals and Systems Periodic SignalsPeriodic Signals )()( 4 sin 3 cos)( 21 txtxtttx +=+= ππ ( ) .6 2 cos3/cos)( 111 ==→== π ωπ Ttttxwhere 26Sopapun Suwansawang ( ) .8 4/ 2 sin)4/sin()( .6 3/ cos3/cos)( 222 111 ==→== ==→== π π ωπ π ωπ Ttttx Ttttxwhere numberrationalais T T 4 3 8 6 2 1 == x(t) is periodic with fundamental period T0 = 4T1=3T2=24. Note : Least Common Multiplier of (6,8) is 24
  27. 27. 6552111 Signals and Systems6552111 Signals and Systems Even and Odd Signals:Even and Odd Signals: A signal x(t) or x[n] is referred to as an even signal if ][][ )()( nxnx txtx −= −= 27Sopapun Suwansawang A signal x(t) or x[n] is referred to as an odd signal if ][][ )()( nxnx txtx −=− −=−
  28. 28. 6552111 Signals and Systems6552111 Signals and Systems Even and Odd Signals:Even and Odd Signals: 28Sopapun Suwansawang
  29. 29. 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, 6552111 Signals and Systems6552111 Signals and Systems Even and Odd Signals:Even and Odd Signals: )()()( txtxtx oe += 29Sopapun Suwansawang ][][][ nxnxnx oe oe += { } { }][][ 2 1 ][ )()( 2 1 )( nxnxnx txtxtx e e −+= −+= { } { }][][ 2 1 ][ )()( 2 1 )( nxnxnx txtxtx o o −−= −−= even part odd part
  30. 30. 6552111 Signals and Systems6552111 Signals and Systems Even and Odd Signals:Even and Odd Signals: Example 4 Find the even and odd components of the signals shown in figure below 30Sopapun Suwansawang Solve even part { })()()(2 tftftfe −+= 2fe(t)
  31. 31. Example 4 : continue Odd part 6552111 Signals and Systems6552111 Signals and Systems Even and Odd Signals:Even and Odd Signals: { })()()(2 tftftfo −−= 2fo(t) 31Sopapun Suwansawang 2fo(t)
  32. 32. Example 4 : continue Check )()()( tftftf oe += 6552111 Signals and Systems6552111 Signals and Systems Even and Odd Signals:Even and Odd Signals: { })()( 2 1 )( tftftfo −−={ },)()( 2 1 )( tftftfe −+= 32 )()()( tftftf oe += Sopapun Suwansawang 2 fo(t) fe(t)
  33. 33. Note that the product of two even signals or of two odd signals is an even signal and that the product of an even signal and an odd signal is an odd signal. (even)(even)=even 6552111 Signals and Systems6552111 Signals and Systems Even and Odd Signals:Even and Odd Signals: (even)(even)=even (even)(odd)=odd (odd)(even)=odd (odd)(odd)=even 33Sopapun Suwansawang
  34. 34. Example 5 Show that the product of two even signals or of two odd signals is an even signal and that the product of an even and an odd signaI is an odd signal. 6552111 Signals and Systems6552111 Signals and Systems Even and Odd Signals:Even and Odd Signals: Let )()()( txtxtx = 34Sopapun Suwansawang Let )()()( 21 txtxtx = If x1(t) and x2(t) are both even, then )()()()()()( 2121 txtxtxtxtxtx ==−−=− If x1(t) and x2(t) are both even, then )()()())()(()()()( 212121 txtxtxtxtxtxtxtx ==−−=−−=−
  35. 35. A deterministic signal is a signal in which each value of the signal is fixed and can be determined by a mathematical expression, rule, or table. Because of this the future values of the signal can be calculated from past values with 6552111 Signals and Systems6552111 Signals and Systems Deterministic and Random Signals:Deterministic and Random Signals: signal can be calculated from past values with complete confidence. A random signal has a lot of uncertainty about its behavior. The future values of a random signal cannot be accurately predicted and can usually only be guessed based on the averages of sets of signals 35Sopapun Suwansawang
  36. 36. 6552111 Signals and Systems6552111 Signals and Systems Deterministic and Random Signals:Deterministic and Random Signals: Deterministic 36Sopapun Suwansawang Random
  37. 37. RightRight--Handed and LeftHanded and Left--Handed SignalsHanded Signals A right-handed signal and left-handed signal are those signals whose value is zero between a given variable and positive or negative infinity. Mathematically speaking, A right-handed signal is defined as any signal 6552111 Signals and Systems6552111 Signals and Systems A right-handed signal is defined as any signal where f(t) = 0 for A left-handed signal is defined as any signal where f(t) = 0 for 37Sopapun Suwansawang ∞<< 1tt −∞>> 1tt
  38. 38. RightRight--Handed and LeftHanded and Left--Handed SignalsHanded Signals 6552111 Signals and Systems6552111 Signals and Systems Right-Handed 1t 38Sopapun Suwansawang Left-Handed 1 1t
  39. 39. Causal vs.Anticausal vs. NoncausalCausal vs.Anticausal vs. Noncausal Causal signals are signals that are zero for all negative time. Anticausal signals are signals that are zero for all positive time. 6552111 Signals and Systems6552111 Signals and Systems for all positive time. Noncausal signals are signals that have nonzero values in both positive and negative time. 39Sopapun Suwansawang
  40. 40. Causal vs.Anticausal vs. NoncausalCausal vs.Anticausal vs. Noncausal 6552111 Signals and Systems6552111 Signals and Systems Causal 40Sopapun Suwansawang Anticausal Noncausal
  41. 41. Energy and Power SignalsEnergy and Power Signals 6552111 Signals and Systems6552111 Signals and Systems Consider : 41Sopapun Suwansawang Rti R tv titvtp )( )( )()()( 2 2 = = ⋅= ∫∫ ∫ ∞ ∞− ∞ ∞− ∞ ∞− == = )()()()( 1 )( 22 tdtitdtv R dttpE Power Energy
  42. 42. Total energy E and average power P on a per-ohm basis are Energy and Power SignalsEnergy and Power Signals 6552111 Signals and Systems6552111 Signals and Systems dttiE ∫ ∞ ∞− = )(2 Joules 42Sopapun Suwansawang dtti T P T TT ∫ −∞→ ∞− = )( 2 1 2 lim Watts
  43. 43. For an arbitrary continuous-time signal x(t), the normalized energy content E of x(t) is defined as ∫∫ −∞→ ∞ ∞− == T TT dttxdttxE 22 )()( lim Energy and Power SignalsEnergy and Power Signals 6552111 Signals and Systems6552111 Signals and Systems The normalized average power P of x(t) is defined as 43 −∞→∞− TT ∫ −∞→ = T TT dttx T P 2 )( 2 1 lim Sopapun Suwansawang
  44. 44. Similarly, for a discrete-time signal x[n], the normalized energy content E of x[n] is defined as Energy and Power SignalsEnergy and Power Signals 6552111 Signals and Systems6552111 Signals and Systems ∑∑ −=∞→ ∞ −∞= == N NnNn nxnxE 22 ][lim][ The normalized average power P of x[n] is defined as 44Sopapun Suwansawang −=∞→−∞= NnNn ∑ −=∞→ + = N NnN nx N P 2 ][ 12 1 lim
  45. 45. x(t) (or x[n]) is said to be an energy signal (or sequence) if and only if 0 < E < ∞∞∞∞, and P = 0. x(t) (or x[n]) is said to be a power signal (or sequence) if and only if 0 < P < ∞∞∞∞, thus implying that E = ∞∞∞∞. Energy and Power SignalsEnergy and Power Signals 6552111 Signals and Systems6552111 Signals and Systems implying that E = ∞∞∞∞. Note that a periodic signal is a power signal if its energy content per period is finite, and then the average power of this signal need only be calculated over a period. 45Sopapun Suwansawang ∫= 0 0 2 0 )( 1 T dttx T P
  46. 46. Exercise Determine whether the following signals are energy signals, power signals, or neither. 1. Energy and Power SignalsEnergy and Power Signals 6552111 Signals and Systems6552111 Signals and Systems )cos()( 0 θω += tAtx1. 2. 3. 46Sopapun Suwansawang )cos()( 0 θω += tAtx tj eAtx 0 )( ω = )()( 3 tuetv t− =
  47. 47. Solve Ex.1 The signal x(t) is periodic with T0=2π/ω0. Energy and Power SignalsEnergy and Power Signals 6552111 Signals and Systems6552111 Signals and Systems dttA T dttx T P TT ∫∫ +== 00 0 0 22 00 2 0 )(cos 1 )( 1 θω )cos()( 0 θω += tAtx 47Sopapun Suwansawang 0000 dtt T A P T ∫ ++= 0 0 0 0 2 )22cos(1( 2 1 θω         ++= ∫ ∫ dttdt T A P T T0 0 0 0 00 2 )22(cos1 2 θω 0 2 2 A = ∞< Thus, x(t) is power signal.
  48. 48. Energy and Power SignalsEnergy and Power Signals 6552111 Signals and Systems6552111 Signals and Systems Solve Ex.2 tjAtAeAtx tj 00 sincos)( 0 ωωω +== The signal x(t) is periodic with T0=2π/ω0. Note that periodic signals are, in general, power signals. ∫ T 21 ∫ T 21 48Sopapun Suwansawang ∫= T x dttx T P 0 2 )( 1 ∫= T tj dtAe T 0 2 0 1 ω 2 2 2 0 0 2 1 T T x A A A dt dt T T T T P A W = = = ⋅ = ∫ ∫
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