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- 1. Unit-1 Signal Analysis 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 1 Prepared by: MR . Arun Kumar (Asst.Prof. SISTec-E EC dept.) MR . Shivendra Tiwari (Asst.Prof. SISTec-E EC dept.)
- 2. Content • Periodic Function • Fourier Series • Complex Form of the Fourier Series • Impulse Train • Analysis of Periodic Waveforms • Half-Range Expansion 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 2
- 3. Periodic Function • Any function that satisfies ( ) ( )f t f t T where T is a constant and is called the period of the function. 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 3
- 4. Example: Find its period. 4 cos 3 cos)( tt tf )()( Ttftf )( 4 1 cos)( 3 1 cos 4 cos 3 cos TtTt tt Fact: )2cos(cos m m T 2 3 n T 2 4 mT 6 nT 8 24T smallest T 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 4
- 5. Example: Find its period.tttf 21 coscos)( )()( Ttftf )(cos)(coscoscos 2121 TtTttt mT 21 nT 22 n m 2 1 2 1 must be a rational number 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 5
- 6. Example: Is this function a periodic one? tttf )10cos(10cos)( 10 10 2 1 not a rational number 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 6
- 7. Fourier Series 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 7
- 8. Introduction • Decompose a periodic input signal into primitive periodic components. A periodic sequence T 2T 3T t f(t) 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 8
- 9. Synthesis T nt b T nt a a tf n n n n 2 sin 2 cos 2 )( 11 0 DC Part Even Part Odd Part T is a period of all the above signals )sin()cos( 2 )( 0 1 0 1 0 tnbtna a tf n n n n Let 0=2 /T. 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 9
- 10. Orthogonal Functions • Call a set of functions { k} orthogonal on an interval a < t < b if it satisfies nmr nm dttt n b a nm 0 )()( 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 10
- 11. Orthogonal set of Sinusoidal Functions Define 0=2 /T. 0,0)cos( 2/ 2/ 0 mdttm T T 0,0)sin( 2/ 2/ 0 mdttm T T nmT nm dttntm T T 2/ 0 )cos()cos( 2/ 2/ 00 nmT nm dttntm T T 2/ 0 )sin()sin( 2/ 2/ 00 nmdttntm T T andallfor,0)cos()sin( 2/ 2/ 00 We now prove this one 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 11
- 12. Proof dttntm T T 2/ 2/ 00 )cos()cos( 0 )]cos()[cos( 2 1 coscos dttnmdttnm T T T T 2/ 2/ 0 2/ 2/ 0 ])cos[( 2 1 ])cos[( 2 1 2/ 2/0 0 2/ 2/0 0 ])sin[( )( 1 2 1 ])sin[( )( 1 2 1 T T T T tnm nm tnm nm m n ])sin[(2 )( 1 2 1 ])sin[(2 )( 1 2 1 00 nm nm nm nm 0 04/26/2014 prepared by Arun Kumar & Shivendra Tiwari 12
- 13. Proof dttntm T T 2/ 2/ 00 )cos()cos( 0 )]cos()[cos( 2 1 coscos dttm T T 2/ 2/ 0 2 )(cos 2/ 2/ 0 0 2/ 2/ ]2sin 4 1 2 1 T T T T tm m t m = n 2 T ]2cos1[ 2 1 cos2 dttm T T 2/ 2/ 0 ]2cos1[ 2 1 nmT nm dttntm T T 2/ 0 )cos()cos( 2/ 2/ 00 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 13
- 14. Proof dttntm T T 2/ 2/ 00 )cos()cos( 0 )]cos()[cos( 2 1 coscos dttm T T 2/ 2/ 0 2 )(cos 2/ 2/ 0 0 2/ 2/ ]2sin 4 1 2 1 T T T T tm m t m = n 2 T ]2cos1[ 2 1 cos2 dttm T T 2/ 2/ 0 ]2cos1[ 2 1 nmT nm dttntm T T 2/ 0 )cos()cos( 2/ 2/ 00 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 14
- 15. Orthogonal set of Sinusoidal Functions Define 0=2 /T. 0,0)cos( 2/ 2/ 0 mdttm T T 0,0)sin( 2/ 2/ 0 mdttm T T nmT nm dttntm T T 2/ 0 )cos()cos( 2/ 2/ 00 nmT nm dttntm T T 2/ 0 )sin()sin( 2/ 2/ 00 nmdttntm T T andallfor,0)cos()sin( 2/ 2/ 00 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 15
- 16. Decomposition dttf T a Tt t 0 0 )( 2 0 ,2,1cos)( 2 0 0 0 ntdtntf T a Tt t n ,2,1sin)( 2 0 0 0 ntdtntf T b Tt t n )sin()cos( 2 )( 0 1 0 1 0 tnbtna a tf n n n n 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 16
- 17. Proof Use the following facts: 0,0)cos( 2/ 2/ 0 mdttm T T 0,0)sin( 2/ 2/ 0 mdttm T T nmT nm dttntm T T 2/ 0 )cos()cos( 2/ 2/ 00 nmT nm dttntm T T 2/ 0 )sin()sin( 2/ 2/ 00 nmdttntm T T andallfor,0)cos()sin( 2/ 2/ 00 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 17
- 18. Example (Square Wave) 11 2 2 0 0 dta ,2,10sin 1 cos 2 2 00 nnt n ntdtan ,6,4,20 ,5,3,1/2 )1cos( 1 cos 1 sin 2 2 00 n nn n n nt n ntdtbn 2 3 4 5--2-3-4-5-6 f(t) 1 ttttf 5sin 5 1 3sin 3 1 sin 2 2 1 )( 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 18
- 19. Harmonics T nt b T nt a a tf n n n n 2 sin 2 cos 2 )( 11 0 DC Part Even Part Odd Part T is a period of all the above signals )sin()cos( 2 )( 0 1 0 1 0 tnbtna a tf n n n n 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 19
- 20. Harmonics tnbtna a tf n n n n 0 1 0 1 0 sincos 2 )( T f 2 2 00 Define , called the fundamental angular frequency. 0nnDefine , called the n-th harmonic of the periodic function. tbta a tf n n nn n n sincos 2 )( 11 0 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 20
- 21. Harmonics tbta a tf n n nn n n sincos 2 )( 11 0 )sincos( 2 1 0 tbta a nnn n n 1 2222 220 sincos 2 n n nn n n nn n nn t ba b t ba a ba a 1 220 sinsincoscos 2 n nnnnnn ttba a )cos( 1 0 n n nn tCC 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 21
- 22. Amplitudes and Phase Angles )cos()( 1 0 n n nn tCCtf 2 0 0 a C 22 nnn baC n n n a b1 tan harmonic amplitude phase angle 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 22
- 23. Fourier Series Complex form of the Fourier Series 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 23
- 24. Complex Exponentials tnjtne tjn 00 sincos0 tjntjn eetn 00 2 1 cos 0 tnjtne tjn 00 sincos0 tjntjntjntjn ee j ee j tn 0000 22 1 sin 0 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 24
- 25. Complex Form of the Fourier Series tnbtna a tf n n n n 0 1 0 1 0 sincos 2 )( tjntjn n n tjntjn n n eeb j eea a 0000 11 0 22 1 2 1 0 00 )( 2 1 )( 2 1 2 n tjn nn tjn nn ejbaejba a 1 0 00 n tjn n tjn n ececc )( 2 1 )( 2 1 2 0 0 nnn nnn jbac jbac a c 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 25
- 26. Complex Form of the Fourier Series 1 0 00 )( n tjn n tjn n ececctf 1 1 0 00 n tjn n n tjn n ececc n tjn nec 0 )( 2 1 )( 2 1 2 0 0 nnn nnn jbac jbac a c 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 26
- 27. Complex Form of the Fourier Series 2/ 2/ 0 0 )( 1 2 T T dttf T a c )( 2 1 nnn jbac 2/ 2/ 0 2/ 2/ 0 sin)(cos)( 1 T T T T tdtntfjtdtntf T 2/ 2/ 00 )sin)(cos( 1 T T dttnjtntf T 2/ 2/ 0 )( 1 T T tjn dtetf T 2/ 2/ 0 )( 1 )( 2 1 T T tjn nnn dtetf T jbac )( 2 1 )( 2 1 2 0 0 nnn nnn jbac jbac a c 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 27
- 28. Complex Form of the Fourier Series n tjn nectf 0 )( dtetf T c T T tjn n 2/ 2/ 0 )( 1 )( 2 1 )( 2 1 2 0 0 nnn nnn jbac jbac a c If f(t) is real, * nn cc nn j nnn j nn ecccecc ||,|| * 22 2 1 |||| nnnn bacc n n n a b1 tan ,3,2,1n 00 2 1 ac 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 28
- 29. Complex Frequency Spectra nn j nnn j nn ecccecc ||,|| * 22 2 1 |||| nnnn bacc n n n a b1 tan ,3,2,1n 00 2 1 ac |cn| amplitude spectrum n phase spectrum 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 29
- 30. Example 2 T 2 T TT 2 d t f(t) A 2 d dte T A c d d tjn n 2/ 2/ 0 2/ 2/0 0 1 d d tjn e jnT A 2/ 0 2/ 0 00 11 djndjn e jn e jnT A )2/sin2( 1 0 0 dnj jnT A 2/sin 1 0 02 1 dn nT A T dn T dn T Ad sin 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 30
- 31. T dn T dn T Ad cn sin 8 2 5 1 T , 4 1 , 20 1 0 T d Td Example 40 80 120-40 0-120 -80 A/5 5 0 10 0 15 0-5 0-10 0-15 0 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 31
- 32. T dn T dn T Ad cn sin 4 2 5 1 T , 2 1 , 20 1 0 T d Td Example 40 80 120-40 0 -120 -80 A/10 10 0 20 0 30 0-10 0-20 0-30 0 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 32
- 33. Example dte T A c d tjn n 0 0 d tjn e jnT A 00 0 1 00 11 0 jn e jnT A djn )1( 1 0 0 djn e jnT A 2/0 sin djn e T dn T dn T Ad TT d t f(t) A 0 )( 1 2/2/2/ 0 000 djndjndjn eee jnT A 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 33
- 34. Fourier Series Impulse Train 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 34
- 35. Dirac Delta Function 0 00 )( t t t and 1)( dtt 0 t Also called unit impulse function. 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 35
- 36. Property )0()()( dttt )0()()0()0()()()( dttdttdttt (t): Test Function 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 36
- 37. Impulse Train 0 tT 2T 3TT2T3T n T nTtt )()( 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 37
- 38. Fourier Series of the Impulse Train n T nTtt )()( T dtt T a T T T 2 )( 2 2/ 2/ 0 T dttnt T a T T Tn 2 )cos()( 2 2/ 2/ 0 0)sin()( 2 2/ 2/ 0 dttnt T b T T Tn n T tn TT t 0cos 21 )( 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 38
- 39. Complex Form Fourier Series of the Impulse Train T dtt T a c T T T 1 )( 1 2 2/ 2/ 0 0 T dtet T c T T tjn Tn 1 )( 1 2/ 2/ 0 n tjn T e T t 0 1 )( n T nTtt )()( 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 39
- 40. Fourier Series Analysis of Periodic Waveforms 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 40
- 41. Waveform Symmetry • Even Functions • Odd Functions )()( tftf )()( tftf 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 41
- 42. Decomposition • Any function f(t) can be expressed as the sum of an even function fe(t) and an odd function fo(t). )()()( tftftf oe )]()([)( 2 1 tftftfe )]()([)( 2 1 tftftfo Even Part Odd Part 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 42
- 43. Example 00 0 )( t te tf t Even Part Odd Part 0 0 )( 2 1 2 1 te te tf t t e 0 0 )( 2 1 2 1 te te tf t t o 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 43
- 44. Half-Wave Symmetry )()( Ttftf and 2/)( Ttftf TT/2T/2 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 44
- 45. Quarter-Wave Symmetry Even Quarter-Wave Symmetry TT/2T/2 Odd Quarter-Wave Symmetry T T/2T/2 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 45
- 46. Hidden Symmetry • The following is a asymmetry periodic function: Adding a constant to get symmetry property. A TT A/2 A/2 TT 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 46
- 47. Fourier Coefficients of Symmetrical Waveforms • The use of symmetry properties simplifies the calculation of Fourier coefficients. – Even Functions – Odd Functions – Half-Wave – Even Quarter-Wave – Odd Quarter-Wave 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 47
- 48. Fourier Coefficients of Even Functions )()( tftf tna a tf n n 0 1 0 cos 2 )( 2/ 0 0 )cos()( 4 T n dttntf T a 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 48
- 49. Fourier Coefficients of Even Functions )()( tftf tnbtf n n 0 1 sin)( 2/ 0 0 )sin()( 4 T n dttntf T b 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 49
- 50. FourierCoefficientsfor Half-Wave Symmetry )()( Ttftf and 2/)( Ttftf TT/2T/2 The Fourier series contains only odd harmonics. 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 50
- 51. FourierCoefficientsfor Half-Wave Symmetry )()( Ttftf and 2/)( Ttftf )sincos()( 1 00 n nn tnbtnatf oddfor)cos()( 4 evenfor0 2/ 0 0 ndttntf T n a T n oddfor)sin()( 4 evenfor0 2/ 0 0 ndttntf T n b T n 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 51
- 52. Fourier Coefficients for Even Quarter-Wave Symmetry TT/2T/2 ])12cos[()( 0 1 12 tnatf n n 4/ 0 012 ])12cos[()( 8 T n dttntf T a 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 52
- 53. Fourier Transform and Applications By Njegos Nincic Fourier 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 53
- 54. Overview • Transforms – Mathematical Introduction • Fourier Transform – Time-Space Domain and Frequency Domain – Discret Fourier Transform 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 54
- 55. Transforms • Transform: – In mathematics, a function that results when a given function is multiplied by a so-called kernel function, and the product is integrated between suitable limits. (Britannica) • Can be thought of as a substitution 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 55
- 56. Transforms • Example of a substitution: • Original equation: x + 4x² – 8 = 0 • Familiar form: ax² + bx + c = 0 • Let: y = x² • Solve for y • x = √y 4 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 56
- 57. Fourier Transform • Property of transforms: – They convert a function from one domain to another with no loss of information • Fourier Transform: converts a function from the time (or spatial) domain to the frequency domain4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 57
- 58. Time Domain and Frequency Domain • Time Domain: – Tells us how properties (air pressure in a sound function, for example) change over time: • Amplitude = 100 • Frequency = number of cycles in one second = 200 Hz 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 58
- 59. Time Domain and Frequency Domain • Frequency domain: – Tells us how properties (amplitudes) change over frequencies: 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 59
- 60. Time Domain and Frequency Domain • Example: – Human ears do not hear wave-like oscilations, but constant tone • Often it is easier to work in the frequency domain 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 60
- 61. Time Domain and Frequency Domain • In 1807, Jean Baptiste Joseph Fourier showed that any periodic signal could be represented by a series of sinusoidal functions In picture: the composition of the first two functions gives the bottom one4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 61
- 62. Time Domain and Frequency Domain 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 62
- 63. Fourier Transform • Because of the property: • Fourier Transform takes us to the frequency domain: 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 63
- 64. Fourier Series Half-Range Expansions 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 64
- 65. Non-Periodic Function Representation • A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ). 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 65
- 66. Without Considering Symmetry • A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ). T 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 66
- 67. Expansion Into Even Symmetry • A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ). T=2 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 67
- 68. Expansion Into Odd Symmetry • A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ). T=2 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 68
- 69. Expansion Into Half-Wave Symmetry • A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ). T=2 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 69
- 70. ExpansionInto Even Quarter-Wave Symmetry • A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ). T/2=2 T=4 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 70
- 71. Expansion Into Odd Quarter-Wave Symmetry • A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ). T/2=2 T=4 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 71
- 72. What is a System? • (DEF) System : A system is formally defined as an entity that manipulates one or more signals to accomplish a function, thereby yielding new signals. system output signal input signal 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 72
- 73. Some Interesting Systems • Communication system • Control systems • Remote sensing system • Biomedical system(biomedical signal processing) • Auditory system 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 73
- 74. Some Interesting Systems • Communication system 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 74
- 75. Some Interesting Systems • Control systems 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 75
- 76. Some Interesting Systems Papero 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 76
- 77. Some Interesting Systems • Remote sensing system Perspectival view of Mount Shasta (California), derived from a pair of stereo radar images acquired from orbit with the shuttle Imaging Radar (SIR-B). (Courtesy of Jet Propulsion Laboratory.) 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 77
- 78. Some Interesting Systems • Biomedical system(biomedical signal processing) 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 78
- 79. Some Interesting Systems • Auditory system 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 79
- 80. Classification of Signals • Continuous and discrete-time signals • Continuous and discrete-valued signals • Even and odd signals • Periodic signals, non-periodic signals • Deterministic signals, random signals • Causal and anticausal signals • Right-handed and left-handed signals • Finite and infinite length 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 80
- 81. Continuous and discrete-time signals • Continuous signal - It is defined for all time t : x(t) • Discrete-time signal - It is defined only at discrete instants of time : x[n]=x(nT) 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 81
- 82. Continuous and Discrete valued singals • CV corresponds to a continuous y-axis • DV corresponds to a discrete y-axis Digital signal 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 82
- 83. Even and odd signals • Even signals : x(-t)=x(t) • Odd signals : x(-t)=-x(t) • Even and odd signal decomposition xe(t)= 1/2·(x(t)+x(-t)) xo(t)= 1/2·(x(t)-x(-t)) 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 83
- 84. 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 84
- 85. Periodic signals, non-periodic signals • Periodic signals - A function that satisfies the condition x(t)=x(t+T) for all t - Fundamental frequency : f=1/T - Angular frequency : = 2 /T • Non-periodic signals 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 85
- 86. Deterministic signals, random signals Deterministic signals -There is no uncertainty with respect to its value at any time. (ex) sin(3t) Random signals - There is uncertainty before its actual occurrence. 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 86
- 87. Causal and anticausal Signals • Causal signals : zero for all negative time • Anticausal signals : zero for all positive time • Noncausal : nozero values in both positive and negative time causal signal anticausal signal noncausal signal 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 87
- 88. Right-handed and left-handed Signals • Right-handed and left handed-signal : zero between a given variable and positive or negative infinity 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 88
- 89. Finite and infinite length • Finite-length signal : nonzero over a finite interval tmin< t< tmax • Infinite-length singal : nonzero over all real numbers 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 89
- 90. Unit-2 Modulation Techniques 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 90
- 91. Amplitude Modulation prepared by Arun Kumar & Shivendra Tiwari4/26/2014 91
- 92. Content • What is Modulation • Amplitude Modulation (AM) • Demodulation of AM signals prepared by Arun Kumar & Shivendra Tiwari 4/26/2014 92
- 93. What is Modulation • Modulation – In the modulation process, some characteristic of a high- frequency carrier signal (bandpass), is changed according to the instantaneous amplitude of the information (baseband) signal. • Why Modulation – Suitable for signal transmission (distance…etc) – Multiple signals transmitted on the same channel – Capacitive or inductive devices require high frequency AC input (carrier) to operate. – Stability and noise rejection prepared by Arun Kumar & Shivendra Tiwari 4/26/2014 93
- 94. About Modulation • Application Examples – broadcasting of both audio and video signals. – Mobile radio communications, such as cell phone. prepared by Arun Kumar & Shivendra Tiwari • Basic Modulation Types – Amplitude Modulation: changes the amplitude. – Frequency Modulation: changes the frequency. – Phase Modulation: changes the phase. 4/26/2014 94
- 95. AM Modulation/Demodulation prepared by Arun Kumar & Shivendra Tiwari Modulator Demodulator Baseband Signal with frequency fm (Modulating Signal) Bandpass Signal with frequency fc (Modulated Signal) Channel Original Signal with frequency fm Source Sink fc >> fm Voice: 300-3400Hz GSM Cell phone: 900/1800MHz 4/26/2014 95
- 96. Amplitude Modulation • The amplitude of high-carrier signal is varied according to the instantaneous amplitude of the modulating message signal m(t). Carrier Signal: or Modulating Message Signal: or The AM Signal: cos(2 ) cos( ) ( ): cos(2 ) cos( ) ( ) [ ( )]cos(2 ) c c m m AM c c f t t m t f t t s t A m t f t prepared by Arun Kumar & Shivendra Tiwari 4/26/2014 96
- 97. * AM Signal Math Expression* • Mathematical expression for AM: time domain • expanding this produces: • In the frequency domain this gives: prepared by Arun Kumar & Shivendra Tiwari ( ) (1 cos ) cosAM m cS t k t t ( ) cos cos cosc cAM mS t t k t t )cos()cos(coscos:using 2 1 BABABA 2 2( ) cos cos( ) cos( )c c c k k AM m mS t t t t freque ncy k/2k/2 Carrier, A=1. upper sideband lower sideband Amplitud e fcfc-fm fc+fm 4/26/2014 97
- 98. AM Power Frequency Spectrum • AM Power frequency spectrum obtained by squaring the amplitude: • Total power for AM: prepared by Arun Kumar & Shivendra Tiwari . 2 2 2 2 4 4 1 2 k k A k freq k2/4k2/4 Carrier, A2=12 = 1 Power fcfc-fm fc+fm 4/26/2014 98
- 99. Amplitude Modulation • The AM signal is generated using a multiplier. • All info is carried in the amplitude of the carrier, AM carrier signal has time-varying envelope. • In frequency domain the AM waveform are the lower-side frequency/band (fc - fm), the carrier frequency fc, the upper-side frequency/band (fc + fm). prepared by Arun Kumar & Shivendra Tiwari 4/26/2014 99
- 100. AM Modulation – Example • The information signal is usually not a single frequency but a range of frequencies (band). For example, frequencies from 20Hz to 15KHz. If we use a carrier of 1.4MHz, what will be the AM spectrum? • In frequency domain the AM waveform are the lower-side frequency/band (fc - fm), the carrier frequency fc, the upper- side frequency/band (fc + fm). Bandwidth: 2x(25K-20)Hz. prepared by Arun Kumar & Shivendra Tiwari frequen cy 1.4 MHz 1,385,000Hz to 1,399,980Hz 1,400,020Hz to 1,415,000Hz fc 4/26/2014 100
- 101. Modulation Index of AM Signal m c A k A )2cos()( tfAtm mm Carrier Signal: cos(2 ) DC:c Cf t A Modulated Signal: ( ) [ cos(2 )]cos(2 ) [1 cos(2 )]cos(2 ) AM c m m c c m c S t A A f t f t A k f t f t prepared by Arun Kumar & Shivendra Tiwari For a sinusoidal message signal Modulation Index is defined as: Modulation index k is a measure of the extent to which a carrier voltage is varied by the modulating signal. When k=0 no modulation, when k=1 100% modulation, when k>1 over modulation. 4/26/2014 101
- 102. prepared by Arun Kumar & Shivendra Tiwari Modulation Index of AM Signal 4/26/2014 102
- 103. prepared by Arun Kumar & Shivendra Tiwari Modulation Index of AM Signal 4/26/2014 103
- 104. prepared by Arun Kumar & Shivendra Tiwari Modulation Index of AM Signal 4/26/2014 104
- 105. High Percentage Modulation • It is important to use as high percentage of modulation as possible (k=1) while ensuring that over modulation (k>1) does not occur. • The sidebands contain the information and have maximum power at 100% modulation. • Useful equation Pt = Pc(1 + k2/2) Pt =Total transmitted power (sidebands and carrier) Pc = Carrier power prepared by Arun Kumar & Shivendra Tiwari 4/26/2014 105
- 106. Demodulation of AM Signals prepared by Arun Kumar & Shivendra Tiwari Demodulation extracting the baseband message from the carrier. •There are 2 main methods of AM Demodulation: • Envelope or non-coherent detection or demodulation. • Synchronised or coherent demodulation. 4/26/2014 106
- 107. Envelope/Diode AM Detector prepared by Arun Kumar & Shivendra Tiwari If the modulation depth is > 1, the distortion below occurs K>1 4/26/2014 107
- 108. Synchronous or Coherent Demodulation prepared by Arun Kumar & Shivendra Tiwari This is relatively more complex and more expensive. The Local Oscillator (LO) must be synchronised or coherent, i.e. at the same frequency and in phase with the carrier in the AM input signal. 4/26/2014 108
- 109. Synchronous or Coherent Demodulation prepared by Arun Kumar & Shivendra Tiwari If the AM input contains carrier frequency, the LO or synchronous carrier may be derived from the AM input. 4/26/2014 109
- 110. Synchronous or Coherent Demodulation prepared by Arun Kumar & Shivendra Tiwari If we assume zero path delay between the modulator and demodulator, then the ideal LO signal is cos( ct). 4/26/2014 110
- 111. Unit-3 Angle Modulation 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 111
- 112. Angle Modulation • Introduction • Types of Angle Modulation – FM & PM • Definition – FM & PM • Signal Representation of FM & PM • Generation of PM using FM • Generation of FM using PM 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 112
- 113. Angle Modulation Consider again the general carrier cccc φ+tωV=tv cos cc φ+tω represents the angle of the carrier. There are two ways of varying the angle of the carrier. a) By varying the frequency, c – Frequency Modulation. b)By varying the phase, c – Phase Modulation 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 113
- 114. Phase Modulation • One of the properties of a sinusoidal wave is its phase, the offset from a reference time at which the sine wave begins. • We use the term phase shift to characterize such changes. • If phase changes after cycle k, the next sinusoidal wave will start slightly later than the time at which cycle k completes. 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 114
- 115. Introduction to Angle Modulation • High degree of noise immunity by bandwidth expansion. • They are widely used in high-fidelity music broadcasting. • They are of constant envelope, which is beneficial when amplified by nonlinear amplifiers. 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 115
- 116. Introduction to Angle Modulation 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 116
- 117. FM and PM 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 117
- 118. Types of FM • Basically 2 types of FM: – NBFM (Narrow Band Frequency Modulation) – WBFM (Wide Band Frequency Modulation) 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 118
- 119. Generation of FM • Mainly there are 2 methods to generate FM Signal. They are: 1. Direct Method 1. Hartley Oscillator 2. Basic Reactance Modulator 2. Indirect Method 1. Amstrong Modulator (Using NB Phase Modulator) 2. Frequency Multiplier 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 119
- 120. Generation of FM • Basically two methods: 1. Direct method • Build a voltage controlled oscillator (VCO) where the frequency is varied in response to an applied modulating voltage by using a voltage-variable capacitor • The main difficulty is that it is very difficult to maintain the stability of the carrier frequency of the VCO when used to generate wide-band FM. 2. Indirect method • Use a narrow-band FM modulator followed by frequency multiplier and mixer for up conversion. • Allows to decouple the problem of carrier frequency stability from the FM modulation.4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 120
- 121. Edwin Howard Armstrong (1890 - †1954) Edwin Howard Armstrong received his engineering degree in 1913 at the Columbia University. He was the inventor of the following basic electronic circuits underlying all modern radio, radar, and television: Regenerative Circuit (1912) Superheterodyne Circuit (1918) Superregenerative Circuit (1922) FM System (1933). 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 121
- 122. Indirect Method – Amstrong Modulator 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 122
- 123. Indirect Method 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 123
- 124. Narrow Band Phase Modulator (NBPM) 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 124
- 125. Detection of FM • Types of FM Detectors: 1. RL Discriminator 2. Tuned FM Discriminator 3. Balanced Slope Detector 4. Centre Tuned Discriminator / Phase Discriminator / Foster – Seeley Discriminator 5. Phase Locked Loop (PLL) Demodulator 6. Ratio Detector 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 125
- 126. Unit-4 Radio Transmitters and Receiver 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 126
- 127. Transmitters and Receivers • Generalized Transmitters • AM PM Generation • Inphase and Quadrature Generation • Superheterodyne Receiver • Frequency Division Multiplexing 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 127
- 128. Generalized Transmitters Re cos cos sin Where cj t c c c j t v t g t e R t t t v t x t t y t t g t R t e x t jy t Any type of modulated signal can be represented by The complex envelope g(t) is a function of the modulating signal m(t) Transmitter Modulating signal Modulated signal Example: ( ) Type of Modulation g(m) AM : [1 ( )] PM : p c jD m t c A m t A e 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 128
- 129. Generalized Transmitters R(t) and θ(t) are functions of the modulating signal m(t) as given in TABLE 4.1 • Two canonical forms for the generalized transmitter: cos cv t R t t t 1. AM- PM Generation Technique: Envelope and phase functions are generated to modulate the carrier as Generalized transmitter using the AM–PM generation technique. 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 129
- 130. Generalized Transmitters x(t) and y(t) are functions of the modulating signal m(t) as given in TABLE 4.1 ttyttxtv cc sincos 2. Quadrature Generation Technique: Inphase and quadrature signals are generated to modulate the carrier as Fig. 4–28 Generalized transmitter using the quadrature generation technique. 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 130
- 131. IQ (In-phase and Quadrature-phase) Detector 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 131
- 132. Generalized Receivers Receivers Tuned Radio Frequency (TRF) Receiver: Composed of RF amplifiers and detectors. No frequency conversion It is not often used. Difficult to design tunable RF stages. Difficult to obtain high gain RF amplifiers Superheterodyne Receiver: Downconvert RF signal to lower IF frequency Main amplifixcation takes place at IF Two types of receivers: 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 132
- 133. Tuned Radio Frequency (TRF) Receivers Active Tuning Circuit Detector Circuit Local Oscillator Bandpass Filter Baseband Audio Amp Composed of RF amplifiers and detectors. No frequency conversion. It is not often used. Difficult to design tunable RF stages. Difficult to obtain high gain RF amplifiers 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 133
- 134. Heterodyning (Upconversion/ Downconversion) Subsequent Processing (common) All Incoming Frequencies Fixed Intermediate Frequency Heterodyning 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 134
- 135. Superheterodyne Receivers Superheterodyne Receiver Diagram4/26/2014 prepared by Arun Kumar & Shivendra 135
- 136. Superheterodyne Receiver 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 136
- 137. Superheterodyne Receivers The RF and IF frequency responses H1(f) and H2(f) are important in providing the required reception characteristics. 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 137
- 138. Superheterodyne Receivers fI F fI F RF Response IF Response 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 138
- 139. Superheterodyne Receivers 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 139
- 140. Superheterodyne Receiver Frequencies 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 140
- 141. Superheterodyne Receiver Frequencies 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 141
- 142. Frequency Conversion Process 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 142
- 143. Image frequency not a problem. Image Frequencies Image frequency is also received 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 143
- 144. AM Radio Receiver 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 144
- 145. Superheterodyne Receiver Typical Signal Levels 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 145
- 146. Double-conversion block diagram. 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 146
- 147. Unit-5 Noise 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 147
- 148. prepared by Arun Kumar & Shivendra Tiwari Noise is the Undesirable portion of an electrical signal that interferes with the intelligence 4/26/2014 148
- 149. prepared by Arun Kumar & Shivendra Tiwari Why is it important to study the effects of Noise? a) Today’s telecom networks handle enormous volume of data b) The switching equipment needs to handle high traffic volumes as well c) our ability to recover the required data without error is inversely proportional to the magnitude of noise What steps are taken to minimize the effects of noise? a) Special encoding and decoding techniques used to optimize the recovery of the signal b) Transmission medium is chosen based on the bandwidth, end to end reliability requirements, anticipated surrounding noise levels and the distance to destination c) Elaborate error detection and correction mechanisms utilized in the communications systems 4/26/2014 149
- 150. prepared by Arun Kumar & Shivendra Tiwari The decibel (abbreviated dB) is the unit used to measure the intensity of a sound.! The smallest audible sound (near total silence) is 0 dB. A sound 10 times more powerful is 10 dB. A sound 1,000 times more powerful than near total silence is 30 dB. Here are some common sounds and their decibel ratings: Normal conversation - 60 dB A rock concert - 120 dB It takes approximate 4 hours of exposure to a 120-dB sound to cause damage to your ears, however 140-dB sound can result in an immediate damage 4/26/2014 150
- 151. prepared by Arun Kumar & Shivendra Tiwari Signal to Noise ratio It is a ratio of signal power to Noise power at some point in a Telecom system expressed in decibels (dB) It is typically measured at the receiving end of the communications system BEFORE the detection of signal. SNR = 10 Log (Signal power/ Noise power) dB SNR = 10 Log (Vs/VN)2 = 20 Log (Vs/VN) 4/26/2014 151
- 152. prepared by Arun Kumar & Shivendra Tiwari 1) The noise power at the output of receiver’s IF stage is measured at 45 µW. With receiver tuned to test signal, output power increases to 3.58 mW. Compute the SNR SNR = 10 Log (Signal power/ Noise power) dB = 10 Log (3.58 mW/ 45 µW) = 19 dB 2) A 1 kHZ test tone measured with an oscilloscope at the input of receiver’s FM detector stage. Its peak to peak voltage is 3V. With test tone at transmitter turned off, the noise at same test point is measure with an rms voltmeter. Its value is 640 mV. Compute SNR in dB. SNR = 20 Log (Vs/Vn) = 20 Log ((.707 x Vp-p/2)/Vn) = 20 Log (1.06V/640 mV) = 4.39 dB 4/26/2014 152
- 153. prepared by Arun Kumar & Shivendra Tiwari Noise Factor (F) It is a measure of How Noisy A Device Is Noise figure (NF) = Noise factor expressed in dB F = (Si/Ni) / (So/No) NF = 10 Log F 4/26/2014 153
- 154. prepared by Arun Kumar & Shivendra Tiwari Noise Types • Atmospheric and Extraterrestrial noise • Gaussian Noise • Crosstalk • Impulse Noise 4/26/2014 154
- 155. prepared by Arun Kumar & Shivendra Tiwari Atmospheric and Extraterrestrial Noise • Lightning: The static discharge generates a wide range of frequencies • Solar Noise: Ionised gases of SUN produce a wide range of frequencies as well. • Cosmic Noise: Distant stars radiate intense level of noise at frequencies that penetrate the earth’s atmosphere. 4/26/2014 155
- 156. prepared by Arun Kumar & Shivendra Tiwari Gaussian Noise: The cumulative effect of all random noise generated over a period of time (it includes all frequencies). Thermal Noise: generated by random motion of free electrons and molecular vibrations in resistive components. The power associated with thermal noise is proportional to both temperature and bandwidth Pn = K x T x BW K = Boltzmann’s constant 1.38x10 -23 T = Absolute temperature of device BW = Circuit bandwidth 4/26/2014 156
- 157. prepared by Arun Kumar & Shivendra Tiwari Shot Noise Results from the random arrival rate of discrete current carriers at the output electrodes of semiconductor and vaccum tube devices. Noise current associated with shot noise can be computed as In = √ 2qIf In = Shot noise current in rms q = charge of an electron I = DC current flowing through the device f = system bandwidth (Hz) 4/26/2014 157
- 158. prepared by Arun Kumar & Shivendra Tiwari Crosstalk: electrical noise or interference caused by inductive and capacitive coupling of signals from adjacent channels In LANs, the crosstalk noise has greater effect on system Performance than any other types of noise Problem remedied by using UTP or STP. By twisting the cable pairs together, the EMF surrounding the wires cancel out each other. 4/26/2014 158
- 159. prepared by Arun Kumar & Shivendra Tiwari Near end crosstalk: Occurs at transmitting station when strong signals radiating from transmitting pair of wires are coupled in to adjacent weak signals traveling in opposite direction Far end crosstalk: Occurs at the far end receiver as a result of adjacent signals traveling in the same direction 4/26/2014 159
- 160. prepared by Arun Kumar & Shivendra Tiwari Minimizing crosstalk in telecom systems 1) Using twisted pair of wires 2) Use of shielding to prevent signals from radiating in to other conductors 3) Transmitted and received signals over long distance are physically separated and shielded 4) Differential amplifiers and receivers are used to reject common-mode signals 5) Balanced transformers are used with twisted pair media to cancel crosstalk signals coupled equally in both lines 6) Maximum channels used within a cable are limited to a certain value 4/26/2014 160
- 161. prepared by Arun Kumar & Shivendra Tiwari Impulse Noise: Noise consisting of sudden bursts of irregularly shape pulses and lasting for a few Microseconds to several hundred milliseconds. What causes Impulse noise? a) Electromechanical switching relays at the C.O. b) Electrical motors and appliances, ignition systems c) Lightning 4/26/2014 161
- 162. Noise factor • IEEE Standards: “The noise factor, at a specified input frequency, is defined as the ratio of (1) the total noise power per unit bandwidth available at the output port when noise temperature of the input termination is standard (290 K) to (2) that portion of (1) engendered at the input frequency by the input termination.” sourcetoduenoiseoutputavailable powernoiseoutputavailable F 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 162
- 163. Noise factor (cont.) • It is a measure of the degradation of SNR due to the noise added - • Implies that SNR gets worse as we process the signal • Spot noise factor • The answer is the bandwidth7/1/2013163 i a o o i i oi iai NfG N S N N S SN SNfGN F )( 1 ))(( 1 o i SNR SNR F kT N F a 1 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 163
- 164. Noise factor (cont.) • Quantitative measure of receiver performance wrt noise for a given bandwidth • Noise figure – Typically 8-10 db for modern receivers • Multistage (cascaded) system )log(10 FNF 12121 3 1 2 1 1 ... 11 n n GGG F GG F G F FF 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 164
- 165. Thank you 4/26/2014 prepared by Arun Kumar & Shivendra Tiwari 165

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