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- 1. 2/9/2014 Communication Systems 1 Communication Systems Instructor: Engr. Dr. Sarmad Ullah Khan Assistant ProfessorAssistant Professor Electrical Engineering Department CECOS University of IT and Emerging Sciences Sarmad@cecos.edu.pk Chapter 2 Dr. Sarmad Ullah Khan Signals and Signal Space 2
- 2. 2/9/2014 Communication Systems 2 Outlines • Signals and systems • Size of signal Dr. Sarmad Ullah Khan Size of signal • Classification of signals • Signal operations • The unit impulse function • Signals versus Vectors • Correlation O th l i l• Orthogonal signals • Trigonometric Fourier Series • Exponential Fourier Series 3 Outlines • Signals and systems • Size of signal Dr. Sarmad Ullah Khan Size of signal • Classification of signals • Signal operations • The unit impulse function • Signals versus Vectors • Correlation O th l i l• Orthogonal signals • Trigonometric Fourier Series • Exponential Fourier Series 4
- 3. 2/9/2014 Communication Systems 3 Signal and system • Signal Signal is a set of Information / Data Dr. Sarmad Ullah Khan Signal is a set of Information / Data For example, Telephone, Telegraph, Stock Market Price Signals are function of independent variable time In electric charge distribution over surface signalIn electric charge distribution over surface, signal is function of space rather than time 5 Signal and system • System System process the received signal Dr. Sarmad Ullah Khan System process the received signal System might modify or extract information from signal For example, Anti aircraft missile launcher may want to know the future location of targetwant to know the future location of target Anti aircraft missile launcher gets information from radar (INPUT) Radar provides target past location and velocity 6
- 4. 2/9/2014 Communication Systems 4 Signal and system • System Anti aircraft missile launcher calculates the future Dr. Sarmad Ullah Khan Anti aircraft missile launcher calculates the future location (OUTPUT) using the received information • Definition System gets a set of signals as INPUT and yield aSystem gets a set of signals as INPUT and yield a set of signals as OUTPUT after proper processing System might be a physical device or it might be an algorithm 7 Signal and system Dr. Sarmad Ullah Khan 8
- 5. 2/9/2014 Communication Systems 5 Outlines • Signals and systems • Size of signal Dr. Sarmad Ullah Khan Size of signal • Classification of signals • Signal operations • The unit impulse function • Signals versus Vectors • Correlation O th l i l• Orthogonal signals • Trigonometric Fourier Series • Exponential Fourier Series 9 Size of Signal • Size of an entity is a quantity that shows largeness or strength of entity Dr. Sarmad Ullah Khan g g y • It is a single value/number measure • How signal (amplitude and duration) can be represented by a single number measure? • For example, person’s width ‘r’ and height ‘h’ • To be more precise, single value measure of a person is its volume 10
- 6. 2/9/2014 Communication Systems 6 Size of Signal • Signal Energy Area under a signal g(t) is its SIZE Dr. Sarmad Ullah Khan g g( ) Signal Size takes two values “Amplitude” and “Duration” This measuring approach is defective for large signals having positive and negative portions 11 Size of Signal • Signal Energy Positive portion is cancelled by negative portion Dr. Sarmad Ullah Khan p y g p resulting in small signal This can be solve by calculating area under g2(t) 12
- 7. 2/9/2014 Communication Systems 7 Size of Signal • Signal Energy If signal is complex signal, then Dr. Sarmad Ullah Khan g p g , h ibl h i d | ( )| Other possible approach is area under |g(t)| Energy measure is more desirable 13 Size of Signal • Signal Power Signal energy must be finite for it to be Dr. Sarmad Ullah Khan g gy meaningful Necessary conditions to make it finite Amplitude goes to zero as time approaches infinity Signal must converge IfIf Amplitude does not go to zero Then Energy is infinite 14
- 8. 2/9/2014 Communication Systems 8 Size of Signal Dr. Sarmad Ullah Khan Does this mean that a 60 hertz sine wave feeding into your headphones is as strong as the 60 hertz sine wave coming out of your outlet? Obviously not. This is what leads us to the idea of signal power. 15 Size of Signal • Signal Power To make energy finite and meaningful, time Dr. Sarmad Ullah Khan gy g , average of signal energy is taken into consideration For comple signals For complex signals Signal power is time average of signal amplitude 16
- 9. 2/9/2014 Communication Systems 9 Size of Signal Dr. Sarmad Ullah Khan • Square root of signal power is the Root Mean Square (RMS) value of signal 17 Size of Signal • Are all energy signals also power signals? • No. In fact, any signal with finite energy will have Dr. Sarmad Ullah Khan zero power. • Are all power signals also energy signals? • No, any signal with non-zero power will have infinite energy. • Are all signals either energy or power signals? • No. Any infinite-duration, increasing-magnitude function will not be either. (e.g. f(t)=t is neither) 18
- 10. 2/9/2014 Communication Systems 10 Size of Signal • Remark: Dr. Sarmad Ullah Khan • The terms energy and power are not used in their conventional sense as electrical energy or power, but only as a measure for the signal size. 19 Example 2.1 • Determine the suitable measures of the signals given below: Dr. Sarmad Ullah Khan • The signal (a) amp 0 as t infinity Therefore• The signal (a) amp. 0 as t infinity .Therefore, the suitable measure for this signal is its energy, given by gE 8444)2()(= 0 0 1 22 =+=+= ∞ − − ∞ ∞− dtedtdttg t 20
- 11. 2/9/2014 Communication Systems 11 Example 2.1 The signal in the fig. Below does not --- 0 as t . However it is periodic, therefore its power exits. ∞ Dr. Sarmad Ullah Khan 21 Example 2.2 Dr. Sarmad Ullah Khan (a) 22
- 12. 2/9/2014 Communication Systems 12 Example 2.2 Dr. Sarmad Ullah Khan Remarks: A sinusoid of amplitude C has power of regardless of its frequency and phase . 23 Example 2.2 Dr. Sarmad Ullah Khan 24
- 13. 2/9/2014 Communication Systems 13 Example 2.2 Dr. Sarmad Ullah Khan We can extent this result to a sum of any number of sinusoids with distinct frequencies. 25 Example 2.2 Dr. Sarmad Ullah Khan Recall that ThereforeTherefore The rms value is 26
- 14. 2/9/2014 Communication Systems 14 Outlines • Signals and systems • Size of signal Dr. Sarmad Ullah Khan Size of signal • Classification of signals • Signal operations • The unit impulse function • Signals versus Vectors • Correlation O th l i l• Orthogonal signals • Trigonometric Fourier Series • Exponential Fourier Series 27 Classification of signals • Continuous-time and discrete-time signals A l d di it l i l Dr. Sarmad Ullah Khan • Analog and digital signals • Periodic and Aperiodic signals • Energy and power signals • Deterministic and random signals • Causal vs Non-causal signals• Causal vs. Non-causal signals • Right Sided and Left Sided Signals • Even and Odd Signals 28
- 15. 2/9/2014 Communication Systems 15 Classification of signals • Continuous-time Signal Dr. Sarmad Ullah Khan A signal that is specified for every value of time t e.g. Audio and video signals 29 Classification of signals • Discrete-time signal A i l h i ifi d f di l f Dr. Sarmad Ullah Khan A signal that is specified for discrete value of time t = nT, e.g. Stock Market daily average 30
- 16. 2/9/2014 Communication Systems 16 Classification of signals • Analog signal A l i l d i i i l Dr. Sarmad Ullah Khan Analog signal and continuous-time signal are two different signals Analog signal whose amplitude can have any value over a continuous range Analog continuous time signal x(t) Analog discrete time signal x[n] 31 Classification of signals • Digital signal Di i l i l d di i i l Dr. Sarmad Ullah Khan Digital signal and discrete-time signal are two different signals Discrete signal whose amplitude can have only a finite number of value Digital continuous time signal Digital discrete time signal 32
- 17. 2/9/2014 Communication Systems 17 Classification of signals • Periodic signal A i l ( ) i id b i di if h i Dr. Sarmad Ullah Khan A signal g(t) is said to be periodic if there exist a positive constant T0, such that g(t) = g(t+T0) for all t Oth i i di i l Otherwise aperiodic signal 33 Classification of signals • Properties of Periodic Signal P i di i l i Dr. Sarmad Ullah Khan Periodic signal must start at time t = - Periodic signal shifted by integral multiple of T0 remains unchanged A periodic signal g(t) can be generated by periodic extension of any segment of g(t) with duration T0duration T0 34
- 18. 2/9/2014 Communication Systems 18 Classification of signals • Energy Signal A i l i h fi i i ll d i l Dr. Sarmad Ullah Khan A signal with finite energy is called energy signal • Power Signal A signal with finite power is called power signal 35 Classification of signals • Remarks: Dr. Sarmad Ullah Khan A signal with finite energy has zero power. A signal can be either energy signal or power signal, not both. Every signal in daily life is energy signal, NOT power signal Power signal in practice is not possible because of infinite duration and infinite energy 36
- 19. 2/9/2014 Communication Systems 19 Classification of signals • Deterministic Signal: Dr. Sarmad Ullah Khan A signal whose physical description is know completely, either mathematically or graphically is called deterministic signal • Random Signal: A signal which is known in terms of probabilistic description such as mean value, mean squared value and distribution 37 Classification of signals • Casual Signal: Dr. Sarmad Ullah Khan A signal which is zero prior to zero time Signal amplitude A=0 for T = -t • Non Casual Signal: A signal which is zero after zero time Signal amplitude A=0 for T = +t 38
- 20. 2/9/2014 Communication Systems 20 Classification of signals • Right sided and Left sided Signal: A i ht id d i l i f t < T d l ft id d i l Dr. Sarmad Ullah Khan A right-sided signal is zero for t < T and a left-sided signal is zero for t > T where T can be positive or negative. 39 Classification of signals • Even and Odd Signal: E i l ( ) d dd i l ( ) d fi d Dr. Sarmad Ullah Khan Even signals xe(t) and odd signals xo(t) are defined as xe(t) = xe(−t) and xo(−t) = −xo(t). 40
- 21. 2/9/2014 Communication Systems 21 Classification of signals • Even and Odd Signal: If h i l i i i d f i If h Dr. Sarmad Ullah Khan If the signal is even, it is composed of cosine waves. If the signal is odd, it is composed out of sine waves. If the signal is neither even nor odd, it is composed of both sine and cosine waves. 41 Outlines • Signals and systems • Size of signal Dr. Sarmad Ullah Khan Size of signal • Classification of signals • Signal operations • The unit impulse function • Signals versus Vectors • Correlation O th l i l• Orthogonal signals • Trigonometric Fourier Series • Exponential Fourier Series 42
- 22. 2/9/2014 Communication Systems 22 Signal Operation Dr. Sarmad Ullah Khan Time Shifting A signal g(t) is said to be time shifted if g(t) is delayed or advanced by time T If signal g(t) is delayed by time T, then 43 If signal g(t) is advanced by time T, then Signal Operation Time Scaling Dr. Sarmad Ullah Khan The compression or expansion of signal g(t) in time is known as Time Scaling Compression g(t)=g(at) Expansion g(t)=g(t/a) 44
- 23. 2/9/2014 Communication Systems 23 Signal Operation Time Inversion Dr. Sarmad Ullah Khan In time inversion, signal g(t) is multiplied by a factor a = -1 in time domain g(t) = g(at) if a=1 Time inverted signal g(t) = g(at) if a=-1 45 Signal Operation • Example: 2.4 F th i l (t) h i th fi B l k t h Dr. Sarmad Ullah Khan • For the signal g(t), shown in the fig. Below , sketch g(-t) 46
- 24. 2/9/2014 Communication Systems 24 Outlines • Signals and systems • Size of signal Dr. Sarmad Ullah Khan Size of signal • Classification of signals • Signal operations • The unit impulse function • Signals versus Vectors • Correlation O th l i l• Orthogonal signals • Trigonometric Fourier Series • Exponential Fourier Series 47 Unit Impulse Signal • The Dirac delta function or unit impulse or often referred to as the delta function is the function that Dr. Sarmad Ullah Khan referred to as the delta function, is the function that defines the idea of a unit impulse in continuous-time • It is infinitesimally narrow, infinitely tall, yet integrates to one • simplest way to visualize this as a rectangular pulsep y g p from a -D/2 to a +D/2 with a height of 1/D • The impulse function is often written as 48
- 25. 2/9/2014 Communication Systems 25 Unit Impulse Signal Dr. Sarmad Ullah Khan =0 for al t ≠ 0 49 Unit Impulse Signal • Multiplication of a Function by an Impulse Dr. Sarmad Ullah Khan If a function g(t) is multiplied by impulse function we get impulse value of g(t) 50
- 26. 2/9/2014 Communication Systems 26 Unit Impulse Signal Dr. Sarmad Ullah Khan 51 Unit Impulse Signal Dr. Sarmad Ullah Khan 52
- 27. 2/9/2014 Communication Systems 27 Unit Impulse Signal • Unit Step Function u(t) Dr. Sarmad Ullah Khan 53 Outlines • Signals and systems • Size of signal Dr. Sarmad Ullah Khan Size of signal • Classification of signals • Signal operations • The unit impulse function • Signals versus Vectors • Correlation O th l i l• Orthogonal signals • Trigonometric Fourier Series • Exponential Fourier Series 54
- 28. 2/9/2014 Communication Systems 28 Signals versus Vectors • A Vector can be represented as a sum of its components Dr. Sarmad Ullah Khan components • A Signal can also be represented as a sum of its• A Signal can also be represented as a sum of its components 55 Signals versus Vectors • A Signal defined over a finite number of time instants can be written as a Vector Dr. Sarmad Ullah Khan can be written as a Vector • Consider a signal g(t) defined over interval [a,b] • Uniformly divide interval [a,b] in N points T1 = a, T2 = a+ϵ, T3 = a+2ϵ, …. TN = a+(N-1)ϵ • Where Step Sizeϵ = 56
- 29. 2/9/2014 Communication Systems 29 Signals versus Vectors • Signal vector g can be written a N-dimensional vector Dr. Sarmad Ullah Khan • Signal vector g grows as N increases g = [g(t1) g(t2) … g(tN)] • Signal transforms into continuous time signal g(t) 57 Signals versus Vectors • Signal transforms into continuous time signal g(t) Dr. Sarmad Ullah Khan • Continuous time signals are straightforward generalization of finite dimension vectors • Vector properties can be applied to signals 58
- 30. 2/9/2014 Communication Systems 30 Signals versus Vectors • A vector is represented by bold-face type Dr. Sarmad Ullah Khan • Specified by its magnitude and its direction • For example, Vector x have magnitude | x | and Vector g have magnitude | g | • Inner product (dot or scalar) of two real valued vectors ‘g’ and ‘x’ is 59 Signals versus Vectors • Component of a Vector Dr. Sarmad Ullah Khan Consider two vectors ‘x’ and ‘g’ ‘cx’ (projection) is component of ‘g’ along ‘x’ What is the mathematical significance of a vector along another vector? g = cx + eg However, this is not a unique way of vector decomposition 60
- 31. 2/9/2014 Communication Systems 31 Signals versus Vectors • Component of a Vector Oth t ‘ ’ i Dr. Sarmad Ullah Khan Other ways to express ‘g’ is g is represented in terms of x plus another vector which is called the error vector e 61 Signals versus Vectors • Component of a Vector Dr. Sarmad Ullah Khan If we approximate e = g – cx Geometrically component of g along x is Hence 62
- 32. 2/9/2014 Communication Systems 32 Signals versus Vectors • Component of a Vector Dr. Sarmad Ullah Khan Based on definition of inner product, multiply both side by |x| 63 Signals versus Vectors • Component of a Vector Dr. Sarmad Ullah Khan If g and x are orthogonal, then 64
- 33. 2/9/2014 Communication Systems 33 Signals versus Vectors • Component of Signal Dr. Sarmad Ullah Khan Vector component and orthogonality can be extended to continuous time signals Consider approximating a real signal g(t) in terms of another real signal x(t) And 65 Signals versus Vectors • Component of Signal Dr. Sarmad Ullah Khan As energy is one possible measure of signal size. To minimize the effect of error signal we need to minimize its size-----which is its energy over the interval [t1 , t2] 66
- 34. 2/9/2014 Communication Systems 34 Signals versus Vectors • Component of Signal Dr. Sarmad Ullah Khan But ‘e’ is a function of ‘c’, not ‘t’, hence 67 Signals versus Vectors • Component of Signal Dr. Sarmad Ullah Khan 68
- 35. 2/9/2014 Communication Systems 35 Signals versus Vectors • Component of Signal Dr. Sarmad Ullah Khan 69 Signals versus Vectors • Component of Signal Dr. Sarmad Ullah Khan Two signals g(t) and x(t) are said to be orthogonal if there is zero contribution from one signal to other signal For N dimensional vectors ‘g’ and ‘x’ 70
- 36. 2/9/2014 Communication Systems 36 For the square signal g(t), find the component of g(t) of the form sint or in other words approximate g(t) in terms of sint Example 2.5 Dr. Sarmad Ullah Khan sint or in other words approximate g(t) in terms of sint tctg sin)( ≅ π20 ≥≤ t 71 ttx sin)( = and Example 2.5 Dr. Sarmad Ullah Khan )( From equation for signals dttxtg E c t tx = 2 1 )()( 1 72 πππ π π π π 4 sinsin 1 sin)( 1 0 22 = −+== tdttdttdttgc o ttg sin 4 )( π ≅
- 37. 2/9/2014 Communication Systems 37 Signals versus Vectors • Orthogonality in Complex Signals Dr. Sarmad Ullah Khan For complex function g(t), its approximation by another complex function x(t) over finite interval ‘c’ and ‘e’ are complex functions 73 Signals versus Vectors • Orthogonality in Complex Signals Dr. Sarmad Ullah Khan Energy of a complex signal x(t) over finite interval Choose ‘c’ such that it reduces ‘Ee’ 74
- 38. 2/9/2014 Communication Systems 38 Signals versus Vectors • Orthogonality in Complex Signals Dr. Sarmad Ullah Khan We know that After certain manipulations ( )( ) ∗∗∗∗ ++=++=+ uvvuvuvuvuvu 222 222 222 2 1 2 1 2 1 )()( 1 )()( 1 )( ∗∗ −+−= t tx x t tx t t e dttxtg E Ecdttxtg E dttgE dttxtg E c t tx )()( 1 2 1 ∗ = 75 Signals versus Vectors • Orthogonality in Complex Signals Dr. Sarmad Ullah Khan So, two complex functions are orthogonal over an interval, if 0)()( 21 2 1 =∗ dttxtx t t 0)()( 21 2 1 = ∗ dttxtx t t or 76
- 39. 2/9/2014 Communication Systems 39 Signals versus Vectors • Energy of the Sum of Orthogonal Signals Dr. Sarmad Ullah Khan Sum of the two orthogonal vectors is equal to the sum of the lengths of the squared of two vectors. z = x+y then Sum of the energy of two orthogonal signals is equal to the 222 yxz += gy g g q sum of the energy of the two signals. If x(t) and y(t) are orthogonal signals over the interval, and if z(t) = x(t)+ y(t) then yxz EEE += 77 Outlines • Signals and systems • Size of signal Dr. Sarmad Ullah Khan Size of signal • Classification of signals • Signal operations • The unit impulse function • Signals versus Vectors • Correlation O th l i l• Orthogonal signals • Trigonometric Fourier Series • Exponential Fourier Series 78
- 40. 2/9/2014 Communication Systems 40 Correlation of Signals • Correlation addresses the question: “to what degree is signal A similar to signal B” Dr. Sarmad Ullah Khan signal A similar to signal B • Two vectors ‘g’ and ‘x’ are similar if ‘g’ has a large component along ‘x’ • If ‘c’ has a large value, then the two vectors will be similar ‘ ’ ld b id d h i i f• ‘c’ could be considered the quantitative measure of similarity between ‘g’ and ‘x’ But such a measure could be defective. The amount of similarity should be independent of the lengths of g and x 79 Correlation of Signals • Doubling g should not change the similarity between g and x Dr. Sarmad Ullah Khan and x • Similarity between the vectors is indicated by angle However: Doubling g doubles the value of c Doubling x halves the value of c c is faulty measure for similarity y y g between the vectors. • The smaller the angle , the largest is the similarity, and vice versa • Thus, a suitable measure would be , given by 80
- 41. 2/9/2014 Communication Systems 41 Correlation of Signals • Where Dr. Sarmad Ullah Khan • This similarity measure is known as correlation co-efficient Independent of the lengths of g and x • And 81 Correlation of Signals • Same arguments for defining a similarity index (correlation co efficient) for signals Dr. Sarmad Ullah Khan (correlation co-efficient) for signals • Consider signals over the entire time interval • To establish a similarity index independent of energies (sizes) of g(t) and x(t), normalize c by normalizing the two signals to have unit energiesg g g 82
- 42. 2/9/2014 Communication Systems 42 Correlation of Signals • Best Friends Dr. Sarmad Ullah Khan • Opposite personalities (Enemies) • Complete Strangers 83 Example 2.6 Find the correlation co-efficient between the pulse x(t) and the pulses nc 6,5,4,3,2,1,)( == itgi Dr. Sarmad Ullah Khan p i 84 5)( 5 0 5 0 2 === dtdttxEx 51 =gE 1 55 1 5 0 = × = dtcndttxtg EE c xg n ∞ ∞− = )()( 1 Similarly Maximum possible similarity
- 43. 2/9/2014 Communication Systems 43 Example 2.6 (cont…) Dr. Sarmad Ullah Khan 5)( 5 0 5 0 2 === dtdttxEx 25.12 =gE 85 1)5.0( 525.1 1 5 0 = × = dtcn dttxtg EE c xg n ∞ ∞− = )()( 1 Maximum possible similarity……independent of amplitude Example 2.6 (cont…) Dr. Sarmad Ullah Khan 5)( 55 2 === dtdttxEx 51 =gESimilarly 86 00 1g 1)1)(1( 55 1 5 0 −=− × = dtcndttxtg EE c xg n ∞ ∞− = )()( 1 y
- 44. 2/9/2014 Communication Systems 44 Example 2.6(cont…) Dr. Sarmad Ullah Khan )1( 2 1 )( 22 2 aT T at T at e a dtedteE −−− −=== 5)( 5 0 5 0 2 === dtdttxEx 87 200 a 5 1 =a 5=T 1617.24 =gE 961.0 1617.25 1 5 0 5 = × = − dtec t n Here Reaching Maximum similarity Outlines • Signals and systems • Size of signal Dr. Sarmad Ullah Khan Size of signal • Classification of signals • Signal operations • The unit impulse function • Signals versus Vectors • Correlation O th l i l• Orthogonal signals • Trigonometric Fourier Series • Exponential Fourier Series 88
- 45. 2/9/2014 Communication Systems 45 Orthogonal Signal Sets • A signal can be represented as a sum of orthogonal set of signals Dr. Sarmad Ullah Khan set of signals • Orthogonal set of signals form a basis for specific signal space • For example, a vector is represented as a sum of orthogonal set of vectors I f di f• It forms a coordinate system for vector space 89 Orthogonal Signal Sets • Orthogonal Vector Space C t i i d ib d b th t ll th l Dr. Sarmad Ullah Khan Cartesian space is described by three mutually orthogonal vectors x1, x2, and x3 If a three dimensional vector g is approximated by two orthogonal vectors x1 and x2, then And 90
- 46. 2/9/2014 Communication Systems 46 Orthogonal Signal Sets • Orthogonal Vector Space If th di i l t i t d b th Dr. Sarmad Ullah Khan If a three dimensional vector g is represented by three orthogonal vectors x1 , x2 and x3, then And e = 0 in this case x1,x2 and x3 is complete set of orthogonal space, No x4 exist 91 Orthogonal Signal Sets • Orthogonal Vector Space Th th l t ll d b i t Dr. Sarmad Ullah Khan These orthogonal vectors are called basis vectors Complete set of vectors is called complete orthogonal basis of a vector A set of vector {xi} is mutually{ i} y orthogonal if 92
- 47. 2/9/2014 Communication Systems 47 Orthogonal Signal Sets • Orthogonal Signal Space Lik t th lit f i l t (t) (t) Dr. Sarmad Ullah Khan Like vector, orthogonality of signal set x1(t), x2(t), ….. xN(t) over time interval [t1, t2] is defined as If all signal energies are equal En = 1 then set is normalized and is called an orthogonal set An orthogonal set can be normalized by dividing xN(t) by 93 Orthogonal Signal Sets • Orthogonal Signal Space N i l (t) [t t ] b t f N th l Dr. Sarmad Ullah Khan Now signal g(t) over [t1, t2] by a set of N-orthogonal signals x1(t), x2(t), ….. xN(t) is Energy of error signal e(t) can be minimized if 94
- 48. 2/9/2014 Communication Systems 48 Outlines • Signals and systems • Size of signal Dr. Sarmad Ullah Khan Size of signal • Classification of signals • Signal operations • The unit impulse function • Signals versus Vectors • Correlation O th l i l• Orthogonal signals • Trigonometric Fourier Series • Exponential Fourier Series 95 Trigonometric Fourier Series • Like vector, signal can be represented as a sum of its orthogonal signal (Basis signals) Dr. Sarmad Ullah Khan g g ( g ) • There are number of such basis signals e.g. trigonometric function, exponential function, Walsh function, Bessel function, Legendre polynomial, Laguerre functions, Jaccobi polynomial • Consider a periodic signal of period T0 • Consider a signal set• Consider a signal set {1+Cos w0t+Cos 2w0t+……Cos nw0t…. Sin w0t+Sin 2w0t……Sin nw0t….} 96
- 49. 2/9/2014 Communication Systems 49 Trigonometric Fourier Series • nw0 is called the nth harmonic of sinusoid of angular frequency w where n is an integer Dr. Sarmad Ullah Khan frequency w0 where n is an integer • A sinusoid of frequency w0 is called the fundamental tone/anchor of the set 97 Trigonometric Fourier Series • This set is orthogonal over any interval of duration because: o o w T π2= Dr. Sarmad Ullah Khan because: = 2 0 coscos o To oo Ttdtmwtnw omn mn ≠= ≠ mn≠ = 0 sinsin oo Ttdtmwtnw omn ≠= 2 o To oo T 0cossin = tdtmwtnw To oo for all nand m and 98
- 50. 2/9/2014 Communication Systems 50 Trigonometric Fourier Series • The trigonometric set is a complete set. Dr. Sarmad Ullah Khan • Each signal g(t) can be described by a trigonometric Fourier series over the interval To : ...2sinsin 21 +++ twbtwb oo ...2coscos)( 21 +++= twatwaatg ooo oTttt +≤≤ 11 ∞ = ++= 1 sincos)( n onono tnwbtnwaatg oTttt +≤≤ 11 or o n T w π2 = 99 Trigonometric Fourier Series • We determine the Fourier co-efficient as: Dr. Sarmad Ullah Khan + + = o o Tt t o o Tt t n tdtnw tdtnwtg C 1 1 1 1 2 cos cos)( d oTt +1 )( 1 ,......3,2,1=n 100 dttg T a to = 1 )( 1 0 tdtnwtg T a o Tt to n o cos)( 2 1 1 + = tdtnwtg T b o Tt to n o sin)( 2 1 1 + =
- 51. 2/9/2014 Communication Systems 51 Compact Trigonometric Fourier Series • Consider trigonometric Fourier series Dr. Sarmad Ullah Khan • It contains sine and cosine terms of the same frequency. We can represents the above equation in a i l t f th f i th ...2sinsin 21 +++ twbtwb oo ...2coscos)( 21 +++= twatwaatg ooo oTttt +≤≤ 11 single term of the same frequency using the trigonometry identity )cos(sincos nononon tnwCtnwbtnwa θ+=+ 22 nnn baC += − = − n n n a b1 tanθ oo aC = oTttt +≤≤ 11 101 ∞ = ++= 1 0 )cos()( n non tnwCCtg θ Example 2.7 Find the compact trigonometric Fourier series for the following function Dr. Sarmad Ullah Khan 102
- 52. 2/9/2014 Communication Systems 52 Example 2.7 Solution: We are required to represent g(t) by the trigonometric Fourier series over the interval andπ≤≤ t0 π=T Dr. Sarmad Ullah Khan series over the interval andπ≤≤ t0 πoT sec 2 2 rad T w o o == π T i i f f F i i 103 Trigonometric form of Fourier series: ??,?,0 nn baa ntbntaatg n n no 2sin2cos)( 1 ++= ∞ = π≤≤ t0 Example 2.7 500 1 2 == − dtea t π Dr. Sarmad Ullah Khan 50.0 0 0 == dtea π + == − 2 0 2 161 2 504.02cos 2 n dtntea t n π π + == − 2 0 2 161 8 504.02sin 2 n n ntdteb t n π π 22 nnn oo baC aC += = 104 0 Compact Fourier series is given by )cos()( 1 0 no n n tnwCCtg θ++= ∞ = π≤≤ t0
- 53. 2/9/2014 Communication Systems 53 Example 2.7 2644 504.0 2 n aC oo == Dr. Sarmad Ullah Khan ( ) ) 161 2 (504.0 )161( 64 161 4 504.0 22222 22 nn n n baC nnn + = + + + =+= ( ) nn a b n n n 4tan4tantan 11 −=−= − = −− θ ( )4t2 2 50405040)( 1− ∞ tt π≤≤ t0 105 ( ) .......)42.868cos(063.0)24.856cos(084.0 )87.824cos(25.1)96.752cos(244.0504.0 4tan2cos 161 504.0504.0)( 1 1 2 +−+−+ −+−+= − + += = oo oo n tt tt nnt n tg π≤≤ t0 π≤≤ t0 Example 2.7 n 0 1 2 3 4 5 6 7 Dr. Sarmad Ullah Khan Cn 0.504 0.244 0.125 0.084 0.063 0.054 0.042 0.063 Өn 0 -75.96 -82.87 -85.24 -86.42 -87.14 -87.61 -87.95 l d d h f f h 106 Amplitudes and phases for first seven harmonics
- 54. 2/9/2014 Communication Systems 54 Trigonometric Fourier Series • Periodicity of the trigonometric Fourier series Dr. Sarmad Ullah Khan The co-efficient of the of the Fourier series are calculated for the interval [t1, t1+T0] ∞ = ∞ = +++=+ ++= 1 00 1 ])([cos()( )cos()( n nono n nono TtnwCCTt tnwCCt θφ θφ for all t )( )cos( )2cos( 1 1 t nwtCC nnwtCC no n no no n no φ θ θπ = ++= +++= ∞ = ∞ = for all t 107 Trigonometric Fourier Series • Periodicity of the trigonometric Fourier series Dr. Sarmad Ullah Khan tdtnwtg T a o To n o 2 cos)( 2 = dttg T a To o o )( 1 = n= 1,2,3,…… tdtnwtg T b o To n o sin)( 2 = n= 1,2,3,…… 108
- 55. 2/9/2014 Communication Systems 55 Trigonometric Fourier Series • Fourier Spectrum Dr. Sarmad Ullah Khan Consider the compact Fourier series This equation can represents a periodic signal g(t) of f i )cos()( 1 0 no n n tnwCCtg θ++= ∞ = d )(frequencies: Amplitudes: Phases: oooo nwwwwdc ,.....,3,2,),(0 nCCCCC ,......,3210 ,,, nθθθθ ,.....,,,0 321 109 Trigonometric Fourier Series • Fourier Spectrum F d i d i ti f )( Dr. Sarmad Ullah Khan nc vs w (Amplitude spectrum) wvsθ (phase spectrum) Frequency domain description of )( tφ Time domain description of )( tφ 110
- 56. 2/9/2014 Communication Systems 56 Example 2.8 Dr. Sarmad Ullah Khan Find the compact Fourier series for the periodic square wave w(t) shown in figure and sketch amplitude and phase spectrum( ) g p p p F i i 111 ∞ = ++= 1 sincos)( n onono tnwbtnwaatw Fourier series: 2 11 4 4 0 == dt T a o o T To W(t)=1 only over (-To/4, To/4) and w(t)=0 over the remaining segmentdttg T a oTt to + = 1 1 )( 1 0 Example 2.8 Dr. Sarmad Ullah Khan == 2 sin 2 cos 2 4 π π n n dttnw T a oT on − 2 4 πnT oTo − = π π n n 2 2 0 ...15,11,7,3 ...13,9,5,1 = = − n n evenn 112 πn ,,, 0sin 2 4 4 == ntdt T b o o T To n 0= nb All the sine terms are zero
- 57. 2/9/2014 Communication Systems 57 Example 2.8 Dr. Sarmad Ullah Khan +−+−+= ....7cos 7 1 5cos 5 1 3cos 3 1 cos 2 2 1 )( twtwtwtwtw oooo π 7532 π The series is already in compact form as there are no sine terms Except the alternating harmonics have negative amplitudes The negative sign can be accommodated by a phase of radians asπ )cos(cos π−=− xx 113 Series can be expressed as: ++−++−++= ....9cos 9 1 )7cos( 7 1 5cos 5 1 )3cos( 3 1 cos 2 2 1 )( twtwtwtwtwtw ooooo ππ π Example 2.8 Dr. Sarmad Ullah Khan 2 1 =oC = πn C n 2 0 oddn evenn − − − = π θ 0 n for all n 3,5,7,11,15,….. for all n = 3,5,7,11,15,….. ≠ We could plot amplitude and phase spectra using these values…. In this special case if we allow Cn to take negative values we do not need a phase of to account for sign.π− 114 Means all phases are zero, so only amplitude spectrum is enough
- 58. 2/9/2014 Communication Systems 58 Example 2.8 Dr. Sarmad Ullah Khan Consider figure )5.0)((2)( −= twtwo 115 +−+−= ....7cos 7 1 5cos 5 1 3cos 3 1 cos 4 )( twtwtwtwtw oooo π Outlines • Signals and systems • Size of signal Dr. Sarmad Ullah Khan Size of signal • Classification of signals • Signal operations • The unit impulse function • Signals versus Vectors • Correlation O th l i l• Orthogonal signals • Trigonometric Fourier Series • Exponential Fourier Series 116
- 59. 2/9/2014 Communication Systems 59 Exponential Fourier Series • According to Euler’s theorem, each Sin function can be represented as a sum of ejwt and e-jwt Dr. Sarmad Ullah Khan be represented as a sum of ejwt and e jwt • Also a set of exponentials ejnwt is orthogonal over any time interval T=2*pi/w A i l (t) l b t d• A signal g(t) can also be represented as an exponential Fourier series over an interval T0 117 Exponential Fourier Series Dr. Sarmad Ullah Khan • Where the coefficient Dn can be calculated as • Exponential Fourier series is an another form of trigonometric Fourier series 118
- 60. 2/9/2014 Communication Systems 60 Exponential Fourier Series • Exponential Fourier series is an another form of trigonometric Fourier series Dr. Sarmad Ullah Khan trigonometric Fourier series 119 Exponential Fourier Series • The compact trigonometric Fourier series of a periodic signal g(t) is given by Dr. Sarmad Ullah Khan periodic signal g(t) is given by 120
- 61. 2/9/2014 Communication Systems 61 Exponential Fourier Series • Exponential Fourier Spectra Dr. Sarmad Ullah Khan To draw Dn we need to find its spectra As Dn is a complex quantity having real and imaginary value, thus we need two plots (real and imaginary parts OR amplitude and angle of Dn) Amplitude and phase is prefered because of its close ti ith th di t fconnection with the corresponding components of trigonometric Fourier series We plot |Dn| vs ω and ∟Dn vs ω 121 Exponential Fourier Series • Exponential Fourier Spectra Dr. Sarmad Ullah Khan Comparing exponential with trigonometric Fourier spectrum yields For real periodic signal, Dn and D n are conjugate, thusp g , n -n j g , 122
- 62. 2/9/2014 Communication Systems 62 Exponential Fourier Series • Exponential Fourier Spectra Dr. Sarmad Ullah Khan 123 Exponential Fourier Series • Exponential Fourier Spectra Dr. Sarmad Ullah Khan sec 2 2 rad T w o o == π ∞ −∞= = n ntj n eDt 2 )(ϕ dteedtet T D ntj t ntj T o n o 2 0 22 1 )( 1 − − − == π π ϕ == +− π )2 2 1 (1 dte tn π=oT 124 π 0 nj 41 504.0 + =
- 63. 2/9/2014 Communication Systems 63 Exponential Fourier Series • Exponential Fourier Spectra Dr. Sarmad Ullah Khan sec 2 2 rad T w o o == π π=oT ntj n e nj t 2 41 1 504.0)( ∞ −∞= + =ϕ and + + + + + + + = 111 ... 121 1 81 1 41 1 1 504.0 642 tjtjtj e j e j e j Dn are complex Dn and D-n are conjugates 125 + + + − +− − −− ... 121 1 81 1 41 1 642 tjtjtj e j e j e j Exponential Fourier Series • Exponential Fourier Spectra Dr. Sarmad Ullah Khan sec 2 2 rad T w o o == π π=oTnnn CDD 2 1 == − nnD θ=< nnD θ−=< −and thus nj nn eDD θ = nj nn eDD θ− − = 126 o o j j o e j D e j D D 96.75 1 96.75 1 122.0 41 504.0 122.0 41 504.0 504.0 − − − − = + = = o o D D 96.75 96.75 1 1 =< −=< −
- 64. 2/9/2014 Communication Systems 64 Exponential Fourier Series • Exponential Fourier Spectra Dr. Sarmad Ullah Khan sec 2 2 rad T w o o == π π=oT o o j j e j D e j D 87.82 2 87.82 2 625.0 81 504.0 625.0 81 504.0 − − − − = + = o o D D 87.82 87.82 1 1 =< −=< − 127 And so on…. Exponential Fourier Series • Exponential Fourier Spectra Dr. Sarmad Ullah Khan 128

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