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Digital Signal Processing Tutorial:Chapt 1 signal and systems

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  • 1. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 1Chapt 01Signal and SystemsDigital Signal ProcessingPrePrepared byIRDC India
  • 2. 10/7/2009 2Copyright© with Authors. All right reservedFor education purpose.Commercialization of this material isstrictly not allowed without permissionfrom author.e-TECHNote from IRDC Indiainfo@irdcindia.com
  • 3. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 3Signal• Definition: It is the function that describes the variation of aphysical variable with respect to independent variable( time,space etc.) in a physical process.Mathematically,S=f(x)Where s is signal /function and x independent variableExamples:1) Change in temperature sensed by the sensor placed in boiler.T=f(t)Here, time t is independent variable.t(msec) 1 2 3 4 5Temp(0C) 29 31 35 45 45
  • 4. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 4Contd..2) The digital data to be sent over a transmission channelt(msec) 0 1 2 3 4 5Digitaldata0 1 1 0 1 03) Pixel intensities in an image2 5 7 4 12 1 6 2 4 910 7 4 4 4 2 9 3 1 28 11 14 4 1 9 7 5 11 121 2 3 2 3 6 8 13 4 56 7 8 15 0 10 7 9 6 56 6 7 8 4 3 11 8 9 07 8 5 6 4 5 13 4 4 47 8 9 8 9 8 7 5 7 77 5 4 3 3 4 5 6 14 114 12 12 1 5 7 2 1 9 9Image, I=f(x,y)In this example intensity in image is thefunction / or signal which is dependent onindependent variable spatial coordinate xand y.
  • 5. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 5Contd..4) The voltage across capacitorRCteV /−=5) share value fluctuation in stock market during year ( month wise)Month Jan Feb Mar Apr May Jun Jul Aug SepSharevalue(Rs)200 214 214 245 198 200 210 200 234
  • 6. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 6Common Discrete time signalsIt is general practice to represent complex signals by using commonor standards signals.In descrete time signals independent variable,t, can be equivalently seen as nT where T is samling period andn is sample number. Thuis , n becomes independent variable indescrete time signal. Some of the common signals are given here.a)Unit impulse function01;0;0≠=nn=)(nδb) Unit step function u(n):0≥n0<nIt is defined as u(n) = 100≥n0<nc)Unit ramp function ( r(n)):r(n) = n0
  • 7. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 7e-TECHNoteThis PPT is sponsored byIRDC Indiawww.irdcindia.com
  • 8. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 8Contd..d) Exponential function:It is expressed as asanAenf =)(When a ia negative when a is positive)sin()( φω += nanf)cos()( φω += nanffπω 2=φe) Sinusoidal functionsine function ,cosine functionis the angular frequency of functionwhere A is the amplitude of functionis the phase of the function.f) Sgn functionIt is defined as1, n>0Sgn(n)= 0,n=0-1,n<0g) Decaying/growing sinusoidal
  • 9. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 9Signal RepresentationDiscrete time signals can be represented as followsGraphical Representation0x(n)Sequence Representation{ }3021)( =nx ;......2)1(;1)0( == xx{ }25212310)( −−↑nx....2)2(,1)1(,2)0(,3)1(,1)2(,0)3(,0)4( ==−==−=−=−=− xxxxxxxFunctional representation=02/2)( nnnxelsewherenn8540≤≤≤≤=nnnx3)(2evennoddn==
  • 10. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 10ProblemsSketch following signals=02/2)( nnnxelsewherenn8540≤≤≤≤=nnnx3)(2evennoddn==−−=nnnx5320)(4422>≤≤≤nnn123
  • 11. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 11Signal Operations• Addition/Subtraction – corresponding samples from both signals would beadded, subtracted• Amplitude Scaling- each sample of the signal would be scaled by scalingfactor• Delaying• Advancing•Time reversing•Rate Changing
  • 12. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 12x(n)Delaying- AdvancingOriginal signalx(n)1234n0Delaying1234n0x(n)x(n-1)x(n)1234n0x(n+1)Advancing
  • 13. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 13Time ReversingOriginal signal x(n)1234n0TR & Delayingx(n)x(-n+1)x(-n-1)TR & AdvancingTime Reversed x(n)1234n0x(-n)x(n)1234n0x(n)1234n0
  • 14. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 14Rate ChangingRate changing – Multirate Signal Processing– changing the sampling rate of signalUp samplingDown samplingOriginal signalx(n)1234n0x(n)x(n)1234n0x(n/2)x(n)13n0x(2n)
  • 15. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 15e-TECHNoteThis PPT is sponsored byIRDC Indiawww.irdcindia.com
  • 16. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 16Signal Classification• Periodic and Aperiodic Signals• Even and Odd signals• Energy and Power signal• Deterministic and stochastic signal
  • 17. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 17Periodic and Aperiodic Signals•A signal is periodic if it satisfies periodicity property x(n+kN)=x(n)where N is a fundamental period and k is any integer•If signal is periodic with fundamental period N, it sis also periodic for anyinteger multiple of NtTptA signal which doesnt satisfy periodicity property is called aperiodic signal
  • 18. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 18Calculation of Periodicity• Sum of two or more periodic signals is also a periodic signal• If xa[n] and xb[n] are two periodic singals with funfadmentalperiod Na and Nb respectively, then singal y[n]=xa[n]+xb[n] is aperiodic singal with fundamental period N given byN= Na*Nb/(GCD(Na,Nb)where GCD greatest common divisor of Na & Nb
  • 19. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 19Even and Odd signals• Signal which satisfiesx(n)=x(-n) for all values for nis called as even signaln0• Signal which satisfiesx(n)=-x(-n) for all values for nis called as odd signal n0Any signal can be expressed in termsof odd and even signal)()()( nxnxnx oddeven +=2)()( nxnxxeven−+=2)()( nxnxxodd−−=where and
  • 20. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 20Energy and Power Signal•The signal x(n) is said to be an energy signal if its energy, ascalculated by following equation , is finite and non-zero.∑−=∞→=NNnNnxEnergy2)(lim•The signal x(n) is said to be an power signal if its power, ascalculated by following equation , is finite and non-zero.∑−=∞→ +=NNnNnxNPower2)(121lim
  • 21. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 21Problem(1)• identify signals if it is power or energy signala) u(n) b) 0.5nu(n)∑∑∞∞∞=−∞===0221)(nnxE21112221lim1121lim1121lim)(121lim=++=++=+=+=∞→∞→−=∞→−=∞→∑∑NNNNNNnNNNnNNNNnxNPa)∑=+−−=212111NNnNNnaaaa 1≠aWe know ,wherePower signal
  • 22. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 22e-TECHNoteThis PPT is sponsored byIRDC Indiawww.irdcindia.com
  • 23. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 23Solution3475.0125.010125.05.0)(02022==−−==−== ∑∑∑∞∞∞∞=−nnxEEnergy signal∑=+−−=212111NNnNNnaaaa 1≠aWe know ,where
  • 24. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 24Quiz1. Sketch∑∞=−=0)()(kknnx δ2. Can step sequence be represented in terms of impulses? If yes how?3. Signal is to be down sampled by a factor of 2.5. Is it possible? If yes ,how?4. Verify odd-even signal equation for the signal x(n)={1 2 3 4}5 Given DTS as { }23221231)( −−−=↑nxSketch a) x(n-3) , b)x(3-n), c) x(-n-1) , d)x(2n) u(2-n) f)x(n-2)∂(n+2)
  • 25. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 25System•DefinitionSystem is device that follows unique relationship betweenexcitation and response( input and output)A discrete-time system is essentially an algorithm forconverting one sequence (called the input) into another sequence(called the output)∫x(n) y(n)y(n)=f[x(n)]f[.] denotes the specificsystem
  • 26. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 26Examples of system• Differentiator y(n)= x(n)-x(n-1)• Square law modulator y(n)= [ x(n)]2• Interpolator/ upsampler=0)()(mnxnyotherwisefMmultipleson =
  • 27. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 27System Representation• Impulse response (in time domain)h[n]={1 -1} , h[n]= { 1 0.8 0.54 0.24 0.006}•Relation between input and outputas seen in system examples•Difference equationy[n]= x[n]- x[n-1] , y[n]= y[n-1] + ax[n] +bx[n-2]•Transfer function( in z-domain)H(z)= 1/(z+1)
  • 28. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 28e-TECHNoteThis PPT is sponsored byIRDC Indiawww.irdcindia.com
  • 29. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 29System Classification• Continuous /discrete time systems• Time variant/invariant systems• Memory less/memory systems( Static/dynamic system)• Causal/anti-causal/non-causal system• Linear/non-linear systems• Lumped/distributed-parameter systems• Stable/unstable systems• Invertible and non-invertible systems
  • 30. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 30Continuous /discrete time systemsIs there any applicationwhich exist only in digitaland not in analog ?If the system process CTS, then system is said to be continuoustime system.e.g. R-C circuit, transmitting antennaIf the system process DTS, then system is said to be discrete timesystem.e.g. Digital adder
  • 31. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 31Lumped/distributed-parameter systems• If the component used in system has identicalvalues of physical parameters( current, voltages etc)throughout its area and can be considered as asingle point (node) in the system , it is called aslumped parameter system• e.g normal components like resistor, capacitor inlow frequency applications etcV1V1V1V1• If the component used in system has differentvalues of physical parameters ( current, voltages etc)throughout its area and cannot be considered as asingle point (node) in the system , it is called aslumped parameter system•E.g. transmission lines, microwave tubes whichnormally used in high frequencyV1V2V3V4
  • 32. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 32Memory less/memory systems( Static/dynamicsystem)A system for which the output depends only on a present inputand thus , not requires memory, is called as memory less (static )system.e.g. y(n)= a*x(n) , y(n)=x2(n)A system for which the output depends on past and/or futurevalues of the input in addition to the present values of input ,hence it needs memory, delay elements, is called asmemory systeme.g. y(n)=x(n)+x(n+2) , y(n)=x(n-2)
  • 33. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 33Causal/anti-causal/non-causal systemA system for which the output at any instatnt depends only on thepast or present values of the input( not on future samples) iscalled as causal systemy(n)= n*x(n) , y(n)=x(n) +x(n-1)A system for which the output at any instatnt depends also onfuture values of the iput , is called as non-causal systeme.g. y(n)=x(n2) , y(n)=x(-n) , y(n)=x(n)+x(n+1)
  • 34. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 34e-TECHNoteThis PPT is sponsored byIRDC Indiawww.irdcindia.com
  • 35. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 35Stable/unstable systemsThe system is said to be stable if any bounded input signalresults in bounded output signalbounded signals u(n) , e-anThe system is said to be unstable if the system givesunbounded output signal in response to bounded inputsignalunbounded signals r(n) , n*u(n)
  • 36. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 36Invertible/non-invertible systemsThe system whose output can be used to determineinput uniquely and exactly, is called as invertible system.e.g. y=2xbut y=x2 is not an invertible system as it would give twopossible inputs ( +ve and –ve)Hence , system defined by y=x2 is non-invertible system
  • 37. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 37Linear /non-linear SystemsIf system holds superposition property , it is called as linear systemif system violates superposition property, it is called as nonlinear systemHX1(n)X2(n)+abY(n)HHSuperposition property of a system with any two inputs x1(n) and x2(n) isdefined asH{a x1(n) +b x2 (n) }= a H{x1(n)} +b H{x2(n) }=a y1(n) +b y2(n)X1(n)X2(n)ab+ Y(n)
  • 38. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 38Problems1) y(n)=nx(n)y1(n)=nx1(n)y2(n)= nx2(n)ay1(n)+by2(n)= nax1(n)+ nbx2(n)=n[ax1(n)+bx2(n)] …..AH[ax1(n)+bx2(n)]= n[ax1(n)+bx2(n)] …..BA=B Linear system2) y(n)=x2(n)y1(n)=x12(n)y2(n)=x22(n)ay1(n)+by2(n)=ax12(n)+bx22(n) ….AH[ax1(n)+bx2(n)]= [ax1(n)+bx2(n)]2= a2x12(n)+b2x22(n)+2abx1(n)x2(n)………BA != BNon-linear system
  • 39. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 39Time Variant/Invariant SystemsA system is said to be time-invariant if its input-outputrelationship does not change with timeA system is said to be time-variant if its input-outputrelationship changes with timeIn other words if a time shift or delay at the inputproduces identical time shift at the output , then systemis said to be time invariant system.i.e. H{x(n-a)}=y(n-a)Other wise it is said to be time variant
  • 40. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 40e-TECHNoteThis PPT is sponsored byIRDC Indiawww.irdcindia.com
  • 41. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 41Contd..HShift by aH Shift by aX(n) y(n-a)X(n) y(n-a)Output of shifted input , y(n,k)Shifted output, y(n-k)
  • 42. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 42Problems1) Y(n)=x(n)+ x(n-1)o/p of delayed input by ki.e. y(n,k)=x(n-k)+x(n-1-k)Delayed o/p by kY(n-k)= x(n-k)+x(n-k-1)Y(n,k)=y(n-k)System is time-invariant2)Y(n)=nx(n)o/p of delayed input by ki.e. y(n,k)=nx(n-k)Delayed o/p by kY(n-k)= (n-k)x(n-k)Y(n,k) !=y(n-k)System is time-variant3) Y(n)=x(-n)o/p of delayed input by ki.e. y(n,k)=x(-n-k) as x(n) x(n-k) x(-n-k)Delayed o/p by kY(n-k)= x(-(n-k))=x(-n+k)Y(n,k)=y(n-k) System is time-variant
  • 43. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 43DT System with Differential equationsSystem relationship between input and outputOutput is a function of input as well as outputs and can be well describedby differential equation∑ ∑= =−+−−=NkMkkk knxbknyany1 0)()()(where {ak} and {bk} are constant parameters that specify thesystem and are independent of x(n) and y(n)
  • 44. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 44IIR and FIR systemsIIR is recursive structureFIR is non-recursive structure
  • 45. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 45e-TECHNoteThis PPT is sponsored byIRDC Indiawww.irdcindia.com
  • 46. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 46ConvolutionDefinition: It is the tool or operation to determine the response ofthe LTI systemConvolution between two DT signals x(n) and h(n) is expressedas∑∑∞−∞=∞−∞=−=−==kkknxkhorkxknhnxnhny)()()()()(*)()(Example: x(n) is input , h(n) = [ 1 0.8 0.4 0.01]x(n)= { 1 3 2 1 2 2 1 1 3 2} y(n)= {1 }
  • 47. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 47Properties of Convolution• h(n) * x(n) = x(n)*h(n)• h(n)* [ax(n)] = a [h(n)*x(n)] where a is constant• h(n)*[x1(n)+x2(n)]=h(n)*x1(n)+h(n)*x2(n)• h(n)*[x1(n)*x2(n)]=[h(n)*x1(n)]*x2(n)
  • 48. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 48Convolution: Graphical MethodSteps :1) Reverse one of the signal to get h(-k)2) Shift right above signal by n to get h(n-k)3) Multiply (dot product) h(n-k) with x(k) to get sample y(n)4) Repeat step 2 and 3 to get sample y(n) for all values of n
  • 49. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 49Contd..]30112[)( −−=↑nx ]121[)( −=↑nhh(k)kh(-k)kx(k)k0 00
  • 50. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 50Contd..h(-k)k0k0x(k)Shift by 0 to get y(0)0 4 1 0 0 0 = 5 = y(0)
  • 51. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 51Contd..h(-1-k)k0k0x(k)Shift by -1 to get y(-1)0 0 2 0 0 0 0 = 2 = y(-1)
  • 52. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 52Contd..h(-2-k)k0k0x(k)Shift by -2 to get y(-2)0 0 0 0 0 0 0 0 = 0 = y(n) for n < -1
  • 53. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 53e-TECHNoteThis PPT is sponsored byIRDC Indiawww.irdcindia.com
  • 54. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 54Contd..h(1-k)k0k0x(k)Shift by 1 to get y(1)-2 2 -1 0 0 = -1 = y(1)
  • 55. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 55Contd..h(2-k)k0k0x(k)Shift by 2 to get y(2)0 -1 -2 0 0 = -3 = y(2)
  • 56. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 56Contd..h(3-k)k0k0x(k)Shift by 3 to get y(3)0 0 1 0 -3 = -2 = y(3)
  • 57. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 57Contd..h(4-k)k0k0x(k)Shift by 4 to get y(4)0 0 0 0 -6 = -6 = y(4)
  • 58. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 58Contd..h(5-k)k0k0x(k)Shift by 5 to get y(5)0 0 0 0 3 = 3 = y(5)
  • 59. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 59Contd..h(6-k)k0k0x(k)Shift by 6 to get y(6)0 0 0 0 0 0 0 = 0 = y(n) for n>5
  • 60. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 60e-TECHNoteThis PPT is sponsored byIRDC Indiawww.irdcindia.com
  • 61. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 61Contd..]30112[)( −−=↑nx ]121[)( −=↑nhh(k)kx(k)k0 0*= 0]3623152[)( −−−−=↑nylength (x)= N1=3length (h)=N2=5Thus,length (y)=N1+N2-1=7
  • 62. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 62301126022430112−−−−−−Convolution: Tabular method]30112[)( −−=↑nx ]121[)( −=↑nhh(n)x(n)30112 −−121−]3623152[)( −−−−=↑ny
  • 63. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 63Problems• Compute convolution for the following signals:– A)– B)– C)]1231[)(↑=nx ]11[)( =nh]54321[)(↑=nx ]11[)( −=nh]12312[)( −=nx ]1234[)( =nh
  • 64. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 64CorrelationThe cross-correlation of x(n) and y(n) is given by∑∞−∞=−=nxy lnynxlr )()()( ∑∞−∞=+=nxy nylnxlr )()()(or....3,2,1,0 ±±±=lforIf signal x(n) and y(n) are same i.e. y(n)=x(n), then auto-correlation is givenby∑∞−∞=−=nxx lnxnxlr )()()(....3,2,1,0 ±±±=lfor)()( lrlr yxxy −=
  • 65. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 65Contd..Computation of correlation is same as computation of convolutionwithout folding operation e.g.]3217312[)( −−=↑nx ]52142211[)( −−−=↑ny3217312 −−↑xy11224125−−−10 -5 15 35 5 10 -15-4 2 -6 -14 -2 -4 62 -1 3 7 1 2 -38 -4 12 28 4 8 -12-4 2 -6 -14 -2 -4 64 -2 -6 14 2 4 -6-2 1 -3 -7 -1 -2 32 -1 3 7 1 2 -3rxy=[ 10 -9 19 36 -14 33 0 7 13 -18 16 -7 5 -3]
  • 66. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 66e-TECHNoteThis PPT is sponsored byIRDC Indiawww.irdcindia.com
  • 67. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 67Quiz• Find the output of the system if input x(n) and impulse responseh(n) are given by ( May 2003, 4 marks)•Determine the autocorrelation of the following signals (Dec 97 , 5marks)– i) x(n)={ 1 2 1 1} ii) y(n)= { 1 1 2 1}– What is your conclusion?021)(===nxotherwisenn11,0,2−=−= )3()2()1()()( −−−+−−= nnnnnh δδδδ
  • 68. 10/7/2009e-TECHNote from IRDC Indiainfo@irdcindia.com 68End of Chapter 01Queries ???