1. 1 Dr. Uri Mahlab

2. 1.א1.א תוכן עניינים : Introduction of Binary Digital Modulation Schemes 2-10 Probability of error 11-21 Transfer function of the optimum filter 22-26 Matched filter receiver 27-29 Correlation receiver 30-32 Example (the BER average of PSK) 33-35 Binary ASK Signaling Schemes 36-40 Coherent ASK 41-43 Noncoherent ASK 44-49 Binary PSK signaling schemes 50-51 Coherent PSK 52-54 Differentially Coherent PSK 55-57 Binary FSK signaling schemes 58-59 Coherent FSK 60-62 Noncoherent FSK 63-64 Comparison of digital modulation schemes 65 M-ary signaling schemes 66-85 Probability of error of M-ary orthogonal signaling scheme 86-88 Synchronization Methods 89-90 Dr. Uri Mahlab

3. INTRODUCTION In order to transmit digital information over * bandpass channels, we have to transfer the information to a carrier wave of.appropriate frequency We will study some of the most commonly * used digital modulation techniques wherein the digital information modifies the amplitude the phase, or the frequency of the carrier in.discrete steps 2 Dr. Uri Mahlab

4. The modulation waveforms for transmitting :binary information over bandpass channels 3 Dr. Uri Mahlab

5. OPTIMUM RECEIVER FOR BINARY :DIGITAL MODULATION SCHEMS The function of a receiver in a binary communication * system is to distinguish between two transmitted signals.S 1 (t) and S 2 (t) in the presence of noise The performance of the receiver is usually measured * in terms of the probability of error and the receiver is said to be optimum if it yields the minimum.probability of error In this section, we will derive the structure of an optimum * receiver that can be used for demodulating binary.ASK,PSK,and FSK signals 4 Dr. Uri Mahlab

6. Description of binary ASK,PSK, and : FSK schemes -Bandpass binary data transmission system Input + + ּ+ּ+ 5 Dr. Uri Mahlab

7. :Explanation * The input of the system is a binary bit sequence {b k } with a *.bit rate r b and bit duration T b The output of the modulator during the Kth bit interval *.depends on the Kth input bit b k The modulator output Z(t) during the Kth bit interval is * a shifted version of one of two basic waveforms S 1 (t) or S 2 (t) and :Z(t) is a random process defined by.1 6 Dr. Uri Mahlab

8. The waveforms S 1 (t) and S 2 (t) have a duration * of T b and have finite energy,that is,S 1 (t) and S 2 (t) =0 if and Energy :Term 7 Dr. Uri Mahlab

9. :The received signal + noise 8 Dr. Uri Mahlab

10. Choice of signaling waveforms for various types of digital* modulation schemes S 1 (t),S 2 (t)=0 for.The frequency of the carrier f c is assumed to be a multiple of r b Type of modulation ASK PSK FSK Dr. Uri Mahlab

11. :Receiver structure output 10 Dr. Uri Mahlab

12. :{Probability of Error-{P e* The measure of performance used for comparing * !!!digital modulation schemes is the probability of error The receiver makes errors in the decoding process * !!! due to the noise present at its input The receiver parameters as H(f) and threshold setting are * !!!chosen to minimize the probability of error 11 Dr. Uri Mahlab

13. :The output of the filter at t=kT b can be written as * 12 Dr. Uri Mahlab

14. :The signal component in the output at t=kT b h( ) is the impulse response of the receiver filter* ISI=0* 13 Dr. Uri Mahlab

15. Substituting Z(t) from equation 1 and making* change of the variable, the signal component :will look like that 14 Dr. Uri Mahlab

16. :The noise component n 0 (kT b ) is given by *.The output noise n 0 (t) is a stationary zero mean Gaussian random process :The variance of n 0 (t) is* :The probability density function of n 0 (t) is* 15 Dr. Uri Mahlab

17. The probability that the kth bit is incorrectly decoded* :is given by.2 16 Dr. Uri Mahlab

18. :The conditional pdf of V 0 given b k = 0 is given by* :It is similarly when b k is 1*.3 17 Dr. Uri Mahlab

19. Combining equation 2 and 3, we obtain an* :expression for the probability of error- P e as.4 18 Dr. Uri Mahlab

20. :Conditional pdf of V 0 given b k :The optimum value of the threshold T 0 * is* 19 Dr. Uri Mahlab

21. Substituting the value of T* 0 for T 0 in equation 4* we can rewrite the expression for the probability :of error as 20 Dr. Uri Mahlab

22. The optimum filter is the filter that maximizes* the ratio or the square of the ratio (maximizing eliminates the requirement S 01 <S 02 ) 21 Dr. Uri Mahlab

23. :Transfer Function of the Optimum Filter* The probability of error is minimized by an * appropriate choice of h(t) which maximizes Where And 22 Dr. Uri Mahlab

24. If we let P(t) =S 2 (t)-S 1 (t), then the numerator of the* :quantity to be maximized is Since P(t)=0 for t<0 and h( )=0 for <0* :the Fourier transform of P 0 is 23 Dr. Uri Mahlab

25. :Hence can be written as* (*) We can maximize by applying Schwarz’s* :inequality which has the form (**) 24 Dr. Uri Mahlab

26. Applying Schwarz’s inequality to Equation(**) with- and We see that H(f), which maximizes,is given by- !!! Where K is an arbitrary constant (***) 25 Dr. Uri Mahlab

27. Substituting equation (***) in(*), we obtain- :the maximum value of as :And the minimum probability of error is given by- 26 Dr. Uri Mahlab

28. :Matched Filter Receiver* If the channel noise is white, that is, G n (f)= /2,then the transfer - :function of the optimum receiver is given by From Equation (***) with the arbitrary constant K set equal to /2- :The impulse response of the optimum filter is 27 Dr. Uri Mahlab

29. Recognizing the fact that the inverse Fourier * of P*(f) is P(-t) and that exp(-2 jfT b ) represent :a delay of T b we obtain h(t) as :Since p(t)=S 1 (t)-S 2 (t), we have* 28 Dr. Uri Mahlab

30. :Impulse response of the Matched Filter * 2 \T b 1 0 0 1- 2 0 TbTb t t t t t (a) (b) (c) 2 \T b 0 2 (d) 2 \T b 0 TbTb 2 (e) Dr. Uri Mahlab

31. :Correlation Receiver* The output of the receiver at t=T b* Where V( ) is the noisy input to the receiver Substituting and noting * : that we can rewrite the preceding expression as (# #) 30 Dr. Uri Mahlab

32. Equation(# #) suggested that the optimum receiver can be implemented * as shown in Figure 1.This form of the receiver is called A Correlation Receiver - + 31 Dr. Uri Mahlab

33. In actual practice, the receiver shown in Figure 1 is actually *.implemented as shown in Figure 2 In this implementation, the integrator has to be reset at the - (end of each signaling interval in order to ovoid (I.S.I + c Figure 2 The bandwidth of the filter preceding the integrator is assumed * !!! to be wide enough to pass z(t) without distortion 32 Dr. Uri Mahlab

34. Example: A band pass data transmission scheme uses a PSK signaling scheme with The carrier amplitude at the receiver input is 1 mvolt and the psd of the A.W.G.N at input is watt/Hz. Assume that an ideal correlation receiver is used. Calculate the.average bit error rate of the receiver 33 Dr. Uri Mahlab

35. :Solution Data rate =5000 bit/sec Receiver impulse response Threshold setting is 0 and 34 Dr. Uri Mahlab

36. =Probability of error = Pe * :Solution Continue 35 Dr. Uri Mahlab

37. The binary ASK waveform can be described as Where and We can represent :Z(t) as 36 * Binary ASK signaling schemes: Dr. Uri Mahlab

38. Where D(t) is a lowpass pulse waveform consisting of.rectangular pulses :The model for D(t) is 37 Dr. Uri Mahlab

39. :The power spectral density is given by The autocorrelation function and the power spectral density :is given by 38 Dr. Uri Mahlab

40. :The psd of Z(t) is given by 39 Dr. Uri Mahlab

41. If we use a pulse waveform D(t) in which the individual pulses g(t) have the shape 40 Dr. Uri Mahlab

42. Coherent ASK We start with The signal components of the receiver output at the :of a signaling interval are 41 Dr. Uri Mahlab

43. :The optimum threshold setting in the receiver is :The probability of error can be computed as 42 Dr. Uri Mahlab

44. :The average signal power at the receiver input is given by We can express the probability of error in terms of the :average signal power The probability of error is sometimes expressed in * : terms of the average signal energy per bit, as 43 Dr. Uri Mahlab

45. Noncoherent ASK :The input to the receiver is * 44 Dr. Uri Mahlab

46. Non-coherent ASK Receiver 45 Dr. Uri Mahlab

47. :The pdf is 46 Dr. Uri Mahlab

48. pdf’s of the envelope of the noise and the signal * :pulse noise 47 Dr. Uri Mahlab

49. :The probability of error is given by 48 Dr. Uri Mahlab

50. 49 Dr. Uri Mahlab

51. BINARY PSK SIGNALING SCHEMES :The waveforms are * :The binary PSK waveform Z(t) can be described by *.D(t) - random binary waveform * 50 Dr. Uri Mahlab

52. :The power spectral density of PSK signal is 51 Dr. Uri Mahlab

53. Coherent PSK :The signal components of the receiver output are 52 Dr. Uri Mahlab

54. :The probability of error is given by 53 Dr. Uri Mahlab

55. 54 Dr. Uri Mahlab

56. DIFFERENTIALLY COHERENT * :PSK DPSK modulator 55 Dr. Uri Mahlab

57. DPSK demodulator Filter to limit noise power Delay Lowpass filter or integrator Threshold device (A/D) 56 Dr. Uri Mahlab

58. Differential encoding & decoding 57 Dr. Uri Mahlab

59. * BINARY FSK SIGNALING SCHEMES : :The waveforms of FSK signaling :Mathematically it can be represented as 58 Dr. Uri Mahlab

60. Power spectral density of FSK signals Power spectral density of a binary FSK signal with 59 Dr. Uri Mahlab

61. Coherent FSK :The local carrier signal required is The input to the A/D converter at sampling time 60 Dr. Uri Mahlab

62. The probability of error for the correlation receiver is :given by 61 Dr. Uri Mahlab

63. .Which are usually encountered in practical system :We now have 62 :When Dr. Uri Mahlab

64. Noncoherent FSK 63 Dr. Uri Mahlab

65. Noncoharenr demodulator of binary FSK ENVELOPE DETECTOR ENVELOPE DETECTOR THRESHOLD DEVICE (A/D) + - 64 Dr. Uri Mahlab

66. Probability of error for binary digital modulation * :schemes 65 Dr. Uri Mahlab

67. :M-ARY coherent PSK The M possible signals that would be transmitted :during each signaling interval of duration Ts are :The digital M-ary PSK waveform can be represented 66 M-ARY SIGNALING SCHEMES Dr. Uri Mahlab

68. :In four-phase PSK (QPSK), the waveform are 67 Dr. Uri Mahlab

69. Phasor diagram for QPSK That are derived from a coherent local carrier reference 68 Dr. Uri Mahlab

70. If we assume that S 1 was the transmitted signal :during the signaling interval (0,T s ),then we have 69 Dr. Uri Mahlab

71. QPSK receiver scheme 70 Dr. Uri Mahlab

72. :The outputs of the correlators at time t=T S are 71 Dr. Uri Mahlab

73. 72 Probability of error of QPSK: Dr. Uri Mahlab

74. 73 Dr. Uri Mahlab

75. Phasor diagram for M-ary PSK ; M=8 74 Dr. Uri Mahlab

76. The average power requirement of a binary PSK :scheme are given by 75 Dr. Uri Mahlab

77. * COMPARISION OF POWER-BANDWIDTH :FOR M-ARY PSK Value of M 4 8 16 32 0.5 0.333 0.25 0.2 0.34 dB 3.91 dB 8.52 dB 13.52 dB 76 Dr. Uri Mahlab

78. RECEIVER FOR FOUR PHASE DIFFERENTIAL PSK Integrate and dump filter Integrate and dump filter Z(t) 77 * M-ary for four-phase Differential PSK: Dr. Uri Mahlab

79. :The probability of error in M-ary differential PSK :The differential PSK waveform is 78 Dr. Uri Mahlab

80. :Transmitter for differential PSK* Serial to parallel converter Diff phase mod. Envelope modulator BPF (Z(t 79 Dr. Uri Mahlab

81. Let us consider an FSK scheme witch have the : following properties 80 * M-ary Wideband FSK Schemas: Dr. Uri Mahlab

82. :Orthogonal Wideband FSK receiver MAXIMUM SELECTOR Z(t) 81 Dr. Uri Mahlab

83. :The filter outputs are 82 Dr. Uri Mahlab

84. :N 0 is given by :The probability of correct decoding as :In the preceding step we made use of the identity 83 Dr. Uri Mahlab

85. The joint pdf of Y2,Y3,…,YM * :is given by 84 Dr. Uri Mahlab

86. where 85 Dr. Uri Mahlab

87. Probability of error for M-ary orthogonal * : signaling scheme 86 Dr. Uri Mahlab

88. The probability that the receiver incorrectly * decoded the incoming signal S 1 (t) is P e1 = 1-P e1 The probability that the receiver makes * an error in decoding is P e = P e1 We assume that, and We can see that increasing values of M lead to smaller power requirements and also to more complex transmitting receiving equipment. 87 Dr. Uri Mahlab

89. In the limiting case as M the probability of error P e satisfies The maximum errorless r b at W data can be transmitted using an M- ary orthogonal FSK signaling scheme The bandwidth of the signal set as M 88 Dr. Uri Mahlab

90. :Synchronization Methods For optimum demodulation of ASK,FSK,and PSK waveforms timing information is needed at the receiver There are three general methods used for synchronization in :digital nodulation schemes.Use of primary or secondary time standard.Utilization of a separate synchronization signal Extraction of clock information from the modulated waveform.itself, referred to as self - synchronization.1.2.3 89 Dr. Uri Mahlab

91. Open loop carrier recovery scheme Closed loop carrier recovery scheme (Extraction of local carrier for coherent demodulation of PSK signals) Squaring circuit Frequency divider BPF Squaring circuit Frequency doubler Loop Filter VCO Recovered carrier cos (w c t) 90 Dr. Uri Mahlab