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  2. 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 1.א 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 fortransmitting :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 .S1(t) and S2(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 {bk} with a * .bit rate r b and bit duration Tb The output of the modulator during the Kth bit interval * .depends on the Kth input bit bk The modulator output Z(t) during the Kth bit interval is * a shifted version of one of two basic waveforms S1(t) or S2(t) and :Z(t) is a random process defined by .1 6 Dr. Uri Mahlab

  8. The waveforms S1(t) and S2(t) have a duration * of Tb and have finite energy,that is,S1(t) and S2(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 S1(t),S2(t)=0 for .The frequency of the carrier fc is assumed to be a multiple of rb Type of modulation ASK PSK FSK Dr. Uri Mahlab

  11. :Receiver structure output 10 Dr. Uri Mahlab

  12. :{Probability of Error-{Pe* 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=kTb can be written as * 12 Dr. Uri Mahlab

  14. :The signal component in the output at t=kTb 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 n0(kTb) is given by * .The output noise n0(t) is a stationary zero mean Gaussian random process :The variance of n0(t) is* :The probability density function of n0(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 V0 given bk = 0 is given by* .3 :It is similarly when bk is 1* 17 Dr. Uri Mahlab

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

  20. :Conditional pdf of V0 given bk :The optimum value of the threshold T0* is* 19 Dr. Uri Mahlab

  21. Substituting the value of T*0 for T0 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 S01<S02) 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) =S2(t)-S1(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 P0 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, Gn(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 jfTb) represent :a delay of Tb we obtain h(t) as :Since p(t)=S1(t)-S2(t) , we have* 28 Dr. Uri Mahlab

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

  31. :Correlation Receiver* The output of the receiver at t=Tb* 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. :Solution Continue =Probability of error = Pe * 35 Dr. Uri Mahlab

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