Weiyao Lin. Shanghai Jiao Tong University. Chapter 5: Digital Transmission through Baseband slchannels Textbook: Ch

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1 Principles of Communications Weiyao Lin Shanghai Jiao Tong University Chapter 5: Digital Transmission through Baseband slchannels Textbook: Ch /2010 Meixia SJTU 1

2 Topics to be Covered Source A/D converter Channel Detector D/A converter User Digital waveforms over baseband channels Band-limited channel and Inter-symbol interference Signal design for band-limited channels System design and channel equalization 2009/2010 Meixia SJTU 2

3 5.1 Baseband Signalling Waveforms To send the encoded digital data over a baseband channel, we require the use of format or waveform for representing the data System requirement on digital waveforms Easy to synchronize High spectrum utilization efficiency Good noise immunity No dc component and little low frequency component Self-error-correction capability 2009/2010 Meixia SJTU 3

4 Basic Waveforms Many formats available. Some examples: On-off or unipolar signaling Polar signaling Return-to-zero signaling Bipolar signaling useful because no dc Split-phase or Manchester code no dc 2009/2010 Meixia SJTU 4

5 On-off (unipolar) polar Return to zero bipolar Manchester 2009/2010 Meixia SJTU 5

6 Spectra of Baseband Signals Consider a random binary sequence g 0 (t) - 0,g 1 (t) - 1 The pulses g 0 (t)g 1 (t)occur independently with probabilities given by p and 1-p p,respectively. The duration of each pulse is given by Ts. g ( 2 ) 0 t + T S s(t ) g ( 2 ) 1 t T S ( t ) = s T s n ( t s ) n = s n () t g 0 ( t nts ), with prob. P = g1 ( t nts ), with prob. 1 P 2009/2010 Meixia SJTU 6

7 Power Spectral Density PSD of the baseband signal s(t) is m m m δ s s m= s s s S( f) = p(1 p) G ( f) G ( f) + pg ( ) + (1 p) G ( ) ( f ) T T T T T 1 st term is the continuous freq. component 2 nd term is the discrete freq. component 2009/2010 Meixia SJTU 7

8 g () t = g () t = g() t For polar signalling with 0 1 and p=1/2 1 2 S( f) = G( f) T For unipolar signalling with g () t = 0 g 0 1 () t = g () t and p=1/2, and g(t) is a rectangular pulse sinπ ft G( f) = T π ft T sinπ ft 1 Sx ( f) = δ ( f) 4 + π ft 4 For return-to-zero unipolar signalling τ = T/2 2 2 T sin π ft / m Sx ( f) = ( ) ( ) 2 16 δ f δ f π ft / odd m T [ mπ ] 2009/2010 Meixia SJTU 8

9 PSD of Basic Waveforms T 4 unipolar 1 T 2 T T 4 τ = T / 2 2 T 4 T Return-to-zero polar T 16 τ = T / 2 1 T 2 T 3 T 4 T Return-to-zero unipolar 2009/2010 Meixia SJTU 9

10 5.2 Bandlimited Channel A bandlimited channel can be modeled as a linear filter with frequency response limited to certain frequency range The filtering effect will cause a spreading (or smearing out) of individual data symbols passing through a channel 2009/2010 Meixia SJTU 10

11 For consecutive symbols, this spreading causes part of the symbol energy to overlap with neighbouring symbols, causing intersymbol interference (ISI). 2009/2010 Meixia SJTU 11

12 Baseband Signaling through Bandlimited Channels Input to tx filter x () t = Aδ ( t it) Output of tx filter Output of rx filter s i= x () t = Ah ( t it) t i T i= i 2009/2010 Meixia SJTU 12

13 Pulse shape at the receiver filter output t Impulse response of the cascade connection of tx, channel, and rx filters Overall frequency response Receiving filter output ( t kt ) n ( ) v( t) = Ak p + o t k = 2009/2010 Meixia SJTU 13

14 Intersymbol Interference The receiving filter output v(t) is sampled at t m =mt (to detect A m ) Desired signal Gaussian noise intersymbol interference (ISI) ISI can significantly ifi degrade d the ability of the data detector. t 2009/2010 Meixia SJTU 14

15 Eye Diagrams A visual method to investigate the problem of ISI Generated by connecting the received waveforms to a conventional oscilloscope Oscilloscope is re-triggered at every symbol period or multiple of symbol periods using a timing recovery signal. Segments of the received waveforms are then superimposed on one another The resulting display is called an eye pattern 2009/2010 Meixia SJTU 15

16 Eye Diagrams (cont d) Distorted binary wave Eye pattern T b 2009/2010 Meixia SJTU 16

17 Eye Diagrams (cont d) Example eye diagrams for different distortions, each has a distinctive effect on the appearance of the eye opening The width of the eye opening defines the time interval over which the wave can be sampled. ped The best sampling time is the instant when the eye is open widest The height of the eye opening defines the margin over noise 2009/2010 Meixia SJTU 17

18 ISI Minimization Choose transmitter and receiver filters which shape the received pulse function so as to eliminate or minimize interference between adjacent pulses, hence not to degrade the bit error rate performance of the link 2009/2010 Meixia SJTU 18

19 5.3 Signal Design for Bandlimited Channel Zero ISI The effect of ISI can be completely negated if it is possible to obtain a received pulse shape, p(t), such that Echos made to be zero at sampling points or This is the Nyquist condition for Zero ISI If p(t) satisfies the above condition, the receiver output simplifies to 2009/2010 Meixia SJTU 19

20 Nyquist Condition: Ideal Solution Nyquist s s first method for eliminating ISI is to use p( t) = ( π t / T ) sin π t /TT = t sinc T P(f) brick wall filter 1 p(t) p(t-t) -1/2T 0 1/2T f Let = called the Nyquist bandwdith, The minimum transmission bandwidth for zero ISI. A channel with bandwidth B 0 can support a max. transmission rate of 2B 0 symbols/sec 2009/2010 Meixia SJTU 20

21 Achieving Nyquist Condition Difficult to design such p(t) or P(f) P(f) is physically unrealizable due to the abrupt transitions at ±B 0 p(t) decays slowly for large t, resulting in little margin of error in sampling times in the receiver. This demands accurate sample point timing - a major challenge in modem / data receiver design. Inaccuracy in symbol timing is referred to as timing jitter. 2009/2010 Meixia SJTU 21

22 Practical Solution: Raised Cosine Spectrum Let P(f) decrease toward to zero gradually rather than abruptly. P(f) is made up of 3 parts: passband, stopband, and transition band. The transition band is shaped like a cosine wave. 1 0 f < f 1 1 π ( f f ) 1 P ( f ) 1 + cos < 2B f1 f 0 f 2 2B0 2 f1 0 f 2B 0 f1 = 1 P(f) 1 2B 0 -f 1 0 f 1 B 0 2B 0 f 2009/2010 Meixia SJTU 22

23 Raised Cosine Spectrum P(f) 1 α =0 Roll-off factor α = 0.5 α = f 1 1 B 0.5 α =1 0 0 B 0 1.5B 0 2B 0 f The sharpness of the filter is controlled by. When α = 0 this reduces to the brick wall filter. The bandwidth required by using raised cosine spectrum increased from its minimum value B 0 to actual bandwidth B = B 0 (1+α) 2009/2010 Meixia SJTU 23

24 Time-Domain Pulse Shape Inverse Fourier transform of raised cosine spectrum cos(2 παbt) pt () = sinc(2 Bt) α Bt 0 Ensures zero crossing at desired sampling instants 0 T b 2T b α=1 Decreases as 1/t 2, reduces the tail of the pulses such that the data receiving is relatively insensitive to sampling time error α=0.5 α= t/t b 2009/2010 Meixia SJTU 24

25 Choice of Roll-offoff Factor Benefits of small a Higher bandwidth efficiency Benefits of large a simpler filter with fewer stages hence easier to implement less sensitive to symbol timing accuracy 2009/2010 Meixia SJTU 25

26 Signal Design with Controlled ISI Partial Response Signals Relax the condition of zero ISI and allow a controlled amount of ISI Then we can achieve the max. symbol rate of 2W symbols/sec The ISI we introduce is deterministic or controlled ; hence it can be taken into account at the receiver 2009/2010 Meixia SJTU 26

27 Duobinary Signal Let {ak} be the binary sequence to be transmitted. The pulse duration is T. Two adjacent pulses are added together, i.e. b k = a k + a k 1 Ideal LPF The resulting sequence {b k } is called duobinary signal 2009/2010 Meixia SJTU 27

28 Characteristics of Duobinary Signal Frequency domain ( j2π ft) G( f) = 1 + e H ( f) L 2 j π ft Te cos π ft ( f 1/2 T ) = 0 ( otherwise) H L T ( f 1/2 T ) ( f) = 0 ( otherwise) 2009/2010 Meixia SJTU 28

29 Time domain Characteristics [ δ δ ] g() t = () t + ( t T) h () t gt () t t T = sinc + sinc T T is called a duobinary signal pulse It is observed that: g(0)= g 0 = 1 g ( T )= g = 1 1 git ( )= g = 0 ( i 0, 1) i (The current symbol) (ISI to the next symbol) L sin πt/ T sin π ( t T) / T = + π t / T π ( t T )/ T 2 T sin πt/ T = πtπ t ( T t ) Decays as 1/t 2, and spectrum within 1/2T 2009/2010 Meixia SJTU 29

30 Decoding Without noise, the received signal is the same as the transmitted signal { } a k y = ag = a + a = b A 3-level sequence = 1 k i k i i= 0 k k k When is a polar sequence with values +1 or -1: 2 ( ak = ak 1 = 1) y = b = 0 ( a = 1, a = 1 or a = 1, a = 1) k k k k 1 k k 1 2 ( a = k a = k 1 1) { } a k When is a unipolar sequence with values 0 or 1 0 ( ak = ak 1 = 0) y = b = 1 ( a = 0, a = 1 or a = 1, a = 0) k k k k 1 k k 1 2 ( ak = ak 1 = 1) 2009/2010 Meixia SJTU 30

31 To recover the transmitted sequence, we can use aˆ = b aˆ = y aˆ k k k 1 k k 1 Main drawback: the detection of the current symbol relies on the detection of the previous symbol => error propagation will occur How to solve the ambiguity problem and error propagation? Precoding: Apply differential encoding on { } so thatt = Then the output of the duobinary signal system is bk = ck + ck 1 a k ck ak ck /2010 Meixia SJTU 31

32 Block Diagram of Precoded Duobinary Signal { a k } { c k } { } b k H ( ) L f yt () { } c k /2010 Meixia SJTU 32

33 Modified Duobinary Signal Modified duobinary signal H bk = ak ak 2 ( f ) After LPF L, the overall response is G f e H f j4π ft ( ) = (1 ) L ( ) = 0 otherwise j2π ft 2Tje sin2 π ft ( f 1/2 T) gt () sin πt/ T sin π ( t 2 T) / T 2 2T sin πt/ T = = πt/ T π( t 2 T)/ T πt( t 2 T) 2009/2010 Meixia SJTU 33

34 2009/2010 Meixia SJTU 34

35 Properties The magnitude spectrum is a half-sin wave and hence easy to implement No dc component and small low freq. component At sampling interval T, the sampled values are g (0) = g = 1 g ( T ) = g = 0 g (2 T ) = g = g ( it ) = g = 0, i 0,1,2 i 2 g () t also delays as 1 / t. But at t = T, the timing offset may cause significant problem. 2009/2010 Meixia SJTU 35

36 Decoding of modified duobinary signal To overcome error propagation, precoding is also needed. The coded signal is c k = a k c k 2 g b k = c k c k 2 a k ck bk H ( ) L f ck /2010 Meixia SJTU 36

37 Update Now we have discussed: d Pulse shapes of baseband signal and their power spectrum ISI in bandlimited channels Signal design for zero ISI and controlled ISI We next discuss system design in the presence of channel distortion Optimal transmitting and receiving filters Channel equalizer 2009/2010 Meixia SJTU 37

38 5.4 Optimum Transmit/Receiver Filter Recall that when zero-isi condition is satisfied by p(t) with raised cosine spectrum P(f), then the sampled output of the receiver filter is (assume ) Consider binary PAM transmission: Variance of N m = with Error Probability can be minimized through a proper choice of H R (f) and H T (f) so that is maximum (assuming H C (f) fixed and P(f) given) 2009/2010 Meixia SJTU 38

39 Optimal Solution Compensate the channel distortion equally between the transmitter and receiver filters Then, the transmit signal energy is given by Hence By Parseval s theorem 2009/2010 Meixia SJTU 39

40 Noise variance at the output of the receive filter is Special case: Performance loss due to channel distortion This is the ideal case with flat fading No loss, same as the matched filter receiver for AWGN channel 2009/2010 Meixia SJTU 40

41 Exercise Determine the optimum transmitting and receiving filters for a binary communications system that transmits data at a rate R=1/T = 4800 bps over a channel with a frequency 1 response H c( f) = ; f W where W= 4800 Hz f 1+ ( ) W The additive noise is zero-mean white Gaussian with spectral density /2010 Meixia SJTU 41

42 Solution Since W = 1/T = 4800, we use a signal pulse with a raised cosine spectrum and a roll-off factor = 1. Thus, Therefore One can now use these filters to determine the amount of transmit energy required to achieve a specified error probability 2009/2010 Meixia SJTU 42

43 Performance with ISI If zero-isi condition is not met, then Let Then 2009/2010 Meixia SJTU 43

44 Often only 2M significant terms are considered. Hence with Finding the probability of error in this case is quite difficult. Various approximation can be used (Gaussian approximation, Chernoff bound, etc). What is the solution? 2009/2010 Meixia SJTU 44

45 Monte Carlo Simulation ~ V m X Threshold = γ t m = mt Let error occurs I( x)= 1 0 else 0 1 L () l Pe I( X ) = L L l = 1 where X (1), X (2),..., X (L) are i.i.d. (independent and identically distributed) ) random samples 2009/2010 Meixia SJTU 45

46 If one wants P e to be within 10% accuracy, how many independent simulation runs do we need? If P e ~ 10-9 (this is typically the case for optical communication systems), and assume each simulation run takes 1 msec, how long will the simulation take? 2009/2010 Meixia SJTU 46

47 We have shown that By properly designing the transmitting and receiving filters one can guarantee zero ISI at sampling instants, thereby minimizing Pe. Appropriate when the channel is precisely known and its characteristics do not change with time In practice, the channel is unknown or time-varying We now consider: channel equalizer A receiving filter with adjustable frequency response With channel measurement, one can adjust the frequency response of the receiving filter so that the overall filter response is near optimum 2009/2010 Meixia SJTU 47

48 5.5 Equalizer Two main types of equalizers Preset equalizers Adaptive equalizers Preset equalizers For channels whose frequency response characteristics are unknown but time-invariant We may measure the channel characteristics, adjust the parameters of the equalizer Once adjusted, the equalizer parameters remain fixed during the transmission of data Such equalizers are called preset equalizers 2009/2010 Meixia SJTU 48

49 Equalizer (cont d) Adaptive equalizers Update their parameters on a periodic basis during the transmission of data This is often done by sending a known signal through the channel and allowing the equalizer to adjust its parameters in response to this known signal (which h is known as Training i sequence) Adaptive equalizers are useful when the channel characteristics are unknown or if they change slowly with time. 2009/2010 Meixia SJTU 49

50 Equalizer Configuration Transmitting i Channel Filter H T (f) H C (f) Equalizer H E (f) v(t) t=mt v m Overall frequency response: To guarantee zero ISI, Nyquist criterion must be satisfied Ideal zero-isi equalizer is an inverse channel filter with 2009/2010 Meixia SJTU 50

51 Linear Transversal Filter As ISI is limited to a finite number of samples, the channel equalizer can be approximated by a finite impulse response (FIR) filter or a transversal filter Unequalized input c -N c -N+1 c N-1 c N (Here τ=t) (2N+1)-tap FIR equalizer t=nt Output {c n } are the adjustable 2N+1 equalizer coefficients N is chosen sufficiently large so that the equalizer spans the length of ISI 2009/2010 Meixia SJTU 51

52 Zero-Forcing Equalizer Let p c (t) denote the received pulse from a channel to be equalized Tx & Channel At sampling time t = mt To suppress 2N adjacent interference terms 2009/2010 Meixia SJTU 52

53 Rearrange to matrix form where (2N+1) x (2N+1) channel response matrix Thus, given p c (t), one can determine the (2N+1) unknown coefficients We have exactly N zeros on both sides of main pulse response 2009/2010 Meixia SJTU 53

54 Example Consider the channel response as shown below Find the coefficients of a Five-tap transversal filter equalizer which will force two zeros on each side of the main pulse response p c (t) 1.0 4T + Δt 2T + Δt T + Δt 3T + Δt 3T + Δt T + Δt Δt 2T + Δt 4T + Δt 2009/2010 Meixia SJTU 54

55 Solution By inspection The channel response matrix is [ P c ] = /2010 Meixia SJTU 55

56 The inverse of this matrix, by numerical methods, is found to be [ P c ] 1 = The coefficient vector is the center column of [P c ] -1. Therefore, c 1 =0.117, c -1 =-0.158, c 0 = 0.937, c 1 = 0.133, c 2 = The sample values of the equalized pulse response It can be verified that 2009/2010 Meixia SJTU 56

57 Note that values of p eq (n) () for n < -2 or n > 2 are not zero. For example: p eq () 3 = 0117)( ) + ( )(. 0 02) + ( )( 0. 05) )(.) + ( )( 01.) = p eq ( 3) = ( )( 0. 2) + ( )( 01. ) + ( )( 0. 05) + ( )( 01. ) + ( )( ) = /2010 Meixia SJTU 57

58 Minimum Mean-Square Error Equalizer Drawback of ZF equalizer Ignores the additive noise, may result in significant noise enhancement in certain frequency range Alternatively, Relax zero ISI condition Select equalizer characteristics such that the combined power in the residual ISI and additive noise at the output t of the equalizer is minimized A channel equalizer that t is optimized i based on the minimum mean-square error (MMSE) criterion is called MMSE equalizer 2009/2010 Meixia SJTU 58

59 MMSE Criterion Output from the channel z ( t) c y( t nt ) = N n= N n The output is sampled at t = mt: Let A m = desired equalizer output MSE [( z( mt ) A ) 2 ] = Minimum = E m 2009/2010 Meixia SJTU 59

60 where Expectation is taken over random sequence Am and the additive noise MMSE solution is obtained by 2009/2010 Meixia SJTU 60

61 MMSE Equalizer vs. ZF Equalizer Both can be obtained by solving similar equations ZF equalizer does not take into consideration effects of noise MMSE equalizer designed so that mean-square error (consisting of ISI terms and noise at the equalizer output) is minimized Both equalizers are known as linear equalizers 2009/2010 Meixia SJTU 61

62 Decision Feedback Equalizer (DFE) DFE is a nonlinear equalizer which attempts to subtract from the current symbol to be detected the ISI created by previously detected symbols Input Feedforward Filter + - Symbol Detection Output Feedback Filter 2009/2010 Meixia SJTU 62

63 Example of Channels with ISI 2009/2010 Meixia SJTU 63

64 Frequency Response Channel B will tend to significantly enhance the noise when a linear equalizer is used (since this equalizer will have to introduce a large gain to compensate channel null). 2009/2010 Meixia SJTU 64

65 Performance of MMSE Equalizer 31-taps Proakis & Salehi, 2 nd 2009/2010 Meixia SJTU 65

66 Performance of DFE Proakis & Salehi, 2 nd 2009/2010 Meixia SJTU 66

67 Maximum Likelihood Sequence Estimation (MLSE) t=mt MLSE Transmitting Channel Receiving (Viterbi Filter H T (f) H C (f) Filter H R (f) y m Algorithm) Let the transmitting filter have a square root raised cosine frequency response The receiving filter is matched to the transmitter filter with The sampled output from receiving filter is 2009/2010 Meixia SJTU 67

68 MLSE Assume ISI affects finite number of symbols, with Then, the channel is equivalent to a FIR discrete-time ti filter T T T T Finite-state it t t machine 2009/2010 Meixia SJTU 68

69 Performance of MLSE Proakis & Salehi, 2 nd 2009/2010 Meixia SJTU 69

70 Equalizers Preset Adaptive Blind Equalizer Equalizer Equalizer Linear Non-linear ZF MMSE DFE MLSE 2009/2010 Meixia SJTU 70

71 Homework 3 Textbook Chapter 7: 7.4, 7.18 Textbook Chapter 9: 9.4, 9.5, 9.12, 9.24 Due: In-class submission on November 7th (Monday) 2009/2010 Meixia SJTU

72 Schedule -1 Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7 Week 8 Ch01:Introduction Ch02:Signal, random process, and spectra Ch03:Analog modulation Ch04: Analog to Digital Conversion Ch05: Digital transmission through baseband channels 2000/2010 Meixia SJTU 6

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