Principles of Communications
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1 Principles of Communications Chapter 5: Digital Transmission through Bandlimited Channels Selected from Chapter of Fundamentals of Communications Systems, Pearson Prentice Hall 2005, by Proakis & Salehi 1
2 Topics to be Covered Source A/D converter Channel Detector D/A converter User Digital waveforms over bandlimited baseband channels Band-limited channel and Inter-symbol interference Signal design for band-limited channels System design Channel equalization 2
3 Bandlimited Channel 20MHz 5MHz, 10MHz 15MHz, 20MHz Modeled as a linear filter with frequency response limited to certain frequency range 3
4 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 4
5 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 5
6 On-off (unipolar) polar Return to zero bipolar Manchester 6
7 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,respectively. The duration of each pulse is given by Ts. g ( 2 ) 0 t+ T S s(t) g ( 2 ) 1 t T S s ( t) = s n ( t) n= s () t n T s g0( t nts ), with prob. P = g1( t nts ), with prob. 1 P 7
8 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 8
9 For polar signalling with 1 S( f) = G( f) T g() t = g() t = gt () and p=1/2 For unipolar signalling with and g(t) is a rectangular pulse sinπ ft G( f) = T π ft g () t = 0 g 0 1 () t = gt () For return-to-zero unipolar signalling τ = T/2 2 2 and p=1/2, T sinπ ft 1 Sx ( f) = δ ( f) 4 + π ft 4 T sin π ft / m Sx ( f) = ( ) ( ) 2 16 δ f δ f π ft / odd m T [ mπ ] 9
10 PSD of Basic Waveforms unipolar Return-to-zero polar Return-to-zero unipolar 10
11 Intersymbol Interference The filtering effect of the bandlimited channel will cause a spreading of individual data symbols passing through For consecutive symbols, this spreading causes part of the symbol energy to overlap with neighbouring symbols, causing intersymbol interference (ISI). 11
12 Baseband Signaling through Bandlimited Channels Source Σ Input to tx filter x () t = Aδ ( t it ) Output of tx filter Output of rx filter s i= x () t = A h ( t it ) t i T i= i 12
13 Source Σ Pulse shape at the receiver filter output Overall frequency response Receiving filter output ( t kt ) n ( ) v( t) = Ak p + o t k = 13
14 Source Σ Sample the rx filter output at (to detect A m ) Desired signal Gaussian noise intersymbol interference (ISI) 14
15 Distorted binary wave Eye Diagram Eye pattern 15 T b
16 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 16
17 Signal Design for Bandlimited Channel Zero ISI Nyquist condition for Zero ISI for pulse shape Echos made to be zero at sampling points or With the above condition, the receiver output simplifies to 17
18 Nyquist Condition: Ideal Solution Nyquist s first method for eliminating ISI is to use p( t) = ( π t / T ) sin π t / T = sinc t T brick wall filter P(f) 1 p(t) p(t-t) -1/2T 0 1/2T f = 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 18
19 Achieving Nyquist Condition Challenges of designing such or is physically unrealizable due to the abrupt transitions at ±B 0 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. 19
20 Practical Solution: Raised Cosine Spectrum is made up of 3 parts: passband, stopband, and transition band. The transition band is shaped like a cosine wave. ( ) < + < = cos ) ( f B f f B f f f B f f f f f P π f 2B 0 0 P(f) B 0 1 f 1 2B 0 -f 1 20
21 Raised Cosine Spectrum P(f) α = 0 α = 0.5 α = 1 Roll-off factor α = f 1 1 B 0 0 B 0 1.5B 0 2B 0 f The sharpness of the filter is controlled by. Required bandwidth 21
22 Time-Domain Pulse Shape Taking inverse Fourier transform cos(2 παbt) pt ( ) = sinc(2 Bt) α Bt 0 Ensures zero crossing at desired sampling instants 0 T b 2T b α=1 α=0.5 α=0 Decreases as 1/t 2, such that the data receiving is relatively insensitive to sampling time error t/t b 22
23 Choice of Roll-off Factor Benefits of small Higher bandwidth efficiency Benefits of large simpler filter with fewer stages hence easier to implement less sensitive to symbol timing accuracy 23
24 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 24
25 Duobinary Signal Let {ak} be the binary sequence to be transmitted. The pulse duration is T. Two adjacent pulses are added together, i.e. bk = ak + ak 1 Ideal LPF The resulting sequence {b k } is called duobinary signal 25
26 Characteristics of Duobinary Signal Frequency domain ( j2π ft ) G( f) = 1 + e H ( f) L H L ( f) T ( f 1/2 T) = 0 ( ot her wi se) jπ ft 2Te cos π ft ( f 1/ 2 T ) = 0 ( ot her wi se) 26
27 [ δ δ ] gt () = () 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 gt ( )= g 1 = 1 Time domain Characteristics g( it )= g = 0 ( i 0, 1) i (The current symbol) (ISI to the next symbol) Decays as 1/t 2, and spectrum within 1/2T L sin πt/ T sin π( t T) / T = + πt/ T π( t T)/ T 2 T sin πt/ T = πt ( T t) 27
28 Decoding Without noise, the received signal is the same as the transmitted signal When { a k } y = ag = a + a 1 = b k i k i i= 0 k k k is a polar sequence with values +1 or -1: 2 ( ak = ak 1 = 1) yk = bk = 0 ( ak = 1, ak 1 = 1 or ak = 1, ak 1 = 1) 2 ( ak = ak 1 = 1) { } a k When is a unipolar sequence with values 0 or 1 0 ( ak = ak 1 = 0) yk = bk = 1 ( ak = 0, ak 1 = 1 or ak = 1, ak 1 = 0) 2 ( ak = ak 1 = 1) A 3-level sequence 28
29 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 { a k } so that k k k 1 Then the output of the duobinary signal system is bk = ck + ck 1 c = a c 29
30 Block Diagram of Precoded Duobinary Signal { } a k { } k c { } + + Transformer b k 0, 1, 2-2, 0, 2 H ( ) L f yt () T { } c k 1 30
31 Modified Duobinary Signal Modified duobinary signal After LPF bk = ak ak 2, the overall response is H ( ) L f G f e H f j4π ft ( ) = (1 ) L( ) = 0 otherwise j2π ft 2Tje sin 2 π ft ( f 1/ 2 T ) gt () sin πt/ T sin π( t 2 T) / T = πt/ T π( t 2 T)/ T = 2 2T sin π / t T πtt ( 2 T) 31
32 32
33 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 0 gt ( ) = g = 0 g(2 T) = g = g( it ) = g = 0, i 0,1, 2 i 2 gt () also delays as 1 / t. But at t = T, the timing offset may cause significant problem. 33
34 Decoding of modified duobinary signal To overcome error propagation, precoding is also needed. ck = ak ck The coded signal is 2 bk = ck ck T
35 Update Now we have discussed: 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 35
36 Optimum Transmit/Receive 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) 36
37 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 37
38 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 38
39 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 f 1+ ( W response H c( f) = ; f W where W= 4800 Hz 2 ) The additive noise is zero-mean white Gaussian with spectral density 39
40 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 40
41 Performance with ISI If zero-isi condition is not met, then Let Then 41
42 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? 42
43 Monte Carlo Simulation ~ V m X Threshold = γ t m = mt Let I( x) = 1 0 error occurs else P = e 1 L L l= 1 ( l) ( ) I X where X (1), X (2),..., X (L) are i.i.d. (independent and identically distributed) random samples 43
44 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? 44
45 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 45
46 What is Equalizer A receiving filter with adjustable frequency response to minimize/eliminate inter-symbol interference 46
47 Equalizer Configuration Transmitting Filter H T (f) Channel H C (f) Equalizer H E (f) v(t) t=mt v m Overall frequency response: Nyquist criterion for zero-isi Ideal zero-isi equalizer is an inverse channel filter with 47
48 Linear Transversal Filter Finite impulse response (FIR) filter Unequalized input (τ=t) c -N c -N+1 c N-1 c N (2N+1)-tap FIR equalizer t=nt Output are the adjustable equalizer coefficients is sufficiently large to span the length of ISI 48
49 Zero-Forcing Equalizer : the received pulse from a channel to be equalized Tx & Channel At sampling time To suppress 2N adjacent interference terms 49
50 Zero-Forcing Equalizer Rearrange to matrix form channel response matrix or the middle-column of 50
51 Example Find the coefficients of a five-tap FIR filter equalizer to force two zeros on each side of the main pulse response p c (t) 1.0 4T + t 2T + t T + t 3T + t 3T + t T + t t 2T + t 4T + t 51
52 Solution By inspection The channel response matrix is [ P c ] =
53 Solution The inverse of this matrix (e.g by MATLAB) 1 [ P c ] = Therefore, c 1 =0.117, c -1 =-0.158, c 0 = 0.937, c 1 = 0.133, c 2 = Equalized pulse response It can be verified that 53
54 p eq ( 3) = ( )( ) + ( )( 0. 02) + ( )( 0. 05) + ( )( 01. ) + ( )( 01. ) = p eq ( 3) = ( )( 0. 2) + ( )( 01. ) + ( )( 0. 05) + ( )( 01. ) + ( )( 0. 01) = Solution Note that values of for or are not zero. For example: 54
55 Minimum Mean-Square Error Equalizer Drawback of ZF equalizer Ignores the additive noise Alternatively, Relax zero ISI condition Minimize the combined power in the residual ISI and additive noise at the output of the equalizer MMSE equalizer: a channel equalizer that is optimized based on the minimum meansquare error (MMSE) criterion 55
56 MMSE Criterion Output from the channel z ( t) c y( t nt ) = N n= N n The output is sampled at : Let A m = desired equalizer output MSE [( z( mt ) A ) 2 ] = Minimum = E m 56
57 MMSE Criterion where E is taken over additive noise and the MMSE solution is obtained by 57
58 MMSE Equalizer vs. ZF Equalizer Both can be obtained by solving similar equations ZF equalizer does not consider 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 Next: non-linear equalizer 58
59 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 59
60 Example of Channels with ISI 60
61 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). 61
62 Performance of MMSE Equalizer 31-taps Proakis & Salehi, 2 nd 62
63 Performance of DFE Proakis & Salehi, 2 nd 63
64 Transmitting Filter H T (f) Maximum Likelihood Sequence Estimation (MLSE) Channel H C (f) Receiving Filter H R (f) t=mt y m MLSE (Viterbi 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 64
65 MLSE Assume ISI affects finite number of symbols, with Then, the channel is equivalent to a FIR discrete-time filter T T T T Finite-state machine 65
66 Performance of MLSE Proakis & Salehi, 2 nd 66
67 Equalizers Preset Equalizer Adaptive Equalizer Blind Equalizer Linear Non-linear ZF MMSE DFE MLSE 67
68 Suggested Reading Chapter of Fundamentals of Communications Systems, Pearson Prentice Hall 2005, by Proakis & Salehi 68
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