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Digital Transmission Over Baseband Channels Principles of Communications Chapter 5: Digital Transmission through Bandlimited Channels Selected from Chapter 9.1-9.4 of Fundamentals of Communications Systems, Pearson Prentice Hall 2005, by Proakis & Salehi Meixia Tao @ SJTU 1 Topics to be Covered A/D D/A Source Channel Detector User converter converter . Digital waveforms over bandlimited baseband channels . Band-limited channel and Inter-symbol interference . Signal design for band-limited channels . System design . Channel equalization Meixia Tao @ SJTU 2 Bandlimited Channel 5MHz, 10MHz 20MHz 15MHz, 20MHz . Modeled as a linear filter with frequency response limited to certain frequency range Meixia Tao @ SJTU 3 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 . … Meixia Tao @ SJTU 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 Meixia Tao @ SJTU 5 0 1 1 0 1 0 0 0 1 1 On-off (unipolar) polar Return to zero bipolar Manchester Meixia Tao @ SJTU 6 Spectra of Baseband Signals . Consider a random binary sequence g0(t) - 0，g1(t) - 1 . The pulses g0(t）g1(t）occur independently with probabilities given by p and 1-p，respectively. The duration of each pulse is given by Ts. s(t) gt0 (+ 2) TS gt1(− 2) TS ∞ Ts s(t) = ∑ sn (t) n=−∞ g0 ( t− nTs ), with prob. P stn ()= g1( t−− nTs ), with prob. 1 P Meixia Tao @ SJTU 7 Power Spectral Density . PSD of the baseband signal s(t) is 2 ∞ 112 m mm = − − + +− δ − S() f p (1) p G01 () f G () f 2 ∑ pG0( ) (1) p G 1 ( ) ( f ) Ts Tsm=−∞ T s TT ss . 1st term is the continuous freq. component . 2nd term is the discrete freq. component Meixia Tao @ SJTU 8 =−= . For polar signalling with g 01 () t g () t gt () and p=1/2 1 2 Sf()= Gf () T = . For unipolar signalling with gt 0 () = 0 g 1 () t gt () and p=1/2, and g(t) is a rectangular pulse 2 sinπ fT Tsinπ fT 1 = Gf() T Sfx ()= + δ ()f π fT 44π fT . For return-to-zero unipolar signalling τ = T/2 2 Tsinπ fT / 2 1 1 1 m = ++δδ − Sfx () ()f ∑ 2 ()f 16π fT / 2 16 4 odd m [mπ ] T Meixia Tao @ SJTU 9 PSD of Basic Waveforms unipolar Return-to-zero polar Return-to-zero unipolar Meixia Tao @ SJTU 10 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). Meixia Tao @ SJTU 11 Baseband Signaling through Bandlimited Channels Source Σ ∞ Input to tx filter xsi() t=∑ Aδ ( t − iT ) i=−∞ ∞ Output of tx filter xt() t=∑ AhiT ( t − iT ) =−∞ Output of rx filter i Meixia Tao @ SJTU 12 Source Σ . Pulse shape at the receiver filter output . Overall frequency response . Receiving filter output ∞ v(t) = ∑ Ak p(t − kT )+ no (t) k =−∞ Meixia Tao @ SJTU 13 Source Σ . Sample the rx filter output at (to detect Am) Gaussian noise Desired signal intersymbol interference (ISI) Meixia Tao @ SJTU 14 Eye Diagram . Distorted binary wave . Eye pattern 15 Tb Meixia Tao @ SJTU 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 Meixia Tao @ SJTU 16 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 Meixia Tao @ SJTU 17 Nyquist Condition: Ideal Solution . Nyquist’s first method for eliminating ISI is to use sin(π t /T ) t p(t) = = sinc π t /T T “brick wall” filter P(f) p(t) p(t-T) 1 -1/2T 0 1/2T f = Nyquist bandwdith, The minimum transmission bandwidth for zero ISI. A channel with bandwidth B0 can support a max. transmission rate of 2B0 symbols/sec Meixia Tao @ SJTU 18 Achieving Nyquist Condition . Challenges of designing such or . is physically unrealizable due to the abrupt transitions at ±B0 . 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. Meixia Tao @ SJTU 19 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. P(f) ≤ < 1 0 | f | f1 1 1 π (| f | − f1 ) P( f ) = 1+ cos f1 ≤| f |< 2B0 − f1 2B0-f1 2 2B0 − 2 f1 f 0 f1 B0 2B 0 0 | f |≥ 2B0 − f1 Meixia Tao @ SJTU 20 Raised Cosine Spectrum P(f) α = 0 Roll-off factor 1 α = 0.5 f α =1− 1 0.5 α = 1 B0 f 0 B0 1.5B0 2B0 . The sharpness of the filter is controlled by . Required bandwidth Meixia Tao @ SJTU 21 Time-Domain Pulse Shape . Taking inverse Fourier transform cos(2παBt ) = 0 pt( ) sinc(2 Bt0 ) 2 22 1− 16α Bt0 Ensures zero crossing at desired sampling instants Decreases as 1/t2, such that the data receiving is relatively insensitive to sampling time error 0 T 2T b b α=1 α=0.5 α=0 0 1 -2 -1 2 t/Tb Meixia Tao @ SJTU 22 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 Meixia Tao @ SJTU 23 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 Meixia Tao @ SJTU 24 Duobinary Signal . Let {ak} be the binary sequence to be transmitted. The pulse duration is T. Two adjacent pulses are added together, i.e. baak= kk + −1 Ideal LPF . The resulting sequence {bk} is called duobinary signal Meixia Tao @ SJTU 25 Characteristics of Duobinary Signal Frequency domain ≤T( fT 1/2 ) − j2π fT = Gf()=( 1 + e) HL () f HfL () 0(ot her wi se ) 2Te− jπ fT cosπ fT ( f≤ 1/ 2 T ) = 0 ()ot her wi se Meixia Tao @ SJTU 26 Time domain Characteristics sinππtT / sin ( t− T ) / T gt()=[δδ () t +−∗ ( t T )] hL () t = + ππtT/ ( t− T )/ T t tT− T2 sinπ tT / =sinc + sinc = ⋅ TT πt() Tt− . gt()is called a duobinary signal pulse . It is observed that: gg(0)=0 = 1 (The current symbol) gT( )= g1 = 1 (ISI to the next symbol) g( iT )= gi = 0 ( i ≠ 0, 1) Decays as 1/t2, and spectrum within 1/2T Meixia Tao @ SJTU 27 Decoding . Without noise, the received signal is the same as the transmitted signal ∞ = yk∑ ag i ki− =+=aakk−1 b k A 3-level sequence i=0 ＋ － . When {} a k is a polar sequence with values 1 or 1: 2 (aakk=−1 = 1) ybkk== 0 ( ak ==−=−= 1, a k−−11 1 or ak1, a k 1) −==−2 (aakk−1 1) . When {} a k is a unipolar sequence with values 0 or 1 0 (aakk=−1 = 0) ybkk= = 1 ( ak= 0, a k−−11 = 1 or a k =1, a k = 0) 2 (aakk=−1 = 1) Meixia Tao @ SJTU 28 . To recover the transmitted sequence, we can use aˆkkk=−=− baˆ −−11 ya kkˆ . 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 cackkk= ⊕ −1 . Then the output of the duobinary signal system is bcck= kk + −1 Meixia Tao @ SJTU 29 Block Diagram of Precoded Duobinary Signal {ak } {c } b k { k } Transformer yt() + HfL () + 0, 1, 2 -2, 0, 2 T {ck −1} Meixia Tao @ SJTU 30 Modified Duobinary Signal . Modified duobinary signal baak= kk − −2 . After LPF , the overall response is HfL () 2Tje− j2π fT sin 2π fT ( f≤ 1/ 2 T ) Gf()= (1 − e− j4π fT ) H () f = L 0 otherwise sinππtT / sin ( t− 2 T ) / T 2T2 sinπ tT / gt()= − = − ππtT/ ( t− 2 T )/ T πtt(− 2) T Meixia Tao @ SJTU 31 Meixia Tao @ SJTU 32 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 gg(0)=0 = 1 gT()= g1 = 0 gT(2 )= g2 = − 1 g( iT )= gi = 0, i ≠ 0,1,2 . gt() also delays as 1 / t 2 . But at tT = , the timing offset may cause significant problem. Meixia Tao @ SJTU 33 Decoding of modified duobinary signal . To overcome error propagation, precoding is also needed. cackkk= ⊕ −2 . The coded signal is bcck= kk − −2 + + + - 2T Meixia Tao @ SJTU 34 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 Meixia Tao @ SJTU 35 Optimum Transmit/Receive Filter .
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