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 Tsinπ fT 1 = Gf() T Sfx ()= + δ ()f π fT 44π fT . For return-to-zero unipolar signalling τ = T/2

2 Tsinπ fT / 2 1 1 1 m = ++δδ − Sfx () ()f ∑ 2 ()f 16π 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 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 .

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. 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 ak 1, 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  Optimal transmitting and receiving filters  Channel equalizer

Meixia Tao @ SJTU 35 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 Nm =

with

Error Probability can be minimized through a proper choice of HR(f) and HT(f) so that is maximum (assuming HC(f) fixed and P(f) given) Meixia Tao @ SJTU 36 Optimal Solution

. Compensate the channel distortion equally between the transmitter and receiver filters

. Then, the transmit signal energy is given by

By Parseval’s theorem . Hence

Meixia Tao @ SJTU 37 . Noise variance at the output of the receive filter is

Performance loss due to channel distortion . Special case: . This is the ideal case with “flat” . No loss, same as the matched filter receiver for AWGN channel

Meixia Tao @ SJTU 38 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 response |H f)| = 1 ; |f| ≤ W where W= 4800 Hz c( f 1+ ( )2 W . The additive noise is zero-mean white Gaussian with spectral density

Meixia Tao @ SJTU 39 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

Meixia Tao @ SJTU 40 Performance with ISI

. If zero-ISI condition is not met, then

. Let

. Then

Meixia Tao @ SJTU 41 . 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?

Meixia Tao @ SJTU 42 Monte Carlo Simulation

~ Vm

X Threshold = γ

tm = mT Let 1 error occurs I(x) =  0 else

L 1 (l) ∴ Pe = ∑ I( X ) L l=1 where X(1), X(2), ... , X(L) are i.i.d. (independent and identically distributed) random samples

Meixia Tao @ SJTU 43 . If one wants Pe to be within 10% accuracy, how many independent simulation runs do we need? -9 . If Pe ~ 10 (this is typically the case for optical communication systems), and assume each simulation run takes 1 msec, how long will the simulation take?

Meixia Tao @ SJTU 44 . 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

Meixia Tao @ SJTU 45 What is Equalizer

. A receiving filter with adjustable frequency response to minimize/eliminate inter-symbol interference

Meixia Tao @ SJTU 46 Equalizer Configuration

t=mT Transmitting Channel Equalizer Filter H (f) H (f) H (f) T C E v(t) vm

. Overall frequency response:

. Nyquist criterion for zero-ISI

. Ideal zero-ISI equalizer is an inverse channel filter with

Meixia Tao @ SJTU 47 Linear Transversal Filter

. Finite impulse response (FIR) filter

Unequalized input c ⊗ c ⊗ c ⊗ ⊗ -N -N+1 N-1 cN t=nT ∑ Output (τ=T) (2N+1)-tap FIR equalizer . are the adjustable equalizer coefficients . is sufficiently large to span the length of ISI

Meixia Tao @ SJTU 48 Zero-Forcing Equalizer

. : the received pulse from a channel to be equalized

Tx & Channel

. At sampling time

To suppress 2N adjacent interference terms

Meixia Tao @ SJTU 49 Zero-Forcing Equalizer

. Rearrange to matrix form

channel response matrix

or the middle-column of

Meixia Tao @ SJTU 50 Example

. Find the coefficients of p (t) a five-tap FIR filter c equalizer to force two zeros on each side of 1.0 the main pulse response

− 4T + ∆t − 2T + ∆t T + ∆t 3T + ∆t

− 3T + ∆t −T + ∆t ∆t 2T + ∆t 4T + ∆t

Meixia Tao @ SJTU 51 Solution . By inspection

. The channel response matrix is

 1.0 0.2 − 0.1 0.05 − 0.02  − −   0.1 1.0 0.2 0.1 0.05  [Pc ] =  0.1 − 0.1 1.0 0.2 − 0.1    − 0.05 0.1 − 0.1 1.0 0.2   0.02 − 0.05 0.1 − 0.1 1.0 

Meixia Tao @ SJTU 52 Solution . The inverse of this matrix (e.g by MATLAB)

 0.966 − 0.170 0.117 − 0.083 0.056   − −   0.118 0.945 0.158 0.112 0.083 −1 [Pc ] = − 0.091 0.133 0.937 − 0.158 0.117     0.028 − 0.095 0.133 0.945 − 0.170 − 0.002 0.028 − 0.091 0.118 0.966  . Therefore,

c1=0.117, c-1=-0.158, c0 = 0.937, c1 = 0.133, c2 = -0.091 . Equalized pulse response

. It can be verified that

Meixia Tao @ SJTU 53 Solution

Note that values of for or are not zero. For example:

peq (3) = (0.117)(0.005) + (−0.158)(0.02) + (0.937)(−0.05) + (0.133)(0.1) + (−0.091)(−0.1) = − 0.027

peq (−3) = (0.117)(0.2) + (−0.158)(−0.1) + (0.937)(−0.05) +(0.133)(0.1) + (−0.091)(−0.01) =0.082

Meixia Tao @ SJTU 54 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 mean- square error (MMSE) criterion

Meixia Tao @ SJTU 55 MMSE Criterion

Output from the channel

N

z(t) = ∑cn y(t − nT ) n=− N The output is sampled at :

Let Am = desired equalizer output

2 MSE = E[(z(mT ) − Am ) ]= Minimum

Meixia Tao @ SJTU 56 MMSE Criterion

where E is taken over and the additive noise

. MMSE solution is obtained by

Meixia Tao @ SJTU 57 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

Meixia Tao @ SJTU 58 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 + Symbol Output Detection Filter -

Feedback Filter

Meixia Tao @ SJTU 59 Example of Channels with ISI

Meixia Tao @ SJTU 60 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).

Meixia Tao @ SJTU 61 Performance of MMSE Equalizer

31-taps Proakis & Salehi, 2nd

Meixia Tao @ SJTU 62 Performance of DFE

Proakis & Salehi, 2nd

Meixia Tao @ SJTU 63 Maximum Likelihood Sequence Estimation (MLSE) t=mT MLSE Transmitting Channel Receiving (Viterbi Filter H (f) H (f) Filter H (f) T C R ym 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

Meixia Tao @ SJTU 64 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

Meixia Tao @ SJTU 65 Performance of MLSE

Proakis & Salehi, 2nd

Meixia Tao @ SJTU 66 Equalizers

Preset Adaptive Blind Equalizer Equalizer Equalizer

Linear Non-linear

ZF DFE MMSE MLSE

Meixia Tao @ SJTU 67 Suggested Reading

. Chapter 9.1-9.4 of Fundamentals of Communications Systems, Pearson Prentice Hall 2005, by Proakis & Salehi

Meixia Tao @ SJTU 68