EE5713 : Advanced Digital Communications Week 12, 13: Inter Symbol Interference (ISI) Nyquist Criteria for ISI Pulse Shaping and Raised-Cosine Filter Eye Pattern Equalization (On Board) 20-May-15 Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713) 1
Baseband Communication System We have been considering the following baseband system he transmitted signal is created by the line coder according to s ( t) = a g ( t ) n= where a n is the symbol mapping and g(t) is the pulse shape Problems with Line Codes n n b One big problem with the line codes is that they are not bandlimited he absolute bandwidth is infinite he power outside the 1st null bandwidth is not negligible. hat is, the power in the sidelobes can be quite high 20-May-15 Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713) 2
Intersymbol Interference (ISI) If the transmission channel is bandlimited, then high frequency components will be cut off Hence, the pulses will spread out If the pulse spread out into the adjacent symbol periods, then it is said that intersymbol interference (ISI) has occurred Intersymbol Interference (ISI) Intersymbol interference (ISI) occurs when a pulse spreads out in such a way that it interferes with adjacent pulses at the sample instant Causes Channel induced distortion which spreads or disperses the pulses Multipath effects (echo) 20-May-15 Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713) 3
Pulse spreading Due to improper filtering (@ x and/or Rx), the received pulses overlap one another thus making detection difficult Example of ISI Assume polar NRZ line code 20-May-15 Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713) 4
Inter Symbol Interference Input data stream and bit superposition he channel output is the sum of the contributions from each bit 20-May-15 Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713) 5
ISI Note: ISI can occur whenever a non-bandlimited line code is used over a bandlimited channel ISI can occur only at the sampling instants Overlapping pulses will not cause ISI if they have zero amplitude at the time the signal is sampled 20-May-15 Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713) 6
ISI Baseband Communication System Model where h h C h R ( t) = Impulse response of the transmitter, ( t) = Impulse response of the channel, ( t) = Impulse response of the receiver s ( t) = a h ( t n ), n n= r( t) = a n n = n e n= g ( t n ) + y t) = a h ( t n) + n ( t) e n( t), where g ( t) = h ( t) * h C ( t), = 1 / f ( where h ( t) h ( t)* h ( t)* h ( t), n e = ( t) n( t) * h ( t) * h ( t) e = C C R R s 20-May-15 Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713) 7
ISI Baseband Communication System Model Note that h e (t) is the equivalent impulse response of the receiving filter o recover the information sequence {a n }, the output y(t) is sampled at t = k, k = 0, 1, 2, he sampled sequence is n= = y ( k) = a h ( k n) + n ( k) or equivalently n e e AWGN term y k = n= a n h k n + nk = h0ak + anhk n + n=, n k n k where h = h ( k), n = n ( k), k = 0, ± 1, ± 2,.. k Desired symbol scaled by gain parameters h 0 e h 0 is an arbitrary constant k e Effect of other symbols at the sampling instants t=k 20-May-15 Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713) 8
Signal Design for Bandlimited Channel Zero ISI y( k ) = h 0 ak + anhe ( k n ) + ne ( k ) n=, n k o remove ISI, it is necessary and sufficient to make the term h e Nyquist Criterion ( k n ) = 0, for n k and h 0 0 Pulse amplitudes can be detected correctly despite pulse spreading or overlapping, if there is no ISI at the decisionmaking instants 20-May-15 Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713) 9
Nyquist s three criteria Nyquist Criteria Pulse amplitudes can be detected correctly despite pulse spreading or overlapping, if there is no ISI at the decisionmaking instants 1: At sampling points, no ISI 2: At threshold, no ISI 3: Areas within symbol period is zero, then no ISI Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713)
1st Nyquist Criterion: ime domain p(t): impulse response of a transmission system (infinite length) p(t) 1 shaping function 0 1 = 2 f N t 0 2t 0 t no ISI! -1 Equally spaced zeros, interval 1 = 2 f n Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713)
1st Nyquist Criterion: ime domain Suppose 1/ is the sample rate he necessary and sufficient condition for p(t) to satisfy p ( n ) = 1, 0, ( n = 0) ( n 0) Is that its Fourier transform P(f) satisfy m= P( f + m ) = Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713)
1st Nyquist Criterion: Frequency domain m= P ( f + m ) = 0 fa = 2 fn 4 f N (limited bandwidth) f Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713)
Fourier ransform At t= p = = = p ( 2m+ 1) Proof ( n ) = P( f ) exp( j2πfn ) m= 1 2 1 1 2 1 1 m = ( 2m 1) p ( t) P( f ) exp( j2πft ) 2 2 = df ( n ) P( f ) exp( j2πfn ) 2 m = 2 1 2 B 2 P P = df ( f + m ) exp( j2πfn ) ( f + m ) exp( j2πfn ) ( f ) exp( j2πfn ) df B df df df ( f ) = P( f + m ) m= Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713)
Proof B ( f ) = P( f + m ) m= B b n ( f ) = b exp( j2πnf ) = n= 1 2 1 2 n B ( f ) exp( j2πnf ) b n ( n ) = p ( n 0) b n = = 0 n 0 ( ) ( f ) B = P( f + m ) = m= Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713)
Nyquist Criterion: ime domain Pulse shape that satisfy this criteria is Sinc(.) function, e.g., t h e ( t) or p( t) = sinc = sinc(2wt ) he smallest value of for which transmission with zero ISI is possible is Problems with Sinc(.) function It is not possible to create Sinc pulses due to 1 = 2 W Infinite time duration Sharp transition band in the frequency domain Sinc(.) pulse shape can cause ISI in the presence of timing errors If the received signal is not sampled at exactly the bit instant, then ISI will occur 20-May-15 Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713) 16
Nyquist Criterion: ime domain p(t) 1 shaping function 0 1 = 2 f s t 0 2t 0 t no ISI! -1 Equally spaced zeros, 1 interval = 2 f s 20-May-15 Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713) 17
Sample rate vs. bandwidth W is the channel bandwidth for P(f) When 1/ > 2W, there is no way, we can design a system with no ISI P(f) 20-May-15 Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713) 18
Sample rate vs. bandwidth When 1/ = 2W (he Nyquist Rate), rectangular function satisfy Nyquist condition sinπt πt, ( f < W ) p( t) = = sinc ; P( f ) = π t ( f ) = rect rect ( f ); 1 f P = 2W 2W 0, ( otherwise), W 20-May-15 Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713) 19
Sample rate vs. bandwidth When 1/ < 2W, numbers of choices to satisfy Nyquist condition Raised Cosine Filter Duobinary Signaling (Partial Response Signals) Gaussian Filter Approximation he most typical one is the raised cosine function 20-May-15 Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713) 20
Raised Cosine Pulse he following pulse shape satisfies Nyquist s method for zero ISI he Fourier ransform of this pulse shape is 2 2 2 2 2 2 4 1 cos sinc 4 1 cos sin ) ( t r t r t t r t r t r t r t p = = π π π π 20-May-15 Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713) 21 he Fourier ransform of this pulse shape is where r is the roll-off factor that determines the bandwidth + + + = r f r f r r f r r f f P 2 1 0, 2 1 2 1, 2 1 cos /2 1 2 1 0, ) ( π
Responses for different roll-off factors (a) Frequency response. (b) ime response
Rolloff and bandwidth Bandwidth occupied beyond 1/2 is called the excess bandwidth (EB) EB is usually expressed as a %tage of the Nyquist frequency, e.g., Rolloff factor, r = 1/2 ===> excess bandwidth is 50 % Rolloff factor, r = 1 ===> excess bandwidth is 100 % RC filter is used to realized Nyquist filter since the transition band can be changed using the roll-off factor he sharpness of the filter is controlled by the parameter r When r = 0 this corresponds to an ideal rectangular function Bandwidth B occupied by a RC filtered signal is increased from its minimum value B 1 2 So the bandwidth becomes: s = min B = B ( 1+ r) 20-May-15 Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713) 23 min
Rolloff and bandwidth Benefits of large roll off factor Simpler filter fewer stages (taps) hence easier to implement with less processing delay Less signal overshoot, resulting in lower peak to mean excursions of the transmitted signal Less sensitivity to symbol timing accuracy wider eye opening r = 0 corresponds to Sinc(.) function 20-May-15 Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713) 24
Partial Response Signals o improve the bandwidth efficiency Widen the pulse, the smaller the bandwidth. But there is ISI. For binary case with two symbols, there is only few possible interference patterns. By adding ISI in a controlled manner, it is possible to achieve a signaling rate equal to the Nyquist rate i.e. Duobinary and Polibinary Signaling (Covered in the previous lectures) 20-May-15 Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713) 25
2 nd Nyquist Criterion Values at the pulse edge are distortion-less
Nyquist s hree Criteria Nyquist hree Criteria Pulse amplitudes can be detected correctly despite pulse spreading or overlapping, if there is no ISI at the decisionmaking instants 1: At sampling points, no ISI 2: At threshold, no ISI 3: Areas within symbol period from other symbols is zero, then no ISI Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713)
Example (wo Criteria)
3 rd Nyquist Criterion Within each symbol period, the integration of signal (area) is proportional to the integration of the transmit signal (area) A 1, p( t) dt= 0, = 2 n 2 + 1 2 n 2 1 n n= 0 0
Eye Patterns An eye pattern is obtained by superimposing the actual waveforms for large numbers of transmitted or received symbols Perfect eye pattern for noise-free, bandwidth-limited transmission of an alphabet of two digital waveforms encoding a binary signal (1 s and 0 s) Actual eye patterns are used to estimate the bit error rate and the signal to- noise ratio 20-May-15 Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713) 30
Eye Patterns Concept of the eye pattern 20-May-15 Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713) 31
Eye Diagram he eye diagram is created by taking the time domain signal and overlapping the traces for a certain number of symbols. he open part of the signal represents the time that we can safely sample the signal with fidelity
Vertical and Horizontal Eye Openings he vertical eye opening or noise margin is related to the SNR, and thus the BER A large eye opening corresponds to a low BER he horizontal eye opening relates the jitter and the sensitivity of the sampling instant to jitter he red brace indicates the range of sample instants with good eye opening At other sample instants, the eye opening is greatly reduced, as governed by the indicated slope
Interpretation of Eye Diagram
Cosine rolloff filter: Eye pattern 2nd Nyquist 1st Nyquist: 2nd Nyquist: 1st Nyquist: 2nd Nyquist: 1st Nyquist 1st Nyquist: 2nd Nyquist: 1st Nyquist: 2nd Nyquist: 20-May-15 Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713) 35