ISI Example 5T5T 0 t Sequence of three pulses (1, 0, 1) sent at a rate 1/T sequence sent sequence received 1 1(!) 1 Signal received Threshold 4T4T 3T3T 2T2T T 0-T-T-2T-3T
Baseband binary data transmission system. ISI arises when the channel is dispersive Frequency limited -> time unlimited -> ISI Time limited -> bandwidth unlimited -> bandpass channel -> time unlimited -> ISI p(t)
ISI First term : contribution of the i-th transmitted bit. Second term : ISI – residual effect of all other transmitted bits. We wish to design transmit and receiver filters to minimize the ISI. When the signal-to-noise ratio is high, as is the case in a telephone system, the operation of the system is largely limited by ISI rather than noise.
ISI Nyquist three criteria –Pulse amplitudes can be detected correctly despite pulse spreading or overlapping, if there is no ISI at the decision- making instants u 1: At sampling points, no ISI u 2: At threshold, no ISI u 3: Areas within symbol period is zero, then no ISI –At least 14 points in the finals u 4 point for questions u 10 point like the homework
1st Nyquist Criterion: Time domain p(t): impulse response of a transmission system (infinite length) Equally spaced zeros, interval t 0 1 p(t) shaping function no ISI !
1st Nyquist Criterion: Time domain Suppose 1/T is the sample rate The necessary and sufficient condition for p(t) to satisfy Is that its Fourier transform P(f) satisfy
1st Nyquist Criterion: Frequency domain (limited bandwidth)
Proof At t=T Fourier Transform
Sample rate vs. bandwidth W is the bandwidth of P(f) When 1/T > 2W, no function to satisfy Nyquist condition. P(f)
Sample rate vs. bandwidth When 1/T = 2W, rectangular function satisfy Nyquist condition
Sample rate vs. bandwidth When 1/T < 2W, numbers of choices to satisfy Nyquist condition A typical one is the raised cosine function
Cosine rolloff/Raised cosine filter Slightly notation different from the book. But it is the same : rolloff factor if
ECE 4371 Fall 2008 Raised cosine shaping Wω P(ω)P(ω) r=0 r = 0.25 r = 0.50 r = 0.75 r = t0 p(t)p(t) 2w Tradeoff: higher r, higher bandwidth, but smoother in time.
Figure 4.10 Responses for different rolloff factors. (a) Frequency response. (b) Time response.
2 nd Nyquist Criterion Values at the pulse edge are distortionless p(t) =0.5, when t= -T/2 or T/2; p(t)=0, when t=(2k-1)T/2, k≠0,1 -1/T ≤ f ≤ 1/T
3 rd Nyquist Criterion Within each symbol period, the integration of signal (area) is proportional to the integration of the transmit signal (area)
Eye Diagram Eye diagram is a means of evaluating the quality of a received “digital waveform” –By quality is meant the ability to correctly recover symbols and timing –The received signal could be examined at the input to a digital receiver or at some stage within the receiver before the decision stage Eye diagrams reveal the impact of ISI and noise Two major issues are 1) sample value variation, and 2) jitter and sensitivity of sampling instant Eye diagram reveals issues of both Eye diagram can also give an estimate of achievable BER Check eye diagrams at the end of class for participation
Interpretation of Eye Diagram
Raised Cosine Eye Diagram The larger , the wider the opening. The larger , the larger bandwidth (1+ )/Tb But smaller will lead to larger errors if not sampled at the best sampling time which occurs at the center of the eye.
Eye Diagram Setup Eye diagram is a retrace display of data waveform –Data waveform is applied to input channel –Scope is triggered by data clock –Horizontal span is set to cover 2-3 symbol intervals Measurement of eye opening is performed to estimate BER –BER is reduced because of additive interference and noise –Sampling also impacted by jitter
Partial Response Signals Previous classes: Sy(w)=|P(w)|^2 Sx(w) –Control signal generation methods to reduce Sx(w) –Raise Cosine function for better |P(w)|^2 This class: 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 (2W symbols/sec) in a channel of bandwidth W Hertz.
Example Duobinary Pulse –p(nTb)=1, n=0,1 –p(nTb)=1, otherwise Interpretation of received signal –2: 11 –-2: 00 –0: 01 or 10 depends on the previous transmission
Duobinary signaling Duobinary signaling (class I partial response)
Duobinary signal and Nyguist Criteria Nyguist second criteria: but twice the bandwidth
Differential Coding The response of a pulse is spread over more than one signaling interval. The response is partial in any signaling interval. Detection : –Major drawback : error propagation. To avoid error propagation, need deferential coding (precoding).
Modified duobinary signaling –In duobinary signaling, H(f) is nonzero at the origin. –We can correct this deficiency by using the class IV partial response.
Modified duobinary signaling Spectrum
Modified duobinary signaling Time Sequency: interpretation of receiving 2, 0, and -2?
Pulse Generation Generalized form of correlative-level coding (partial response signaling)
Tradeoffs Binary data transmission over a physical baseband channel can be accomplished at a rate close to the Nyquist rate, using realizable filters with gradual cutoff characteristics. Different spectral shapes can be produced, appropriate for the application at hand. However, these desirable characteristics are achieved at a price : –A large SNR is required to yield the same average probability of symbol error in the presence of noise.
Other types of partial response signals Typer0r1r2r3r4 p(t)P(W)Levels ideal1 2 I11 3 II121 5 III21 6 IV10 3 V paper