1. ECE 4371, Fall, 2015 Introduction to Telecommunication Engineering/Telecommunication Laboratory Zhu Han Department of Electrical and Computer Engineering Class 12 Oct. 6 st, 2015

2. Outline Inter-Symbol-Interference Nyguist Three Criteria Eye Diagram

3. ISI Example 5T5T 0 t Sequence of three pulses (1, 0, 1) sent at a rate 1/T sequence sent 1 0 1 sequence received 1 1(!) 1 Signal received Threshold 4T4T 3T3T 2T2T T 0-T-T-2T-3T

4. 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)

5. 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.

6. 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

7. 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 !

8. 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

9. 1st Nyquist Criterion: Frequency domain (limited bandwidth)

10. Proof At t=T Fourier Transform

11. Proof

12. Sample rate vs. bandwidth W is the bandwidth of P(f) When 1/T > 2W, no function to satisfy Nyquist condition. P(f)

13. Sample rate vs. bandwidth When 1/T = 2W, rectangular function satisfy Nyquist condition

14. Sample rate vs. bandwidth When 1/T < 2W, numbers of choices to satisfy Nyquist condition A typical one is the raised cosine function

15. Cosine rolloff/Raised cosine filter Slightly notation different from the book. But it is the same : rolloff factor if

16. ECE 4371 Fall 2008 Raised cosine shaping Wω P(ω)P(ω) r=0 r = 0.25 r = 0.50 r = 0.75 r = 1.00 0 t0 p(t)p(t) 2w Tradeoff: higher r, higher bandwidth, but smoother in time.

17. Figure 4.10 Responses for different rolloff factors. (a) Frequency response. (b) Time response.

18. Cosine rolloff filter: Bandwidth efficiency Vestigial spectrum 2nd Nyquist (r=1) r=0 

19. 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

20. Example

21. 3 rd Nyquist Criterion Within each symbol period, the integration of signal (area) is proportional to the integration of the transmit signal (area)

22. Cosine rolloff filter: Eye pattern 2nd Nyquist 1st Nyquist   2nd Nyquist: 1st Nyquist:   2nd Nyquist: 1st Nyquist:   2nd Nyquist: 1st Nyquist:   2nd Nyquist: 1st Nyquist:

23. Eye Diagram The eye diagram is created by taking the time domain signal and overlapping the traces for a certain number of symbols. The open part of the signal represents the time that we can safely sample the signal with fidelity

24. Vertical and Horizontal Eye Openings The vertical eye opening or noise margin is related to the SNR, and thus the BER –A large eye opening corresponds to a low BER The horizontal eye opening relates the jitter and the sensitivity of the sampling instant to jitter –The 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

25. Interpretation of Eye Diagram

26. Jitter in Circuit design Circuit design

27. 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.

28. 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

29. 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

30. Figure 4.34 (a) Eye diagram for noiseless quaternary system. (b) Eye diagram for quaternary system with SNR  20 dB. (c) Eye diagram for quaternary system with SNR  10 dB.

31. Figure 4.35 (a) Eye diagram for noiseless band-limited quaternary system: cutoff frequency fo  0.975 Hz. (b) Eye diagram for noiseless band-limited quaternary system: cutoff frequency fo  0.5 Hz.

32. Eye Diagram In Phase

33. Linear Modulation with Nyquist Impulse Shaping QPSK diagram under limited bandwidth conditions  if system (tx and rx filter) meets 1st Nyquist : 4 sharp signal points (right diagram)