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ECE 4331, Fall, 2009

ECE 4331, Fall, 2009. Zhu Han Department of Electrical and Computer Engineering Class 14 Oct. 13 th , 2009. Midterm. Distribution Mean 79 Variance 8.9. Receiver Structure. Matched filter: match source impulse and maximize SNR g rx to maximize the SNR at the sampling time/output

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ECE 4331, Fall, 2009

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  1. ECE 4331, Fall, 2009 Zhu Han Department of Electrical and Computer Engineering Class 14 Oct. 13th, 2009

  2. Midterm • Distribution • Mean 79 • Variance 8.9

  3. Receiver Structure • Matched filter: match source impulse and maximize SNR • grx to maximize the SNR at the sampling time/output • Equalizer: remove ISI • Timing • When to sample. Eye diagram • Decision • d(i) is 0 or 1 Noisena(t) gTx(t) gRx(t) d(i) ?

  4. Matched Filter: optimal receive filter for maximized Matched filter example • Received SNR is maximized at time T0 example: Receive filter (mathed filter) transmit filter

  5. Error Rate Due to the Noise

  6. Error Rate Due to the Noise Figure 4.5 Noise analysis of PCM system. (a) Probability density function of random variable Y at matched filter output when 0 is transmitted. (b) Probability density function of Y when 1 is transmitted.

  7. Expressions with and Bit error rate with error function complement  antipodal: unipolar Q function

  8. Bit error rate for unipolar and antipodal transmission • BER vs. SNR • Coherent (antipodal) and noncoherent (unipolar) detection theoretical -1 simulation 10 unipolar -2 10 BER antipodal -3 10 -4 10 -2 0 2 4 6 8 10

  9. 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)

  10. (Polar form) TX Filter Channel RX Filter

  11. 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.

  12. sequence sent 1 0 1 sequencereceived 1 1(!) 1 Signal received Threshold t 0 5T T 2T 4T -3T -2T -T 0 3T Sequence of three pulses (1, 0, 1)sent ata rate 1/T ISI Example

  13. 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 • 1: At sampling points, no ISI • 2: At threshold, no ISI • 3: Areas within symbol period is zero, then no ISI • At least 14 points in the finals • 4 point for questions • 10 point like the homework

  14. no ISI ! Equally spaced zeros, interval 1st Nyquist Criterion: Time domain p(t):impulse response of a transmission system (infinite length) p(t) 1  shaping function 0 t -1

  15. 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

  16. 1st Nyquist Criterion: Frequency domain (limited bandwidth)

  17. Proof Fourier Transform At t=T

  18. Proof

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

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

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

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

  23. P(ω) r=0 r = 0.25 r = 0.50 r = 0.75 r = 1.00 W ω p(t) 0 0 t Raised cosine shaping • Tradeoff: higher r, higher bandwidth, but smoother in time. 2w

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

  25. Cosine rolloff filter: Bandwidth efficiency • Vestigial spectrum • Example 7.1   r=0 2nd Nyquist (r=1)

  26. 2nd 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

  27. Example

  28. 3rd Nyquist Criterion • Within each symbol period, the integration of signal (area) is proportional to the integration of the transmit signal (area)

  29.    1st Nyquist: 1st Nyquist: 1st Nyquist: 1st Nyquist: 2nd Nyquist     2nd Nyquist: 2nd Nyquist: 2nd Nyquist: 2nd Nyquist: 1st Nyquist Cosine rolloff filter: Eye pattern

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