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Today’s Schedule

Today’s Schedule. Reading: Lathi 11.5,13.1 Mini-Lecture 1: Go over Quiz 2 Mini-Lecture 2: Bandpass Random Processes -Equivalent filters Mini-Lecture 3: Optimal Threshold Detection. Bandpass Random Process.

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Today’s Schedule

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  1. Today’s Schedule • Reading: Lathi 11.5,13.1 • Mini-Lecture 1: • Go over Quiz 2 • Mini-Lecture 2: • Bandpass Random Processes -Equivalent filters • Mini-Lecture 3: • Optimal Threshold Detection Dickerson EE422

  2. Bandpass Random Process • What happens to a signal at a receiver? How does the PSD of the signal after a BPF correspond to the signal before the BPF? Sx(w) -wc wc w 0 Dickerson EE422

  3. BPF System • Bandpass random process can be written as: • With the impulse response: cos(wct+q) 2cos(wct+q) xc(t) Ideal LPF H0(w) x x x(t) x(t) + sin(wct+q) 2sin(wct+q) xs(t) Ideal LPF H0(w) x x Dickerson EE422

  4. Impulse Response • Impulse Response • Transfer Function • So, xc(t) and xs(t) are low-pass random processes, what else can be deduced? • Assume theta is uniformly distributed phase noise H0(w) 1 2pB 4pB H(w) 1 -wc wc 0 Dickerson EE422

  5. PSD of BP Random Processes • PSD of xc(t) and xs(t) Sx(w) -wc wc w Sx(w-wc) +Sx(w+wc) LPF -2wc 2wc w 0 Sxc(w) or Sxs(w) w -2wc 0 2wc Dickerson EE422

  6. Mean and variance of narrowband noise • In-phase and quadrature components have the same PSD • In-phase and quadrature components of narrowband noise are zero-mean • Noise comes original signal being passed through a narrowband linear filter • Variance of the processes is the same (area under PSD same) Dickerson EE422

  7. Properties • If the narrowband noise is Gaussian, then the in-phase (nI) and quadrature (nQ) are jointly Gaussian. • If the narrow band noise is wide-sense stationary (WSS), then the in-phase and quadrature components are jointly WSS. Dickerson EE422

  8. Cross Correlations • Definition Cross Correlation • Definition Jointly Stationary Dickerson EE422

  9. Jointly Stationary Properties • Properties • Uncorrelated: • Orthogonal: • Independent: if x(t1) and y(t2) are independent (joint distribution is product of individual distributions) Dickerson EE422

  10. Activity • Working with a partner come up with a list of communications systems that will need bandpass analysis for performance assessment. • The PSD of a BP white noise process is N/2. What is the PSD and variance of the in-phase and quadrature components? Dickerson EE422

  11. Example White noise process • The PSD of a BP white noise process is N/2. What is the PSD and variance of the in-phase and quadrature components? Dickerson EE422

  12. Variance of White Noise Process From the SNR calculations, it is clear that: Dickerson EE422

  13. Sinusoid in Gaussian Noise • Signal is a sinusoid mixed with narrow-band additive white Gaussian noise (AWGN) • Can be written as: Dickerson EE422

  14. Example (continued) • In-phase and Quadrature terms of noise Gaussian with variance s2 • In a similar transformation to that used for calculating the dart board example, the joint density can be found in polar coordinates Dickerson EE422

  15. Marginal Density of E • Rician Density • Approaches a Gaussian if A>>s Dickerson EE422

  16. Digital Communications Systems in Noise • Analog Comm: Goal is to reproduce the waveform accurately • Figure of Merit: output signal to noise ratio • Digital Communications: Goal is to decide which waveform was transmitted accurately • Figure of Merit: Probability of error in making this decision at the receiver Dickerson EE422

  17. Detection: Bipolar Signaling s(t) • If “1”, send p(t) • If “0”, send –p(t) • Received signal: r(t) = +/- p(t) + n(t) • Optimal threshold? • Noise is additive, Gaussian noise Peak Detect and Sample Detector + n(t) Dickerson EE422

  18. PE Calculations • P(1/0) – prob of detecting a ‘1’ when ‘0’ sent • P(n>Ap) • P(0/1) – prob of detecting a ‘0’ when ‘1’ sent • P(n<-Ap) Dickerson EE422

  19. Bipolar • Total Error Probability • If P(1)=P(0)=0.5 Q(x) 0.5 x Dickerson EE422

  20. Minimize PE • To minimize PE, must maximize r since Q decreases monotonically as r increases • Ap is the signal amplitude and sn is the rms noise. • Goal: filter signal to enhance signal and reduce noise power Dickerson EE422

  21. Linear Filtering • Recall: Linear Network h(t) H(f) y(t)=+/- po(t)+no(t) Y(f) x(t)=+/- p(t)+n(t) X(f) Dickerson EE422

  22. Signal Amplitude • The goal is maximize (same as maximizing r) +/- po(t)+no(t) +/- p(t)+n(t) t=tm h(t) H(f) Threshold Detector p(t)+n(t) p(t)+n(t) p(t) p(t) TS TS Dickerson EE422

  23. Signal and Noise Calculation • Signal output: • Output noise power or variance • So, the ratio becomes: Dickerson EE422

  24. Next Time • Reading: Lathi 13.1, 13.2 • Mini-Lecture 1: • Optimum Threshold Detection • Mini-Lecture 2: • Optimum Binary receivers • Mini-Lecture 3: • Optimum Binary receivers Dickerson EE422

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