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TELECOMMUNICATIONS

TELECOMMUNICATIONS. Dr. Hugh Blanton ENTC 4307/ENTC 5307. POWER SPECTRAL DENSITY. Summary of Random Variables. Random variables can be used to form models of a communication system Discrete random variables can be described using probability mass functions

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TELECOMMUNICATIONS

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  1. TELECOMMUNICATIONS Dr. Hugh Blanton ENTC 4307/ENTC 5307

  2. POWER SPECTRAL DENSITY

  3. Summary of Random Variables • Random variables can be used to form models of a communication system • Discrete random variables can be described using probability mass functions • Gaussian random variables play an important role in communications • Distribution of Gaussian random variables is well tabulated using the Q-function • Central limit theorem implies that many types of noise can be modeled as Gaussian Dr. Blanton - ENTC 4307 - Correlation 3

  4. Random Processes • A random variable has a single value. However, actual signals change with time. • Random variables model unknown events. • A random process is just a collection of random variables. • If X(t) is a random process then X(1), X(1.5), and X(37.5) are random variables for any specific time t. Dr. Blanton - ENTC 4307 - Correlation 4

  5. Terminology • A stationary random process has statistical properties which do not change at all with time. • A wide sense stationary (WSS) process has a mean and autocorrelation function which do not change with time. • Unless specified, we will assume that all random processes are WSS and ergodic. Dr. Blanton - ENTC 4307 - Correlation 5

  6. Spectral Density Although Fourier transforms do not exist for random processes (infinite energy), but does exist for the autocorrelation and cross correlation functions which are non-periodic energy signals. The Fourier transforms of the correlation is called power spectrum or spectral density function (SDF). Dr. Blanton - ENTC 4307 - Correlation 6

  7. Review of Fourier Transforms Definition: A deterministic, non-periodic signal x(t) is said to be an energy signal if and only if Dr. Blanton - ENTC 4307 - Correlation 7

  8. The Fourier transform of a non-periodic energy signal x(t) is The original signal can be recovered by taking the inverse Fourier transform Dr. Blanton - ENTC 4307 - Correlation 8

  9. Remarks and Properties The Fourier transform is a complex function in whaving amplitude and phase, i.e. Dr. Blanton - ENTC 4307 - Correlation 9

  10. Example 1 Let x(t) = eat u(t), then Dr. Blanton - ENTC 4307 - Correlation 10

  11. Autocorrelation • Autocorrelation measures how a random process changes with time. • Intuitively, X(1) and X(1.1) will be more strongly related than X(1) and X(100000). • Definition (for WSS random processes): • Note that Power = RX(0) Dr. Blanton - ENTC 4307 - Correlation 11

  12. Power Spectral Density • P(w) tells us how much power is at each frequency • Wiener-Klinchine Theorem: • Power spectral density and autocorrelation are a Fourier Transform pair! Dr. Blanton - ENTC 4307 - Correlation 12

  13. Properties of Power Spectral Density • P(w)  0 • P(w) = P(-w) Dr. Blanton - ENTC 4307 - Correlation 13

  14. Gaussian Random Processes • Gaussian Random Processes have several special properties: • If a Gaussian random process is wide-sense stationary, then it is also stationary. • Any sample point from a Gaussian random process is a Gaussian random variable • If the input to a linear system is a Gaussian random process, then the output is also a Gaussian process Dr. Blanton - ENTC 4307 - Correlation 14

  15. Linear System • Input: x(t) • Impulse Response: h(t) • Output: y(t) x(t) h(t) y(t) Dr. Blanton - ENTC 4307 - Correlation 15

  16. Computing the Output of Linear Systems • Deterministic Signals: • Time Domain: y(t) = h(t)* x(t) • Frequency Domain: Y(f)=F{y(t)}=X(f)H(f) • For a random process, we still relate the statistical properties of the input and output signal • Time Domain: RY()= RX()*h() *h(-) • Frequency Domain: PY()= PX()|H(f)|2 Dr. Blanton - ENTC 4307 - Correlation 16

  17. Power Spectrum or Spectral Density Function (PSD) • For deterministic signals, there are two ways to calculate power spectrum. • Find the Fourier Transform of the signal, find magnitude squared and this gives the power spectrum, or • Find the autocorrelation and take its Fourier transform • The results should be the same. • For random signals, however, the first approach can not be used. Dr. Blanton - ENTC 4307 - Correlation 17

  18. Let X(t) be a random with an autocorrelation of Rxx(t) (stationary), then and Dr. Blanton - ENTC 4307 - Correlation 18

  19. Properties: • SXX(w) is real, and SXX(0)  0. • Since RXX(t) is real, SXX(-w) = SXX(w), i.e., symmetrical. • Sxx(0) = Dr. Blanton - ENTC 4307 - Correlation 19

  20. RXX(t) sX2d(t) Special Case For white noise, Thus, SXX(w) sX2  t w Dr. Blanton - ENTC 4307 - Correlation 20

  21. Example 1 Random process X(t) is wide sense stationary and has a autocorrelation function given by: Find SXX. Dr. Blanton - ENTC 4307 - Correlation 21

  22. Example 1 RXX(t) sX2 t Dr. Blanton - ENTC 4307 - Correlation 22

  23. Dr. Blanton - ENTC 4307 - Correlation 23

  24. Example 2 Let Y(t) = X(t) + N(t) be a stationary random process, where X(t) is the actual signal and N(t) is a zero mean, white gaussian noise with variance sN2 independent of the signal. Find SYY. Dr. Blanton - ENTC 4307 - Correlation 24

  25. Correlation in the Continuous Domain • In the continuous time domain Dr. Blanton - ENTC 4307 - Correlation 25

  26. v1(t) 1.0 t T 2T 3T v2(t) 1.0 t T 2T 3T -1.0 • Obtain the cross-correlation R12(t) between the waveform v1 (t) and v2 (t) for the following figure. Dr. Blanton - ENTC 4307 - Correlation 26

  27. The definitions of the waveforms are: and Dr. Blanton - ENTC 4307 - Correlation 27

  28. We will look at the waveforms in sections. • The requirement is to obtain an expression for R12(t) • That is, v2(t), the rectangular waveform, is to be shifted right with respect to v1(t) . Dr. Blanton - ENTC 4307 - Correlation 28 t

  29. v(t) T/2 1.0 T -1.0 t+T/2 t The situation for is shown in the figure. The figure show that there are three regions in the section for which v2(t) has the consecutive values of -1, 1, and -1, respectively. The boundaries of the figure are: t Dr. Blanton - ENTC 4307 - Correlation 29

  30. Dr. Blanton - ENTC 4307 - Correlation 30

  31. Dr. Blanton - ENTC 4307 - Correlation 31

  32. Dr. Blanton - ENTC 4307 - Correlation 32

  33. v(t) The situation for is shown in the figure. The figure show that there are three regions in the section for which v2(t) has the consecutive values of 1, -1, and 1, respectively. The boundaries of the figure are: T/2 1.0 t T t-T/2 -1.0 t Dr. Blanton - ENTC 4307 - Correlation 33

  34. Dr. Blanton - ENTC 4307 - Correlation 34

  35. Dr. Blanton - ENTC 4307 - Correlation 35

  36. Dr. Blanton - ENTC 4307 - Correlation 36

  37. 0.25 T/2 T t -0.25 Dr. Blanton - ENTC 4307 - Correlation 37

  38. Let X(t) denote a random process. The autocorrelation of X is defined as Dr. Blanton - ENTC 4307 - Correlation 38

  39. Properties of Autocorrelation Functions for Real-Valued, WSS Random Processes • 1. Rx(0) = E[X(t)X(t)] = Average Power • 2. Rx(t) = Rx(-t). The autocorrelation function of a real-valued, WSS process is even. • 3. |Rx(t)| Rx(0). The autocorrelation is maximum at the origin. Dr. Blanton - ENTC 4307 - Correlation 39

  40. y(t) (t+t)/2 t/2 t t 2 2-t Autocorrelation Example Dr. Blanton - ENTC 4307 - Correlation 40

  41. y(t) (t+t)/2 t/2 t t 2 0 2-t Dr. Blanton - ENTC 4307 - Correlation 41

  42. Dr. Blanton - ENTC 4307 - Correlation 42

  43. y(t) 1 0 1 6 7 2 3 4 5 -1 Correlation Example t Dr. Blanton - ENTC 4307 - Correlation 43

  44. t=0:.01:2; y=(t.^3./24.-t./2.+2/3); plot(t,y) Dr. Blanton - ENTC 4307 - Correlation 44

  45. Dr. Blanton - ENTC 4307 - Correlation 45

  46. t=0:.01:2; y=(-t.^3./24.+t./2.+2/3); plot(t,y) Dr. Blanton - ENTC 4307 - Correlation 46

  47. Dr. Blanton - ENTC 4307 - Correlation 47

  48. Dr. Blanton - ENTC 4307 - Correlation 48

  49. tint=0; tfinal=10; tstep=.01; t=tint:tstep:tfinal; x=5*((t>=0)&(t<=4)); subplot(3,1,1), plot(t,x) axis([0 10 0 10]) h=3*((t>=0)&(t<=2)); subplot(3,1,2),plot(t,h) axis([0 10 0 10]) axis([0 10 0 5]) t2=2*tint:tstep:2*tfinal; y=conv(x,h)*tstep; subplot(3,1,3),plot(t2,y) axis([0 10 0 40]) Dr. Blanton - ENTC 4307 - Correlation 49

  50. Matched Filter Dr. Blanton - ENTC 4307 - Correlation 50

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