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

2. POWER SPECTRAL DENSITY

3. Dr. Blanton - ENTC 4307 - Correlation 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

4. Dr. Blanton - ENTC 4307 - Correlation 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.

5. Dr. Blanton - ENTC 4307 - Correlation 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.

6. Dr. Blanton - ENTC 4307 - Correlation 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).

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

8. Dr. Blanton - ENTC 4307 - Correlation 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

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

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

11. Dr. Blanton - ENTC 4307 - Correlation 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 = R X (0)

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

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

14. Dr. Blanton - ENTC 4307 - Correlation 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

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

16. Dr. Blanton - ENTC 4307 - Correlation 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: R Y (  )= R X (  )*h(  ) *h(-  ) Frequency Domain: P Y (  )= P X (  )  |H(f)| 2

17. Dr. Blanton - ENTC 4307 - Correlation 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.

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

19. Dr. Blanton - ENTC 4307 - Correlation 19 Properties: (1)S XX (  ) is real, and S XX (0)  0. (2)Since R XX (t) is real, S XX (-  ) = S XX (  ), i.e., symmetrical. (3)Sxx(0) = (4)

20. Dr. Blanton - ENTC 4307 - Correlation 20 Special Case For white noise, Thus,  R XX (  )  X   S XX (  ) XX  

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

22. Dr. Blanton - ENTC 4307 - Correlation 22 Example 1 R XX (  ) XX 

23. Dr. Blanton - ENTC 4307 - Correlation 23

24. Dr. Blanton - ENTC 4307 - Correlation 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  N 2 independent of the signal. Find S YY.

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

26. Dr. Blanton - ENTC 4307 - Correlation 26 Obtain the cross-correlation R 12 (  ) between the waveform v 1 (t) and v 2 (t) for the following figure. T2T3T v 1 (t) t 1.0 v 2 (t) t T2T 3T 1.0

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

28. Dr. Blanton - ENTC 4307 - Correlation 28 We will look at the waveforms in sections. The requirement is to obtain an expression for R 12 (  ) That is, v 2 (t), the rectangular waveform, is to be shifted right with respect to v 1 (t). t

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

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31. Dr. Blanton - ENTC 4307 - Correlation 31

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

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

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

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

40. Dr. Blanton - ENTC 4307 - Correlation 40 Autocorrelation Example t2 2-t  y(    t 

41. Dr. Blanton - ENTC 4307 - Correlation 41  t2 2-t y(    t  0

42. Dr. Blanton - ENTC 4307 - Correlation 42 

43. Dr. Blanton - ENTC 4307 - Correlation 43 Correlation Example t y(t  01 2345 67 1

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

45. Dr. Blanton - ENTC 4307 - Correlation 45

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

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49. Dr. Blanton - ENTC 4307 - Correlation 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])

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

51. Dr. Blanton - ENTC 4307 - Correlation 51 Matched Filter A matched filter is a linear filter designed to provide the maximum signal-to-noise power ratio at its output for a given transmitted symbol waveform. Consider that a known signal s(t) plus a AWGN n(t) is the input to a linear time- invariant (receiving) filter followed by a sampler.

52. Dr. Blanton - ENTC 4307 - Correlation 52 At time t = T, the sampler output z(t) consists of a signal component a i and noise component n 0. The variance of the output noise (average noise power) is denoted by  0 2, so that the ratio of the instantaneous signal power to average noise power, (S/N) T, at time t = T is

53. Dr. Blanton - ENTC 4307 - Correlation 53 Random Processes and Linear Systems If a random process forms the input to a time-invariant linear system, the output will also be a random process. The input power spectral density G X (f) and the output spectral density G Y (f) are related as follows:

54. Dr. Blanton - ENTC 4307 - Correlation 54 We wish to find the filter transfer function H 0 (f) that maximizes We can express the signal a i (t) at the filter output in terms of the filter transfer function H(f) and the Fourier transform of the input signal, as

55. Dr. Blanton - ENTC 4307 - Correlation 55 If the two-sided power spectral density of the input noise is N 0 /2 watts/hertz, then we can express the output noise power as Thus, (S/N) T is

56. Dr. Blanton - ENTC 4307 - Correlation 56 Using Schwarz’s inequality, and

57. Dr. Blanton - ENTC 4307 - Correlation 57 Or where

58. Dr. Blanton - ENTC 4307 - Correlation 58 The maximum output signal-to-noise ratio depends on the input signal energy and the power spectral density of the noise. The maximum output signal-to-noise ratio only holds if the optimum filter transfer function H0(f) is employed, such that

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60. Dr. Blanton - ENTC 4307 - Correlation 60 Since s(t) is a real-valued signal, we can use the fact that and

61. Dr. Blanton - ENTC 4307 - Correlation 61 to show that Thus, the impulse response of a filter that produces the maximum output signal-to-noise ratio is the mirror image of the message signal s(t), delayed by the symbol time duration T.

62. Dr. Blanton - ENTC 4307 - Correlation 62 T t s(t) -T t s(-t)h(t)=s(T-t) t T Signal waveformMirror image of signal waveform Impulse response of matched filter

63. Dr. Blanton - ENTC 4307 - Correlation 63 The impulse response of the filter is a delayed version of the mirror image (rotated on the t = 0 axis) of the signal waveform. If the signal waveform is s(t), its mirror image is s(-t), and the mirror image delayed by T seconds is s(T-t).

64. Dr. Blanton - ENTC 4307 - Correlation 64 The output of the matched filter z(t) can be described in the time domain as the convolution of a received input wavefrom r(t) with the impulse response of the filter.

65. Dr. Blanton - ENTC 4307 - Correlation 65 Substituting ks(T-t) with k chosen to be unity for h(t) yields. When T = t

66. Dr. Blanton - ENTC 4307 - Correlation 66 The integration of the product of the received signal r(t) with a replica of the transmitted signal s(t) over one symbol interval is known as the correlation of r(t) with s(t).

67. Dr. Blanton - ENTC 4307 - Correlation 67 The mathematical operation of a matched filter (MF) is convolution; a signal is convolved with the impulse response of a filter. The mathematical operation of a correlator is correlation; a signal is correlated with a replica of itself.

68. Dr. Blanton - ENTC 4307 - Correlation 68 The term matched filter is often used synonymously with correlator. How is that possible when their mathematical operations are different?

69. Dr. Blanton - ENTC 4307 - Correlation 69 s 0 (t) s 1 (t) TbTb TbTb A A -A

70. Dr. Blanton - ENTC 4307 - Correlation 70 h 0 =s 0 (T b -t) h 0 = s 1 (T b -t) TbTb TbTb A A -A

71. Dr. Blanton - ENTC 4307 - Correlation 71 y 0 (t) TbTb A2TbA2Tb 2T b y 0 (t) TbTb 2T b