ECE 4371, Fall, 2009 Introduction to Telecommunication Engineering

1. ECE 4371, Fall, 2009Introduction to Telecommunication Engineering Zhu Han Department of Electrical and Computer Engineering Class 2 Aug. 27nd, 2009

2. Outline • Chapter 2 and Some background suppose to be known, quick review, please read by yourself • Definition of Random Process • Stationary and Ergotic • Mean Correlation and Covariance • Power Spectral Density • Some Typical Random Process • Representation of Random Process

3. fX(x) x(t) time, t Deterministic and random processes • Deterministic processes: physical process is represented by explicit mathematical relation. • Random processes: result of a large number of separate causes. Described in probabilistic terms and by properties which are averages. • The probability density function describes the general distribution of the magnitude of the random process, but it gives no information on the time or frequency content of the process

4. Stationarity and Ergodicity • Ensemble averaging : properties of the process are obtained by averaging over a collection or ‘ensemble’ of sample records using values at corresponding times • Stationary random process : Ensemble averages do not vary with time. Example 1.1 • Time averaging : properties are obtained by averaging over a single record in time • Ergodic process : Stationary process in which averages from a single record are the same as those obtained from averaging over the ensemble. • In other word, average over different random process with the same distribution is equal to the average of one random process over time.

5. x(t) x T time, t Mean • The mean value,x , is the height of the rectangular area having the same area as that under the function x(t) • Can also be defined as the first moment of the p.d.f.

6. x x(t) x T time, t Mean square value, variance, standard deviation • Mean square value • Variance: (average of the square of the deviation of x(t) from the mean value,x) • Standard deviation, x, is the square root of the variance

7. x(t)  T time, t Autocorrelation • The autocorrelation, or autocovariance, describes the general dependency of x(t) with its value at a short time later, x(t+) • The value of x() at  equal to 0 is the variance How the future is correlated with now Normalized auto-correlation : R()=x()/x2

8. 1 R() 0 Time lag,  Autocorrelation • The autocorrelation for a random process eventually decays to zero at large  • The autocorrelation for a sinusoidal process (deterministic) is a cosine function which does not decay to zero • Example 1.2, 1.3 • Power spectrum density is the Fourier transform of autocorrelation function. Example 1.5, 1.6, 1.7

9. x(t) x  T time, t y(t) y T time, t Cross-correlation • The cross-correlation function describes the general dependency of x(t) with another random process y(t+), delayed by a time delay,  • Example 1.4

10. Covariance • The covariance is the cross correlation function with the time delay, , set to zero • Note that here x'(t) and y'(t) are used to denote the fluctuating parts of x(t) and y(t) (mean parts subtracted) • The correlation coefficient, , is the covariance normalized by the standard deviations of x and y • When x and y are identical to each other, the value of  is +1 (full correlation) • When y(t)=x(t), the value of  is  1 • In general,  1<  < +1

11. Correlation • indicates the strength and direction of a linear relationship between two random variables

12. Gaussian Process • Distribution • Multivariate Gaussian • If input of a stable linear filter is Gaussian, output is Gaussian

13. Central Limit Theorem • Xi are statistically independent • Xi has the same mean and variance • Sn = X1 + ... + Xn • Then the distribution of Zn converges towards the standard normal distribution N(0,1) as n approaches ∞ • This implies if a random process is affected by many factors, the resulting distribution is probably Gaussian.

14. Shot Noise • Shot Noise: type of electronic noise that occurs when the finite number of particles that carry energy, such as electrons in an electronic circuit or photons in an optical device, is small enough to give rise to detectable statistical fluctuations in a measurement. It is important in electronics, telecommunications, and fundamental physics. • Photon noise is the dominant source of noise in the images that are collected by most digital cameras on the market today

15. Thermal Noise • Electronicnoise generated by the thermal agitation of the charge carriers (usually the electrons) inside an electrical conductor at equilibrium, which happens regardless of any applied voltage. • Thermal noise is approximately white, meaning that the power spectral density is equal throughout the frequency spectrum. Additionally, the amplitude of the signal has very nearly a Gaussianprobability density function. • Example 1.10 1.11

16. Narrowband Noise Representation • The noise process appearing at the output of a narrowband filter is called narrowband noise. • With the spectral components of narrowband noise concentrated about some midband frequency as in Figure 1.18a. • We find that a sample function n(t) of such a process appears somewhat similar to a sine wave of frequency as in Figure 1.18b. • Representations of narrowband noise • A pair of component called the in-phase and quadrature components. • Two other components called the envelop and phase.

17. Representation of Narrowband Noise in Terms of In-Phase and Quadrature Components • Consider a narrowband noise of bandwidth 2B centered on frequency ,as illustrated in Figure 1.18. • We may represent n(t) in the canonical (standard) form: where, is in-phase component of and is quadrature component of .

18. and have important properties: • and have zero mean. • is Gaussian, then and are jointly Gaussian. • is stationary, then and are jointly stationary. • Both and have the same power spectral density. • and have the same variance as

20. Representation of Narrowband Noise in Terms of Envelope and Phase Components

22. Figure 1.21 Illustrating the coordinate system for representation of narrowband noise: (a) in terms of in-phase and quadrature components, and (b) in terms of envelope and phase.

25. Sine Wave Plus Narrowband Noise

28. : Rician distribution reduced to the Rayleigh distribution The envelope distribution is approximately Gaussian when a is large

29. Questions?