ANALOG COMMUNICATION (8)

Transcription

1 /5/3 DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING ANALOG COMMUNICATION (8) Fall 3 Original slides by Yrd. Doç. Dr. Burak Kellei Modified by Yrd. Doç. Dr. Didem Kivan Tureli OUTLINE Random Variables Statistial Averages Random Proesses Mean, Correlation, and Autoorrelation Transmission of a Random Proess Through a Linear Filter Power Spetral Density Gaussian Proess Noise Narrowband Noise

2 /5/3 RANDOM SIGNALS Random signal: it is not possible to predit its preise value in advane. It may be possible to desribe its statistial properties, suh as average power, average spetral distribution of its power. The mathematial disipline that deals with the statistial haraterization of random signals is probability theory. Probability theory is rooted in phenomena that an be modeled by an experiment with an outome that is subjet to hane. RANDOM VARIABLES While the meaning of the outome of a random experiment is lear, suh outomes are often not the most onvenient representation for mathematial analysis. We use the expression random variable to desribe this proess of assigning a number to the outome of a random experiment. Random variables may be disrete and take only a finite number of values, suh as oin-tossing experiment. Alternatively, random variables may be ontinuous and take real values. Probability density funtion (pdf) is a funtion that desribes the relative likelihood for this random variable to our at a given point.

3 /5/3 PROBABILITY DENSITY FUNCTION The probability for the random variable to fall within a partiular region is given by the integral of this variable s density over the region. The probability density funtion is nonnegative everywhere, and its integral over the entire spae is equal to one. STATISTICAL AVERAGES The expeted value or mean of a random variable x is defined by x E xf x E donates the statistial expetation operator. Funtion of Random Variable: Y g x E Y E yf Y y gx gx f x dy x 3

4 /5/3 EAMPLE Let Y=g()=os() where is a random variable uniformly distributed in the interval (-p, p) f x p p x p otherwise What is the expeted value of Y p p p osx sinx E Y p p xp MOMENTS For the speial ase of g()= n, we obtain nth moment of the probability distribution of the random variable. E n n x f x n= gives the mean and n= gives the meansquare value of E x f x 4

5 /5/3 CENTRAL MOMENTS Central moments are simply the moments of the differene between a random variable and its mean. Thus, the nth entral moment is E n n x x f x n= is zero, whereas n= is defined as the variane of the random variable. var E The square-root of the variane is alled the standard deviation of the random variable. x x f x x x EAMPLE Let Y=g()=os() where is a random variable uniformly distributed in the interval (-p, p) f x p What is the variane of Y var var E osx p p p p p x p otherwise x f x osx p p x p x xp p x 5

6 /5/3 RANDOM PROCESS The random signals in ommuniation are funtions of time, defined on some observation interval. The sample spae or ensemble omprised of funtions of time is alled a random or stohasti proess. MEAN, CORRELATION The mean of the random proess (t) is obtained by observing the proess at some time t, x E t xf x The autoorrelation funtion of the proess (t) as the expetation of the produt of two random variables (t ) and (t ), obtained by observing (t) at time t and t. R t t, t E t t x x f t t x, x 6

7 /5/3 PROPERTIES OF THE AUTOCORRELATION FUNCTION For onveniene we redefine the autoorrelation funtion of (t) R t E t t t for all t Mean square value of the proess is R E t The autoorrelation funtion is an even funtion R t R t The autoorrelation funtion has its maximum magnitude at t= R t R POWER SPECTRAL DENSITY The Fourier transform of the autoorrelation funtion is alled the power spetral density S (f) of the random proess (t) S R f t R S t e f e jpft jpft dt dt 7

8 /5/3 GAUSSIAN PROCESS The random variable Y has Gaussian distribution if its probability density funtion has the form f Y y For the speial ase when the Gaussian random variable Y is normalized to have a mean of zero and a variane of one N Most of the physial phenomena are modeled using Gaussian distribution. y Y Y e p, f y e Y Y p y NOISE Noise is the unwanted signals that disturb the transmission and proessing of ommuniation signals. The soures of noise may be external to the system, suh as atmospheri noise, galati noise, man-made noise. Internally generated noise arises from spontaneous flutuations of urrent and voltage in eletrial iruits. This type of noise limits the transmission and quality of the ommuniation signals. Most ommon noise of this type is thermal noise. 8

9 /5/3 THERMAL NOISE Thermal noise arises from the random motion of eletrons in a ondutor. The mean-square value of the thermal noise voltage V TN appearing aross the terminals of a resistor, measured in a bandwidth of Df Hertz is V 4kTRDf E TN where k is Boltzmann onstant equal to.38x -3 joules per degree Kelvin, T is the absolute temperature in degrees Kelvin and R is the resistane in ohms. WHITE NOISE The noise analysis of ommuniation systems is based on idealized form of noise alled white noise, whih has onstant power spetral density aross frequeny. N is in watts per Hertz and referened to the input stage of the reeiver. N kt e k is Boltzmann onstant and Te is the equivalent noise temperature of the reeiver. 9

10 /5/3 EAMPLE - IDEAL LOW-PASS FILTERED WHITE NOISE Suppose that a white Gaussian noise w(t) of zero mean and power spetral density N / is applied to an ideal low-pass filter of Bandwidth B and passband response of one. Calulate the autoorrelation of the noise at the filter output. What is the noise power at the filter output? H f B f B f B EAMPLE - SOLUTION The autoorrelation funtion of n(t) is the inverse Fourier transform of the power spetral density. B N jpft RN t e df B N Bsin Bt N S N f B f B f B The power at the filter output an be alulated Integrate the power spetral density R N () is also the noise power. N B

11 /5/3 NARROWBAND NOISE Usually there is a bandpass filter before the frontend. This bandpass filter is just large enough to pass the desired hannel and suppress the rest of the band However, noise within the desired band also passes this filter. This noise has narrow bandwidth omparing white noise is alled narrowband noise. NARROWBAND NOISE Depending on the appliation there are two different representation of this noise In-phase and Quadrature Envelope and phase

12 /5/3 NARROWBAND NOISE IN TERM OF IN- PHASE AND QUADRATURE COMPONENTS Consider a narrowband noise n(t) of bandwidth B entered on frequeny f. Using omplex notation, the narrowband noise n t n t osp f t n t sinp f t I where ni(t) is in-phase omponent of n(t) and nq(t) is quadrature omponent of n(t). This type of representation is very helpful to analyze the noise on digital ommuniation systems. Q PROPERTIES OF THE IN-PHASE AND QUADRATURE COMPONENTS OF NARROWBAND NOISE The in-phase omponent n I (t) and quadrature omponent n Q (t) of narrowband noise has zero mean. n(t) is Gaussian n I (t) and n Q (t) are jointly Gaussian n(t) is stationary n I (t) and n Q (t) are jointly stationary They have the same power spetral density, whih is related to the S N (f) of the narrowband noise n(t) S NI f S NQ f S N f f S f f B f B otherwise They have the same variane as the narrowband noise n(t) N

13 /5/3 EAMPLE IDEAL BAND-PASS FILTERED WHITE NOISE Consider a white Gaussian noise of zero mean and power spetral density N / is passed through an ideal band-pass filter H f f B f f otherwise B H(f) Calulate the autoorrelation of the noise at the filter output Calulate the noise power at the filter output Calulate the in-phase and quadrature noise autoorrelations EAMPLE - SOLUTION The autoorrelation funtion of n(t) is the inverse Fourier transform of the power spetral density R N t f B f B N N Bsin e N Bsin jpft df f B f B N jpft jpft Bt e e Bt ospf t e jpft df Total noise power at the output is N B 3

14 /5/3 EAMPLE - SOLUTION The spetral density of the noise at the filter output is symmetri about ±f. Therefore the spetral density of n I (t) and n Q (t) is R t R t N Bsin Bt NI NQ NARROWBAND NOISE IN TERM OF ENVELOPE AND PHASE COMPONENTS We may represent the noise n(t) in terms of its envelope and phase omponents. n r t rt ospf t t t n t n t t tan n n t t r(t) and (t) are statistially independent. I Q I Q 4