Chapter Review of of Random Processes

Transcription

1 Chapter.. Review of of Random Proesses Random Variables and Error Funtions Conepts of Random Proesses 3 Wide-sense Stationary Proesses and Transmission over LTI 4 White Gaussian Noise

2 Random Variables and Error Funtions Means, Varianes, and moments of of Random Variables Let be a random variable with the density funtion f (. Mean of Variane of The seond of moment of Relationship μ E[ ] f ( d If is a disrete random variable, then the above integrals are replaed by the summations. σ Var[ ] E[( μ σ m μ ] ( μ f ( d m E[ ] f ( d

3 Correlation and Covariane Let and be two random variables, f, (, y be their joint density funtion, μ,, E[ ] μ E[ ] their respetive means, σ Var [ ] σ Var[ ] varianes,, and Z g(, is well-defined. Then Z is also a random variable, and the mean of Z is given by E [ g(, ] g(, y f, (, y d If g (,, then E [ g(, ] E[ ] is the orrelation of and. If g, ( μ ( μ, then E[( μ ( μ ] is the ovariane of and. If g, ( μ ( μ /( σ then ( Cov(, is the orrelation oeffiient. Relationship: E[ g(, ] ( ρ(, E[ g(, ] Cov(, ρ(, σ σ If ρ 0, then and are said to be unorrelated. σ

4 Guassian Random Variables N(μ, σ: is a gaunssian r.v. with mean μ and variane σ. The pdf of : f ( t e πσ The df of : ( t μ σ, < t < Unit or standard gaussian: μ 0 and σ. The df of the unit gaussian: Φ( f(t e π t / dt μ 0, σ F ( f ( t dt π σ f(t Φ( the shaded area t μ t Computation of F(: F ( Φ μ σ

5 Jointly Gaussian Random Variables Jointly Gaussian Random Variables Let and be gaussian random variables with means μ and μ y, varianes σ and σ y. We say that and have a bivariate Gaussian pdf if the joint pdf of and is given by S y f ( ep, (, ρ ρ σ πσ where y y S σ σ μ μ ρ σ μ σ μ ( ( +

6 Property. If f, (, y is defined by the above, then the onditional distributions of given that y, and given that, respetively, are all gaussian distributions with the following parameters listed in (a. (a f ( y : ( ~ (, ~ N μ y σ where ~ μ ( y μ σ + ρ σ ~ σ σ ( ρ ( y μ What are the parameters of f ( y? (b The parameter ρ is equal to the orrelation oeffiient of and, i.e., ρ ρ(, Cov(, σ σ ( and are independent if and only if and are unorrelated. In other word, and are independent if and only if ρ 0.

7 Q( Funtion f(t The df of the unit Gaussian random variable, reprodued here: Φ( e π t / dt Q( funtion is defined by Q( Φ( Φ( the shaded area f(t t whih is alled the tail probability. Q( the shaded area t

8 The error funtion: Relation: t dt e erf 0 ( π t dt e erf ( π What are the relationships among Φ(, Q(, erf(, or erf(? ( erf Q Φ ( erf + erf + σ μ ( erf F Error Funtion and the Complementary Error Funtion Error Funtion and the Complementary Error Funtion ( erf erf is a gaunssian r.v. with mean μ and variane σ, then the df F( of an be epressed in terms of erf(: The omplementary error funtion:

9 A few on the of of few remarks on the definition random proesses: A random proess is a olletion of random variables: {(t, t T} where (t is a random variable whih maps an outome ξ to a number (t, ξ for every outome of an eperiment. We will use (t to represent a random proess omitting, as in the ase of random variables, its dependene on ξ. Conepts of Random Proesses (t has the following interpretations: It is a family (or ensemble of funtions (t, ξ. Both t and ξ are variables It is a single time funtion (or a sample funtion, the realization of the proess. t is variable and ξ is fied (t is a random variable equal to the state of the given proess at time t. t is fied and ξ is variable If t and ξ are fied, then (t is a number

10 In In a ommuniation system, there are two types of of random proesses: For the transmitted signals, they have two properties: - the signals are funtions of time - the signals are random in the sense that before onduting an eperiment, it is not possible to desribe eatly the waveform that will be observed. Thus a transmitted signal is a random proess. The noise /interferene introdued by hannel is a random 0

11 Eample.. Eperiment: to transmit 3 binary digits to the reeiver 0 T T T: symbol duration (t is a random proess or a random signal whih is a olletion of all the following eight waveforms. { t,..., ( } 0( 7

12 t t 0 Sample funtions T T 3T (t 00 (t 00 (t 0 3 (t 00 4 (t 0 5 (t 0 6 (t 7 (t

13 Eample.. The voltage of an a generator with random amplitude A and phase Θ is given by ( t Aos(πf t + Θ where Θ is a random variable uniformly distributed over (0, π. (t is a random proess whih onsists of a family of osine waves and a single sample is the funtion: ( t, θ Aos(πf t + θ The following figures showed samples funtions for θ being equal to 0, θ/, -θ /4, and θ/4.

14 θ 0 θ π/ t t θ - π/4 θ π/ t t

15 Differene between the two eamples The first eample onsists of a family of funtions that annot be desribed in terms of a formula. In the seond eample, the random proess onsists of a family of osine waves and it is ompletely speified in terms of the random variable 5

16 Means, autoorrelation and rossorrelation Mean of the random proess (t is the mean of random variable (t at time instant t: μ + ( t E[ ( t] f ( t ( d where f (t( is the pdf of (t at time instant t. Autoorrelation funtion of (t is a funtion of two variables t t and t t + τ, R ( t, t + τ E[ ( t ( t + τ ] This is a measure of the degree to whih two time samples of the same random proess are related. Crossorrelation funtion of two random proesses (t abd (t is a funtion of two variables t t, and t t + τ,, defined by R, ( t, t + τ E[ ( t ( t + τ ]

17 3 Wide-Sense Stationary (WSS Proesses and Transmission over LTI What is a WSS Proess? A proess (t is WSS if the following onditions are satisfied: ( i μ ( t E[ ( t] ( ii R ( t + τ, t R ( τ onstant independent of t. depends only on the time differene τ and not on the variables t t + τ. and t t. In other words, a random proess (t is WSS if its two statistis, its mean and autoorrelation, do not vary with a shift in the time 7

18 Eample.3. Find the mean and autoorrelation funtion of the random proess (t ( in E.g.., pdf ( t Aos(πf t + Θ where Θ is uniformly distributed over [0, π]. : f Θ Is (t a WSS? Solution. /(π, 0 θ π ( θ 0, otherwise Aording to the definitions, E[ Aos(πf t + Θ Aos(πf ( t + τ + Θ] A E os(πf τ + os(πf (t + τ + Θ R A ( t, t + τ os(πf τ Sine both the mean and autoorrelation of (t do not depend on time t, then (t is a WSS proess. E[ ( t] π 0 Aos(πf t + θ 0 dθ π

19 Properties of Autoorrelation Funtion of a WSS proess (t. R ( τ R ( τ Symmetri in τ about zero. R ( τ R (0 for all τ, Maimum value ourred at the 3. R ( τ S ( f 4. origin Autoorrleation and psd form a pair of the Fourier transform R (0 E[ ( t ] The value at origin is equal to the average power of the 9

20 Power Spetral Density (PSD of a WSS Random Proess For a given WSS proess (t, the psd of (t is the Fourier transform of its autoorrelation, i.e., S ( f F ( ( τ R + R ( τ e dτ j πfτ R + j πfτ ( τ F ( S ( f S ( f e 0

21 For the random proess in Eample.3, we have R ( τ A os(πf Hene, the psd of (t is the Fourier transform of the autoorrelation of (t, given by τ A S ( f δ + 4 [ δ ( f f + ( f f ] A /4 S (f A /4 - f f f

22 Eample. 4 Let ( t ( tos(πf t + Θ where (t is a WSS proess with psd S (f, Θ is uniformly distributed over [0, π], and (t is independent of Θ and os( πf t + Θ Find the psd of (t. Solution. First we need to show that (t is WSS. Mean: m ( t E[ ( t] E[ ( tos(πf t + Θ] E[ ( t] E[os(πf t + Θ] ( t 0 m (by independene 0 (by

23 Autoorrelation of (t: R ( t + τ, t E[ ( t ( t + τ ] E[ ( tos(πf t + Θ ( t + τ os(πf ( t + τ + Θ] E[ ( t ( t + τ ] E[os(πf t + Θos(πf ( t + τ + Θ] R ( τ os(πf τ R (τ By Eample.3. Hene, (t is WSS. Therefore S ( t F[ ( τ ] R 4 j πfτ j ( ( πfτ τ e + e F[ R 4 [ S ( f f + S ( f + f ] ]

24 Properties of of PSD. S ( f 0. S ( f S ( f 3. S ( f ( τ R always real valued for (t real-valued a pair of Fourier transform 4. P R (0 S ( f df + Relationship between average power and 4

25 Transmission over LTI Systems Response of LTI system to a random input (t: Properties of the output: If (t is WSS, so does (t. (t h(t (t Mean: μ μ H(0 ( t ( t h( t ( t τ h( τ dτ 3 Autoorrelation: R 4 PSD: ( τ R ( τ h( τ h( τ S ( f S ( f H ( 5

26 (t R (τ h(t (t R y (τ F F - S (f H(f S 6

27 4 White Gaussian Noise Proesses Definition. A random proess N(t is a white gaussian noise if N(t is a WSS proess satisfying ( E[ N( t] 0 (zero mean μ N ( For any time instants t < t < < t k, N(t, N(t,, N(t k are independent gaussian random variables. N( t N( t N( t k t t 7

28 The harateristis of N(t ( N(t is a Gaussian random variable for any time instane t. ( Zero mean: (3 The autoorrelation funtion: N τ δ ( τ, < τ < 0 R N ( R N (τ μ N 0 (4 The power spetral density is flat: N f, < f 0 S N ( < where N 0 is the power per unit bandwidth of N(t. N 0 / S N (f N 0 / τ f (5 The average power is infinity: P N E[ N ( 8

29 Power spetrum of thermal noise S N (f N 0 / f The above spetrum ahieves its maimum at f 0. The spetrum goes to infinity, but the rate of onvergene to zero is very slow. From this, we onlude that the thermal noise, though not preisely white, for all pratial purpose an be modeled as a white gaussian noise 9

30 Eample. 5. Sum with a White Gaussian Noise. Suppose that N(t is a white Gaussian noise with power spetrum N 0 /. We also assume that (t and N(t are joint WSS and independent. Find the mean and the power spetral density of (t. (t Solution. Mean: + N(t (t (t + N(t μ ( t E[ ( t N( t] + E [ ( t] + E[ N( t] μ + 0 μ Autoorrelation: ( t, t + τ E[ ( t ( t + τ ] R [( ( t + N( t( ( t + τ + N( + τ ] E t R ( t, τ + R ( t, τ + R ( t, τ R ( t, τ N N + R ( τ + R ( τ R ( τ N (by independene and joint WSS Thus, (t is WSS. The psd of (t: S ( f F( R ( τ F( R ( τ + R ( τ S N0 ( f + SN ( f S ( f + N N

31 Eample.6 Effet of an Ideal Filter on White Gaussian Noise. A white gaussian noise N(t with psd the ideal low-pass filter shown below. N S N f ( 0 is the input to H(f -f f f Find the psd and the autoorrelation of the output proess (t. Is the output proess (t a white Gaussian 3

32 Solution. The psd is given by S ( f S ( f H ( f N N 0 0 for f otherwise f The autoorrelation is the inverse Fourier transform of the psd, thus R ( τ F ( S ( f N0 f sin ( f τ Remark. The ideal low-pass filter transforms the autoorrelation (delta funtion of white noise into a sin funtion. After filtering, we no longer have white noise. The output signal will have zero orrelation with shifted opies of itself, only at shifts of τ n /( f ( n not 3

33 Eample.6 Effet of an RC Filter on White Gaussian Noise A white gaussian noise N(t with psd RC filter with impulse response N S N f ( 0 is the input to the h( t e 0 at t 0 otherwise where a /(RC. Find the psd and the autoorrelation of the output proess 33

34 Solution. The frequeny response is: H ( f F( h( t H ( f a + jπf a + (πf Thus S N a a + (πf 0 ( f S N ( f H ( f R ( τ F ( S ( f N 0RC τ / RC e 4 Again, we no longer have white noise after filtering. The RC filter transforms the input autoorrelation funtion of white noise into an eponential 34