Bài giảng Lý thuyết tín hiệu - Chương III: Tín hiệu ngẫu nhiên - Võ Thị Thu Sương

TÍN HIỆU NGẪU NHIÊN  Định nghĩa  Tín hiệu không đóan được trước khi nó xuất hiện  Không thể mô tả bởi biểu thức tóan học  Được mô tả bằng lý thuyết xác xuất  Được gọi là “quá trình ngẫu nhiên”  Quá trình ngẫu nhiên gồm một số hữu hạn các biến ngẫu nhiên  Ví dụ: ( ) 5 c o s ( 2 ) , w h e re i s ra n d o m c x t f t    CuuDuongThanCong.com https://fb.com/tailieudientucntt 1.5 Random Signals  1.5.1 Biến ngẫu nhiên X(A)  Biến ngẫu nhiên là một đại lượng thực mà trị của nó phụ thuộc vào biến cố ngẫu nhiên. (để biến cố NN có thể được mô tả một cách định lượng) Ví du độ lệch của viên đạn so với mục tiêu là một đại lượng phụ thuộc vào kết qủa của lần bắn. Sự phụ thuộc này được được biểu diễn bởi quy luật xác suất gọi chung là phân bố  Sự phân bố của biến NN được mô tả bởi hàm mật độ xác suất PX(x). 2 1 - 1 2 . n o n -n e g a t iv e : ( ) 0 . n o rm a liz e d : ( ) 1 . e v e n t p ro b a b i li ty : ( )= ( 2 ) 1 3 X X x X x p x p x d x P x X x p x d x         CuuDuongThanCong.com https://fb.com/tailieudientucntt  Discrete pdf  has the same properties (change integration to summation)  Two important random variables and their pdf ( ) i p X x 0 1 ( . U n i fo rm ra n d o m v a r ia b le 1 c o n t in u o u s ( ) , f o r 1 d i s c re te : ( ) , f o r { , , } . G a u s s ia n (n o rm a l) r a n d o m 1 2 v a r ia b le 1 ( ) 2 X i M x m X X p x a x b b a p X x X x x M p x e               2 2 ) 2 X X  CuuDuongThanCong.com https://fb.com/tailieudientucntt  Các thông số  Example:  Data bits are modeled as uniform random variable with two values  Symbols are modeled as uniform random variable with M values  Noise is modeled as Gaussian random variable with zero mean and non-zero variance 2 2 2 2 . m e a n : { } ( ) . v a r ia n c e : { ( ) } { } (v a r ia n c e = m e a n s q 1 2 u a re v a lu e - m e a n v a lu e s q u a re ) X X X X X m E X xp x d x E X m E X m          CuuDuongThanCong.com https://fb.com/tailieudientucntt 1.5 Random Signals  1.5.2 Random process: X(A,t)  Là một hàm hai biến A, t time-domain signal waveform with some random event  Usually written as X(t) by embedding A  Stationary random process  Average parameters do not depend on time  We consider stationary random process (signal) only  Can usually be described conveniently only by average parameters  event time S ta t io n a ry . m e a n : ( ) { ( )} c o n s ta n t . a u to c o rre la t io n ( s ta t io n a ry c a s e ) : 1 2 ( ) { ( ) ( )} X X X m t E X t m R E X t X t          CuuDuongThanCong.com https://fb.com/tailieudientucntt  Example (Note: expectation/integration is conducted with random variable, not t) 2 0 F in d th e m e a n a n d a u to c o r re la t io n o f th e r a n d o m p ro c e s s ( ) 5 c o s ( 2 ) , w h e re [ 0 , 2 ) i s u n i fo rm ra n d o m . : 1 { ( )} ( ) ( ) 5 c o s ( 2 ) 0 2 ( ) { ( ) ( )} S o lu t io n c X c X x t f t m E x t x t f d f t d R E x t x t                           2 0 = ( ) ( ) ( ) 1 = 5 c o s ( 2 )5 c o s ( 2 2 ) 2 2 5 = c o s ( 2 ) 2 c c c c x t x t f d f t f t f d f                      CuuDuongThanCong.com https://fb.com/tailieudientucntt  1.5.2.3 Autocorrelation  Defined by matching of a signal with a delayed version of itself  Measure how closely a signal matches a shifted copy of itself  Is a function of delay , not time t  Note for figure: Random process cos(2πfct+θ) does not look like noise. CuuDuongThanCong.com https://fb.com/tailieudientucntt  1.5.4 Power Spectral Density (PSD)  PSD is FT{autocorrelation}  The only way for frequency-domain description of random signal (since FT{x(t)} does not exist) ( ) ( ) IF T x x F T G f R     2 5 E x a m p le : F o r ( ) c o s ( 2 ) , th e P S D is 2 2 5 ( ) { ( )} [ ( ) ( ) ] 4 x c x x c c R f G f F T R f f f f             PSD of random process 5cos(2πfct+θ) CuuDuongThanCong.com https://fb.com/tailieudientucntt  1.5.3 Parameters and their physical meaning  Mean & variance of random variable  Mean, autocorrelation, PSD of random process - 2 2 2 1 . : d c le v e l o f th e s ig n a l 2 . { ( )} , ( 0 ) , ( ) : a v e ra g e s ig n a l p o w e r 3 . : a v e ra g 4 . F o r s ig n a ls w i th o u t d c z e ro -m e a n e p o w e r o f A C c o m p o n e n t s ig n a ls i ) 0 . i i ) { X X X X X X m E X t R G E f m f d         2 ( t ) } e q u a ls a v e ra g e s ig n a l p o w e rX CuuDuongThanCong.com https://fb.com/tailieudientucntt  1.5.5 Noise in communication systems  AWGN: additive white Gaussian noise  Additive: Noise is added (not multiplied) to the signal  White: has constant PSD (equal power for all frequency)  Gaussian: in every time-instant (sampling instant), the noise is Gaussian random variable  Noise is usually assumed zero- mean AWGN x(t) n(t) y(t) 2 2 0 0 2 S ig n a l m o d e l: ( ) ( ) ( ) P S D : ( ) w a t t s /H z 2 A u to c o r re la t io n : z e ro -m e a n A W G N ( ) p ro p e r t ie s ( ) ( ) 2 1 p d f : ( ) : i ) i i ) i i i ) 2 n n n y t x t n t N G f N R t p n n e             CuuDuongThanCong.com https://fb.com/tailieudientucntt  AWGN is a useful abstract noise model, although it is not practical due to infinite power  In sampled process (discrete process), since δ(0)=1, we still have  Discrete zero-mean AWGN: power & variance are both N0/2 AWGN PSD & Auto- correlation 2 2 0 { } 2 N E X   CuuDuongThanCong.com https://fb.com/tailieudientucntt 1.6 Signal transmission through linear systems  1.6.1 Deterministic signals x(t) h(t) y(t) X(f) Y(f)H(f) )()()( )()()(*)()( fHfXfY dthxthtxty        1.6.2 Random signals  No Y(f), X(f) exist! But can use PSD. 2 ( ) ( ) * ( ) ( ) ( ) ( ) ( ) ( )y x y t h t x t x h t d G f G f H f           CuuDuongThanCong.com https://fb.com/tailieudientucntt  1.6.3 Distortionless transmission & ideal filter  Distortionless transmission  Time-domain: only constant magnitude change & a delay  Frequency domain: constant magnitude response and linear phase response  Ideal filter: distortionless in passband )( )()( fj efHfH   where 0 p a s s b a n d ( ) 0 s to p b a n d ( ) 2 K H f f f t           Example. Input: AWGN with PSD . System: ideal lowpass filter with unit magnitude response in passband fu.. Then the output PSD is 0 ( ) / 2 n G f N 0 2 0 ( ) ( ) , ( ) ( ) j f t y t K x t t Y f K e X f     0 / 2 , fo r ( ) 0 , O th e rw is e u u y N f f f G f       CuuDuongThanCong.com https://fb.com/tailieudientucntt Review: Analog Communications  Amplitude modulation  4 main types, share similar modulator/demodulator x(t) tf c2cos B.P.F y(t) modulator AM: amplitude modulation DSB: double-sideband modulation SSB: single-sideband modulation VSB: vestigial sideband modulation y(t) tf c2cos L.P x(t) demodulator  Frequency modulation (FM,PM) CuuDuongThanCong.com https://fb.com/tailieudientucntt  1.7.1 DSB (Page 45-47, Page 1022) D S B s ig n a l: ( ) ( ) c o s ( 2 ) D S B s p e c t ru m : 1 ( ) [ ( ) ( ) ] 2 ( ) , ( ) : m e s s a g e s ig n a l a n d s p e c t ru m D S B s ig n a l b a n d w i th = 2 * m e s s a g e b a n d w id t h c c c c c x t x t f t X f X f f X f f x t X f          ( ) 2 D S B x t W W CuuDuongThanCong.com https://fb.com/tailieudientucntt  DSB demodulation  DSB is a main digital passband modulation technique y(t) tf c2cos L.P x(t) demodulator lo w p a s s lo w p a s s lo w p a s s ( ) ( ) : r e c e iv e d s ig n a l D e m o d u la t io n o u tp u t i s : ˆ ( ) ( ) c o s ( 2 ) = ( ) c o s ( 2 ) c o s ( 2 ) 1 = ( ) [1 c o s ( 4 ) ] 2 ( ) = 2 c c c c c y t x t x t y t f t x t f t f t x t f t x t        CuuDuongThanCong.com https://fb.com/tailieudientucntt Tín hiệu dừng (t) là tín hiệu dừng chặt nếu:      )(),...(),()(),...(),( 2121   nn tttfEtttfE (t) là tín hiệu dừng rộng nếu:    consttE      2121 ;, ttRttR   (t) là tín hiệu Egodic nếu: (t) là TH dừng rộng và          T T T dtttR  * 2 1 lim CuuDuongThanCong.com https://fb.com/tailieudientucntt