TÍN HIỆU NGẪU NHIÊN (lý THUYẾT TÍNH HIỆU SLIDE)

TÍN HIỆU NGẪU NHIÊN  Định nghĩa  Tín hiệu khơng đóan trước xuất  Khơng thể mơ tả biểu thức tóan học  Được mô tả lý thuyết xác xuất  Được gọi “q trình ngẫu nhiên” ●  Ví Q trình ngẫu nhiên gồm số hữu hạn biến ngẫu nhiên dụ: x(t ) = 5cos(2π f c t + θ ), where θ is random 1.5 Random Signals  1.5.1 Biến ngẫu nhiên X(A)  Biến ngẫu nhiên đại lượng thực mà trị phụ thuộc vào biến cố ngẫu nhiên (để biến cố NN mơ tả cách định lượng) Ví du độ lệch viên đạn so với mục tiêu đại lượng phụ thuộc vào kết qủa lần bắn Sự phụ thuộc được biểu diễn quy luật xác suất gọi chung phân bố  Sự phân bố biến NN mô tả hàm mật độ xác suất PX(x) non-negative: p X ( x) ≥ normalized: ∞ ∫-∞ p X ( x)dx = event probability: P( x1 ≤ X < x2 )= ∫ x2 x1 p X ( x)dx  Discrete ● pdf p ( X = x ) i has the same properties (change integration to summation)  Two important random variables and their pdf Uniform random variable , for a ≤ x ≤ b b−a discrete: p ( X = xi ) = , for X ∈ {x0 ,L , xM −1} M Gaussian (normal) random variable continuous p X ( x) = p X ( x) = 2π σ X − e ( x − mX )2 2σ X2  Các thông số mean: variance: ∞ m X = E{ X } = xp X ( x)dx −∞ σ X2 = E{( X − m X ) } = E{ X } − m 2X ∫ (variance = mean square value - mean value square)  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 1.5 Random Signals event  time 1.5.2 Random process: X(A,t)    Là 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 not depend on time We consider stationary random process (signal) only Can usually be described conveniently only by average parameters mean: Stationary m X (t ) = E{ X (t )}  → mX constant autocorrelation (stationary case): RX (τ ) = E{ X (t ) X (t + τ )} ● Example (Note: expectation/integration is conducted with random variable, not t) Find the mean and autocorrelation of the random process x(t ) = 5cos(2π f c t + θ ), where θ ∈ [0, 2π ) is uniform random Solution: m X = E{x(t )} = ∫ x(t ) fθ (θ )dθ = ∫ 2π 5cos(2π f c t + θ ) RX (τ ) = E{x(t ) x(t + τ )} dθ = 2π = ∫ x(t ) x(t + τ ) fθ (θ ) dθ =∫ 2π = 5cos(2π f c t + θ )5cos(2π f c t + 2π f cτ + θ ) 25 cos(2π f cτ ) dθ 2π  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 τ Note for figure: Random process cos(2πfct+θ) does not look like noise  1.5.4 Power Spectral Density (PSD) IFT  → Rx (τ ) G PSD is FT{autocorrelation}   x( f )¬ FT  The only way for frequency-domain description of random signal (since FT{x(t)} does not exist) 25 Example: For Rx (τ ) = cos(2π f cτ ), the PSD is 25 Gx ( f ) = FT {Rx (τ )} = [δ ( f − f c ) + δ ( f + f c )] PSD of random process 5cos(2πfct+θ)  1.5.3 Parameters and their physical meaning  Mean & variance of random variable  Mean, autocorrelation, PSD of random process m X : dc level of the signal ∞ E{ X (t )}, RX (0), ∫ G X ( f ) df : average signal power -∞ σ X2 : average power of AC component For signals without dc ⇔ zero-mean signals i) m X = ii) σ X2 = E{ X (t)} equals average signal power  1.5.5 Noise in communication systems  AWGN: additive white Gaussian noise ● ● ●  Additive: Noise is added (not Signal model: y (t ) = x(t ) + n(t ) multiplied) to the signal zero-mean AWGN n(t ) properties: White: has constant PSD (equal power for all frequency) N0 i) PSD: G ( f ) = watts/Hz n Gaussian: in every time-instant (sampling instant), the noise is N Gaussian random variable ii) Autocorrelation: R (τ ) = δ (τ ) n Noise is usually assumed zeromean AWGN iii) pdf: p ( n) = 2π σ − n2 e 2σ  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 N0 σ = E{ X } = Discrete zero-mean AWGN: power2& variance are both N0/2 ● AWGN PSD & Autocorrelation through linear systems  1.6.1 Deterministic signals ∞ y (t ) = x(t ) * h(t ) = ∫ x(τ )h(t − τ )dτ −∞ Y ( f ) = X ( f )H ( f )  1.6.2 Random signals  No Y(f), X(f) exist! But can use PSD ∞ x(τ )h(t − τ ) dτ −∞ y (t ) = h(t ) * x(t ) = ∫ G y ( f ) = Gx ( f ) H ( f )  1.6.3 Distortionless transmission & ideal filter  Distortionless transmission ● ●  Time-domain: only constant magnitude change & a delay − j 2π ft0 y ( t ) = Kx ( t − t ), Y ( f ) = Ke X ( f ) response Frequency domain: constant magnitude and linear phase response   K passband − jθ ( f ) Ideal filter: distortionless in passband H ( f ) =   H( f ) = H( f )e where  0 stopband θ ( f ) = 2π ft   Gn PSD ( f ) = N0 / Example Input: AWGN with System: ideal lowpass filter with unit magnitude  N / 2, for − fu ≤ f ≤ fu response in passband fGu.y.( fThen PSD is ) =  the output  0, Otherwise Review: Analog Communications Amplitude modulation   main types, share similar modulator/demodulator AM: amplitude modulation DSB: double-sideband modulation SSB: single-sideband modulation VSB: vestigial sideband modulation  Frequency modulation (FM,PM)  1.7.1 DSB (Page 4547, Page 1022) • DSB signal: xc (t ) = x(t ) cos(2π f c t ) • DSB spectrum: [ X ( f − f c ) + X ( f + f c )] x(t ), X ( f ) : message signal and spectrum Xc ( f ) = • DSB signal bandwith=2*message bandwidth WDSB = 2Wx (t )  DSB demodulation y (t ) = xc (t ) : received signal Demodulation output is: xˆ (t ) = y (t ) cos(2π f c t ) lowpass = x(t ) cos(2π f c t ) cos(2π f c t ) lowpass = x(t ) [1 + cos(4π fc t )] lowpass =  x(t ) DSB is a main digital passband modulation technique Tín hiệu dừng ξ(t) tín hiệu dừng chặt nếu: E [ f {ξ (t1 ), ξ (t ), ξ (t n )} ] = E [ f {ξ (t1 + ε ), ξ (t + ε ), ξ (t n + ε )} ] ξ(t) tín hiệu dừng rộng nếu: E [ξ ( t ) ] = const R( t1 ,t ) = R (τ ) ;τ = t1 −t ξ(t) tín hiệu Egodic nếu: ξ(t) TH dừng rộng T R(τ ) = lim ∫ ξ ( t )ξ * ( t − τ ) dt T →∞ −T ...1.5 Random Signals  1.5.1 Biến ngẫu nhiên X(A)  Biến ngẫu nhiên đại lượng thực mà trị phụ thuộc vào biến cố ngẫu nhiên (để biến cố NN mơ tả cách định lượng) Ví du độ lệch... passband modulation technique Tín hiệu dừng ξ(t) tín hiệu dừng chặt nếu: E [ f {ξ (t1 ), ξ (t ), ξ (t n )} ] = E [ f {ξ (t1 + ε ), ξ (t + ε ), ξ (t n + ε )} ] ξ(t) tín hiệu dừng rộng nếu: E [ξ (... ε )} ] ξ(t) tín hiệu dừng rộng nếu: E [ξ ( t ) ] = const R( t1 ,t ) = R (τ ) ;τ = t1 −t ξ(t) tín hiệu Egodic nếu: ξ(t) TH dừng rộng T R(τ ) = lim ∫ ξ ( t )ξ * ( t − τ ) dt T →∞ −T