Lecture 7: Wireless Channels and Diversity Advanced Digital Communications (EQ2410) 1

Transcription

1 Wireless : Wireless Advanced Digital Communications (EQ2410) 1 Thursday, Feb. 11, :00-12:00, B24 1 Textbook: U. Madhow, Fundamentals of Digital Communications, / 15

2 Wireless Lecture 1-6 Equalization (signal processing) Channel Coding (information and coding theory) : Wireless / 15

3 Wireless Examples for wireless communications Radio and TV broadcast Point-to-point microwave links Satellite communications Cellular communications Wireless local area networks (WLANs), bluetooth, etc. Sensor networks Important characteristic: broadcast nature All users which are close enough can listen. Interference from other users Coordination required (TDMA, FDMA, CDMA) Frequency planing 3 / 15

4 Wireless Statistical models are defined based on channel measurements. Algorithm and system development based on channel models. Complex baseband model with transmitted signal u(t) and received signal y(t) y(t) = A k e jφ k u(t τ k )e j2πfc τ k Multipath propagation, M paths Amplitude of the k-th path: A k Changes in the phase (e.g., due to scattering): φ k Delay on the k-th path: τ k Phase lag due to transmission delay: 2πf cτ k Impulse response and transfer function of the complex baseband channel h(t) = A k e jθ k δ(t τ k ), and H(f ) = A k e jθ k e j2πf τ k with θ k = (φ k 2πf cτ k mod 2π), uniformly distributed in [0, 2π] 4 / 15

5 Wireless Channel transfer function is approximately constant over the signal band which is used; i.e., the channel impulse response is reduced to one impulse with gain h H(f 0) = A k e jγ k with Re(h) = A k cos(γ k ) and Im(h) = A k sin(γ k ) with γ k = (θ k 2πf 0τ k mod 2π) and the center frequency f 0. Central limit theorem: for large M, Re(h) and Im(h) can be modeled as jointly Gaussian with mean E[Re(h)] = E[Re(h)] = 0 variance var[re(h)] = var[im(h)] = 1 A 2 2 k and covariance cov[re(h), Im(h)] = 0 h CN(0, A 2 k) Re(h) N(0, 1 A 2 k) and Im(h) N(0, 1 A 2 k) / 15

6 Wireless Rayleigh fading: for zero-mean Gaussian Re(h) and Im(h) it follows with σ 2 = var[re(h)] = var[im(h)] that g = h 2 is exponentially distributed p G (g) = 1 2σ 2 exp( g/(2σ2 ))I {g 0} r = h is Rayleigh distributed p R (r) = r σ 2 exp( r 2 /(2σ 2 ))I {r 0} Rice fading: one dominant multipath (line-of-sight, LOS) component, A 1e jγ 1, i.e., we have with h diffuse CN(0, M A 2 k). k=2 h = A 1e jγ 1 + h diffuse Accordingly, h CN(A 1e jγ 1, M A 2 k), and r = h is Rician distributed. k=2 6 / 15

7 Signal with bandwidth W ; signal-spaced sampling with T s = 1/W Wireless Tapped delay line (TDL) model (compare model for ISI channel) h(t) = α i δ(t i W ) and α i = {α i } is zero-mean, proper complex Gaussian. Power-delay profile (PDP, for τ 0) P(τ) = 1 τ ms e τ τms with the root mean squared delay τ MS E[ α i 2 ] = (i+1)/w P(τ)dτ k:τ k i W A k e jθ k i/w Applications: (among others) GSM channel models 7 / 15

8 Frequency-selective vs. Wireless Delay spread and coherence bandwidth Delay spread: T m, maximum τ for which P(τ) > ɛ (ɛ 0) Coherence bandwidth: B m, maximum bandwidth for which the channel is approximately constant in f. B m 1/T m Transmitted signal s(t) with bandwidth W W B m frequency-flat fading (only scaling and phase-shift, no filtering ) W B m frequency-selective fading (linear filtering, ISI) 8 / 15

9 Model: TDL with time-varying coefficients {α i } Wireless Moving receiver with speed v max Doppler shift f D = f cv/c; i.e., a sinusoid with frequency f c will be shifted to frequencies f c ± f D. Clarke s Model Time varying complex gain X (t) = k e j(2πf k t+θ k ) y x y(t) = X (t) u(t) Doppler shift of the k-th component f k = f D cos(β k ) X (t): zero-mean proper complex Gaussian Power spectral density for rich scattering and omnidirectional antennas 1 S X (f ) = πf D 1 (f /fd ) 2 9 / 15

10 Fast/Slow Wireless Doppler spread: f D, a frequency impulse (sinusoid) is broadened to bandwidth f D. Coherence time: T D, the channel is approximately constant in time for T D seconds. T D 1/f D Transmitted signal s(t) with bandwidth W W f D slow fading (no Doppler spread) W f D fast fading (Doppler spread) 10 / 15

11 Wireless Assumption: uncoded transmission over a slow fading channel y(t) = h s(t) + n(t) Normalized fading: h CN(0, 1) G = h 2 p G (g) = exp( g)i g 0 R = G p R (r) = 2r exp( r 2 )I g 0 Instantaneous and average SNR: S = E b /N 0 = G S, and S = Ē b /N 0 Error Probability (averaged over fading) P e = E[P e(g)] = P e(g)p G (g)dg = E[P e(r)] = P e(r)p R (r)dr Noncoherent FSK in Rayleigh fading P e(g) = 1/2 exp( G S/2) P e = (2 + Ēb/N 0 ) 1 Binary DPSK in Rayleigh fading P e(g) = 1/2 exp( G S) P e = (2 + 2Ē b /N 0 ) 1 Coherent FSK in Rayleigh fading P e(r) = Q(R S) P e = 1 2 (1 (1 + 2N 0/Ē b ) 1/2 ) Coherent BPSK in Rayleigh fading P e(r) = Q(R 2 S) P e = 1 2 (1 (1 + N 0/Ēb) 1/2 ) 11 / 15

12 Wireless Fast fading (ideal interleaving and long blocks) Model: y[n] = h[n] s[n] + w[n] Normalized fading: h[n] CN(0, 1) Transmit power constraint: E[ x[n] 2 ] P AWGN: w[n] CN(0, 2σ 2 ) Signal-to-noise ratio: SNR = E[ h 2 ]P/(2σ 2 ) Ergodic capacity for Gaussian s[n] CN(0, P) (averaged over G) C Rayleigh = E[log(1 + G SNR)] = Slow fading Model: y[n] = h s[n] + w[n] for a given G = h 2 0 log(1 + g SNR)p G (g)dg C(h) = log(1 + h 2 SNR) = log(1 + g SNR) = C(g) can become C(h) = 0. No rate R which guarantees error-free transmission for all h. For a given R, system outage if C(h) < R outage probability Pr(C(h) < R) Outage capacity and outage probability C out = log(1 + G out SNR) with Pr(G < G out) = ɛ 12 / 15

13 Wireless Problem with fading: strong SNR fluctuations The channel can be bad for some time! Solution: diversity; i.e., provide the receiver with different copies of the same signal (create parallel channels). Spatial diversity, multiple antennas Temporal diversity (e.g., repetition coding in time) Frequency diversity (e.g., select carriers with independent fading) Model: N received branches y 1,..., y N for the same symbol s with y i = h i s + n i Independent zero-mean complex AWGN terms n i with variance σ 2 Complex fading gains h i Diversity combining Selection combining (choose the strongest path) Maximum ratio combining (optimal linear combination of branches) Equal gain combining (sum of all branches) Switched diversity (pick one branch at random) 13 / 15

14 Wireless Coherent maximum ratio combining (with h = (h 1,..., h N ) and y = (y 1,..., y N )) L(y s, h) = N L(y i s, h i ) with ( 1 L(y i s, h i ) = exp σ 2 [Re( y i, h i s ) 1 ) 2 h i s 2 ] Decision variable (ignoring the energy term) Z = N Re( y i, h i s ) = Coherent matched filter N hi y i, s 14 / 15

15 Wireless Matched filter output N r = hi y i = ( N N h i 2 ) s + ( hi n i ) SNR= ( N h i 2 ) 2 P/( N h i 2 σ 2 ); i.e., SNR gain G = N h i 2 Error probability for BPSK (averaged over all realizations of h) Pe = Pr(r < 0 s = +1) = E[P h (h)] ( ) N N 1 ( ) 1 µ ( N 1 + i 1 + µ = 2 i 2 i=0 with µ = S/(1 + S) and the average SNR S High SNR: Pe = K(N)(1/(4 S)) N, with K(N) = ( ) 2N 1 N Diversity gain: ln P e lim S ln S = N Similar analysis for selection combining, equal gain combining etc. 15 / 15 ) i