Fading Channels I: Characterization and Signaling

Transcription

1 Fading Channels I: Characterization and Signaling Digital Communications Jose Flordelis June, 3, 2014

2 Characterization of Fading Multipath Channels

3 Characterization of Fading Multipath Channels In addition to the time spread introduced by the multipath medium, in this chapter we consider variations in time of the nature of the multipath. Each path n has an associated propagation delay τ n and attenuation factor α n. The time-variant impulse response of the equivalent lowpass channel is given by c(τ; t) = n α n (t)e j2πfcτn(t) δ[τ τ n (t)] for the discrete-time channel, and c(τ; t) = α(τ; t)e j2πfct for the continuous-time channel. Note that c(τ; t) represents the response of the channel at time t due to an impulse applied at time t τ.

4 Characterization of Fading Multipath Channels The received signal to an unmodulated carrier transmission s l (t) = 1 is given by r l (t) = n = n α n (t)e j2πfcτn(t) α n (t)e jθn(t) where θ n (t) = 2πf c τ n (t) and f c is the carrier frequency. θ n (t) changes much faster than α n (t). Signal fading is a result of the time variations in the phases {θ n (t)}. A statistical treatment of the channel is suitable. Some models are Rayleigh fading channel. Rice fading channel. Nakagami-m fading channel.

5 Channel Correlation Functions and Power Spectra Assume that c(τ; t) is wide-sense-stationary. The associated autocorrelation function is R c (τ 2, τ 1 ; t) = E[c (τ 1 ; t)c(τ 2 ; t + t)] = R c (τ 1 ; t)δ(τ 2 τ 1 ) where, in the last step, uncorrelated scattering is assumed. R c (τ) R c (τ; 0) is called the multipath intensity profile or the power delay spectrum of the channel. The support of R c (τ) is called the multipath spread of the channel and is denoted by T m.

6 Channel Correlation Functions and Power Spectra Equivalently, in the Fourier domain we have the time-variant transfer function C(f ; t) defined as C(f ; t) = We define the autocorrelation function c(τ; t)e j2πf τ dτ R C (f 2, f 1 ; t) = E[C (f 1 ; t)c(f 2 ; t + t)] = R C ( f ; t) R C ( f ; t) is called the spaced-frequency, spaced-time correlation function of the channel. The spaced-frequency correlation function is defined as R C ( f ) R C ( f ; 0), and can also be computed as R C ( f ) = R c(τ; t)e j2π f τ dτ R C ( f ) provides a measure of the coherence bandwidth of the channel, ( f ) c 1 T m.

7 Channel Correlation Functions and Power Spectra

8 Channel Correlation Functions and Power Spectra In order to characterize time variations we define S C ( f ; λ) = R C ( f ; t)e j2πλ t d t. Furthermore, S C (λ) S C (0; λ) is called the Doppler power spectrum of the channel. The support of S C (λ) is called the Doppler spread B d of the channel. The coherence time ( t) c 1 B d is related to the spaced-time correlation function R C ( t) R C (0; t). Finally, we define the scattering function of the channel S(τ; λ) as S(τ; λ) = S C ( f ; λ)e j2πτ f d f.

9 Channel Correlation Functions and Power Spectra

10 Channel Correlation Functions and Power Spectra

11 Channel Correlation Functions and Power Spectra

12 Statistical Models for Fading Channels Large number of scatterers contributing n X n, with X n N(0, σ 2 ) i.i.d, then, by the central limit theorem, phase uniformly distributed in [0, 2π] and envelope follows a Rayleigh probability distribution where Ω = E(R 2 ). p R (r) = 2r Ω e r 2 /Ω, r 0 Nakagami-m distribution, which includes the Rayleigh distribution as a special case (m = 1). Rice distribution, which includes the power of the non-fading, specular components.

13 Propagation Models for Mobile Radio Channels

14 The Effect of Signal Characteristics on the Choice of a Channel Model The effect of the channel c(τ; t) on the transmitted signal s l (t) is a function of our choice of signal bandwidth W and signal duration T. W 1 T m ( f ) c, frequency-selective channel. W 1 T m ( f ) c, frequency-nonselective channel or flat fading. Signaling interval T 1 B d ( t) c, slowly fading channel. Signaling interval T 1 B d ( t) c, fast fading channel. For a frequency-nonselective channel r l (t) = C(0; t)s l (t), with C(0; t) = α(t)e jφ(t). E.g. C(0; t) zero-mean complex-valued Gaussian stochastic process. Define the spread factor of the channel T m B d. Channel is underspread if T m B d < 1.

15 Frequency-Nonselective, Slowly Fading Channel Channel modeled as a one-tap filter, with filter coefficient C(0; t) = α(t)e jφ(t) constant for at least T. Therefore, r l (t) = αe jφ s l (t) + z(t), 0 t T. Assume phase shift φ estimated withour error, hence, coherent detection at the output of a matched filter demodulator applies. For PSK (4.3-13), P b (γ b ) = Q( 2γ b ) where γ b = α 2 E b /N 0 and α is fixed. Similarly, for FSK (4.2-42), P b (γ b ) = Q( γ b ). Finally, P b = 0 P b (γ b )p b (γ b ) dγ b.

16 Frequency-Nonselective, Slowly Fading Channel, with Rayleigh Statistics Assume α is Rayleigh-distributed. Thus, γ b is chi-square-distributed with p(γ b ) = 1 γ b e γ b/ γ b, γ b 0 and γ b = E b N 0 E(α 2 ) is the average signal-to-noise ratio. The bit error probability is then (see problem ) P b = ( 1 ( 1 γb 1 + γ b γb 2 + γ b ), for BPSK ), for BFSK

17 Frequency-Nonselective, Slowly Fading Channel, with Rayleigh Statistics The error probabilities for high SNR, i.e. γ b 1 become 1/4 γ b, for coherent PSK 1/2 γ b, for coherent, orthogonal FSK P b 1/2 γ b, for DPSK 1/ γ b, for noncoherent, orthogonal FSK The error rate P b decreases only inversely with SNR. Compare to a nonfading channel (exponential decrease with SNR). DPSK performs in pair with coherent, orthogonal FSK.

18 Frequency-Nonselective, Slowly Fading Channel, with Rayleigh Statistics

19 Frequency-Nonselective, Slowly Fading Channel, with Nakagami-m Statistics When envelope α is characterized by the Nakagami-m distribution, the distribution of γ = α 2 E b /N 0 is p(γ) = where γ = E/N 0 E(α 2 ). mm Γ(m) γ m γm 1 e mγ/ γ, m = 1 corresponds to Rayleigh fading. m > 1 corresponds to fading less severe than Rayleigh. m < 1 corresponds to fading more severe than Rayleigh.

20 Diversity Techniques for Fading Multipath Channels The idea is to supply to the receiver several replicas of the same information transmitted over independently fading channels. For L independently fading replicas with probability p, the probability of all replicas being in a fading deep is p L. How to achieve diversity? Frequency diversity, by transmitting on L carriers with separation between successive carriers greater than ( f ) c. Time diversity, by transmitting on L time slots with separation between successive time slots greater than ( t) c. Space diversity, by transmitting on L antennas with minimum separation between antennas greater than ( d) c. Angle-of-arrival diversity. Polarization diversity. Frequency diversity can also be obtained by using a wideband signal such that W > ( f )c. The achievable diversity order is L W /( f )c, which corresponds to the number of resolvable signal components.

21 Diversity Techniques for Fading Multipath Channels with Binary Signals Determine the bit error probability P b for a binary digital communication system with L diversity channels, each frequency-nonselevtive and slowly fading. The fading processes {C k (0; t)} are assumed mutually independent. The noise processes {z k (t)} are assumed mutually independent, with identical autocorrelation functions. r lk (t) = α k e jφ k s km (t) + z k (t), k = 1, 2,..., L, m = 1, 2

22 Diversity Techniques for Fading Multipath Channels with Binary Signals The optimum demodulator consists of a bank of matched filters and a maximal ratio combiner (MRC). b k1 (t) = sk1 (T t) b k2 (t) = sk2 (T t) Compensate for the phase shift in the channel. Weight the signal by a factor proportional to the signal strenght. For PSK, at the output of the MRC we have U = Re ( 2E ) L L αk 2 + α k N k = 2E k=1 and N k = e jφ k k=1 T 0 z k(t)sk (t) dt. L L αk 2 + α k Re(N k ) k=1 k=1

23 Diversity Techniques for Fading Multipath Channels For PSK modulation and a fixed set of {α k } we have that P b (γ b ) = Q(E 2 (U)/σ 2 U ) = Q( 2γ b ), where the SNR per bit, γ b, is given as γ b = E N 0 L αk 2 = k=1 L k=1 and γ k = Eα 2 k /N 0 is the instantaneous SNR on the kth channel. The probability density function p(γ b ) is that of a chi-square-dsitributed r.v. with 2L degrees of freedom: γ k p(γ b ) = 1 (L 1)! γ L c γ L 1 b e γ b/ γ c where γ c = E(α 2 k )E/N 0 is the average SNR per channel.

24 Diversity Techniques for Fading Multipath Channels Finally, we need to average over the fading channel statistics P b = 0 P b (γ b )p b (γ b ) dγ b. The closed-form solution to this integral is P b = [ 1 2 (1 L 1 ( ) L 1 + k µ)]l [ 1 (1 + µ)]k k 2 k=0 γc where, by definition µ = 1 + γ c We note that P b,psk = 1 2 (1 µ), 1 P b,psk = 1 (1 + µ) 2 where P b,psk is the probability error for a single fading channel.

25 Diversity Techniques for Fading Multipath Channels When the average SNR per channel γ c is greater that 10 db, we have that P b ( 1 ( ) 2L 1 ) L. 4 γ c L The error rate decreases inversely with the Lth power of the SNR. Actually ( 1 ) L( ) 2L 1 4 γ L, for BPSK c ( 1 ) L( ) 2L 1 2 γ L, for BFSK c P b ( 1 ) L( ) 2L 1 2 γ L, for DPSK c ( 1 γ ) L( ) 2L 1, for noncoherent BFSK c L

26 Diversity Techniques for Fading Multipath Channels

27 Diversity Techniques for Fading Multipath Channels, Nakagami Fading A K-channel system transmitting in a Nakagami fading channel with independent fading is equivalent to an L = Km channel diversity in a Rayleigh fading channel.

28 A Tapped Delay Line Model The W -bandlimited, time-variant frequency-selective channel can be modeled as a tapped delay line with tap spacing 1/W and tap weight coefficients c n (t) c(τ; t) = n= c n (t)δ(τ n/w ) (13.5 8) with c n (t) = 1 W c( n ; t), and the corresponding time-variant W transfer function is C(f ; t) = n= c n (t)e j2πfn/w. For all practical purposes truncation can be applied at L = T m W + 1 taps. c n (t) are complex-valued stationary, mutually uncorrelated (US) random processes.

29 A Tapped Delay Line Model

30 The RAKE Demodulator Assume binary signaling with T T m, then r l (t) = L c k (t)s li (t k/w ) + z(t) k=1 = v i (t) + z(t), 0 t T, i = 1, 2. The optimal demodulator consists of two filters matched to v 1 (t) and v 2 (t), and has decision variables [ T U m = Re = Re 0 [ L k=1 ] r l (t)vm(t) dt T 0 r l (t)c k (t)s m(t k/w ) dt ], m = 1, 2. The tapped delay line demodulator attempts to collect the signal energy from all received signal paths that fall within the span of the delay line and carry the same information.

31 The RAKE Demodulator

32 The RAKE Demodulator

33 Performance of RAKE Demodulator Assume that {c k (t)} are estimated perfectly, and are constant within any one signaling interval. The decision variables are [ L ] T U m = Re r l (t)s lm (t k/w ) dt, m = 1, 2 k=1 c k 0 with the received signal (assume, say, s l1 (t) is transmitted) r l (t) = L c n s l1 (t n/w ) + z(t), 0 t T. n=1 This gives [ L U m = Re + Re ck k=1 n=1 [ L c k L ] T c n s l1 (t n/w )slm (t k/w ) dt 0 T z(t)s lm (t k/w ) dt ], m = 1, 2.

34 Performance of RAKE Demodulator For pseudorandom sequences s l1 (t) and s l2 (t), the decission variables simplify to [ L ] T U m = Re c k 2 s l1 (t k/w )slm (t k/w ) dt + Re k=1 [ L k=1 c k T 0 0 z(t)s lm (t k/w ) dt ], m = 1, 2. When the binary signals are antipodal, a single decision variable suffices ( ) L L U 1 = Re 2E αk 2 + α k N k k=1 k=1 where α k = c k and N k = e jφ k T 0 z(t)s l (t k/w ) dt. We have already seen this expression!

35 Performance of RAKE Demodulator The RAKE demodulator, with perfect (noiseless) estimates of the channel tap weights is equivalent to a maximal ratio combiner in a system with Lth order diversity. Let s consider this time binary antipodal signals subjected to distinct {E(αk 2 )}. The error probability is given by ] P b = 1 L γ k (1 ρ r ) π k [ γ k (1 ρ r ) k=1 with π k = L γ k i=1 and γ k = E E(αk 2 γ k γ i N ). 0 For large values of the average SNR for all k taps, i.e., γ k 1 ( ) L 2L 1 1 P b L 2 γ k (1 ρ r ). k=1

36 Performance of RAKE Demodulator How to estimate {c k (t)}? Assume sufficiently slow channel fading, e.g. ( t) c /T 100. See Fig to Fig

37 Performance of RAKE Demodulator

38 Performance of RAKE Demodulator

39 Generalized RAKE Demodulator Addresses communication scenarios in which additive interference from other users of the channels results in coloured additive Gaussian noise. See Fig and Fig Assumes knowledge of the channel coefficients {c i } and the time delays {τ i }. In CDMA systems, an unmodulated spread spectrum signal is used. Our problem is to estimate the weights {w i } at the L g fingers, with L g > L. U = w H y y = gb + z and z contains additive Gaussian noise plus interference from other users plus ISI from channel multipath. The ML detection solution is given by (linear MMSE estimator) w = R 1 z g

40 Performance of RAKE Demodulator

41 Performance of RAKE Demodulator

42 Receiver Structures for Channels with Intersymbol Interference In the event that T b T m does not hold, the RAKE demodulator output will be corrupted by ISI. An equalizer in needed. RAKE sampled at bit rate Tb, followed by equalizer, i.e., MLSE or DFE (Fig ). Chip equalizer at chip rate T c, with LT c = T b (Fig ).

43 Receiver Structures for Channels with Intersymbol Interference

44 Receiver Structures for Channels with Intersymbol Interference

45 Multicarrier Modulation (OFDM) OFDM is especially vulnerable to Doppler spread, which results into intercarrier interference (ICI). We analyze the performance degration due to Doppler spread in such a system, and some ICI suppression techniques.

46 Performance Degradation of an OFDM System due to Doppler Spreading OFDM system with N subcarriers {e j2πfkt }, M-ary QAM or PSK, symbol duration T and f k = k/t, k = 1, 2,... N with { 1 T e j2πf i t e j2πfkt 1 k = i dt = T 0 0 k i Frequency-selective time-varying channel with impulse response c(τ; t), but non-selective for within each subcarrier band, i.e. c k (τ; t) = α k (t)δ(t), k = 0, 1,..., N 1, and {α k (t)} complex-valued, jointly stationary, Gaussian stochastic processes with zero mean and cross-covariance R αk α i (τ) = E[α k (t + τ)α i (t)].

47 Performance Degradation of an OFDM System due to Doppler Spreading Furthermore R αk α i (τ) = R 1 (τ)r 2 (k i). R 1 (τ) = J 0 (2πf m τ), with J 0 (x) the zero-order Bessel function of the first kind. Equivalenty, 1 f f m S(f ) = πf m 1 (f /fm ) 2 0 otherwise as is Jakes (1974). R 2 (k) = R C (k/t ) and, R C (f ) = β β + j2πf R c(τ) = βe βτ.

48 Performance Degradation of an OFDM System due to Doppler Spreading Use the two-term Taylor series expansion on {α k (t)}, i.e. α k (t) = α k (t 0 ) + α k (t 0)(t t 0 ), t 0 = T 2, 0 t T. Results in channel c k (τ; t) = α k (t)δ(t) = α k (t 0 )δ(t) + (t t 0 )α k (t 0)δ(t). The baseband signal s(t) = 1 N 1 s k e j2πfkt, T k=0 0 t T with E[ s k 2 ] = 2E avg is filtered with c(τ; t).

49 Performance Degradation of an OFDM System due to Doppler Spreading The received signal is r(t) = 1 N 1 α k (t 0 )s k e j2πfkt + 1 N 1 (t t 0 )α k (t 0)s k e j2πfkt +n(t) T T k=0 k=0 The output of the ith correlator at the sampling instant is ŝ i = 1 T T 0 = α i (t 0 )s i + T 2πj r(t)e j2πf i t dt N 1 k=0,k i α (t 0 )s k k i + n i, where the terms are the desired signal, ICI and additive noise, resp.

50 Performance Degradation of an OFDM System due to Doppler Spreading After some computation we arrive at S I = (Tf m ) N 1 k=0,k i. 1 (k i) 2 For a large number of subcarriers N the distribution of ICI is approximately Gaussian. ICI severely degrades the performance of an OFDM system.

51 Performance Degradation of an OFDM System due to Doppler Spreading

52 Suppression of ICI in OFDM Systems ICI in an OFDM system analogous to ISI in a single-carrier system: apply (linear) MMSE criterion. Estimate symbol s k (m) as ŝ k (m) = b H k (m)r(m), k = 0, 1,..., N 1 in order to minimize E[ s k (m) ŝ k (m) 2 ] = E[ s k (m) b H k (m)r(m) 2 ] where R(m) denotes the output of the DFT processor.

53 Suppression of ICI in OFDM Systems The optimum coefficient vector is b k (m) = [G(m)G H (m) + σ 2 I N ] 1 g k (m), k = 0, 1,..., N 1 where E[R(m)R H (m)] = G(m)G H (m) + σ 2 I N E[R(m)s k (m)] = g k(m) and G(m) = W H H(m)W, and W is the orthonormal IDFT transformation matrix. Knowledge of the channel impulse response is required.