Lecture 7 Wireless Channels and Diversity Ming Xiao CommTh/EES/KTH Lecture 7: Wireless Channels and Diversity Overview Advanced Digital Communications (EQ2410) 1 Channel Modeling Narrowband Fading Frequency-Selective Fading Ming Xiao Time-Varying CommTh/EES/KTH Channels Performance for Fading Channels Capacity Receive Diversity Thursday, Feb. 11, 2016 Coherent Diversity 10:00-12:00, B24 Combining 1 Textbook: U. Madhow, Fundamentals of Digital Communications , 2008 1 / 15
Overview Lecture 1-6 Lecture 7 • Equalization (signal processing) Wireless Channels and Diversity • Channel Coding (information and coding theory) Ming Xiao CommTh/EES/KTH Overview Lecture 7: Wireless Channels and Diversity Channel Modeling Narrowband Fading 1 Overview Frequency-Selective 2 Channel Modeling Fading 3 Narrowband Fading Time-Varying Channels 4 Frequency-Selective Fading Performance for 5 Time-Varying Channels Fading Channels 6 Performance for Fading Channels Capacity 7 Capacity Receive Diversity 8 Receive Diversity Coherent Diversity Combining 9 Coherent Diversity Combining 2 / 15
Overview Examples for wireless communications Lecture 7 Wireless Channels • Radio and TV broadcast and Diversity • Point-to-point microwave links Ming Xiao CommTh/EES/KTH • Satellite communications Overview • Cellular communications Channel Modeling • Wireless local area networks (WLANs), bluetooth, etc. Narrowband Fading • Sensor networks Frequency-Selective Fading Time-Varying Channels Important characteristic: broadcast nature Performance for Fading Channels • All users which are close enough can listen. Capacity • Interference from other users Receive Diversity • Coordination required (TDMA, FDMA, CDMA) Coherent Diversity Combining • Frequency planing 3 / 15
Channel Modeling • Statistical models are defined based on channel measurements. • Algorithm and system development based on channel models. Lecture 7 • Complex baseband model with transmitted signal u ( t ) and received Wireless Channels and Diversity signal y ( t ) M � A k e j φ k u ( t − τ k ) e − j 2 π f c τ k y ( t ) = Ming Xiao CommTh/EES/KTH k =1 Overview Multipath propagation, M paths Channel Modeling • Amplitude of the k -th path: A k Narrowband Fading • Changes in the phase (e.g., due Frequency-Selective Fading to scattering): φ k Time-Varying • Delay on the k -th path: τ k Channels Performance for • Phase lag due to transmission Fading Channels delay: 2 π f c τ k Capacity Receive Diversity • Impulse response and transfer function of the complex baseband Coherent Diversity channel Combining M M A k e j θ k δ ( t − τ k ) , A k e j θ k e − j 2 π f τ k � � h ( t ) = and H ( f ) = k =1 k =1 with θ k = ( φ k − 2 π f c τ k mod 2 π ), uniformly distributed in [0 , 2 π ] 4 / 15
Narrowband Fading • Channel transfer function is approximately constant over the signal band which is used; i.e., the channel impulse response is reduced to Lecture 7 one impulse with gain Wireless Channels M and Diversity � A k e j γ k h ≈ H ( f 0 ) = with Ming Xiao CommTh/EES/KTH k =1 M M � � Re( h ) = A k cos( γ k ) and Im( h ) = A k sin( γ k ) Overview Channel Modeling k =1 k =1 with γ k = ( θ k − 2 π f 0 τ k mod 2 π ) and the center frequency f 0 . Narrowband Fading Frequency-Selective • Central limit theorem: for large M , Re( h ) and Im( h ) can be Fading Time-Varying modeled as jointly Gaussian with Channels • mean E[Re( h )] = E[Re( h )] = 0 Performance for M Fading Channels • variance var[Re( h )] = var[Im( h )] = 1 A 2 � 2 k Capacity k =1 • and covariance cov[Re( h ) , Im( h )] = 0 Receive Diversity Coherent Diversity M Combining � A 2 h ∼ CN (0 , k ) k =1 M M Re( h ) ∼ N (0 , 1 Im( h ) ∼ N (0 , 1 � � A 2 A 2 k ) and k ) 2 2 k =1 k =1 5 / 15
Narrowband Fading • Rayleigh fading: for zero-mean Gaussian Re( h ) and Im( h ) it follows with σ 2 = var[Re( h )] = var[Im( h )] that Lecture 7 Wireless Channels • g = | h | 2 is exponentially distributed and Diversity Ming Xiao 1 CommTh/EES/KTH 2 σ 2 exp( − g / (2 σ 2 )) I { g ≥ 0 } p G ( g ) = Overview • r = | h | is Rayleigh distributed Channel Modeling p R ( r ) = r σ 2 exp( − r 2 / (2 σ 2 )) I { r ≥ 0 } Narrowband Fading Frequency-Selective Fading • Rice fading: one dominant multipath (line-of-sight, LOS) Time-Varying component, A 1 e j γ 1 , i.e., we have Channels Performance for h = A 1 e j γ 1 + h diffuse Fading Channels Capacity M Receive Diversity A 2 with h diffuse ∼ CN (0 , � k ). Coherent Diversity k =2 Combining M → Accordingly, h ∼ CN ( A 1 e j γ 1 , A 2 � k ), and r = | h | is Rician distributed. k =2 6 / 15
Frequency-Selective Fading • Signal with bandwidth W ; signal-spaced sampling with T s = 1 / W Lecture 7 • Tapped delay line (TDL) model (compare model for ISI channel) Wireless Channels and Diversity ∞ Ming Xiao α i δ ( t − i CommTh/EES/KTH � � A k e j θ k h ( t ) = W ) and α i = i =1 k : τ k ≈ i Overview W Channel Modeling → { α i } is zero-mean, proper complex Gaussian. Narrowband Fading Frequency-Selective • Power-delay profile (PDP, for τ ≥ 0) Fading Time-Varying 1 Channels τ ms e − τ P ( τ ) = τ ms Performance for Fading Channels with the root mean squared delay τ MS Capacity Receive Diversity ( i +1) / W Coherent Diversity � Combining E[ | α i | 2 ] = → P ( τ ) d τ i / W • Applications: (among others) GSM channel models 7 / 15
Frequency-selective vs. Narrowband Fading Delay spread and coherence bandwidth Lecture 7 Wireless Channels and Diversity • Delay spread: T m , maximum τ for which P ( τ ) > ǫ ( ǫ → 0) Ming Xiao CommTh/EES/KTH • Coherence bandwidth: B m , maximum bandwidth for which the channel is approximately constant in f . Overview Channel Modeling B m ≈ 1 / T m Narrowband Fading Frequency-Selective Fading Time-Varying Channels Transmitted signal s ( t ) with bandwidth W Performance for • W ≪ B m ⇒ frequency-flat fading (only scaling and phase-shift, no Fading Channels “filtering”) Capacity Receive Diversity • W ≫ B m ⇒ frequency-selective fading (linear filtering, ISI) Coherent Diversity Combining 8 / 15
Time-Varying Channels • Model: TDL with time-varying coefficients { α i } • Moving receiver with speed v → max Doppler shift f D = f c v / c ; i.e., Lecture 7 Wireless Channels a sinusoid with frequency f c will be shifted to frequencies f c ± f D . and Diversity • Clarke’s Model Ming Xiao CommTh/EES/KTH • Time varying complex gain Overview � e j (2 π f k t + θ k ) X ( t ) = Channel Modeling Narrowband Fading k Frequency-Selective → y ( t ) = X ( t ) · u ( t ) Fading Time-Varying • Doppler shift of the k -th Channels component f k = f D cos( β k ) Performance for Fading Channels • X ( t ): zero-mean proper Capacity complex Gaussian 5 Receive Diversity 4.5 y • Power spectral density for rich 4 Coherent Diversity 3.5 Combining scattering and omnidirectional 3 antennas 2.5 2 1.5 1 S X ( f ) = 1 � π f D 1 − ( f / f D ) 2 0.5 x 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 9 / 15
Fast/Slow Fading Lecture 7 Wireless Channels • Doppler spread: f D , a frequency impulse (sinusoid) is broadened to and Diversity bandwidth f D . Ming Xiao CommTh/EES/KTH • Coherence time: T D , the channel is approximately constant in time Overview for T D seconds. Channel Modeling T D ≈ 1 / f D Narrowband Fading Frequency-Selective Fading Time-Varying Channels Transmitted signal s ( t ) with bandwidth W Performance for Fading Channels • W ≫ f D ⇒ slow fading (no Doppler spread) Capacity • W ≪ f D ⇒ fast fading (Doppler spread) Receive Diversity Coherent Diversity Combining 10 / 15
Performance for Fading Channels • Assumption: uncoded transmission over a slow fading channel y ( t ) = h · s ( t ) + n ( t ) Lecture 7 • Normalized fading: h ∼ CN (0 , 1) Wireless Channels and Diversity → G = | h | 2 ∼ p G ( g ) = exp( − g ) I g ≥ 0 √ Ming Xiao G ∼ p R ( r ) = 2 r exp( − r 2 ) I g ≥ 0 → R = CommTh/EES/KTH • Instantaneous and average SNR: S = E b / N 0 = G ¯ S , and ¯ S = ¯ E b / N 0 • Error Probability (averaged over fading) Overview Channel Modeling � P e = E[ P e ( G )] = P e ( g ) p G ( g ) dg Narrowband Fading Frequency-Selective � = E[ P e ( R )] = P e ( r ) p R ( r ) dr Fading Time-Varying Channels • Noncoherent FSK in Rayleigh fading Performance for P e ( G ) = 1 / 2 exp( − G ¯ P e = (2 + ¯ E b / N 0 ) − 1 S / 2) ⇒ Fading Channels Capacity • Binary DPSK in Rayleigh fading P e ( G ) = 1 / 2 exp( − G ¯ P e = (2 + 2 ¯ E b / N 0 ) − 1 Receive Diversity S ) ⇒ Coherent Diversity • Coherent FSK in Rayleigh fading Combining P e = 1 � ¯ 2 (1 − (1 + 2 N 0 / ¯ E b ) − 1 / 2 ) P e ( R ) = Q ( R S ) ⇒ • Coherent BPSK in Rayleigh fading P e = 1 � 2 ¯ 2 (1 − (1 + N 0 / ¯ E b ) − 1 / 2 ) P e ( R ) = Q ( R S ) ⇒ 11 / 15
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