' $& %November 17th, 2004Cooperative Diversity forWireless NetworksVijay DiviSignals, Information, and Algorithms Laboratory' $Introduction& %V. Divi, Nov. 17th, 2004• Fading is a major limitation in wireless systems• Different forms of diversity used to overcome fading– Spatial Diversity– Temporal Diversity– Frequency Diversity• Focus on specific form of spatial diversity named cooperative diversity– Other terminals in network act as relays between source and dest.– Relays experience independent fading and act as a virtual antenna array' $Background& %V. Divi, Nov. 17th, 2004• Scalar Fading Channel– Slow/Fast Fading– Transmitter CSI• MIMO Communication System– Diversity Gain– Multiplexing Gain• Connection to Cooperative-Diversity' $Scalar Fading Channel& %V. Divi, Nov. 17th, 2004Input signal experiences Rayleigh fading h and additive Gaussian noise wy = hx+ wwhere x ∼ CN(0,σ2x), w ∼ CN(0,σ2w)SNR:ρ△= SNR = σ2x/σ2wTwo major questions:• How often does the realization of h change?• Does the transmitter have knowledge of h?' $Transmission Cases& %V. Divi, Nov. 17th, 2004Fading:• Slow Fading - Channel does not change within time period of interest• Fast Fading - Channel changes on the order of symbols/codewordsChannel State Information (CSI):• Tx CSI - Transmitter has knowledge of h• No Tx CSI - Transmitter does not know h, only its distributionSlow Fading Fast FadingTx CSI Deterministic Capacity Time Water-fillingC = log(1+ |h|2ρ)No Tx CSI Outage Capacity Ergodic CapacityR, poutC = E[log(1+ |h|2ρ)]' $Slow Fading/No Tx CSI& %V. Divi, Nov. 17th, 2004• Most difficult case• Any non-zero rate chosen by the transmitter may be above the supported rateof the channel• Channel is not changing with time (non-ergodic), the transmission may alwaysfail with some non-zero probability• Shannon capacity is zero, i.e., system cannot guarantee that any amount ofinformation can be transmitted reliably.• Outage capacity: given R and SNR, there is a non-zero probability that thechannel does not support this rate; this probability is known as the outageprobability (pout).• Can use relays to improve transmission, act like a virtual MIMO system' $Multiple Antenna Systems& %V. Divi, Nov. 17th, 2004• The discrete-time MIMO channel model for system with m transmit and n re-ceive antennas isy = Hx+ w.• H is an n× m matrix of channel gains whose entries are i.i.d., CN(0,1)• H drawn from such a distribution causes– Outages in the system– Increases degrees of freedom for communications' $Diversity Gain& %V. Divi, Nov. 17th, 2004• High-SNR regime: the diversity gain d measures the rate at which the errorprobability decays with SNR (1/ρd)d = − limρ→∞logPe(ρ)logρ• For a system with m transmit and n receive antennas, in the best case, theerror can be made to decay with SNR as 1/ρmn• Diversity is a result of the mn independent paths between the transmit andreceive antennas.' $Multiplexing Gain& %V. Divi, Nov. 17th, 2004• Because the path gains are independent, the channel matrix is well-conditionedwith high probability• Can view the MIMO system as min(m,n) independent spatial channels be-tween the transmitter and receiver.• Fast fading ergodic capacity: C(ρ) = min(m,n)logρ+ O(1)• For slow fading, spatial multiplexing gain r measures the rate at which R in-creases with respect logρ and is formally defined as:r = limρ→∞R(ρ)logρ, R.= rlogρ• Multiplexing gain can be thought of as the number of independent spatial chan-nels being used optimally, which is upper-bounded by min(m,n).' $Diversity-Multiplexing Tradeoff& %V. Divi, Nov. 17th, 2004• Natural to talk about a tradeoff between the r and d' $Connection to Cooperative-Diversity& %V. Divi, Nov. 17th, 2004• Scalar fading channel provides a lower bound for the performance of any validcooperative-diversity scheme– Can always ignore use of the relay (non-cooperative protocol) and achievescalar channel performance• 2× 1 MIMO system provides an upper bound– In MIMO system, the transmitter has full control over both antennas andcan code across both of them, allowing more flexibility.– Often referred to as genie-aided protocol' $System Model& %V. Divi, Nov. 17th, 2004SRD• xs[n] and xr[n] are the transmitted signals from the source and relay• yr[n] and yd[n] are the received signals of the relay and destination• Direct source to destination transmission modeled as:yd[n] = as,dxs[n] + zd[n].• Fading ai, j∼ CN(0,σ2i, j) and independent• Noise zj[n] ∼ CN(0,N0), mutually independent' $Model Assumptions& %V. Divi, Nov. 17th, 2004• Half duplex: antennas cannot transmit and receive simultaneously• Channels experience independent flat Rayleigh-fading and white complex Gaus-sian noise.• Slow fading environment• Transmitters have no CSI. Receivers have full CSI.• Each terminal has one antenna with power constraint, P.' $Case Analysis& %V. Divi, Nov. 17th, 20041. Single Relay Systems• Amplify-and-Forward• Decode-and-Forward2. Multiple Terminal System• AF Revisited• DF Revisited• Cooperative Broadcast Channel• Cooperative Multiple-Access Channel' $Single Relay Systems& %V. Divi, Nov. 17th, 2004• Three terminals: source, relay, destination• Relay can either amplify or decode, re-encode, and transmit• Work by Laneman-Wornell-Tse– Additional constraint that only one terminal is transmitting at a time• Work by Azarian-El Gamal-Schniter– Allow multiple terminals to transmit simultaneously– Protocols optimal in certain regimes.' $Basic Amplify-and-Forward& %V. Divi, Nov. 17th, 2004• Relay can only transmit amplified version of received signalSRDn = 1,...,N/2R as,rxs[n] + zr[n]D as,dxs[n] + zd[n]SRDn = N/2+ 1,...,NRD ar,d(βyr[n− N/2]) + zd[n]• Relay power constraint: β ≤rP|as,r|2P+No' $Basic AF Performance& %V. Divi, Nov. 17th, 2004• Mutual information between the input and 2 outputs asIAF=12log(1+ ρ|as,d|2+ f(ρ|as,r|2,ρ|ar,d|2))wheref(x,y) =xyx+ y+ 1.• For an outage event (small fading coefficients), f(x,y) can be approximatedas min(x,y)△= γ.' $Basic AF Performance cont.& %V. Divi, Nov. 17th, 2004• Thus, our outage event becomes (I¡R):12log(ρ|as,d|2+ ργ) < Rlog(ρ) + log(|as,d|2+ γ) < 2rlogρlog(|as,d|2+ γ) < (2r− 1)logρ|as,d|2+ γ < ρ(2r−1)• Since γ and |as,d|2are both exponential, the probability of outage becomes(ρ2r−1)2• Diversity-multiplexing tradeoff of d(r) = 2(1− 2r)'