Cooperative-Diversity in Wireless NetworksVij ay DiviNovember 17, 20041 IntroductionOne major limitation of wireless systems is the fad ing channel that exists between usersand the base station. When a user’s channel realization is bad, i t may be unable to com-municate with the base station. To overcome this problem, many different forms of di-versity have been studied including: spatial, temporal, and frequency diversity. Recently,there has been an increasing popularity in a specific form of spatial diversity, namelycooperative diversity.Spatial diversity relies on the principle that multiple antennas located a sufficientdistance apart experience independent fading channels. Unlike the traditional form ofspatial diversity where a user has access to a physical array of different antennas, coop-erative diversity is based off the relay channel model where different users share theirindividual antennas. In particular, information is transmitted from source to destinationboth directly and through relays in the network; thus the relays and source act as a virtualantenna array. Within this report, we present multiple cooperative-diversity protocolsand analyze their performance via the diversity-multiplexing tradeoff curves.2 BackgroundTo begin, we will overview the scalar fading channel and introduce different channe lmodeling assumptions that will have a key impact on the performance. We then introducethe MIMO communication system in which the source and destination have multipleantennas and study the performance gains with spatial diversity. The analysis of MIMOsystems is closely related to the analysis of cooperative-diversity protocols; in particular,the MIMO systems provide an upper bound for our the cooperative-diversity systemperformance.2.1 Scalar Fading ChannelIn a scalar fading channel, the input signal x experiences Rayleigh fading h and additiveGaussian noise w, modeled asy = hx + w (1)1where the input x ∼ CN(0, σ2x) and n oise w ∼ CN(0, σ2w).The analysis of system performance depends on the assumptions we make on thefading parameter h. There a re two different properties of h that we focus on: h ow oftenthis parameter changes with time and whether its value is known at the transmitter (Tx).These properties define four major cases which are studied in communications. The termsfast-fading and slow-fading refer to the coherence time for which a specific realization ofh is valid. For fast-fading, we assume that the channel fading realization changes often,on the time order of symbols or codewords. For slow-fading, the channel realization doesnot change within the time period of interest. Channel State Information (CSI) is used todescribe whether a terminal knows the value of h; in particular, we focus on whether CSIexists at the transmitter. In practice, CSI is attained a t the transmitter via a feedback linkor when there is 2-way communication. For our discussion, we make the assumption thatCSI exists at the receiver, which is a valid assumption for most communication systems.• Slow Fading/Tx CSI - The channel ha s one realization of the fading coefficient andthis value is known to the transmitter. The capacity of the channel is a well definedrandom quantity since the transmitter can use a rate equal to the max rate supportedby the fad in g realization (see table 1).• Fast Fading/Tx CSI - Unlike the case above, the channel is changing with everysymbol. Thus, an optimal strategy is for the transmitter to vary the amount of powerused for each symbol based up the channel re alization, i.e. a water-filling procedurein time. I t is difficult to find a closed form expression for the capacity in this case.• Slow Fading/No Tx CSI - Because the channel is unknown to the transmitter, anynon-zero rate chosen by the transmitter may be above the supported rate of thechannel. Note, since the channel is not changing with time (non-ergodic), the trans-mission may always fail with some non-zero probability. Thus, the Shannon capac-ity is zero, i.e., the system cannot guarantee that any amount of information canbe transmitted reliably. I nstead, the notion of outage capacity is introduced. For aspecified rate R and SNR, there is a non-zero probability that the channel does notsupport this rate; this probability is known as the outage probability (pout).• Fast Fading/No Tx CSI - For this case, the notion of ergodic capacity is introduced.Although the channel may be bad for certain times, we can bound this amount oftime and pick long codewords which experience enough good channels so that itcan be decoded without error; the capacity is the a verage over many channel real-izations. R ephrased, a s we a t longer periods of time, the average mutual informa-tion per symbol converges to the expectation of mutual information.2The cases are summarized in the table below:Slow 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)]Table 1 : Transmission regimes for scalar fading channelThe variable ρ represents the SNR, σ2x/σ2w. For the sequel, we will focus on the case ofslow fading with no CSI (Channel State Information) at the transmitter. Note, this is themost challenging case since we do not know the channel and it does not change.Although transmission directly between the source and destination may be in out-age for a certain rate, using relays increases diversity by providing alternate channelswhich may experience less fading. Cooperative diversity systems use relays as virtualantenna arrays which allow them to pe rform better than the single antenna system. How-ever, virtual antennas may not be as helpful as having extra antennas at the transmitter.Similar to the lower bound provided by the scalar fading channel, we can attain an upperbound on performance of cooperative-diversity systems by looking at the MIMO (multi-ple input multiple output) setup where the transmitter has multiple antennas.2.2 Multiple Antenna SystemsMultiple antennas at the transmitters and/or receivers help increase the performance ofwireless systems. The diversity d ue to multiple paths between the transmitter and re-ceiver results in higher data rates with lower outage probabilities. Recently, the diversity-multiplexing tradeoff has been widely studied as a metric for determining the perfor-mance of a MIMO schemes. This tradeoff measures the high-SNR asymptotic tradeoffbetween outage p robability and