equation 1 where h i and d i are the channel gains and distances between the transmitter and receiver respectively the attenuation exponent is u03b2 e g u03b2 2 for free space propagation the relay operates in half duplex mode where the transmitting and listening modes cannot occur simultaneously in the df protocol the block of symbols with length n is split into two phases in the first phase the relay is in listening mode and receives the signal from the source at the end of this phase the relay decodes the source information message the relay then switches to transmitting mode in the second phase and sends symbols to help the destination decode the source message during the first phase the signal received by the relay is given by equation 2 y r k g 0 x s k v k k 1 2 u03b1 n 2 y k g 1 x s k w k k 1 2 u03b1 n 3 where x s k denotes the codeword that is to be transmitted from the source v k denotes the additive noise introduced in the channel between the source and the relay and w k denotes the additive noise introduced in the channel between the source and the destination u03b1 denotes the proportion of symbols that is to be devoted to the first phase during the second phase i e the relay transmitting phase the signal received by the destination is given by equation.