Ofdm-Based Cooperative Communications in a Single Path Relay Network and a Multiple Path Relay Network Essays

Ofdm-Based Cooperative Communications in a Single Path Relay Network and a Multiple Path Relay Network Essays Ofdm-Based Cooperative Communications in a Single Path Relay Network and a Multiple Path Relay Network Essay Ofdm-Based Cooperative Communications in a Single Path Relay Network and a Multiple Path Relay Network Essay In this thesis, we investigate cooperation by applying OFDM signals to cooperative relay networks. We consider the single path relay network and the multiple path relay network. Using the amplify-and-forward relay algorithm, we derive the input-output relations and mutual informations of both networks. Using a power constraint at each relay, we consider two relay power allocation schemes.The ? rst is constant gain allocation, where the amplifying gain used in the amplify-and-forward algorithm is constant for all subcarriers. The second is equal power allocation, where each subcarrier transmits the same power. The former scheme does not require CSI (channel state information), while the latter one does. We simulate the mutual informations using the two relay power allocation schemes. Results indicate that equal power allocation gives a slightly higher mutual information for the single path relay network. For the multiple path network, the mutual information is practically the same for both schemes.Using the decode-and-forward relay algorithm, we derive the inputoutput relations for both networks. The transmitter and each relay are assumed to have uniform power distributions in this case. We simulate the BER (bit error rate) and WER (word error rate) performance for the two networks using both the amplify-and-forward and decode-and-forward relay algorithms. For the single path relay network, amplify-andforward gives very poor performance, because as we increase the distance between the transmitter and receiver (and thus, add more relays), more noise and channel distortion enter the system. Decode-and-forward gives signi? antly better performance because noise and channel distortion are eliminated at each relay. For the multiple path relay network, decode-and-forward again gives better performance than amplify-and-forward. However, the performance gains are small compared to the single path relay network case.Therefore, amplify-and-forward may be a more attractive choice due to its lower complexity. ix CHAPTER I INTRODUCTION Wireless communication systems inherently su? er from multipath propagation and channel fading. Time diversity, space diversity, frequency diversity [8], and combinations of the three are traditionally used to combat these e? cts. More recently, relays situated between the transmitter and receiver are also being exploited to improve information transfer. The relays are a network of transceiver nodes between the transmitter and receiver that facilitate the transfer of information. Thus, the relay network as a whole is an equivalent channel between the transmitter and receiver. This type of scheme is known as cooperation or cooperative communications in the literature because the relay network is cooperating with the transmitter and receiver to improve performance.In this thesis, we consider cooperation in the context of orthogonal frequency division multiplexing (OFDM) systems. 1. 1 Motivation The motivation for cooperati ve communications is obvious. Cellular phones, laptops and personal digital assistants (PDAs) are just three examples of wireless devices that are very prevalent today. These transceiver devices usually communicate independently from each other. As the authors in [6] note, this is wasting the broadcast nature of the wireless medium.For example, if a base station is communicating with a user’s cellular phone, his/her nearby laptop has the capability to receive the base station’s signals and relay them to the phone, improving the end-to-end performance of the base station-phone link. Unfortunately, laptops and cellular phones today are not designed this way. This illustration is an example of an ad-hoc network, where nodes spontaneously recognize each other and cooperate. In this thesis, we investigate structured networks, where each node knows the existence of all the other nodes a priori.Whether the nodes discover each other through an ad-hoc algorithm or they are pre- programmed to have this knowledge is beyond the scope 1 of this thesis. 1. 2 Related Literature The authors in [10], [11] have considered cooperation between intra-cell users in a code division multiple access (CDMA) cellular network. In this case, cooperation results in higher data rates and leads to lower power requirements for users. As well, the system is less sensitive to channel variations. Relaying of signals, as viewed from the physical layer, is not a trivial issue. The authors in [5], [6], [7] have provided several physical layer relay algorithms.These include amplifyand-forward, decode-and-forward and selection relaying. In amplify-and-forward, a node ampli? es its receive symbol, subject to a power constraint, before re-transmitting to the next node. This algorithm is obviously with low complexity. In decode-and-forward, a node fully decodes a symbol, re-encodes it and then re-transmits it. In other words, this scheme attempts to eliminate channel distortion and noise at each node. In selection relaying, a node only re-transmits a symbol if the measured receiving channel gain is above a certain threshold.If the threshold is not reached, the relay requests a re-transmission from the sender. In networking terminology, this is a type of automatic repeat request (ARQ) scheme. The authors in [6], [7] have investigated cooperation for the classical relay channel introduced in [1], [7]. Outage probability is used to characterize performance. Outage probability is the probability that the mutual information between the transmitter and receiver does not reach a certain throughput threshold. Without cooperation, the outage probability decays proportionally with 1/SNR, where SNR is the signal-to-noise ratio of the channel.Using cooperation and the amplify-and-forward scheme, the outage probability decays proportionally with 1/SNR2 , achieving full diversity. This results in large power savings for the transmitter. The authors in [3], [4] have investigated coo peration for a single path of relays connected in series. The motivation for this network structure is that broader wireless coverage can be achieved, while still maintaining a low power constraint at the transmitter. The authors consider analog relaying and digital relaying as two possible relay algorithms. These are 2 equivalent to the amplify-and-forward and decode-and-forward algorithms, respectively.A power budget is considered where each packet travelling through the network is only allowed to consume a total ? xed amount of power. As well, each node has a certain transmit power limit. The outage probability is then minimized by allocating power among the relay network under these power constraints. This power allocation accounts for the channel conditions in the network in order to achieve the optimal outage probability. Simulations indicate that 2 dB of total power can be saved for 5 relays by using optimal power allocation instead of uniform power allocation. This is for th e decode-and-forward case.However, at high SNR values, the decode-and-forward case approximates the amplify-and-forward case. The authors in [13] have investigated cooperation for multiple paths of relays connected in parallel. In the conventional scheme, all relays participate using amplify-and-forward. This is called all-participate amplify-and-forward (AP-AF). The authors also consider an algorithm where only one relay is selected in the transmission to maximize the mutual information. This is called selection amplify-and-forward (S-AF). S-AF selects the relay which results in the maximum mutual information between transmitter and receiver.Simulations of outage probability indicate that 5 dB of SNR can be saved for 3 relays by using S-AF instead of AP-AF. The authors in [9] derive symbol error probabilities for multiple paths of relays. 1. 3 OFDM in Cooperative Communications In this thesis, we continue to investigate cooperation by applying OFDM signals to cooperative relay netw orks. We consider a single path relay network and a multiple path relay network. Using the amplify-and-forward relay algorithm, we derive the input-output relations and the mutual informations of both networks. Using a power constraint at each relay, we consider two relay power allocation schemes.The ? rst is constant gain allocation, where the amplifying gain used in the amplify-and-forward algorithm is constant for all subcarriers. The second is equal power allocation, where each subcarrier transmits the same power. We simulate the mutual informations using these two relay power allocations. Using the decode-and-forward relay algorithm, we derive input-output relations for both 3 networks. We simulate bit error rates (BERs) and word error rates (WERs) for the two networks using both the amplify-and-forward and decode-and-forward relay algorithms. 1. 4Organization of Thesis The thesis is organized as follows. In Chapter 2, we consider the single path relay network in [3], [4]. In C hapter 3, we consider a modi? ed version of the multiple path relay network in [13] where the transmitter-receiver direct link is removed. Notice that these latter two relay con? gurations are series and parallel analogs of each other. As well, they do not involve a direct link between the transmitter and receiver. Finally, Chapter 4 concludes the thesis and provides future research directions. 4 CHAPTER II SINGLE PATH RELAY NETWORK 2. 1 2. . 1 Amplify-and-Forward System Model Figure 1 shows the single path relay network. In the ? gure, r0 is the transmitter, rm+1 is the receiver, and r1 , . . . , rm are m relay nodes connected in series forming a single path link between the transmitter and receiver. The relays perform amplify-and-forward (AF) relaying. We assume that OFDM with N subcarriers is used in the system. hk , . . . , hk (0) (m) (0) are the complex subchannel gains at the kth subcarrier in the link, for (m) k = 1 to N . nk , . . . , nk are the corresponding noises, which a re assumed to be mutually ndependent, zero-mean, circular symmetric complex Gaussians all with variance N0 B/N , where N0 is the power spectral density of the underlying continuous time noise process and B is the OFDM bandwidth of the system. Let pk = Ptot /N be the transmitter power on (l) the kth subcarrier, where Ptot is the net transmitter power. Let pk be the amplifying gain used in the amplify-and-forward algorithm at the lth relay, for l = 1 to m. The kth (0) receive symbol at rl is ampli? ed by pk before it is forwarded to the next node. (l) Let xk be the kth transmit symbol with zero mean and unit variance.Let yk be the kth receive symbol at the receiver. Using Figure 1, the input-output relation is (0) nk r1 nk hk (1) (m? 1) nk (m) rm+1 r0 hk (0) hk (m? 1) rm hk (m) Receiver Transmitter Figure 1: Single Path Relay Network 5 m yk = i=0 (i) hk (i) pk m xk + j=0 ? ? m (i) hk i=j+1 where we assume r i=q a(i) = 1 for q gt; r and any a(i) . We use this assumption throughout (i) (j) pk ? n k , ? (1) the rest of this paper. If we de? ne m hk = i=0 hk (i) pk , ? k = i=j+1 (i) (j) m hk (i) pk , (i) (2) ?k = and ?k (0)  ·  ·  · ? k (m) , nk = nk (0)  ·  ·  · nk (m) T , (3) wk = ? k nk , then (1) can be written as yk = hk xk + wk .Now, consider the variance of wk . Using (2), (3), and (4), we have Rw k w k ? = E [wk wk ] (4) (5) (6) (7) (8) (i) (i) bk p k ? , = E ? k nk nH ? H k k = ? k E nk nH ? H k k = N0 B N m j=0 ? m where E [ ·] is the expectation operator, ( ·)? is the complex conjugate operator for a scalar, ( ·)H is the Hermitian (complex transpose) operator for a vector or matrix, and bk = hk (i) (i) 2 ? i=j+1 ? (9) , for i = 0 to m. Rwk wk is positive for a nonzero N0 . We de? ne a transformed version of the system in (5) ? yk = hk xk + wk , ? ? (10) 6 ? where yk = yk / Rwk wk , hk = hk / Rwk wk , and wk = wk / Rwk wk .The variances of wk ? ? ? and yk are ? E [wk wk ] = E ? = wk Rwk wk ? wk Rw k w k (11) (12) (13) Rw k w k Rw k w k = 1 and E [? k yk ] = E y ? hk xk + wk ? ? hk xk + wk ? ? (14) (15) ? ? = hk h? + 1 k = 1 Rw k w k m i=0 bk p k (i) (i) + 1, (16) ? ? respectively. The cross terms do not appear in (16) because hk , wk , and xk are mutually independent. Note that the transformed system has unit variance noise. 2. 1. 2 Mutual Information To derive the mutual information, note that the di? erential entropy of a circular symmetric complex Gaussian vector, v, with covariance matrix, K, is h (v) = log2 det (? eK) [2].When the circular symmetric complex Gaussian is a scalar, v, the di? erential entropy is 2 2 h (v) = log2 ? e? v , where ? v is the variance of v. Let Ik be the mutual information between the transmitter and receiver on the kth subcarrier Ik = h (? k ) ? h (wk ) y ? = log2 ? e = log2 1 Rwk wk 1 Rw k w k m i=0 m i=0 (17) bk p k bk p k (i) (i) (i) (i) +1 ? log2 (? e) (18) (19) +1 , where the ? rst equality comes from basic mutual information calculations [1]. The total mutual information betwe en the transmitter and receiver, I, is the sum of all Ik divided by N . That is, after substituting (9) into (19), we have I = 1 N NIk k=1 (20) 7 = 1 N N k=1 log2 ? 1 + SNR ? T ? ? bk (0) (i) (i) m i=1 bk pk (i) (i) m i=j+1 bk pk m j=0 , (21) where SNR = Ptot /N0 B. If we denote b(i) = for i = 0 to m and T b1 (i)  ·  ·  · bN (i) and p(i) = p1 (i)  ·  ·  · pN (i) T , (22) eN = 1  ·Ã‚ ·Ã‚ · 1 N ones , (23) then (21) can be written in matrix form. First, let m zsingle = b(0) ? ? i=1 b(i) ? p(i) where the ? and ? operators both represent element-wise matrix multiplication and the ? /? ? m j=0 ? m i=j+1 b(i) ? p(i) , (24) ?/ operator represents element-wise matrix division. Then, (21) in matrix form is I= 1 T e log2 eN + SNR zsingle , N N (25) here log2 ( ·) of a vector is the vector of the logarithms of the vector’s entries. 2. 1. 3 Relay Power Allocation We assume that the net transmit power at the transmitter and at each each relay is Ptot . At the transmit ter, we assume a uniform power distribution, that is, pk (0) = Ptot /N . To (l) derive the power constraint at each relay and thus, possible power allocations, consider vk , the kth transmit symbol of rl vk = (l) (l) pk ? ? l? 1 i=0 N hk (i) pk (l) 2 (i) l? 1 xk + j=0 ? ? l? 1 i=j+1 hk (i) (i) (j) pk ? n k ? . ? ? (26) The constraint is Ptot = k=1 N E ? vk . Thus, l? 1 j=0 (l) (0) P Ptot = pk ? k tot N k=1 l? 1 i=1 bk p k (i) (i) N0 B + N ? ? l? 1 i=j+1 (i) (i) bk pk (27) 8 or pk ? (0) bk N k=1 N (l) ? l? 1 i=1 (i) (i) bk p k Note that (28) is de? ned recursively. The power constraint for pk depends on pk , . . . , pk pk is the base case in the recursion, which follows from (28), when l = 1. (1) 1 l? 1 ? l? 1 (i) (i) b p = 1. + SNR j=0 i=j+1 k k (l) ? (28) (1) (l? 1) . (l) One power allocation at the lth relay is to set pk constant for all subcarriers. This results in moving pk in (28) out of the summation because it is no longer a function of k pk,ct = pct = (l) (l) (l) N SNRN k=1 ? We call this constant gain allocation (CT). Note that this power allocation does not require each relay to have any CSI (channel state information). The lth relay only has to multiply its entire OFDM receive symbol by a constant, pct , such that the total transmit power is (l) ?SNRb(0) k l? 1 i=1 (i) (i) bk pct l? 1 + j=0 ? ? l? 1 i=j+1 (i) (i) bk pct . (29) Ptot . We call constant gain capacity, Cct , as the mutual information in (25) resulting from this power allocation. A second power allocation is to choose pk such that every subcarrier transmits the same power at the lth relay.The transmit power on the kth subcarrier is the kth summand on the right hand side of (27). Since they are all equal to Ptot /N , we have Ptot (l) (0) P = pk,eq ? bk tot N N pk,eq = SNRbk (0) (l) (l) ? l? 1 i=1 (i) (i) bk pk,eq N0 B + N l? 1 j=0 ? ? l? 1 i=j+1 or (i) (i) bk pk,eq (30) SNR l? 1 i=1 bk pk,eq (i) (i) l? 1 + j=0 ? ? l? 1 i=j+1 We call this equal power allocation (EQ). Note that this power allocation does require each relay to have the CSI of its upstream channels. We call equal power capacity, Ceq , as the mutual information in (25) resulting from this power allocation. 2. 1. 4 Capacity Simulations k pk,eq ? (i) (i) ?. (31) We simulate Cct and Ceq assuming that all distances between any two adjacent transceiver nodes are the same. Therefore, all path loss e? ects are normalized to 0 dB. Shadowing 9 between nodes is assumed to be log-normally distributed. That is, the received power gain due to shadowing in dB is a zero-mean Gaussian with variance of 8 dB, which is typical for cellular land mobile applications [12]. We model frequency selective fading e? ects as Typical Urban (TU) channels and Hilly Terrain (HT) channels [12]. We use an OFDM bandwidth of 800 kHz divided into N = 128 equal blocks.Maintaining OFDM orthogonality, this translates into an OFDM symbol period of Ts = 160  µs. Results are shown in Figures 2 and 3. The plots exhibit the familiar monot onically increasing shape for mutual information in the case of direct transmission between a transmitter and receiver. This is expected if we look at the mutual information in (25). We can think of this con? guration as still being direct transmission where the channel is the single path relay network, characterized by zsingle . Note that zsingle also determines the power allocations in the relays.In other words, (25) is a system level representation of the mutual information. As we increase the distance between the transmitter and receiver (and thus, add more relays), more noise and channel distortion enter the system. Consequently, the mutual information decreases. Equal power allocation results in a slightly higher mutual information than that of constant gain allocation. TU channels and HT channels give very similar results. 2. 2 2. 2. 1 Decode-and-Forward System Model In decode-and-forward (DF), each relay fully recovers the information bits (with possible errors) after receiv ing an OFDM symbol.It then converts the information bits back into an OFDM symbol and then transmits it. The transmitter and all the relays transmit with the same uniform power distribution. That is, pk = pk = for k = 1 to N and for l = 1 to m. (l) (0) Let xk be the kth transmit symbol from the transmitter and xk be the kth transmit (m+1) be the symbol from the lth relay, all with with zero mean and unit variance. Let yk (0) (l) Ptot , N (32) 10 8 7 8 Cct Ceq 7 Cct Ceq Capacity (bits/s/Hz) 6 5 4 3 2 1 0 Capacity (bits/s/Hz) 0 3 6 9 12 15 18 21 24 6 5 4 3 2 1 0 0 3 6 9 12 15 18 21 24 SNR (dB) (a) m=1 8 7 8 SNR (dB) (b) m=2 Cct CeqCct Ceq 7 Capacity (bits/s/Hz) 6 5 4 3 2 1 0 Capacity (bits/s/Hz) 0 3 6 9 12 15 18 21 24 6 5 4 3 2 1 0 0 3 6 9 12 15 18 21 24 SNR (dB) (c) m=3 SNR (dB) (d) m=4 Figure 2: Capacity in a single path relay network with TU channels using AF. N = 128, m = 1, 2, 3, and 4. 11 8 7 8 Cct Ceq 7 Cct Ceq Capacity (bits/s/Hz) 6 5 4 3 2 1 0 Capacity (bits/s/Hz) 0 3 6 9 12 15 18 21 24 6 5 4 3 2 1 0 0 3 6 9 12 15 18 21 24 SNR (dB) (a) m=1 8 7 8 SNR (dB) (b) m=2 Cct Ceq Cct Ceq 7 Capacity (bits/s/Hz) 6 5 4 3 2 1 0 Capacity (bits/s/Hz) 0 3 6 9 12 15 18 21 24 6 5 4 3 2 1 0 0 3 6 9 12 15 18 21 24 SNR (dB) (c) m=3 SNR (dB) (d) m=4Figure 3: Capacity in a single path relay network with HT channels using AF. N = 128, m = 1, 2, 3, and 4. 12 output 1 input output 2 output 3 Figure 4: Convolutional encoder. kth receive symbol at the receiver and yk be the kth receive symbol at the lth relay. Using (l) Figure 1, the input-ouput relation at the lth relay is yk = hk (l) (l? 1) Ptot (l? 1) (l? 1) + nk . x N k (33) The input-output relation at the receiver is yk (m+1) = hk (m) Ptot (m) (m) + nk . x N k (34) 2. 3 BER and WER Simulations We simulate bit error rates (BERs) and word error rates (WERs) for both the amplify-andforward and decode-and-forward cases.At the transmitter (and at the transmitter structure of a relay using decode-and-forward), each information word contains 83 bits. Using the convolutional encoder shown in Figure 4, the information word is encoded into a 255 bit codeword. A zero bit is padded at the end to make 256 bits. The bits are then interleaved and modulated onto N = 128 QPSK (quadrature phase shift keying) subcarriers to form one OFDM symbol. At the receiver (and at the receiver structure of a relay using decodeand-forward), the codeword is recovered (with possible errors) using a matched ? lter and 13 deinterleaving.A Viterbi decoder is used to decode the codeword. Both hard decisions and soft decisions are used. We assume that all distances between any two adjacent transceiver nodes are the same. Therefore, all path loss e? ects are normalized to 0 dB. Shadowing is assumed to be lognormally distributed. That is, the received power gain due to shadowing in dB is a zero-mean Gaussian with variance of 8 dB, which is typical for cellular land mobile applications [12]. We model frequency selective fading as Typical Urban (TU) channels and Hilly Terrain (HT) channels [12]. We use an OFDM bandwidth of 800 kHz divided into N = 128 equal blocks.Maintaining OFDM orthogonality, this translates into an OFDM symbol period of Ts = 160  µs. 2. 3. 1 Amplify-and-Forward The BER versus SNR and WER versus SNR plots for a single path relay network with TU channels using amplify-and-forward are shown in Figures 5 and 6, respectively. The corresponding plots for HT channels are shown in Figures 7 and 8, respectively. As expected, soft decisions in Viterbi decoding give better performance than hard decisions. In particular, there is up to 4 dB of SNR gain for the constant gain allocation and m = 1 case, as shown in Figures 5(a), 6(a), 7(a), and 8(a).In general, using hard decisions with constant gain allocation results in the worst performance. Soft decisions with equal power allocation gives the best performance, except for the m = 1 case, where soft decisions with constant gain allocation is slightly better. As w e increase the distance between the transmitter and receiver (and thus, add more relays), more noise and channel distortion enter the system. Consequently, the error rate (BER and WER) performance becomes worse and as a result, all four curves are very close together at low to medium SNR values. TU channels and HT channels give very similar results. . 3. 2 Decode-and-Forward The BER versus SNR and WER versus SNR plots for a single path relay network with TU channels using decode-and-forward are shown in Figures 9 and 10, respectively. The 14 10 0 10 0 10 ?1 10 ?1 BER 10 ?2 BER hard, constant gain allocation hard, equal power allocation soft, constant gain allocation soft, equal power allocation 0 3 6 9 12 15 18 21 24 10 ?2 10 ?3 10 ?3 10 ?4 10 ?4 hard, constant gain allocation hard, equal power allocation soft, constant gain allocation soft, equal power allocation 0 3 6 9 12 15 18 21 24 SNR (dB) (a) m=1 10 0 SNR (dB) (b) m=2 10 0 10 ?1 10 ?1 BER 10 ?2 BER ard, constant gain allocati on hard, equal power allocation soft, constant gain allocation soft, equal power allocation 0 3 6 9 12 15 18 21 24 10 ?2 10 ?3 10 ?3 10 ?4 10 ?4 hard, constant gain allocation hard, equal power allocation soft, constant gain allocation soft, equal power allocation 0 3 6 9 12 15 18 21 24 SNR (dB) (c) m=3 SNR (dB) (d) m=4 Figure 5: BER in a single path relay network with TU channels using AF. N = 128, m = 1, 2, 3, and 4. 15 10 0 10 0 WER 10 ?1 WER hard, constant gain allocation hard, equal power allocation soft, constant gain allocation soft, equal power allocation 0 3 6 9 12 15 18 21 24 10 1 10 ?2 10 ?2 hard, constant gain allocation hard, equal power allocation soft, constant gain allocation soft, equal power allocation 0 3 6 9 12 15 18 21 24 SNR (dB) (a) m=1 10 0 SNR (dB) (b) m=2 10 0 WER 10 ?1 WER hard, constant gain allocation hard, equal power allocation soft, constant gain allocation soft, equal power allocation 0 3 6 9 12 15 18 21 24 10 ?1 10 ?2 10 ?2 hard, constant gain alloc ation hard, equal power allocation soft, constant gain allocation soft, equal power allocation 0 3 6 9 12 15 18 21 24 SNR (dB) (c) m=3 SNR (dB) (d) m=4 Figure 6: WER in a single path relay network with TU channels using AF.N = 128, m = 1, 2, 3, and 4. 16 10 0 10 0 10 ?1 10 ?1 BER 10 ?2 BER hard, constant gain allocation hard, equal power allocation soft, constant gain allocation soft, equal power allocation 0 3 6 9 12 15 18 21 24 10 ?2 10 ?3 10 ?3 10 ?4 10 ?4 hard, constant gain allocation hard, equal power allocation soft, constant gain allocation soft, equal power allocation 0 3 6 9 12 15 18 21 24 SNR (dB) (a) m=1 10 0 SNR (dB) (b) m=2 10 0 10 ?1 10 ?1 BER 10 ?2 BER hard, constant gain allocation hard, equal power allocation soft, constant gain allocation soft, equal power allocation 0 3 6 9 12 15 18 21 24 10 ?2 10 ?3 10 ?3 10 4 10 ?4 hard, constant gain allocation hard, equal power allocation soft, constant gain allocation soft, equal power allocation 0 3 6 9 12 15 18 21 24 SNR ( dB) (c) m=3 SNR (dB) (d) m=4 Figure 7: BER in a single path relay network with HT channels using AF. N = 128, m = 1, 2, 3, and 4. 17 10 0 10 0 WER 10 ?1 WER hard, constant gain allocation hard, equal power allocation soft, constant gain allocation soft, equal power allocation 0 3 6 9 12 15 18 21 24 10 ?1 10 ?2 10 ?2 hard, constant gain allocation hard, equal power allocation soft, constant gain allocation soft, equal power allocation 0 3 6 9 12 15 18 21 24SNR (dB) (a) m=1 10 0 SNR (dB) (b) m=2 10 0 WER 10 ?1 WER hard, constant gain allocation hard, equal power allocation soft, constant gain allocation soft, equal power allocation 0 3 6 9 12 15 18 21 24 10 ?1 10 ?2 10 ?2 hard, constant gain allocation hard, equal power allocation soft, constant gain allocation soft, equal power allocation 0 3 6 9 12 15 18 21 24 SNR (dB) (c) m=3 SNR (dB) (d) m=4 Figure 8: WER in a single path relay network with HT channels using AF. N = 128, m = 1, 2, 3, and 4. 18 10 0 10 0 hard soft ?1 ? 1 hard soft 10 10 BER 10 ?2 BER 0 3 6 9 12 15 18 21 24 10 ?2 10 ?3 0 ?3 10 ?4 10 ?4 0 3 6 9 12 15 18 21 24 SNR (dB) (a) m=1 10 0 SNR (dB) (b) m=2 10 0 hard soft ?1 ? 1 hard soft 10 10 BER 10 ?2 BER 0 3 6 9 12 15 18 21 24 10 ?2 10 ?3 10 ?3 10 ?4 10 ?4 0 3 6 9 12 15 18 21 24 SNR (dB) (c) m=3 SNR (dB) (d) m=4 Figure 9: BER in a single path relay network with TU channels using DF. N = 128, m = 1, 2, 3, and 4. corresponding plots for HT channels are shown in Figures 11 and 12, respectively. As expected, soft decisions in Viterbi decoding give better performance than hard decisions. In particular, there is up to 5 dB of SNR gain, as shown in the plots.As we increase the distance between the transmitter and receiver (and thus, add more relays), more noise and channel distortion enter the system. However, the error rate (BER and WER) performance su? ers only slightly as m increases. TU channels and HT channels give very similar results. 2. 3. 3 Comparison The BER versus SNR and WER versus SNR plots for a single path relay network with TU channels using amplify-and-forward and decode-and-forward are shown in Figures 13 and 19 10 0 10 0 hard soft hard soft WER 10 ?1 WER 0 3 6 9 12 15 18 21 24 10 ?1 10 ?2 10 ?2