Achievable Rates and Optimal Resource Allocation for Imperfectly-Known Fading Relay Channels

1 Achie v able Rates a nd Optimal Re source Alloca tion for Imper fectly-Kno wn F ading Rela y Channels Junwei Zhang, Mustafa Cenk Gurso y Department of Electrical Engineering Uni v ersity of Nebraska-Lincoln, Lincoln, NE 68588 Email: jzhang13@bigred.unl.edu, gursoy@engr .unl.edu Abstract 1 In this paper, achiev able rates an d optimal r esource allocation strategies fo r imperfectly-known fading relay channels are studied. It is assumed that com munication starts with the network training phase in which the receiv ers estimate the fading coefficients of their respective channels. In the data transmission phase, am plify-and -forward and decode- and-fo rward relaying schem es with different degree s of coope ration are co nsidered, and the correspond ing achiev able rate expressions are ob tained. Three resource allocation problems are addressed: 1) power allocation between data and training symbols; 2) time/bandwid th alloc ation to the relay; 3) power allocation between the source and relay in the presence of total power co nstraints. T he achiev able r ate expressions are em ployed to identify the o ptimal resource alloca tion strategies. Finally , energy efficiency is in vestigated by stu dying the bit energy requ irements in th e low- SNR regime. Index T erms : Relay channel, co operative transmission, ch annel estimatio n, imp erfectly-k nown fading channels, achiev able rates, o ptimal reso urce allocation, energy efficiency in the lo w-power regime. I . I N T R O D U C T I O N In wireless communi cations, deterioration in performance is experienced due t o va rious impediments such as i nterference, fluctuations in power due to reflections and attenuati on, and randomly-varying channel conditions caused b y m obility and changi ng en vironment. Recently , cooperative wireless commu nications has attracted much interest as a techniqu e that can mitigate these degrada tions and provide higher rates or improve the reliability through diversity gains. The relay channel was first i ntroduced by van d er Meulen in [1], and init ial research was primarily conducted to understand the rates achieved in relay channel s [2], 1 This work was supported in part by the NSF CAREER Grant CCF- 05463 84. The material in this paper was presented in part at the 45th Annual Al lerton Conference on Communication, Control and Computing in S ept. 2007. August 23, 2021 DRAFT [3]. More recently , di versity gains of coop erati ve transmission techni ques have been studied in [4]–[7]. In [6], sev eral cooperati ve protocols ha ve been proposed, with amplify-and-forward (AF) and decode-and- forward (DF) being the t wo basi c relayi ng schemes. The performance of these prot ocols are characterized i n terms of outage e vents and outage probabilities. In [8], t hree dif ferent time-division AF and DF cooperativ e protocols with different t he d egrees of broadcasting and receiv e colli sion are studied. In general, the area has seen an explosive growth in t he number of studies (s ee e.g., [9], [10], [11], [12], [13], and references therein). An excellent revie w of cooperative strategies from both rate and diversity improvement perspecti ves is provided in [14] in which the i mpacts of c ooperative s chemes on device architecture and higher -layer wireless networking protocols are als o addressed. Recently , a special i ssue has been dedicated to models , theory , and codes for relaying and cooperation in communication netw orks in [15]. As noted above, st udies on relaying and cooperation are numerous. H o wev er , most work has assumed that the channel conditi ons are perfectly known at the receiver and/o r transmitter s ides. Especially in m obile applications, thi s assump tion is unw arranted as randomly-varying channel conditio ns can be learned by the recei vers on ly imperfectly . Moreover , the performance analysis o f cooperative schemes in such scenarios is especially interesti ng and called for because relaying introduces additional channels and hence i ncreases uncertainty i n the m odel if the chann els are k nown only imperfectly . Recently , W ang et al. in [16] consi dered pilot-assist ed transmi ssion over wireless s ensory relay networks, and analyzed scali ng laws achieved by the amplify-and-forward scheme in the asym ptotic regimes of large nodes, large block length, and sm all SNR values. In this study , the channel conditions are being learned only by the relay nodes. In [17], A v estimehr and Tse studied the outage capacity of slo w f ading relay c hannels. They showed that Bursty Amplify-Forwa rd strategy achi e ves the ou tage capacity in the lo w SNR and low outage probabili ty re gime. Interestingly , they further proved that the optimali ty of Bursty AF is preserved ev en if the receiver s do not h a ve prior knowledge of the channels. In this paper , we study t he achiev able rates of imperfectly-known fading relay channels. W e assu me that transmissio n takes place in two phases: network training phase and data transm ission phase. In the network training phase, a pri ori unknown fading coef ficients are estimated at the rece iv ers with t he ass istance of pilot symbols. Following the training phase, AF and DF relaying techniques with di f ferent degrees of cooperation are employed in the d ata transm ission. W e first obtain achiev able rate e xpressions for dif ferent relaying protocols and subsequently identify optimal resource allocation st rategies that maximi ze the rates. W e consider three types of resource allocation problems: 1) po wer allocation between data and t raining 2 symbols; 2) ti me/bandwidth all ocation t o the relay; 3) po wer allocation between the so urce and relay if there is a total power constraint in the system. Finally , we inv estigate the ener gy efficienc y by findin g the bit ener gy requirements in the lo w- SNR re gime. The organization of t he rest of the paper is as follo ws. In Section II, we describe the channel mod el. Network training and data transm ission phases are explained in Section III. W e obtain the achie vable rate expressions in Section IV and study t he optimal resource allocation strategies in Section V . W e discuss the ener gy ef ficiency in the lo w- SNR regime in Sec tion VI. Finally , we provide conclu sions i n Section VII. I I . C H A N N E L M O D E L W e consider the th ree-node relay network wh ich consist s of a source, destination, and a relay node. ✇ ✇ ✇ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✯ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❥ ✲ Source Relay Destination h sr h r d h sd y d y r d x s y r x r Source-destination, sou rce-relay , and relay-destination channels are modeled as Rayleig h block-fading chan- nels with fading coefficients denot ed by h sr , h sd , and h r d , respectively for eac h channel. Du e to the block- fading assumpti on, the fading coefficients h sr ∼ C N (0 , σ sr 2 ) , h sd ∼ C N (0 , σ sd 2 ) , and h r d ∼ C N (0 , σ r d 2 ) 2 stay constant for a block of m sy mbols before they assume independent realizatio ns for the following block. In thi s sy stem, the source no de tries to send informati on to the destinati on n ode with the help of the intermediate relay node. It is assumed that th e source, relay , and destinati on nodes do not have prior knowledge of the re alizations of the f ading coef ficients. The transmissi on is conducted in two phases: netw ork training phase in which the fading coeffi cients are estim ated at the receiv ers, and data transm ission phase. Overall, the source and relay are subject to t he follo wing power const raints in one block: k x s,t k 2 + E {k x s k 2 } ≤ mP s , (1) k x r,t k 2 + E {k x r k 2 } ≤ mP r . (2) where x s,t and x r,t are the s ource and relay training signal vectors, respecti vely , and x s and x r are the corresponding source and relay data v ectors. 2 x ∼ C N ( d, σ 2 ) i s used to denote a proper complex Gaussian random variable with mean d and v ariance σ 2 . 3 I I I . N E T W O R K T R A I N I N G A N D D A TA T R A N S M I S S I O N A. Network T raining Phas e Each block t ransmission starts with the training phase. In the first symbol p eriod, source transm its a pilot symbol to enable the relay and destinatio n to estimate the channel coeffic ients h sr and h sd , respectively . In the average power l imited case, sending a single pilot i s opt imal because instead of increasing the number of pilot symbo ls, a si ngle pil ot wit h higher p owe r can be used. Th e sign als receiv ed by the relay and destinati on are y r,t = h sr x s,t + n r , and y d,t = h sd x s,t + n d , (3) respectiv ely . Similarly , i n the second symbol period, relay transmits a pilo t sym bol to enable the destination to estimate the channel coef ficient h r d . The sign al received by the destination i s y r d,t = h r d x r,t + n r d . (4) In the above formulations, n r ∼ C N (0 , N 0 ) , n d ∼ C N (0 , N 0 ) , and n r d ∼ C N (0 , N 0 ) represent independ ent Gaussian noise samp les at the relay and t he destination nodes. In the traini ng process, it is assumed that the recei vers em ploy mini mum mean-square- error (MMSE) estimation. W e assume that the source allocates δ s of its total power for t raining while the relay allocates δ r of its t otal po wer for training. As described in [25], the MMSE estimate of h sr is gi ven by ˆ h sr = σ 2 sr √ δ s mP s σ 2 sr δ s mP s + N 0 y r,t , (5) where y r,t ∼ C N (0 , σ 2 sr δ s mP s + N 0 ) . W e denote by ˜ h sr the estimate error which is a zero-mean complex Gaussian random va riable wit h variance v ar ( ˜ h sr ) = σ 2 sr N 0 σ 2 sr δ s mP s + N 0 . Similarly , for th e fading coef ficients h sd and h r d , we h a ve ˆ h sd = σ 2 sd √ δ s mP s σ 2 sd δ s mP s + N 0 y d,t , y d,t ∼ C N (0 , σ 2 sd δ s mP s + N 0 ) , v ar ( ˜ h sd ) = σ 2 sd N 0 σ 2 sd δ s mP s + N 0 , (6) ˆ h r d = σ 2 r d √ δ r mP r σ 2 r d δ r mP r + N 0 y r d,t , y r d,t ∼ C N (0 , σ 2 r d δ r mP r + N 0 ) , v ar ( ˜ h r d ) = σ 2 r d N 0 σ 2 r d δ r mP r + N 0 . (7) W it h t hese estimates, the fading coef ficients can now be expressed as h sr = ˆ h sr + ˜ h sr , h sd = ˆ h sd + ˜ h sd , h r d = ˆ h r d + ˜ h r d . (8) 4 B. Data T ransmission Pha se The practical relay node us ually cannot transm it and recei ve data simultaneous ly . Thus, we assume that the relay works un der half-duplex const raint. Hence, the relay first listens and then t ransmits. As discussed in the previous section , wi thin a block of m symb ols, th e first two symbol s are allocated to network training. In the remaining duration of m − 2 symbo ls, data transm ission t akes place. W e introduce the relay t ransmission parameter α and assume th at α ( m − 2) sy mbols are allocated for relay transmiss ion. Hence, α can be seen as the fraction of total t ime or bandwid th all ocated to the relay . Note that the p arameter α enables us to cont rol the degree of cooperation. W e cons ider se veral transmissi on protocols which can be cl assified into t wo categories by whether or not the source and relay simu ltaneously transmit s inform ation: non-overlapped and overlapped transmission. Note that in both cases, the relay transmits over a duration of α ( m − 2 ) s ymbols. In non-ove rlapped transm ission, source transm its ov er a duration of (1 − α )( m − 2) symbol s and becomes silent as the relay transmits . O n the other hand, in overlapped transmis sion, s ource transmits all the time and sends m − 2 symbols in each block. W e assume that the d ata vectors x s and x r are composed of i ndependent random variables wit h equal ener gy . Hence, t he covariance matrices of x s are gi ven by E { x s x † s } = P ′ s 1 I = (1 − δ s ) mP s ( m − 2)(1 − α ) I , and E { x s x † s } = P ′ s 2 I = (1 − δ s ) mP s ( m − 2) I , (9) in non-ov erlapped and overlapped transmissi ons, respectiv ely . The cov ariance m atrix for x r is E { x r x † r } = P ′ r I = (1 − δ r ) mP r ( m − 2) α I . (10) 1) Non-overlapped transmissio n: W e first consider the two simplest cooperati ve protocols: non-ov erlapped AF , and non-overlapped DF with repetition coding wh ere the relay decodes the message and re-enc odes it using the same cod ebook as the source. In these protocols, since the relay either amplifies the receiv ed signal, or decodes it but uses the same codebo ok as the source when forwarding, source and relay should be allocated equal time slots i n the cooperation phase. Therefore, we in itially ha ve direct transmi ssion from the source to the destination withou t any aid from the relay over a duration of (1 − 2 α )( m − 2) symbols. In this phase, source sends x s 1 and the receive d signal at the destination is given by y d 1 = h sd x s 1 + n d 1 . (11) 5 Subsequently , cooperative t ransmission starts. At first, the so urce t ransmits an α ( m − 2) -dimensio nal sym bol vector x s 2 which is rec eiv ed at the the relay and the destination , respective ly , as y r = h sr x s 2 + n r , and y d 2 = h sd x s 2 + n d 2 . (12) For compact representation , we denote the signal received at the destination directly from t he source by y d = [ y T d 1 y T d 2 ] T where T d enotes the transpose operation. Next, the source becomes silent, and the relay transmits an α ( m − 2) -dimensional sym bol vector x r which is generated from th e pre viously receiv ed y r [6] [7]. This approach corresponds to proto col 2 in [8], which realizes the maxi mum degrees o f broadcasti ng and exhibits no recei ve collis ion. The destination recei ves y r d = h r d x r + n r d . (13) After substituting t he e xpressions in (8) into (11)–(13), we have y d 1 = ˆ h sd x s 1 + ˜ h sd x s 1 + n d 1 , y r = ˆ h sr x s 2 + ˜ h sr x s 2 + n r , y d 2 = ˆ h sd x s 2 + ˜ h sd x s 2 + n d 2 , (14) y r d = ˆ h r d x r + ˜ h r d x r + n r d . (15) W e define the so urce data v ector as x s = [ x T s 1 x T s 2 ] T . Note that we have 0 < α ≤ 1 / 2 for AF and repetition coding DF . Therefore, α = 1 / 2 model s full cooperation while we hav e noncooperati ve comm unications as α → 0 . It should also be noted that α sh ould in general be chosen such that α ( m − 2) is an integer . For non-overlapped transmission, we also consi der DF with parallel channel codin g, in which the relay uses a differe nt codebook to encode the mess age. In this case, the source and relay do not hav e to be allocated the same durati on in the cooperation ph ase. Therefore, source transmits over a duration of (1 − α )( m − 2) symbols while the relay transm its in the remainin g duration of α ( m − 2) symbol s. Clearly , the range of α is now 0 < α < 1 . In t his case, the input-outp ut relati ons are gi ven by (12) and (13). Since there i s no separate direct transmissi on, x s 2 = x s and y d 2 = y d in (12). Moreover , the dimensions of the v ectors x s , y d , y r are now (1 − α )( m − 2) , while x r and y r d are v ectors of dim ension α ( m − 2) . 2) Overl apped transmission: In this category , we consi der a mo re general and compli cated scenario in which the source transm its all the time. In AF and repeti tion DF , similarly as in the non-overlapped model, cooperativ e transmission takes place in the duratio n of 2 α ( m − 2) symbol s. The remaining durati on of (1 − 2 α )( m − 2) sy mbols is a llocated to unaided di rect transmiss ion from the source to t he destinati on. 6 Again, we ha ve 0 < α ≤ 1 / 2 in this setting. In these protocols, the input-output relations are expressed as follows: y d 1 = h sd x s 1 + n d 1 , y r = h sr x s 21 + n r , y d 2 = h sd x s 21 + n d 2 , and y r d = h sd x s 22 + h r d x r + n r d . (16) Above, x s 1 , x s 21 , x s 22 , which hav e respective di mensions (1 − 2 α )( m − 2) , α ( m − 2) and α ( m − 2) , represent the sou rce data v ectors sent in direct transmission, cooperativ e transmis sion when relay is lis tening, and cooperativ e transmiss ion when relay is transmitting , respectively . Note again t hat the sou rce t ransmits all the time. x r is the relay’ s data vector with dim ension α ( m − 2) . y d 1 , y d 2 , y r d are the corresponding receive d vectors at the destination , and y r is the receive d vector at t he relay . The input vector x s now is defined as x s = [ x T s 1 , x T s 21 , x T s 22 ] T and we again denote y d = [ y T d 1 y T d 2 ] T . If we express the f ading coefficients as h = ˆ h + ˜ h in (16), we obtain the following in put-output relations: y d 1 = ˆ h sd x s 1 + ˜ h sd x s 1 + n d 1 , y r = ˆ h sr x s 21 + ˜ h sr x s 21 + n r , y d 2 = ˆ h sd x s 21 + ˜ h sd x s 21 + n d 2 , and (17) y r d = ˆ h sd x s 22 + ˆ h r d x r + ˜ h sd x s 22 + ˜ h r d x r + n r d . (18) I V . A C H I E V A B L E R A T E S In this section, we provide achi e va ble rate expressions for AF and DF relayin g in bot h n on-over lapped and overlapped t ransmission scenarios described in Sec tion III. Achiev able rate expressions are obtained by considering the estimate errors as additio nal sources of Gaussian nois e. Since Gaussian nois e is the worst uncorrelated add itiv e noise for a Gaussian model [20], [21], achie va ble rates given in th is section can be regarded as worst-case rates. W e first consider AF relaying scheme. The ca pacity o f t he AF relay channel is the maximum mutual information between the transmitted signal x s and recei ved signals y d and y r d giv en the estimates ˆ h sr , ˆ h sd , ˆ h r d : C = sup p x s ( · ) 1 m I ( x s ; y d , y r d | ˆ h sr , ˆ h sd , ˆ h r d ) . (19) Note that this formul ation presupposes that the desti nation has t he knowledge of ˆ h sr . Hence, we assume that the value of ˆ h sr is forwarded reliably from the relay to the desti nation over l o w-rate cont rol links. In general, solving the optim ization prob lem i n (19) and obt aining the channel capacity is a difficult t ask. Therefore, we concentrate on finding a lower bound on the capacity . A lower bound is obtain ed by replacing the product of t he estimate error and the transmit ted signal in th e inp ut-output relations with the worst-case noise wit h 7 the same correlation. In non-over lapped transmission, we consider z d 1 = ˜ h sd x s 1 + n d 1 , z r = ˜ h sr x s 2 + n r , z d 2 = ˜ h sd x s 2 + n d 2 , and z r d = ˜ h r d x r + n r d , (20) as the new noise vectors whose covariance matrices, respecti vely , are E { z d 1 z † d 1 } = σ 2 z d 1 I = σ 2 ˜ h sd E { x s 1 x † s 1 } + N 0 I , E { z r z † r } = σ 2 z r I = σ 2 ˜ h sr E { x s 2 x † s 2 } + N 0 I , (21) E { z d 2 z † d 2 } = σ 2 z d 2 I = σ 2 ˜ h sd E { x s 2 x † s 2 } + N 0 I , E { z r d z r d † } = σ 2 z r d I = σ 2 ˜ h r d E { x r x † r } + N 0 I . (22) Similarly , i n ov erlapped transmis sion, we define z d 1 = ˜ h sd x s 1 + n d 1 , z r = ˜ h sr x s 21 + n r , z d 2 = ˜ h sd x s 21 + n d 2 , z r d = ˜ h sd x s 22 + ˜ h r d x r + n r d , (23) as noise vectors with covariance matrices E { z d 1 z † d 1 } = σ 2 z d 1 I = σ 2 ˜ h sd E { x s 1 x † s 1 } + N 0 I , E { z r z † r } = σ 2 z r I = σ 2 ˜ h sr E { x s 21 x † s 1 } + N 0 I , (24) E { z d 2 z † d 2 } = σ 2 z d 2 I = σ 2 ˜ h sd E { x s 21 x † s 21 } + N 0 I , E { z r d z r d † } = σ 2 z r d I = σ 2 ˜ h sd E { x s 22 x † s 22 } + σ 2 ˜ h r d E { x r x † r } + N 0 I . (25) An achiev able rate e xpression is obtained by solving the follo wing optimi zation problem wh ich requires finding the worst-case noise: C > I AF = inf p z d 1 ( · ) ,p z r ( · ) ,p z d 2 ( · ) ,p z r d ( · ) sup p x s ( · ) 1 m I ( x s ; y d , y r d | ˆ h sr , ˆ h sd , ˆ h r d ) . (26) The following results provide I AF for both no n-ove rlapped and overlapped transmissio n scenarios. Theor em 1: An achie vable rate of AF i n n on-over lapped transm ission scheme is given by I AF = 1 m E " (1 − 2 α )( m − 2 ) log 1 + P ′ s 1 | ˆ h sd | 2 σ 2 z d 1 ! + α ( m − 2) log 1 + P ′ s 1 | ˆ h sd | 2 σ 2 z d 2 + f " P ′ s 1 | ˆ h sr | 2 σ 2 z r , P ′ r | ˆ h r d | 2 σ 2 z r d #!# (27) where f ( x, y ) = xy 1 + x + y (28) P ′ s 1 | ˆ h sd | 2 σ 2 z d 1 = P ′ s 1 | ˆ h sd | 2 σ 2 z d 2 = δ s (1 − δ s ) m 2 P 2 s σ 4 sd / (1 − α ) (1 − δ s ) mP s σ 2 sd N 0 / (1 − α ) + ( m − 2)( σ 2 sd δ s mP s + N 0 ) N 0 | w sd | 2 (29) 8 P ′ s 1 | ˆ h sr | 2 σ 2 z r = δ s (1 − δ s ) m 2 P 2 s σ 4 sr / (1 − α ) (1 − δ s ) mP s σ 2 sr N 0 / (1 − α ) + ( m − 2)( σ 2 sr δ s mP s + N 0 ) N 0 | w sr | 2 (30) P ′ r | ˆ h r d | 2 σ 2 z r d = δ r (1 − δ r ) m 2 P 2 r σ 4 r d /α (1 − δ r ) mP r σ 2 r d N 0 /α + ( m − 2 )( σ 2 r d δ r mP r + N 0 ) N 0 | w r d | 2 . (31) In the abo ve equ ations and h enceforth, w sr ∼ C N (0 , 1) , w sd ∼ C N (0 , 1) , w r d ∼ C N (0 , 1) denote indepen- dent, standard Gaussian random va riables. Pr oof : Note that in non-overlapped AF relaying, I ( x s ; y d , y r d | ˆ h sr , ˆ h sd , ˆ h r d ) = I ( x s 1 ; y d 1 | ˆ h sd ) + I ( x s 2 ; y d 2 , y r d | ˆ h sr , ˆ h sd , ˆ h r d ) (32) where the first m utual expression on the righ t-hand side of (32) is for the direct transmission and the second is for the cooperative transmissio n. In the direct transmiss ion, we hav e y d 1 = ˆ h sd x s 1 + z d 1 . (33) In this setting, it is wel l-known t hat the worst-case noise z d 1 is Gaussian [20] and x s 1 with independent Gaussian components achieves inf p z d 1 ( · ) sup p x s 1 ( · ) I ( x s 1 ; y d 1 | ˆ h sd ) = E " (1 − 2 α )( m − 2 ) log 1 + P ′ s 1 | ˆ h sd | 2 σ 2 z d 1 !# . (34) Therefore, we now concentrate on the cooperative phase. For better illustration, we rewrite the channel input-output relationships in (14) and (15) for each symbol: y r [ i ] = ˆ h sr x s 2 [ i ] + z r [ i ] , y d 2 [ i ] = ˆ h sd x s 2 [ i ] + z d 2 [ i ] , (35) for i = 1 + (1 − 2 α )( m − 2) , ..., (1 − α )( m − 2) , and y r d [ i ] = ˆ h r d x r [ i ] + z r d [ i ] , (36) for i = (1 − α )( m − 2) + 1 , ..., m − 2 . In AF , the si gnals recei ved and transmitted b y the relay have following relation: x r [ i ] = β y r [ i − α ( m − 2 )] , where β 6 s E [ | x r | 2 ] | ˆ h sr | 2 E [ | x s 2 | 2 ] + E [ | z r | 2 ] . (37) 9 Now , we can write the channel in the vector form   y d 2 [ i ] y r d [ i + α ( m − 2)]   | {z } ˇ y d [ i ] =   ˆ h sd ˆ h r d β ˆ h sr   | {z } A x s [ i ] +   0 1 0 ˆ h r d β 0 1   | {z } B      z r [ i ] z d 2 [ i ] z r d [ i + α ( m − 2)]      | {z } z [ i ] (38) for i = 1 + (1 − 2 α )( m − 2 ) , ..., ( 1 − α )( m − 2) , W ith the above notatio n, we can write the inp ut-output mutual information as I ( x s 2 ; y d 2 , y r d | ˆ h sr , ˆ h sd , ˆ h r d ) = (1 − α )( m − 2) X i =1+(1 − 2 α )( m − 2) I ( x s [ i ]; ˇ y d [ i ] | ˆ h sr , ˆ h sd , ˆ h r d ) = α ( m − 2 ) I ( x s ; ˇ y d | ˆ h sr , ˆ h sd , ˆ h r d ) (39) where in (39) we removed the dependence on i wit hout loss of generality . Note that ˇ y i s defined in (38). Now , we can calculate the worst-case capacity by proving that Gaussian distribution for z r , z d 2 , and z r d provides t he worst case. W e employ techniques s imilar to that in [20]. Any set of particul ar distributions for z r , z d 2 , and z r d yields an upper bound on the worst case. Let us choose z r , z d 2 , and z r d to be zero mean complex Gaussian distributed. Then as in [6], inf p z r ( · ) ,p z d 2 ( · ) ,p z r d ( · ) sup p x s 2 ( · ) I ( x s ; ˇ y d | ˆ h sr , ˆ h sd , ˆ h r d ) ≤ E log det  I + ( E ( | x s | 2 ) AA † )( B E [ zz † ] B † ) − 1  (40) where the expectation is with respect t o the fading esti mates. T o obtain a lo wer boun d, we comp ute t he mutual informatio n for the channel in (38) assuming th at x s is a zero-mean com plex Gaus sian wi th variance E ( | x s | 2 ) , b ut the di stributions of noise com ponents z r , z d 2 , and z r d are arbitrary . In this case, we have I ( x s ; ˇ y d ; | ˆ h sr , ˆ h sd , ˆ h r d ) = h ( x s | ˆ h sr , ˆ h sd , ˆ h r d ) − h ( x s | ˇ y d , ˆ h sr , ˆ h sd , ˆ h r d ) > log π eE ( | x s | 2 ) − log π e v ar ( x s | ˇ y d , ˆ h sr , ˆ h sd , ˆ h r d ) (41) where t he inequalit y is due t o the fact that Gaussi an dist ribution provides the lar gest entropy and hence h ( x s | ˇ y d , ˆ h sr , ˆ h sd , ˆ h r d ) ≤ log π e v ar ( x s | ˇ y d , ˆ h sr , ˆ h sd , ˆ h r d ) . From [20], we know that v ar ( x s | ˇ y d , ˆ h sr , ˆ h sd , ˆ h r d ) 6 E h ( x s − ˆ x s )( x s − ˆ x s ) † | ˆ h sr , ˆ h sd , ˆ h r d i (42) for any estimate ˆ x s giv en ˇ y d , ˆ h sr , ˆ h sd , and ˆ h r d . If we substitut e the LM MSE est imate ˆ x s = R xy R − 1 y ˇ y d into 10 (41) and (42 ), we o btain 3 I ( x s ; ˇ y d | ˆ h sr , ˆ h sd , ˆ h r d ) ≥ E log det  I + ( E [ | x s | 2 ] AA † )( B E [ zz † ] B † ) − 1  . (43) Since the l owe r bound (43) applies for an y no ise d istribution, we can easily see that inf p z r ( · ) ,p z d 2 ( · ) ,p z r d ( · ) sup p x s 2 ( · ) I ( x s ; ˇ y d | ˆ h sr , ˆ h sd , ˆ h r d ) > E log det  I + ( E [ | x s | 2 ] AA † )( B E [ zz † ] B † ) − 1  . (44) From (40) and (44), we conclude that inf p z r ( · ) ,p z d 2 ( · ) ,p z r d ( · ) sup p x s 2 ( · ) I ( x s ; ˇ y d | ˆ h sr , ˆ h sd , ˆ h r d ) = E log det  I + ( E [ | x s | 2 ] AA † )( B E [ zz † ] B † ) − 1  (45) = E " log 1 + P ′ s 1 | ˆ h sd | 2 σ 2 z d 2 + f " P ′ s 1 | ˆ h sr | 2 σ 2 z r , P ′ r | ˆ h r d | 2 σ 2 z r d #!# . (46) In (46), P s 1 ′ and P ′ r are the p o wers of source and relay sy mbols and are g iv en in (9) and (10). M oreover , σ 2 z d 2 , σ 2 z r , σ 2 z r d are the va riances o f the noi se components d efined in (20). Now , combining (26), (32), (34), and (46), w e obtain the achiev able rate expression in (27). Note that (29)–(31) are obtained by u sing the expressions for the channel estimates in (5)–(7) and no ise v ariances in (21) and (22).  Theor em 2: An achie vable rate of AF i n overlapped transm ission scheme is giv en by I AF = 1 m E " (1 − 2 α )( m − 2 ) log(1 + P ′ s 2 | ˆ h sd | 2 σ 2 z d 1 )+( m − 2) α log 1 + P ′ s 2 | ˆ h sd | 2 σ 2 z d 2 + f  P ′ s 2 | ˆ h sr | 2 σ 2 z r , P ′ r | ˆ h r d | 2 σ 2 z r d  + q  P ′ s 2 | ˆ h sd | 2 σ 2 z d 2 , P ′ s 2 | ˆ h sd | 2 σ 2 z r d , P ′ s 2 | ˆ h sr | 2 σ 2 z r , P ′ r | ˆ h r d | 2 σ 2 z r d  !# (47) where q ( . ) is d efined as q ( a, b, c, d ) = (1+ a ) b (1 + c ) 1+ c + d . Moreov er P ′ s 2 | ˆ h sd | 2 σ 2 z d 1 = P ′ s 2 | ˆ h sd | 2 σ 2 z d 2 = δ s (1 − δ s ) m 2 P 2 s σ 4 sd (1 − δ s ) mP s σ 2 sd N 0 + ( m − 2)( σ 2 sd δ s mP s + N 0 ) N 0 | w sd | 2 (48) P ′ s 2 | ˆ h sr | 2 σ 2 z r = δ s (1 − δ s ) m 2 P 2 s σ 4 sr (1 − δ s ) mP s σ 2 sr N 0 + ( m − 2)( σ 2 sr δ s mP s + N 0 ) N 0 | w sr | 2 (49) P ′ s 2 | ˆ h sd | 2 σ 2 z r d = δ s (1 − δ s ) m 2 P 2 s σ 4 sd ( σ 2 r d δ r mP r + N 0 ) | w sd | 2 ( m − 2)( σ 2 sd δ s mP s + N 0 )( σ 2 r d δ r mP r + N 0 ) N 0 + (1 − δ r ) mP r σ 2 r d N 0 ( σ 2 sd δ s mP s + N 0 ) /α + (1 − δ s ) mP s σ 2 sd N 0 ( σ 2 r d δ r mP r + N 0 ) (50) 3 Here, we use the property that det( I + AB ) = det( I + BA ) . 11 P ′ r | ˆ h r d | 2 σ 2 z r d = δ r (1 − δ r ) m 2 P 2 r σ 4 r d ( σ 2 sd δ s mP s + N 0 ) /α | w r d | 2 ( m − 2)( σ 2 sd δ s mP s + N 0 )( σ 2 r d δ r mP r + N 0 ) N 0 + (1 − δ r ) mP r σ 2 r d N 0 ( σ 2 sd δ s mP s + N 0 ) /α + (1 − δ s ) mP s σ 2 sd N 0 ( σ 2 r d δ r mP r + N 0 ) (51) Pr oof : N ote that the onl y difference between t he o verlapped and non-ov erlapped transmissi ons is that source continues its transmission as the relay transmits. As a result, the power of each source symbol is now P ′ s 2 giv en i n (9). Additionally , when both the source and relay are transmittin g, the recei ved signal at the destination is y r d = ˆ h sd x s 22 + ˆ h r d x r + ˜ h sd x s 22 + ˜ h r d x r + n r d . The input -output mutual information in one block is I ( x s ; y d , y r d | ˆ h sr , ˆ h sd , ˆ h r d ) = I ( x s 1 ; y d 1 | ˆ h sd ) + I ( x s 21 , x s 22 ; y d 2 , y r d | ˆ h sr , ˆ h sd , ˆ h r d ) . (52) The first t erm on the right-hand-sid e o f (52) corresponds to the mutual informati on of the direct transmis- sion and is t he same as that in non-overlapped transmis sion. Hence, the worst-case rate expression obtained in the proo f of Theorem 1 is v alid for this case as well. In the cooperative phase, the input-output relatio n for each sym bol can be written in t he follo wing m atrix form:   y d 2 [ i ] y r d [ i + α ( m − 2)]   | {z } ˇ y d [ i ] =   ˆ h sd 0 ˆ h r d β ˆ h sr ˆ h sd   | {z } A   x s [ i ] x s [ i + α ( m − 2)]   | {z } ˇ x s [ i ] +   0 1 0 ˆ h r d β 0 1   | {z } B      z r [ i ] z d 2 [ i ] z r d [ i + α ( m − 2)]      | {z } z [ i ] (53) where i = 1 + (1 − 2 α )( m − 2) , ..., (1 − α )( m − 2) and β 6 q E [ | x r | 2 ] | ˆ h sr | 2 E [ | x s | 2 ]+ E [ | z r | 2 ] . Note that we ha ve defined x s = [ x T s 1 , x T s 21 , x T s 22 ] T , and the expression in (53) uses t he property that x 21 ( j ) = x s ( j + (1 − 2 α ) ( m − 2)) and x s 22 ( j ) = x s ( j + (1 − α )( m − 2)) for j = 1 , . . . , α ( m − 2) . The input-outpu t m utual information i n the cooperativ e phase can no w be expressed as I ( x s 21 , x s 22 ; y d 2 , y r d | ˆ h sr , ˆ h sd , ˆ h r d ) = (1 − α )( m − 2) X i =1+(1 − 2 α )( m − 2) I ( ˇ x s [ i ]; ˇ y d [ i ] | ˆ h sr , ˆ h sd , ˆ h r d ) = α ( m − 2 ) I ( ˇ x s ; ˇ y d | ˆ h sr ˆ h sd , ˆ h r d ) (54) where in (54) we removed the dependence on i without loss of generality . Note that ˇ x and ˇ y are defined in (53). As shown in proof of Theorem 1, the worst-case achiev able rate for cooperativ e transm ission is inf p z r ( · ) ,p z d 2 ( · ) ,p z r d ( · ) sup p x s 2 ( · ) I ( ˇ x s ; ˇ y d | ˆ h sr , ˆ h sd , ˆ h r d ) = E log det  I + ( E [ ˇ x s ˇ x † s ] AA † )( B E [ zz † ] B † ) − 1  . ( 55) 12 Using th e definitions i n (53) and ev aluating t he log det expression in (55), and combining th e direct transmissio n worst-case achie vable rate, we arri ve to (47). (48)–(51) are obtained by using t he expressions for the channel estimates in (5)–(7) and noise v ariances in (24) and (25).  Next, we cons ider DF relaying schem e. In DF , there are two dif ferent coding approaches [7], namely repetition coding and parallel channel coding . W e first consider repetition channel coding scheme. The following results provide achiev able rate expressions in both no n-ove rlapped and overlapped t ransmission scenarios. Theor em 3: An achiev able rate expression for DF with repetition chann el codi ng for non-overlapped transmissio n scheme is gi ven by I D F r = (1 − 2 α )( m − 2 ) m E " log 1 + P ′ s 1 | ˆ h sd | 2 σ 2 z d 1 !# + α ( m − 2) m min { I 1 , I 2 } (56) where I 1 = E  log  1 + P ′ s 1 | ˆ h sr | 2 σ 2 z r   , and I 2 = E  log  1 + P ′ s 1 | ˆ h sd | 2 σ 2 z d 2 + P ′ r | ˆ h r d | 2 σ 2 z r d   . (57) Moreover , P ′ s 1 | ˆ h sd | 2 σ 2 z d 1 , P ′ s 1 | ˆ h sd | 2 σ 2 z d 2 , P ′ s 1 | ˆ h sr | 2 σ 2 z r , and P ′ r | ˆ h r d | 2 σ 2 z r d are the sam e as defined in (29)–(31). Pr oof : For DF with repetition coding in non-overlapped transmission , an achie vable rate expression is I ( x s 1 ; y d 1 | ˆ h sd ) + min n I ( x s 2 ; y r | ˆ h sr ) , I ( x s 2 ; y d , y r d | ˆ h sd , ˆ h r d ) o . (58) Note that the first and second mut ual inform ation expressions in (58) are for the di rect transmi ssion between the source and destination, and direct transmission b etween the source and relay , respectively . Therefore, as in the proof o f Theorem 1, the w orst-case ac hiev able rates can be immedi ately seen to be equal to the first term on t he right-hand side of (56) and I 1 , respecti vely . In repetition coding , after successfully decodi ng the source in formation, the relay transmits the same code word as the source. As a result, the input-output relation in the cooperati ve phase can be expressed as   y d [ i ] y r d [ i + α ( m − 2)]   | {z } ˇ y d [ i ] =   ˆ h sd ˆ h r d β   | {z } A x s [ i ] +   z d 2 [ i ] z r d [ i + α ( m − 2)]   | {z } z [ i ] . (59) where β ≤ q E [ | x r | 2 ] E [ | x s | 2 ] . From (59), it i s clear that the kno wledge of ˆ h sr is not required at the destination. W e can easily see that (59 ) is a simpler e xpression than (38) in the AF case, therefore we can adopt the same 13 methods as emp loyed in the proof of Th eorem 1 to show that Gaussian noise i s the worst nois e and I 2 is the worst-case rate.  Theor em 4: An achie vable rate expression for DF with repetition channel coding for overlapped trans- mission scheme is given by I D F r = (1 − 2 α )( m − 2 ) m E " log 1 + P ′ s 2 | ˆ h sd | 2 σ 2 z d 1 !# + ( m − 2) α m min { I 1 , I 2 } (60) where I 1 = E " log 1 + P ′ s 2 | ˆ h sr | 2 σ 2 z r !# , I 2 = E " log 1 + P ′ s 2 | ˆ h sd | 2 σ 2 z d 2 + P ′ r | ˆ h r d | 2 σ 2 z r d + P ′ s 2 | ˆ h sd | 2 σ 2 z r d + P ′ s 2 | ˆ h sd | 2 σ 2 z d 2 P ′ s 2 | ˆ h sd | 2 σ 2 z r d ! # . (61) P ′ s 2 | ˆ h sd | 2 σ 2 z d 1 , P ′ s 2 | ˆ h sd | 2 σ 2 z d 2 , P ′ s 2 | ˆ h sr | 2 σ 2 z r , P ′ s 2 | ˆ h sd | 2 σ 2 z r d , P ′ r | ˆ h r d | 2 σ 2 z r d hav e the same e xpressions as i n (48)–(51). Pr oof : Note that in overlapped transm ission, source transm its over the entire duration of ( m − 2) sym bols, and hence t he channel input -output relati on i n t he cooperati ve phase is e xpressed as follows:   y d [ i ] y r d [ i + α ( m − 2)]   | {z } ˇ y d [ i ] =   ˆ h sd 0 ˆ h r d β ˆ h sd   | {z } A   x s [ i ] x s [ i + α ( m − 2)]   | {z } ˇ x s [ i ] +   z d 22 [ i ] z r d 2 [ i + α ( m − 2)]   | {z } z [ i ] . (62) The result i s i mmediately obtained usin g t he same techniques as in t he proof o f Theorem 2.  Finally , we consider DF with parallel channel coding and assume that non-overlapped transmission scheme is adopted. From [11], we note that an achie vable rate expression is min { (1 − α ) I ( x s ; y r | ˆ h sr ) , (1 − α ) I ( x s ; y d | ˆ h sd ) + α I ( x r ; y r d | ˆ h r d ) } . Note that w e do not have s eparate direct transm ission in this relaying scheme. Using si milar metho ds as before, we obt ain t he follo wing resul t. The proof is omitted t o avoid repetition. Theor em 5: An achie vable rate of non-o verlapped DF with parallel channel coding scheme is given by I D F p = min ( (1 − α ) ( m − 2) m E " log 1 + P ′ s 1 | ˆ h sr | 2 σ 2 z r !# , (1 − α ) ( m − 2) m E " log 1 + P ′ s 1 | ˆ h sd | 2 σ 2 z d 2 !# + α ( m − 2) m E " log 1 + P ′ r | ˆ h r d | 2 σ 2 z r d !# ) (63) 14 where P ′ s 1 | ˆ h sd | 2 σ 2 z d 2 , P ′ s 1 | ˆ h sr | 2 σ 2 z r , and P ′ r | ˆ h r d | 2 σ 2 z r d are gi ven in (29)-(31).  V . O P T I M A L R E S O U R C E A L L O C A T I O N Ha ving obtain ed achie v able rate expressions in Section IV, we now identify optimal resource allocation strategies that maximize the rates. W e consider t hree resource al location problems: 1) power allocation between the training and data symbols; 2) ti me/bandwidth allo cation to the relay; 3) power all ocation between the source and relay under a total power constraint. W e first study how much power should be allocated for channel training. In non over lapped AF , i t can be see n th at δ r appears only in P ′ r | ˆ h r d | 2 σ 2 z r d in th e achiev able rate expression (27 ). S ince f ( x, y ) = xy 1+ x + y is a mono tonically i ncreasing function of y for fixed x , (27) is m aximized by maximi zing P ′ r | ˆ h r d | 2 σ 2 z r d . W e can maximize P ′ r | ˆ h r d | 2 σ 2 z r d by maxi mizing the coefficient of the random variable | w r d | 2 in (31), and th e opt imal δ r is gi ven below: δ opt r = − mP r σ 2 r d − αmN 0 + 2 αN 0 + p α ( m − 2)( m 2 P r σ 2 r d αN 0 + m 2 P 2 r σ 4 r d + αmN 2 0 + mP r σ 2 r d N 0 − 2 mP r σ 2 r d αN 0 − 2 N 0 α ) mP r σ 2 r d ( − 1 + αm − 2 α ) . (64) Optimizing δ s is more com plicated as it is related to all t he terms in (27), and h ence obtainin g an analytical solution is un likely . A suboptim al solution is to maximize P ′ s 1 | ˆ h sd | 2 σ 2 z d 1 and P ′ s 1 | ˆ h sr | 2 σ 2 z r separately , and obt ain two solutions δ subopt s, 1 and δ subopt s, 2 , respectiv ely . Note that expressions for δ subopt s, 1 and δ subopt s, 2 are exactly the same as that in (64) wi th P r , α , and σ r d replaced by P s , ( 1 − α ), and σ sd and σ sr , respectively . Wh en the source-relay channel is bett er t han the so urce-destination channel and the fraction of time o ver which direct transmissio n is performed is small, P ′ s 1 | ˆ h sr | 2 σ 2 z r is a more dom inant factor and δ subopt s, 2 is a good choi ce for traini ng power allocation. Otherwise, δ subopt s, 1 might be preferred. No te that i n non-ov erlapped DF with repetition and parallel coding, P ′ r | ˆ h r d | 2 σ 2 z r d is the o nly term t hat includes δ r . Therefore, si milar result s and discussi ons apply . For instance, the optimal δ r has the same expression as that in (64). Figure 1 plots the optimal δ r as a function of σ r d for diffe rent relay power constraints P r when m = 50 and α = 0 . 5 . It is observed in all cases that the allocated trainin g power monot onically decreases wi th improving channel quality and con ver ges to √ α ( m − 2) − 1 αm − 2 α − 1 ≈ 0 . 169 which is in dependent of P r . In ov erlapped transmis sion schem es, bo th δ s and δ r appear in more than one term in the achiev able rate expressions. Therefore, we resort to n umerical results to identi fy the optim al values. Figures 2 and 3 plo t the achiev able rates as a function of δ s and δ r for overlapped AF . In both figures, we have assumed that 15 σ sd = 1 , σ sr = 2 , σ r d = 1 and m = 50 , N 0 = 1 , α = 0 . 5 . While Fig. 2, where P s = 50 and P r = 50 , considers high SNR s, we assume that P s = 0 . 5 and P r = 0 . 5 in Fig. 3. In Fig. 2, we obs erve t hat increasing δ s will increase achie v able rate until δ s ≈ 0 . 1 . Further increase in δ s decreases th e achiev able rates. On the other hand, rates always increase wi th increasing δ r . T his ind icates that cooperation is not beneficial i n terms of achiev able rates and direct transmission s hould be preferred. On the other hand , i n t he low-power regime consid ered in Fig. 3 , t he optimal values of δ s and δ r are approximately 0.18 and 0. 32, re spective ly . Hence, the relay in this case helps to improve the rates. Next, we analyze the effect of t he degree o f cooperation o n t he performance i n AF and repetit ion DF . Figures 4 -7 plot the achie vable rates as a function of α which give s the fraction of to tal t ime/bandwidth allocated to the relay . Achi e vable rates are o btained for different channel qualities given by the s tandard deviations σ sd , σ sr , and σ r d of t he fading coef ficients. W e observe that if the input p owe r i s high, α should be either 0 . 5 or close to zero depending on the channel qual ities. On the other hand, α = 0 . 5 always gives us the best performance at low SNR lev els regardless of t he channel qualities. Hence, while coop eration is beneficial in the low- SNR regime, non cooperativ e transm issions mig ht be opti mal at hig h S NR s. W e note from Fig. 4 t hat cooperation starts being us eful as the source-relay channel variance σ 2 sr increases. Similar resul ts are also ob served in Fig 5. Hence, the source-relay channel quality is one of th e ke y factors in determini ng the usefulness of cooperation in the high SNR regime. In Fig. 8, we plot the achie vable rates of DF parallel channel coding , derive d in Theorem 5. W e can see from the figure th at th e best performance is obtained when the source-relay channel q uality is high (i.e., when σ sd = 1 , σ sr = 10 , σ r d = 2 ). Additionally , we observ e that as th e source-relay channel i mproves, more resources need to be allocated to the relay t o achieve t he best performance. W e n ote that s ignificant improvements with respect t o direct transmission (i.e., the case in which α → 0 ) are obtain ed. Finally , we can see that when compared to AF and DF with repetit ion cod ing, DF with p arallel channel coding achie ves higher rates. On the ot her hand, AF and repetition codin g D F have advantages in the imp lementation. Obviously , the relay , which ampli fies and forwards, has a simpler task than that which decodes and forwards. Moreover , as pointed out in [14], if AF or repetition coding DF is employed in the system, the arc hitecture of the dest ination node is simplified because th e data arriving from the source and relay can be combined rather than stored separately . In certain cases, source and relay are s ubject to a total power constraint. Here, we introduce the power allocation coef ficient θ , and t otal power const raint P . P s and P r hav e the following relation s: P s = θ P , 16 P r = (1 − θ ) P , and P s + P r ≤ P . Next, we in vestigate how dif ferent values of θ , and hence diffe rent po wer allocation strategies, affe ct the achiev able rates. An analytical results for θ that m aximizes the achiev able rates is diffi cult to obtain. Therefore, we again r esort t o numerical analysis. In all numerical resul ts, we assume that α = 0 . 5 which pro vides the maxim um of degree of cooperation. First , we c onsider the AF . The fixed parameters we choose are P = 1 00 , N 0 = 1 , δ s = 0 . 1 , δ r = 0 . 1 . Fig. 9 pl ots the achie vable rates in the overlapped transm ission scenario as a function of θ for dif ferent channel cond itions, i.e., differe nt values of σ sr , σ r d , and σ sd . W e observ e t hat the best performance is achieved as θ → 1 . Hence, eve n in the overlapped scenario, al l the power should be allocated to the source and direct transmissi on should be preferred at th ese hig h SNR leve ls. No te t hat if direct transmissio n is performed, there is no need to learn the relay-destination chann el. Since the tim e all ocated to the training for this channel shoul d b e allocated to data transmissio n, t he real rate of direct transmi ssion is slightly higher than the point that the coo perati ve rates con verge as θ → 1 . For this reason, we also provide the direct transm ission rate separately in Fig. 9. Further numerical analysi s has indi cated that direct transmission ove r performs non-overlapped AF , overlapped and non-overlapped DF with repetition coding as well at this level of i nput po wer . On the other hand, in Fig . 10 which plots t he achie v able rates of non-ov erlapped DF with parallel cod ing as a function of θ , we observe that direct t ransmission rate, which is the same as t hat give n in Fig. 9, i s exceeded if σ sr = 10 and hence the source-relay channel is very st rong. The best performance is achieved when θ ≈ 0 . 7 and therefore 70% of the power is allocated to the source. Figs. 11, 12, and 13 plot the non-overlapped achiev abl e rates wh en P = 1 . In all cases, we o bserve that performance lev els h igher than that of direct transm ission are achieved un less the quali ties of the source-relay and relay-destin ation channels are comparable to that of the source-destinati on channel (e.g., σ sd = 1 , σ sr = 2 , σ r d = 1 ). Moreover , we note that the best p erformances are attained when the sou rce- relay and relay-desti nation channels are both consid erably better th an t he source-destination channel (i.e., when σ sd = 1 , σ sr = 4 , σ r d = 4 ). As expected, hig hest gains are obtained with parallel coding DF although repetition coding i ncur on ly small los ses. Finally , Fig. 14 plo t t he achiev able rates of overlapped AF when P = 1 . Similar conclusions apply also here. Howev er , it i s interesting to note that o verlapped AF rates are smaller than those achie ved by non-overlapped AF . T his beha vior is also observed when DF with repetition coding i s considered. Note that in non-overlapped transmissi on, source transm its in a sho rter durati on of time with high er power . This signaling scheme p rovides b etter performance as expected because it is well-known that flash sig naling achie ves the capacity in the low-S NR re gime in im perfectly known channels [18]. 17 V I . E N E R G Y E FFI C I E N C Y Our analysis has shown that cooperative relaying is generally beneficial i n t he l ow- power re gime, resulti ng in higher achiev able rates when compared to direct transm ission. In this section, we provide an energy ef ficiency perspective and remark that care should also be taken when o perating at very low S NR values. The least am ount of energy required to s end one informati on bit reliably is given by 4 E b N 0 = SNR C ( SNR ) where C ( S NR ) is th e channel capacity in bits/ symbol. In our s etting, the capacity will be replaced by the achiev able rate expressions and hence the resultin g bit energy , denoted by E b,U N 0 , p rovides the least amoun t of no rmalized bit ener gy values in the worst-case scenario and also serves as an upp er bound on the achie va ble bit ener gy lev els in the channel. W e no te that i n finding the bit energy values, we assume that SNR = P / N 0 where P = P r + P s is the total po wer . The next resul t provides the asymptotic beha vior of the bit ener gy as SNR decreases to zero. Theor em 6: The normalized b it energy in all relayi ng schemes grows without boun d as th e si gnal-to-noise ratio decre ases to zero, i.e., E b,U N 0     I =0 = lim SNR → 0 SNR I ( SNR ) = 1 ˙ I (0) = ∞ . (65) Pr oof : The ke y point to prove thi s theorem is to show that when SNR → 0 , the mutual inform ation decreases as SNR 2 , and hence ˙ I (0) = 0 . This can be easily sho wn because when P → 0 , in all the terms P ′ s 1 | ˆ h sd | 2 σ 2 z d 1 , P ′ s 1 | ˆ h sd | 2 σ 2 z d 2 , P ′ s 1 | ˆ h sr | 2 σ 2 z r , P ′ r | ˆ h r d | 2 σ 2 z r d , P ′ s 2 | ˆ h sd | 2 σ 2 z d 1 , P ′ s 2 | ˆ h sd | 2 σ 2 z d 2 , P ′ s 2 | ˆ h sr | 2 σ 2 z r , P ′ s 2 | ˆ h sd | 2 σ 2 z r d , and P ′ r | ˆ h r d | 2 σ 2 z r d in Theorems 1-5, th e denominator g oes to a constant while the numerator decreases as P 2 . Hence, these terms dimin ish as S NR 2 . Since log(1 + x ) = x + o ( x ) for sm all x , we conclude t hat th e achie vable rate expressions also decrease as SNR 2 as SNR vanishes.  Theorem 6 indicates that it is extremely ener gy-inefficient t o operate at very lo w SNR v alues. W e identi fy the most energy-ef ficient operating p oints in num erical results. W e choose the foll owing numerical values for the fixed parameters: δ s = δ r = 0 . 1 , σ sd = 1 , σ sr = 4 , σ r d = 4 , α = 0 . 5 , and θ = 0 . 6 . Fig. 15 plots the bit ener gy curves as a function of SNR fo r di f ferent values of m in the non-overlapped AF case. W e can s ee from the figure that the minimum bit ener gy , whi ch is achie ved at a nonzero va lue of SNR , decreases with increasing m and is achiev ed at a lower SNR v alue. Fig. 16 shows the minimum bit energy for different relaying schemes with overlapped or non-overlapped t ransmission techniques. W e observe that 4 Note that E b N 0 is t he bit energy normalized by the noise power spectral leve l N 0 . 18 the mi nimum bit energy decreases with increasing m in al l cases . W e realize that DF is in general m uch more energy-ef ficient than AF . Moreov er , we note th at emp loying n on-over lapped rather than overlapped transmissio n i mproves the energy efficienc y . W e further remark t hat the performances of non-overlapped DF with repetition coding and parallel coding are very clo se. V I I . C O N C L U S I O N In this paper , we have studied t he imperfectly-known fading relay channels. W e have assu med that the source-destination, source-relay , and relay-destinati on channels are not known by the correspondi ng recei vers a priori, and transm ission starts with the training phase in which the channel f ading coeffi cients are learned with the assistance of pilo t symbol s, albeit imperfectly . Hence, in this sett ing, relaying i ncreases the channel uncertainty in the syst em, and there is increased estim ation cost associated with cooperation. W e hav e in ve stigated the performance of relaying by obtaini ng achiev able rates for AF and DF relaying s chemes. W e hav e cons idered both non-ov erlapped and ov erlapped transmission s cenarios. W e ha ve controlled the degree of cooperation by varying the par ameter α . W e ha ve identified the optimal resource allocation strategies using the achiev able rate expressions. W e ha ve obs erved t hat if th e s ource-relay channel quality is lo w , t hen cooperation i s not beneficial and direct transmissi on should be preferred at high SNR s. On the oth er hand, we h a ve seen that relaying g enerally i mproves the performance at l ow SNR s. W e have noted t hat DF wi th parallel codi ng provides t he high est rates. Additionally , under t otal power const raints, we ha ve identified t he optimal power allocation between the sou rce and relay . W e hav e again poi nted out that relaying degrades the performance at high SNR s unless DF with parallel channel coding is used and the s ource-relay channel quality is high. The benefits of relaying is again demonstrated at low SNR s. W e hav e noted that non-ov erlapped transmissio n is s uperior compared to ov erlapped one in this regime. Finally , we h a ve considered the ener gy ef ficiency in the lo w-po wer r egime, and prov ed that the bit ener gy increases without bound as SNR di minishes. Hence, operation at ve ry low S NR levels sh ould be a voided. From the ener gy efficienc y perspective, we ha ve again observed that non-overlapped trans mission provides better performance than overlapped transmis sion. W e hav e also noted that DF i s more energy ef ficient than AF . R E F E R E N C E S [1] E. C. v an der Meulen, “Three-terminal communication channels, ” Adv . Appl. Probab ., vol. 3, pp. 120-154, 1971. [2] T . M. Cover and A.A. El Gamal, “Capacity theorems for the relay channel, ” IEEE Trans. Inf. Theory , vol. IT -25, no. 5, pp. 572-584, Sep. 1979. 19 [3] A. A. El Gamal and M. Aref, “The capacity of the semideterministic relay ch annel, ” IEEE T rans. Inf. Theory , vol. IT -28, no. 3, pp. 536-536, May 1982. [4] A. S endonaris, E. Erkip, and B. 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Gursoy , “ An energy efficienc y perspecti ve on training for fading channels, ” Proceedings of the IE EE ISIT 2007. 20 0 5 10 15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 σ rd δ r Pr=0.1 Pr=1 Pr=5 Pr=20 Pr=200 Fig. 1. δ r vs. σ r d for different v alues of P r when m = 50 . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 δ r δ s Achievable Rates (bits/symbol) Fig. 2. Overlappe d AF achie va ble rates vs. δ s and δ r when P s = P r = 50 21 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.1 0.2 0.3 0.4 δ r δ s Achievable Rates (bits/symbol) Fig. 3. Overlappe d AF achie va ble rates vs. δ s and δ r when P s = P r = 0 . 5 0 0.1 0.2 0.3 0.4 0.5 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 α Achievable Rates (bits/symbol) σ sd =1 σ sr =10 σ rd =2 σ sd =1 σ sr =6 σ rd =3 σ sd =1 σ sr =4 σ rd =4 σ sd =1 σ sr =2 σ rd =1 Fig. 4. Overlappe d AF achie va ble rate vs. α when P s = P r = 50 , δ s = δ r = 0 . 1 , m = 50 . 22 0 0.1 0.2 0.3 0.4 0.5 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 α Achievable Rates (bits/symbol) σ sd =1 σ sr =10 σ rd =2 σ sd =1 σ sr =6 σ rd =3 σ sd =1 σ sr =4 σ rd =4 σ sd =1 σ sr =2 σ rd =1 Fig. 5. Overlappe d DF with repetition coding achiev able rate vs. α when P s = P r = 50 , δ s = δ r = 0 . 1 , m = 50 . 0 0.1 0.2 0.3 0.4 0.5 0.4 0.5 0.6 0.7 0.8 0.9 1 α Achievable Rates (bits/symbol) σ sd =1 σ sr =10 σ rd =2 σ sd =1 σ sr =6 σ rd =3 σ sd =1 σ sr =4 σ rd =4 σ sd =1 σ sr =2 σ rd =1 Fig. 6. Overlappe d AF achie va ble rate vs. α when P s = P r = 0 . 5 , δ s = δ r = 0 . 1 , m = 50 . 23 0 0.1 0.2 0.3 0.4 0.5 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 α Achievable Rates (bits/symbol) σ sd =1 σ sr =10 σ rd =2 σ sd =1 σ sr =6 σ rd =3 σ sd =1 σ sr =4 σ rd =4 σ sd =1 σ sr =2 σ rd =1 Fig. 7. Overlappe d DF with repetition coding achiev able rate vs. α when P s = P r = 0 . 5 , δ s = δ r = 0 . 1 , m = 50 . 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 α Achievable Rates (bits/symbol) σ sd =1 σ sr =10 σ rd =2 σ sd =1 σ sr =6 σ rd =3 σ sd =1 σ sr =4 σ rd =4 σ sd =1 σ sr =2 σ rd =1 Fig. 8. Non-ov erlapped DF parallel coding achiev able rate vs. α when P s = P r = 0 . 5 , δ s = δ r = 0 . 1 , m = 50 . 24 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 θ Achievable Rates (bits/symbol) σ sd =1, σ sr =10, σ rd =2 σ sd =1, σ sr =6, σ rd =3 σ sd =1, σ sr =4, σ rd =4 σ sd =1, σ sr =2, σ rd =1 Real Rate of Direct Transmission Fig. 9. Overlappe d AF achie va ble rate vs. θ . P = 100 , m = 50 . 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 θ Achievable Rates (bits/symbol) σ sd =1, σ sr =10, σ rd =2 σ sd =1, σ sr =6, σ rd =3 σ sd =1, σ sr =4, σ rd =4 σ sd =1, σ sr =2, σ rd =1 Fig. 10. Non-o verlappe d Parallel coding DF rate vs. θ . P = 100 , m = 50 . 25 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 θ Achievable Rates (bits/symbol) σ sd =1, σ sr =10, σ rd =2 σ sd =1, σ sr =6, σ rd =3 σ sd =1, σ sr =4, σ rd =4 σ sd =1, σ sr =2, σ rd =1 Real Rate of Direct Transmission Fig. 11. Non-o verlappe d AF achiev able rate vs. θ . P = 1 , m = 50 . 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 θ Achievable Rate (bits/symbol) σ sd =1, σ sr =10, σ rd =2 σ sd =1, σ sr =6, σ rd =3 σ sd =1, σ sr =4, σ rd =4 σ sd =1, σ sr =2, σ rd =1 Fig. 12. Non-o verlappe d Repetition coding DF rate vs. θ . P = 1 , m = 50 . 26 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 θ Achievable Rate (bits/symbol) σ sd =1, σ sr =10, σ rd =2 σ sd =1, σ sr =6, σ rd =3 σ sd =1, σ sr =4, σ rd =4 σ sd =1, σ sr =2, σ rd =1 Fig. 13. Non-o verlappe d Parallel coding DF rate vs. θ . P = 1 , m = 50 . 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 θ Achievable Rates (bits/symbol) σ sd =1, σ sr =10, σ rd =2 σ sd =1, σ sr =6, σ rd =3 σ sd =1, σ sr =4, σ rd =4 σ sd =1, σ sr =2, σ rd =1 Fig. 14. Overlapped AF achiev able rate vs. θ . P = 1 , m = 50 . 27 0 0.2 0.4 0.6 0.8 1 −5 0 5 10 15 SNR E b /N 0 (dB) m=20 m=50 m=100 m=10000 Fig. 15. Non-ov erlapped AF E b,U / N 0 vs. S NR 40 50 60 70 80 90 100 −7 −6 −5 −4 −3 −2 −1 0 m minimum E b /N 0 (dB) Overlapped AF Non−overlapped AF non−overlapped DF P non−overlappd DF R overlapped DF Fig. 16. E b,U / N 0 vs. m for different tr ansmission scheme 28