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wireless relay networks are ubiquitous from classical satellite communications to the increasing use of wireless ad hoc networks where mobiles , tablets , or laptops have the ability to act as wireless relays for other similar devices . a common coding strategy deployed in relay networks is for the relays to decode the source message , and then forward a function of the message ( e.g. , some parity bits ) to the destination to facilitate decoding . in this context , one can use a _ partial - decode - forward ( pdf ) _ scheme where the relays decode only a fraction of the source message and forward a function of the decoded part , or a _ complete - decode - forward ( cdf ) _ scheme where the relays fully decode the source message and forward a function of it . another consideration in designing wireless relay networks is whether to use a full - duplex ( communication possible in both directions simultaneously without interference ) or a half - duplex ( communication possible in only one direction at a time ) scheme . the choice of full- versus half - duplex , as well as the choice of cdf or pdf , impacts on the achievable communication rate . achievable rates have been demonstrated for the full - duplex relay channel using cdf @xcite and pdf @xcite , ( * ? ? ? 16.3 ) , and for the half - duplex relay channel using cdf @xcite and pdf @xcite . pdf includes cdf and direct transmission as special cases  the relay decodes all source messages in the former and decodes nothing in the latter . for the additive white gaussian noise ( awgn ) full - duplex relay channel , it has been shown  @xcite that ( i ) when the source - destination link is better than the source - relay link , the maximum pdf rate can be achieved by not using the relay at all , i.e. , direct transmission ; and ( ii ) when the source - relay link is better than the source - destination link , the maximum pdf rate can be achieved by having the relay decoding all source messages , i.e. , cdf . while it is not desirable to have the relay decoding only part of the messages in the awgn full - duplex relay channel , pdf can achieve rates strictly higher than direct transmission and cdf in the half - duplex counterpart .      in this paper , we focus on the half - duplex relay channel , and compare the rates achievable by existing coding schemes . for a given half - duplex relay channel , let @xmath0 and @xmath1 be the maximum rates achievable ( optimized over channel codes under respective coding schemes ) using the complete - decode - forward and the partial - decode - forward schemes respectively . for the trivial case where the source - destination link is better than the source - relay link , @xmath1 can be attained by direct transmission ( similar to the full - duplex case ) . otherwise , we have the following :    [ theorem : main ] consider the awgn half - duplex single - relay channel . if the source - relay link is better than the source - destination link , then @xmath2    this means regardless of channel parameters , partial decoding can achieve at most 12.5% rate gain compared to full decoding . we further show that in the following three asymptotic cases , pdf and cdf actually achieve the same rates : ( i ) the relay is close to the source , ( ii ) the relay is close to the destination , and ( iii ) the signal - to - noise ratio ( snr ) is low . we also show that for the case of high snr , cdf achieves rates within a constant bit gap ( which is a function of the channel gains ) of that achievable by pdf . this gap , when normalized to the cdf rate , diminishes as the snr increases . theoretically , this work gives an upper bound to the performance gain of pdf over cdf . practically , it shows that without a significant percentage loss in transmission rate , one could choose cdf over pdf , as the former is potentially less complex to implement , and is certainly less complex to design . [ fig : src ] denotes the single - relay channel consisting of the source ( denoted by node 0 ) , the relay ( node 1 ) , and the destination ( node 2 ) . we consider the awgn channel where ( i ) the received signal at the relay is given by @xmath3 , and ( ii ) the received signal at the destination is given by @xmath4 . here , @xmath5 is the transmitted signal of node @xmath6 , @xmath7 is the noise at node @xmath8 , and @xmath9 is the channel gain from node @xmath6 to node @xmath8 . each @xmath7 is an independent zero - mean gaussian random variable with variance @xmath10 = n_j$ ] . the channel gains are fixed and are made known a priori to all the nodes . consider a block of @xmath11 channel uses , and let @xmath12 ( and @xmath13 ) denote the transmitted ( and received ) symbol of node @xmath6 at the @xmath14-th channel use . a rate-@xmath15 block code comprises ( i ) a message set @xmath16 , ( ii ) an encoding function for the source @xmath17 for each message @xmath18 , ( iii ) encoding functions for the relay @xmath19 for all @xmath20 , and ( iv ) a decoding function for the destination @xmath21 . here , we have used bold - faced symbols to represent vectors of length @xmath11 , e.g. , @xmath22 . we assume that the message is uniformly distributed in @xmath23 . the rate @xmath15 is achievable if there exists a sequence of rate-@xmath15 block codes such that @xmath24 as @xmath25 .      for the half - duplex channel , the relay can only transmit or receive ( i.e. , listen ) , but not do both , at any time . we set @xmath26 if the relay transmits at the @xmath14-th channel use , and @xmath27 otherwise , i.e. , if the relay listens . note that the definition of block code above is applicable for both full - duplex and half - duplex relay channels with these extra transmit / listen constraints . without loss of optimality , the source always transmits , and the destination always listens . we consider a _ fixed - slot _ structure where the transmit / listen mode of the relay for each channel use is fixed prior to the transmissions and is known to all nodes  @xcite . consider @xmath11 channel uses during which the relay listens in @xmath28 channel uses and transmits in @xmath29 channel uses . we assume the following _ per - symbol _ power constraints  @xcite : @xmath30 \leq p_i$ ] for both @xmath31 and for all @xmath32 , where the expectation operation @xmath33 $ ] is taken over the message @xmath34 and the channel noise @xmath35 . as a result , the source and the relay can not optimize their transmit power for variations in the relay s transmit / listen mode .. ] we denote the snr from node @xmath6 to node @xmath8 by @xmath36 . we assume that @xmath37 for all @xmath6 and @xmath8 . we now summarize the encoding and decoding schemes of pdf for the half - duplex relay channel  @xcite . consider @xmath38 blocks , each of @xmath11 channel uses , and @xmath39 source messages @xmath40 , where each @xmath41 . the source splits each message @xmath42 into two independent parts @xmath43 , where @xmath44 and @xmath45 so that @xmath46 . in block @xmath47 , the source transmits @xmath48 . the relay decodes @xmath49 in block @xmath50 , and transmits it in block @xmath51 . the rate of this code over the entire @xmath38 transmission blocks is @xmath52 bits / channel use . if the destination can decode all @xmath40 reliably ( i.e. , with diminishing error probability as @xmath11 increases ) , then by choosing a sufficiently large @xmath38 , any rate below and arbitrarily close to @xmath15 is achievable .     of the time the relay listens , and in @xmath53 of the time the relay transmits ]    as the encoding and decoding operations repeat themselves over the blocks , we focus on encoding in block @xmath50 and decoding over blocks @xmath50 and @xmath51 , depicted in fig . [ fig : pdf - half ] . during the first @xmath28 channel uses of block @xmath50 ( when the relay listens ) channel uses and transmits in the remaining channel uses of each block . in general , the channel uses in which the relay listens or transmits can be arbitrarily chosen in each block without affecting the achievable rates as long as @xmath54 is fixed . ] , the source transmits @xmath55 . here , we have used the superscript @xmath56 to denote the length of the sub - codeword . assume that the relay has decoded @xmath57 in the previous block , i.e. , block @xmath58 . during the remaining @xmath29 channel uses of block @xmath50 , the relay transmits an independently generated codeword @xmath59 , while the source splits its transmit power into two parts : ( i ) a portion of @xmath60 of the power is used to transmit the same codeword as the relay , i.e. , @xmath61 , so that this portion of the source s signal and the relay s signal add coherently at the destination twice : @xmath55 in block @xmath50 and @xmath62 in block @xmath51 . note that these two codewords are independently generated . ] ; and ( ii ) the remaining portion of @xmath63 of the power is dedicated to a different codeword that is to be decoded by the destination , i.e. , @xmath64 . in brief , the source transmits @xmath65 $ ] in block @xmath50 . the above encoding scheme requires that the relay decodes @xmath49 at the end of block @xmath50 ; the relay can reliably do so if @xmath66 . the decoding of the message @xmath67 at the destination is performed over blocks @xmath50 and @xmath51 . assume that the destination has decoded messages @xmath68 . using the last @xmath29 channel uses of block @xmath50 , the destination can decode @xmath69 if @xmath70 . using the first @xmath28 channel uses of block @xmath50 and the last @xmath29 channel uses of block @xmath51 , the destination can decode @xmath49 if @xmath71 ^ 2 } { ( 1-\beta ) \lambda_{02 } + 1 } \right)$ ] . here , @xmath72 ^ 2 $ ] is power of the coherently combined signals from the relay and the source , @xmath73 is the power of signals carrying @xmath74 which appears as noise when decoding @xmath49 , and the channel noise power @xmath75 $ ] is one . combining these rate constraints , pdf  @xcite achieves rates up to    @xmath76    setting @xmath77 ( which means @xmath78 ; i.e. , the relay decodes all source messages ) , we have cdf  @xcite , which achieves rates up to    @xmath79 ^ 2 ) \big\}. \label{eq : cdf - rate}\end{aligned}\ ] ]    clearly , @xmath80 . furthermore , setting @xmath81 and @xmath82 ( which means @xmath83 ; i.e. , the relay decodes nothing ) , we have direct transmission . when @xmath84 ( the source - destination link is better than the source - relay link ) , @xmath85 , which is attained only at @xmath86 , i.e. , with direct transmission . when @xmath87 ( the source - destination link is as good as the source - relay link ) , we again have @xmath85 , which is attained by @xmath82 and any @xmath88 $ ] . although the relay is used when @xmath89 , pdf still achieves the same rate as that of direct transmission , @xmath90 .    for the rest of this paper , we will consider the non - trivial case of @xmath91 ( when the source - relay link is better than the source - destination link ) . we will show later that @xmath1 can be strictly higher than @xmath0 . furthermore , it is easy to see that , as all @xmath92 , both pdf and cdf strictly outperform direct transmission . we define the gap between the pdf and the cdf rates by    @xmath93    and the normalized ( to @xmath0 ) gap by    @xmath94 first , we have the following lemma :    [ lemma : cdf ] let @xmath95 be the @xmath54 that attains @xmath0 . if @xmath91 , it follows that @xmath96 ^ 2)$ ] . let @xmath97 and @xmath98 ^ 2)$ ] . the function @xmath99 is a continuous and increasing in @xmath54 with @xmath100 and @xmath101 ; @xmath102 is a continuous and decreasing function of @xmath54 with @xmath103 and @xmath104 . so the solution to @xmath105 is at the point where @xmath99 and @xmath102 intersect .    from the above lemma , we can show that    [ lemma : cdf - rate ] if @xmath91 , then @xmath106 ^ 2 ) \log ( 1   + \lambda_{01 } ) } { \log ( 1 + [ \sqrt{\lambda_{02 } } + \sqrt{\lambda_{12}}]^2 ) + \log ( 1 + \lambda_{01 } ) - \log ( 1 + \lambda_{02 } ) } .\end{gathered}\ ] ]    substituting @xmath95 from lemma  [ lemma : cdf ] into gives lemma  [ lemma : cdf - rate ] .    now , by selecting the @xmath107 $ ] that maximizes the two terms on the right - hand side of ( i.e. , @xmath108 and @xmath77 respectively ) , we can upper bound @xmath1 as    @xmath109 ^ 2 ) \big\}\nonumber \\ & \triangleq r_\text{pdf}^\text{ub } , \label{eq : pdf - ub } \end{aligned}\ ] ]    where @xmath110 is an upper bound to @xmath1 .    using the same arguments in the proof of lemma  [ lemma : cdf ] , we obtain the following :    [ lemma : pdf - ub - a ] let @xmath111 be the @xmath54 that attains @xmath110 . if @xmath91 then @xmath112 ^ 2 ) $ ] . hence , the upper bound to @xmath1 is given as follows :    [ lemma : pdf - ub - b ] if @xmath91 , then    @xmath113 ^ 2 ) \log ( 1 + \lambda_{01 } ) } { \log ( 1 + [ \sqrt{\lambda_{02 } } + \sqrt{\lambda_{12}}]^2 ) + \log ( 1 + \lambda_{01 } ) - 2 \log ( 1 + \lambda_{02 } ) } \nonumber \\ & - \frac { \frac{1}{2 } [ \log ( 1 + \lambda_{02})]^2 } { \log ( 1 + [ \sqrt{\lambda_{02 } } + \sqrt{\lambda_{12}}]^2 ) + \log ( 1 + \lambda_{01 } )   - 2 \log ( 1 + \lambda_{02 } ) } .\end{aligned}\ ] ]    substituting @xmath111 in lemma  [ lemma : pdf - ub - a ] into gives lemma  [ lemma : pdf - ub - b ] .      we now derive the following upper bound on the normalized gap   between the pdf and the cdf rates : @xmath114    first , we define the following : @xmath115 , @xmath116 , @xmath117 , @xmath118 , and @xmath119 ^ 2\right)$ ] . note that while @xmath120 , @xmath121 , @xmath122 , and @xmath14 are completely determined by the channel gains , the source s transmit power , and the receiver noise levels at the relay and the destination , @xmath123 is additionally determined by the relay s transmit power ( in addition to the channel gains , the source s transmit power , and the noise levels ) . with the above definitions , we have the normalized gap upper bounded by @xmath124    to prove , we first show the following lemma :    [ lemma : gap ] if @xmath91 , then @xmath125 } . \label{eq : lemma}\ ] ]    we first write @xmath126 where @xmath127 . note that for any fixed channel gains , @xmath128 , @xmath129 , and @xmath130 , the variables @xmath14 , @xmath121 , and @xmath122 are fixed , but the variable @xmath123 can take any value by choosing an appropriate @xmath131 subject to the constraint @xmath132 ^ 2\right )   > \log ( 1 + \lambda_{02 } )   \triangleq u $ ] . this means @xmath123 can be chosen independent of @xmath14 , @xmath121 , and @xmath122 . now , for any @xmath14 , @xmath121 , and @xmath122 , we maximize @xmath133 with respect to @xmath123 . note that as @xmath91 , we must have @xmath134 . so , for the region of interest , i.e. , where @xmath135 , the function @xmath133 is continuously differentiable . by differentiating @xmath133 with respect to @xmath123 , we get @xmath136 ^ 2}$ ] . the denominator is always positive , and the numerator is a quadratic function with roots @xmath137 and @xmath138 . note that @xmath139 since @xmath140 . this means @xmath141 so , the maximum of @xmath133 for @xmath142 occurs at @xmath138 . thus , @xmath143[u + \sqrt{u(u+v)}+v]}.\ ] ] substituting @xmath144 and @xmath145 into the above equation , we get .    with the above lemma , we now prove . first , we define @xmath146 , where @xmath147 because @xmath148 . the right - hand side of can be written as @xmath149 where @xmath150 $ ] . this means @xmath151 $ ] . note that @xmath152 is continuously differentiable for all @xmath147 , and its first derivative is @xmath153 so , @xmath154 . this gives . combining and the trivial lower bound @xmath80 , we have theorem  [ theorem : main ] . @xmath155      suppose that @xmath157 . the optimal @xmath54 for the max - min operation in is close to zero . hence , @xmath158 ^2)$ ] . for , the optimal @xmath54 is close to zero and the optimal @xmath60 is close to one . hence , @xmath159 ^2)$ ] . so the trivial lower bound @xmath80 is ( almost ) tight when @xmath157 . numerical results show that the normalized gap @xmath156 approaches the upper bound of @xmath160 ( in theorem  [ theorem : main ] ) as @xmath161 increases ( for selected @xmath162 and @xmath163 for each @xmath161 ) . for example , setting @xmath164 , @xmath165 , and @xmath166 , the normalized gap is @xmath167 . in this section , we consider the free - space path loss model where the channel gain is given by @xmath168 , where @xmath169 is the distance between the transmitter and the receiver , and so @xmath170 . @xmath171        \big [ \log(1 + [ \sqrt{\lambda_{02 } } + \sqrt{\lambda_{12}}]^2 ) -        \log(1+\lambda_{02 } ) \big ] \log(1+\lambda_{02 } ) } { 2\big[\log ( 1 +        [ \sqrt{\lambda_{02 } } + \sqrt{\lambda_{12}}]^2 ) + \log ( 1 +        \lambda_{01 } ) - 2 \log ( 1 + \lambda_{02})\big ] \big[\log ( 1 +        [ \sqrt{\lambda_{02 } } + \sqrt{\lambda_{12}}]^2 ) + \log ( 1 +        \lambda_{01 } ) - \log ( 1 + \lambda_{02})\big ] } } \triangleq g^\text{ub}. \label{eq : g-2 } % & = \frac{(w - u)(q - u)u}{2(q+v)(q+w - u ) } ,    \end{aligned}\ ] ]      we fix coordinates of the source at @xmath172 and the destination at @xmath173 , and compute the actual @xmath156 for different relay coordinates @xmath174 . we assume that @xmath175 , and @xmath176 . we are interested in networks where @xmath91 , which is equivalent to restricting the relay position to @xmath177 . the result is shown in fig . [ fig:3d ] . while an upper bound derived for @xmath156 in the previous section is @xmath178 for all channel parameters , for this example of equal power and equal noise , the actual gap is only 6.45% or less . , for varying relay positions ( only positions at @xmath179 are shown due to symmetry ) : we have @xmath180 , width=340 ]      now , we present four cases in which @xmath181 or @xmath156 tends to zero , meaning that cdf performs as well as pdf . in the following subsections , unless otherwise stated , the transmit power @xmath128 and @xmath131 , the inter - node distances @xmath182 , @xmath183 , @xmath184 , and the receiver noise @xmath129 , @xmath130 are all positive and finite . first , we find an upper bound on the gap between the pdf and the cdf rates , given by @xmath185 in .      when the relay is close to the source , i.e. , @xmath186 , we have @xmath187 . since @xmath163 and @xmath161 are finite , we see that both @xmath188 and @xmath189 as @xmath186 . similarly , when the relay is close to the destination , i.e. , @xmath190 , we have @xmath191 . since @xmath162 and @xmath163 are finite , we also see that @xmath188 and @xmath189 as @xmath190 . now , we consider fixed node positions , and increase the transmit power of the nodes . we let @xmath192 , @xmath193 , @xmath194 , @xmath195 for some constants @xmath196 and @xmath197 , and investigate the behavior of @xmath181 and @xmath156 when @xmath198 increases . we further define the following , which are independent of @xmath198 : @xmath199 and @xmath200 ^ 2}{\lambda_{02 } } = \frac{\left[\sqrt{d_{02}^{-2}k_0}+ \sqrt{d_{12}^{-2}k_1}\right]^2}{d_{02}^{-2}k_0}$ ] .    for high snr , we use the approximation @xmath201 , and obtain @xmath202}\nonumber \\ & \approx % \frac { \log c_1 \log c_2 } { 2\log ( c_1c_2 ) } =   \frac{1}{2 } \left ( \frac{1}{\log c_1 } + \frac{1}{\log c_2 }   \right)^{-1}. \label{eq : high - snr}\end{aligned}\ ] ]    so , in the high snr regime where the pdf and the cdf rates are high , the gap between the rates is upper bounded by a constant . this constant , when normalized to the pdf or the cdf rates , approaches zero as the snr increases , i.e. , @xmath203 . [ remark : large - g ] from , we see that @xmath204 increases as @xmath205 and @xmath206 increase . this means with a fixed @xmath163 , for sufficiently large @xmath162 and @xmath161 , and a much larger @xmath198 , @xmath185 can be made arbitrarily large . finally , we investigate the case where the snr tends to zero . we follow the settings ( inter - node distances , transmit power , and receiver noise ) as in the high - snr case above , but with @xmath207 .    for low snr , as @xmath207 , both @xmath208 . so , @xmath209 .    using the approximation @xmath210 , we further see that the normalized gap is @xmath211 where @xmath212 , + @xmath213 , and + @xmath214 .    as @xmath91 , we have @xmath215 . so @xmath216 , and it follows that @xmath217 . consequently , @xmath218 if any of the following is true : ( i ) @xmath219 , i.e. , when the relay is close to the source , @xmath186 ; ( ii ) @xmath220 , i.e. , when the relay is close to the destination , @xmath190 ; or ( iii ) @xmath221 , i.e. , when the relay and the destination are about the same distance from the source , @xmath222 [ by the triangle inequality , we have @xmath223 . in this letter , we have shown that there is little to be gained by doing partial decoding rather than complete decoding at the relay for the half - duplex awgn single - relay channel . we have analytically shown that using partial - decode - forward ( pdf ) can increase the rate over complete - decode - forward ( cdf ) by at most one - eighth . we have also identified four scenarios in which the cdf rates asymptotically approach the pdf rates .    while a multiplicative - type bound on the gain of pdf over cdf has been obtained in this paper , we note that the absolute gain _ might _ be unbounded in the high snr regime ( we have shown in remark  [ remark : large - g ] that our upper bound on the absolute gain is unbounded ) . hence , one may attempt pdf in the high - snr regime when the absolute gain in transmission rate ( as opposed to the percentage gain ) outweighs the system complexity .    the results in this letter provide a strong motivation to extend the comparison between these two strategies to the multiple - relay channel to investigate if partial - decode - forward is beneficial when there are more relays in the network , in which partial - decode - forward is much more computationally complex than complete - decode - forward . l.  ong , w.  wang , and m.  motani , `` achievable rates and optimal schedules for half duplex multiple - relay networks , '' in _ proc . 46th allerton conf . control comput . ( allerton conf . ) _ , monticello , usa , sept . 2326 2008 , pp . 13171324 .
this paper compares the partial - decode - forward and the complete - decode - forward coding strategies for the half - duplex gaussian single - relay channel . 
 we analytically show that the rate achievable by partial - decode - forward outperforms that of the more straightforward complete - decode - forward by at most 12.5% . 
 furthermore , in the following asymptotic cases , the gap between the partial - decode - forward and the complete - decode - forward rates diminishes : ( i ) when the relay is close to the source , ( ii ) when the relay is close to the destination , and ( iii ) when the snr is low . 
 in addition , when the snr increases , this gap , when normalized to the complete - decode - forward rate , also diminishes . 
 consequently , significant performance improvements are not achieved by optimizing the fraction of data the relay should decode and forward , over simply decoding the entire source message .    achievable rate , decode - forward , half duplex , partial decode - forward , relay channel .