, or error exponent, of a one-way channel characterizes the rate of decay of the probability of error when communicating one ofmessages as
where is the smallest probability of error that can be achieved by a code of rate with block length . Error exponents have been the subject of intense interest in both the absence [4, 3] and presence of feedback (to be reviewed later). If feedback is available, the transmitter is given access to a (possibly noisy, possibly encoded) function of the received output, that may dramatically increase the error exponents of one-way channels relative to when feedback is absent. Error exponents in the presence of ideal, noiseless feedback were considered by Berlekamp in  for general discrete memoryless channels, and by Pinsker  for the AWGN case. It was shown that while feedback cannot increase the capacity of non-anticipatory channels, it may greatly improve the error exponents achieved. Shannon pointed out this fact for the reliability of discrete memoryless channels with perfect feedback . This was first demonstrated for the AWGN channel in , in which the probability of error decays double-exponentially in the blocklength , and later by  who demonstrated that a decay rate equal to any number of exponential levels is possible. One natural question is whether this increase in error exponents may be attributed to the feedback being noiseless. In recent years, it has been shown that even noisy feedback is useful in improving error exponents (though less dramatically), with a limited number of available results in the one-way setting. In this article we continue this line of work and study error exponents of one-way additive white Gaussian noise (AWGN) channels with noisy AWGN feedback for the transmission of a finite number of messages (zero-rate).
We also extend the study of error exponents with noisy feedback to the two-way channel, in which two terminals exchange independent messages. In this bidirectional model, each transmitter’s encoding function output at time is a function not only of the message, but also of the past available channel outputs. This transmitter’s encoding function is said to be adaptative or interactive, a term used to emphasize how in two-way networks, each terminal’s channel input may adapt to its received channel outputs. The capacity region of the two-way AWGN channel (with independent noise across the terminals) is known, and is a rectangular region where both users may simultaneously attain their interference-free AWGN capacity. That is, adaptation at the transmitters is useless from a capacity perspective. The question we ask and answer here is whether adaptation may improve the error exponents of this two-way AWGN channel, for the transmission of a finite number of messages. This is the first study of two-way error exponents, to the best of our knowledge, and it is of initial interest as the exact error exponent for the one-way channel with noisy feedback is still open in this finite message regime. Error exponents for positive rates are left for future work, and may extend existing work such as [8, 10, 11, 12, 13, 14]. We also initially study error exponents for fixed block length codes rather than variable block length codes, as studied in [15, 16, 17] and the references therein for discrete memoryless channels. We focus on AWGN channels, and will review results that address the problem for the transmission of a finite number of messages for such channels [18, 19, 20, 21, 22] in the coming sections.
The achievable error exponents of AWGN channels are sensitive to the type of power constraint imposed on the channel inputs. In this work, we consider two constraints, defined as the
Almost sure (AS) power constraint:
Expected block (EXP) power constraint:
In (2) and (3), corresponds to the -th channel input of user , and to the power available at the -th terminal. The EXP constraint is less stringent than the AS constraint, and allows very high amplitude transmissions to occur with exponentially small probability. These rare events may correspond to decoding errors, and transmissions may be used to correct such errors, thereby increasing achievable error exponents. Burnashev and Yamamoto  comment that this “trick” can not be used for general discrete memoryless channels, but is useful for channels characterized by additive noise.
We present results on the one-way and two-way AWGN channel error exponents under both the AS and EXP power constraint in two separate sections. Each section includes the problem statement, past related work and our new achievable error exponent (regions). Our contributions are as follows:
Theorem 1, proven in Section IV, demonstrates how under the AS power constraint, the transmission of a finite number of messages , and a feedback link stronger than the forward link, the use of new proposed active feedback scheme in a one-way AWGN channel with noisy AWGN feedback leads to a higher error exponent gain over the non-feedback transmission exponent than that reported for passive feedback by Xiang and Kim in .
Theorem 2, proven in Section V, generalizes an achievable error exponent under the EXP power constraint presented by Kim, Lapidoth and Weissman in  for the transmission of two messages to any finite number of messages , again for the one-way AWGN channel with AWGN noisy feedback. The generalization is based on the use of a simplex code, and is an active feedback scheme.
Theorems 3 and 4 demonstrate new achievable error exponent regions (using passive and active feedback respectively) for the two-way AWGN channel under the AS power constraint, provided that one channel’s signal to noise ratio (SNR) is better than the other. This non-symmetric SNR scenario is of interest since for the one-way AWGN channel under the AS constraint, only a feedback link significantly stronger that the forward has been shown to lead to error exponent gains over feedback-free transmissions. These results follow as a direct application of Theorem 1 for active feedback and the results reported in  for passive feedback.
The achievable error exponents for the transmission of a finite number of messages presented here will all be based on the use of a simplex code for the non-feedback transmissions present in our schemes. The use of simplex codes enable the use of a geometric approach for upper bounding the probability of error, that can be visualized for a small number of messages () and can be extended to any finite . Under the EXP constraint, some feedback and retransmission signals that occur with exponentially small probability employ very high amplitude signals. These rarely occurring transmissions may be used to ensure an exceedingly small probability of error.
I-B Article outline
The remainder of this article is organized as follows. Section II presents a summary of previous results followed by our findings for the one-way AWGN channel at zero-rate: Theorems 1 and 2, which are proven in Sections IV and V respectively. The two-way AWGN channel is studied in Section III, with the main results presented in Theorems 3 and 4, that follow from the use of one-way achievability schemes under the AS constraint, and Theorem 5, which addresses the case of EXP constraint and is proven in Section VI. Section VII address the relation between the number of messages and the block length for our proposed schemes. Numerical simulations are presented in Section VIII. Finally, Section IX presents our conclusions and a discussion of open problems.
Notation. We use to indicate the probability of any event conditioned on the transmission of message , i.e. the probability of error given that has been sent as: . We indicate the length of a sequence using a superscript, i.e. denotes that sequence lasts for channel uses. Subscript indicates the terminal that generated the sequence. Error exponents are denoted by , accompanied by a superscript to denote the forward direction power constraint, and a subscript to denote the feedback link power constraint, if applicable. We use
to denote the expectation operator. Random variables are indicated by upper case letters, taking on instance in lower cases from alphabets in calligraphic font (random variabletakes on ). We use to indicate that as . We use the terms terminal and user interchangeably to refer to the devices involved in the communication process.
Ii Error exponents for the one-way AWGN channel at zero-rate
This section defines error exponents for one-way channels with feedback, and presents existing results on achievable error exponents for the one-way AWGN channel at zero-rate under different power constraints and types of feedback. We also introduce new achievable error exponents for the transmission of a finite number of messages (and this is what we will refer to as zero-rate from now on), under the AS power constraint with active feedback (Theorem 1) and under the EXP power constraint (Theorem 2).
Ii-a Definitions for the One-Way AWGN channel
In the one-way AWGN channel, terminal 1 (transmitter) wishes to transmit a message , selected uniformly from a set of equally likely messages to terminal 2 (receiver) using a code of fixed block length
. Forward and backward directions are characterized by independent AWGN channels. The AWGN noises are of zero mean and of variancesand respectively, both identically distributed and independent across users and channel uses. Figure 1 shows the one-way AWGN channel with active noisy AWGN feedback, characterized by Equations (4) and (5) for the -th channel use:
The model above captures noiseless feedback by taking , and the absence of feedback by taking . Channel inputs for each direction are subject to either an power constraint according to Equations (2) and (3). In the following we will make definitions for a system with feedback; definitions in the absence of feedback should be clear from context and omission of the corresponding feedback-related terms.
Let all be the set of reals and be a block length- code for the transmission of messages consisting of forward and feedback encoding functions,
leading to channel inputs and , and a decoding rule
that determines the best estimate of the transmitted message, denoted by .
Let denote the probability of error attained by a particular code under an power constraint. We define the achievable error exponent for the one-way AWGN channel with feedback under the power constraint as:
where the subscript FB indicates the presence of feedback and will be omitted for the non-feedback case. This notation follows that in  for the transmission of two messages with active noisy feedback.
Ii-B Error exponents for the one-way AWGN channel without feedback and with perfect output feedback
Results presented in this section correspond to the cases of feedback-free and perfect output feedback transmissions for a finite number of transmitted messages. These are introduced as references since they lower and upper bound what can be achievable with noisy feedback, respectively. The feedback-free scenario () was studied by Shannon in , who showed that under the AS power constraint and the transmission of messages, the best achievable error exponent is that of Equation (9) and attainable using a simplex code:
For and , Equation (9) becomes , and . Note that for large , .
The perfect-feedback scenario () was studied by Pinsker in , who showed that for the transmission of messages under the AS power constraint, the error exponent of the non-feedback AWGN channel shown in (9), can be improved up to:
Note that for , this coincides with the achievable exponent in (9) and hence no further improvements over the non-feedback error exponent are possible subject to the AS power constraint, even using perfect feedback [23, 24]. Pinsker’s result in (10) constitutes an upper bound on the error exponent under the AS constraint for the one-way AWGN channel with noisy feedback.
Ii-C Error exponents for the One-Way AWGN channel with noisy feedback: past work
When feedback is over a noisy channel, it is relevant to distinguish between active and passive feedback, roughly defined as follows:
In passive feedback, the forward channel output observed at the receiver is directly sent to the source and no encoding function is used. The transmitter sees a noisy version of the signal obtained by the receiver.
In active feedback, the forward channel output is encoded using a function that takes as argument the sequence of all channel outputs available at the receiver, i.e. , and this is returned to the transmitter at the -th channel use. Active feedback renders the returned transmission more robust against noise. Note that active feedback may mimic passive feedback by taking the encoding function in Figure 1 as a one-to-one mapping of signal for each channel use .
Yamamoto and Burnashev addressed the problem for the AWGN channel and noisy feedback under the AS power constraint for the transmission of a non-exponentially growing number of messages (zero-rate) using fixed block length encoding [21, 25, 26, 22]. Their work extends previous techniques used to demonstrate achievable error exponent results on the Binary Symmetric Channel (BSC) with noisy feedback [27, 28], to the AWGN channel. Under the AS power constraint and for a large (non-exponentially growing with ) number of messages , such that and , the following error exponent is achievable as in :
The number in the subscript of stands for the use of one switching moment
switching moment, or a change in the forward encoding function. Also, when , this becomes:
The result for very small feedback noise (11) was improved in . Then, for with as , an error exponent as that of Equation (13) is attainable after one switching moment. Note that for a very small noise variance in the feedback link, this yields a larger improvement than that of Equation (11) for very large .
Kim, Lapidoth and Weissman [29, 30] addressed the error exponents for the AWGN channel with feedback for the transmission of a small number of messages. They presented bounds on error exponents for active and passive feedback for the transmission of messages [18, 19] over the one-way AWGN channel with noisy AWGN feedback. Results for messages using passive feedback were presented by Xiang and Kim in . We summarize these results next. Several of their techniques have been useful in the extensions to general and to the two-way channel presented here.
Ii-C1 Achievable error exponents for the transmission of two messages
In  Kim et al. presented the following achievable error exponent for the transmission of two messages under the EXP constraint and passive feedback:
The use of active noisy feedback in the transmission of two messages is presented in , considering the channel shown in Figure 1 under the EXP power constraint for the forward channel and both AS and EXP power constraints for the feedback link. The achievable error exponent expressions using active feedback are respectively:
AS power constraint:
EXP power constraint:
Ii-C2 Achievable error exponents for messages under the AS power constraint and passive feedback
Xiang and Kim  studied the reliability function for the transmission of messages under the Peak Energy constraint (PE): (which satisfies the AS constraint as well) on the forward channel, and passive feedback. A block diagram of this scheme is shown in Figure 2 (left), which reflects how in passive feedback, the signal at the receiver is immediately returned to the transmitter at each channel use. Figure 2 (right) shows the scheme we propose for active feedback presented in Section II-D1.
The achievable error exponent derived in  is expressed as the minimum of three terms; the first two related to the forward and feedback transmissions, and the last resulting from a decoding rule that considers both:
Above, is used to characterize the duration of the transmission ( channel uses) and retransmission ( channel uses) stages, and the size of a protection region that determines whether a retransmission is necessary, see [20, Figs. 5-6]. Since the first and third terms in (17) are equal for , the above may be rewritten in a way that explicitly shows the contribution of the passive feedback stage as:
Comparing the above result with (9) for indicates that the first argument of the function yields a gain over the feedback-free error exponent for any (with a maximum of for ). Once is chosen, therefore ensuring that the first argument does provide an error exponent larger than that in the absence of feedback, the improvement is maintained if the second argument is equal to or greater than the first. This leads to:
i.e., if is very small, the feedback link’s SNR needs to be remarkably larger than the forward link’s. Note that the largest error exponent achievable by (18) requires choosing very close to zero, and requires a very high SNR in the feedback link. In contrast, choosing close to leads to the smallest error exponent attainable by the first argument of (18) (which coincides with the feedback-free exponent of Equation (9)). This error exponent is achievable as long as the feedback SNR satisfies , otherwise, the scheme would not even achieve the feedback-free error exponent. The error exponent of [20, Theorem 1 ] shown in Equation (17) can be generalized for all finite (see [20, Appendix]), which may be manipulated into the following form:
Ii-D Error exponents for the one-way AWGN channel with noisy feedback: contributions
In this section we present new achievable error exponents for the one-way AWGN channel for the transmission of a finite number of messages under the AS and EXP power constraints, with active, noisy feedback.
Ii-D1 Achievable error exponents for messages under the AS power constraint and active feedback
The achievability scheme presented in Figure 2 (left) and proposed in  can be slightly modified to employ active feedback as shown in Figure 2 (right) for a finite number of messages . In contrast to the two stage block diagram of passive feedback, active feedback involves three non-overlapping stages. First, a feedback-free transmission based on a simplex code of messages is used to send during channel uses. Once the first stage is complete, the receiver opts for immediate decoding only if the transmitted signal has been received within a protection region (similar to that defined for passive feedback in ) and if so, it ignores all future transmissions until the next message is sent. If the received signal is outside the protection region, the active feedback stage takes place for channel uses. In this stage, the receiver uses a simplex code of messages to inform the transmitter of the most likely pair of codewords it has determined: , where represent the two closest (minimum distance) messages to the received signal . The transmitter decodes message as , and uses this to generate a binary retransmission signal aimed to help the receiver making the right decision based on the true message . The final retransmission stage lasts for channel uses ( ), and corresponds to the transmission of two antipodal signaling codewords that are generated depending on whether the true message is equal to the first or second element in . If the true message is not in , this is counted as an error, and nothing is transmitted to the receiver. This approach leads to the first of our main results:
An achievable error exponent for the transmission of a finite number of messages over a one-way AWGN channel with active noisy feedback under the AS power constraint is given by:
for parameter .
In Equation (21), the second argument of the operator corresponds to the active feedback contribution to the probability of error given by the use of a simplex code -Equation (9)- and the transmission of messages corresponding to all possible unordered codeword pairs, which as in the case of three messages, are labeled in lexicographic order for the sake of identification. A comparison of Equations (21) and (20) shows that the second argument of the function under active feedback does not depend on . Active feedback uses a feedback-free transmission of messages based on a simplex code, and not on the transmission of the exact location of the received signal as it is in passive feedback. In the latter, the probability of error is based on the geometry defined for the protection regions determined by parameter . Section IV presents the complete proof of this result. First, we present the illustrative case of and later generalize this for any finite .
Ii-D2 Achievable error exponents for the transmission of messages under the EXP power constraint and active noisy feedback
This section presents our second contribution that results from a direct generalization of the work of Kim, Lapidoth and Weissman  for messages to an arbitrary but finite number . This is presented in Theorem 2, whose proof is presented in Section V.
An achievable error exponent for the transmission of a finite number of messages over a one-way AWGN channel with active noisy feedback under the EXP power constraint in both the forward and feedback directions is given by:
Iii Error exponents for the Two-Way AWGN channel
The two-way channel was first introduced by Shannon in  and further studied by Han in . The capacity region of the two-way AWGN channel (with independent noise across the terminals) is known, and is a rectangular region where both users may simultaneously attain their interference-free AWGN capacity. The rates achievable for this channel can not be increased by interaction or adaptation between the two terminals. The question pursued here is whether the same is true of error exponents – may they be improved through the use of adaptation/interaction between the terminals in a two-way setting, where feedback and messages must share the same resources. Two-way error exponents have not been studied in the past, to the best of our knowledge. In the two-way AWGN channel each terminal may intuitively perform two types of tasks: 1) transmission of their own message, and 2) transmission of feedback information for the other terminal. Since each terminal may use part of its available power to cooperate with the opposite direction, we have found that gains resulting from interaction may come at the price of a reduction of the error exponent of the terminal providing feedback – there is a tradeoff between the achievable error exponents in the two directions (at least under the presented schemes).111Recall that we are operating at zero-rate, so the tradeoff with rate is not captured in this simplified, yet still challenging setting. However, the error exponent region does generally depend on , the number of messages being transmitted in each direction. Extensions to having a different number of messages be sent in each direction is left for future work.
Iii-a Two-way AWGN channel model description and definitions
The two-way AWGN channel is depicted in Figure 3, comprising two users denoted as terminal for . Terminal transmits message , uniformly selected from to terminal using a fixed block length code of size .
The general two-way AWGN channel model is presented in Equation (23), which characterizes the channel output received at the -th terminal
where, for , is a constant, corresponds to channel inputs satisfying the input block power constraint , to channel outputs and to zero-mean Gaussian noise processes, each independent and identically distributed across channel uses. The model described by (23) can be simplified by noting that each terminal can subtract its own transmission , thus, the two-way AWGN channel may equivalently be represented as in (24), where each link is modeled as an independent and interference free AWGN channel with a noise variance perceived at the -th terminal’s receiver, with for simplicity:
Here, we characterize error exponents for the zero rate operational rate pair, i.e. for , and for different ratios between the SNRs in the two directions. Let be the signal-to-noise ratio for link . Then we will consider both symmetric and non-symmetric channels. The non-symmetric case is of particular importance under the AS power constraint, since as indicated in [20, 25, 22] only a feedback channel stronger that the forward direction is able to attain gains over feedback-free error exponents on the one-way AWGN channel. This condition becomes even more critical for the two-way channel, since the feedback link is also used for messages transmission. Moreover, symmetric SNR channels seem to be unable to achieve an error exponent region greater than those achieved by independent transmissions under the AS constraint.
In Section II, we reviewed how error exponents of the one-way AWGN channel can be improved by the inclusion of (even noisy) feedback. In all cases, the receiver’s resources are dedicated solely to feedback and helping the message transmission. If the receiver transmits its own messages as well, the one-way channel with noisy feedback transforms into the two-way channel and results from the former can be applied in the latter. Since messages flow in two directions, an error exponent pair must be considered now.
Let be a block length- code, consisting of two encoding and two decoding rules as depicted in Figure 3. Each terminal’s encoding rule consists of a set of functions, defined for the -th channel use as:
leading to the -th channel inputs for terminal : . Decoding rules are denoted by , and estimate the received message based on the sequence as:
Let and similarly denote the probability of error in the forward and backward directions simultaneously achieved by a particular code under power constraints.
A pair of error exponents is called achievable for the transmission of a finite number of messages , under the power constraint for the two-way AWGN channel, if there exists a code such that for large , simultaneously
The error exponent region for the two-way AWGN channel for the transmission of messages corresponds to the union over all achievable error exponent pairs , where we will often drop the arguments of for simplicity and sometimes we may refer to as .
Next we present and extend the results initially demonstrated in , with proofs in the upcoming sections.
Iii-B Achievable error exponent for non-interactive terminals
While in the two-way setting, both terminals may adapt their current inputs to past received channel outputs, they need not do so, and may ignore (for the purpose of generating their channel inputs) the received outputs altogether. The following proposition establishes the achievable error exponent region when the terminals do not interact:
The achievable error exponent region for non-interactive terminals is formed by the union over all simultaneously achieved error exponents pairs under both, AS and EXP power constraints for the transmission of messages:
The above equations follow directly from applying Shannon’s result of , given in Equation (9), to each communicating pair, and using a simplex code in each direction. In this scheme, both terminals are concerned about their own message transmission only, and do not allocate any resources to help the other direction.
Iii-C Achievable error exponent region under interactive terminals
In this section we analyze the error exponent achieved by interactive transmission protocols, and have identified certain scenarios (one direction much better than the other) for which under the AS constraint, interaction leads to an improvement over the feedback-free error exponent region characterized by Proposition 1. This improvement comes at the cost of an error exponent decrease linked to the amount of power a terminal is unable to use for its own message transmission, as it was allocated to serve the opposite direction through feedback.
As described for the one-way AWGN channel, the power constraint imposed on the input block codewords and the number of messages to be transmitted, along with channels’ SNRs determine whether error exponent gains are feasible for one or both communication directions through interaction. Consider for example the AS constraint and the transmission of two messages, and observe that interaction can not improve the two-way error exponent as even noiseless feedback is unable to improve the non-feedback error exponent of a binary transmission for a one-way AWGN channel under AS constraint . Therefore, even in the ideal case that a noiseless feedback link is available for each direction (and not used to transmit the true messages), the highest attainable error exponent for each direction coincides with the non-feedback one, which suggests that non-interactive transmissions suffice for . In general, the same upper bound applies for the transmission of messages (Pinsker’s upper bound is independent of ) and error exponent gains over non-feedback are possible for in both directions.
Iii-C1 Achievable error exponents region under the AS power constraint
We study the transmission of messages over an AWGN two-way channel for which , as no gains over Proposition 1 appear (with current known achievability schemes) to be attainable for symmetric SNRs, as it was the case for the one-way AWGN channel under the AS constraint as well [20, 25, 26]. Assuming that the communication direction is noisier than the direction, consider a two-way achievability scheme that takes advantage of this asymmetry: the stronger link can be used during a fraction of the block length to transmit message without the help of feedback using a simplex code, and for another fraction of time, to improve the error exponent of the weaker direction by providing passive or active feedback in the transmission of message . Figure 4 shows these two approaches, for passive and active feedback respectively. The two schemes differ in when the transmission of message occurs. In the case is transmitted with passive feedback, message is sent during the first channel uses (for ) over the stronger link while the weaker channel remains idle, since Terminal 1 may initiate its own transmission (helped by terminal 2) only once the message transmission concludes. In the remaining channel uses, message is transmitted employing passive feedback, as described in Section II for the one-way AWGN. Since terminal 2 has already used part of its available power for the forward transmission, only the remainder can be used to serve the other direction through feedback. Note that in passive feedback, a signal received at Terminal 2 at the -th channel use is immediately fed back to the Terminal 1 without delay, and that both directions are busy at the same time.
The case of active feedback, shown in Figure 4 (right), differs from passive feedback in that the transmission of , and the first stage of the active feedback supported transmission of may occur simultaneously. Note that this is possible since both directions are independent, and also, because the active feedback stage can only start once the first stage transmission is concluded, which leaves room for the transmission of .
An achievable error exponent region for the transmission of messages over a two-way AWGN channel with non-symmetric SNR , under the AS power constraint and passive feedback is the union over all error exponent pairs over parameters , and satisfying:
Equation (31) follows from (9), the use of a non-feedback transmission for message in the first channel uses using a simplex code of symbols. Equation (32) follows from direct application of the noisy feedback aided scheme presented in [20, Section II-B] and reviewed in Section II-C2, which is used for the transmission of message in the remaining channel uses. Note that (32) is presented in the form of (18) since it explicitly shows the error exponent contribution of the forward and feedback transmissions.
The achievable error exponent region for the two-way AWGN channel with passive feedback under the AS power constraint is presented in Figure 5, parametrized by as:
An achievable error exponent region using active feedback, under asymmetric SNRs and the AS power constraint in each direction is presented in Theorem 4:
An achievable error exponent region for the transmission of messages over a two-way AWGN channel with non-symmetric SNR , under an AS power constraint and active feedback is the union over all error exponent pairs over parameters and satisfying:
As in Theorem 3, Equation (37) follows from using (9) for channel uses and a simplex code of symbols for the transmission of . Equation (38) results from the use of encoded feedback. Specifically, message is transmitted in channel uses employing the active noisy feedback-aided scheme for the one-way AWGN channel communication presented in Section II-D1, in Theorem 1, Equation (21).
The error exponent region for the case of active feedback is shown in Figure 6. A portion of this region can be characterized for as the line
The scheme shown in the block diagram of Figure 4 yields the largest error exponent region derived so far for interactive terminals under non-symmetric SNRs under the AS power constraint. It appears that error exponent gains over feedback free transmission are possible only for the weaker direction, which is not surprising given the results for the one-way channel with noisy feedback under an AS power constraint.
Next, we consider the EXP power constraint, which holds for all relative SNR conditions. As described in Section I, the EXP power constraint permits very high amplitude transmissions associated to very rarely occurring events, which are used to correct detected decoding errors. In the following subsections, we show that this unique feature can be leveraged for the two-way AWGN channel as well.
Iii-C2 Achievable error exponents region under the EXP power constraint
This achievability scheme is based on the use of the building block component introduced by Kim et al. in  for both directions. We modified this component, originally designed for the one-way AWGN channel, to operate for a general number of messages as presented in Theorem 2 and shown in Section V. Here, we utilize it for the simultaneous transmission of messages in opposite directions. Note that we can employ a similar approach as that used for the one-way transmission, specifically: an initial transmission of the true message followed by a feedback transmission that returns the message estimated by the receiver, followed (possibly) by a high amplitude retransmission to correct decoding errors if necessary. In the two-way channel, these three stages can occur simultaneously, with the restriction that each transmitter must satisfy the power constraints. This scheme results in the region of Theorem 5, whose proof is presented in Section VI:
An achievable error exponent region for the two-way AWGN channel and the transmission of a finite number of messages under the EXP power constraint for both directions is given by the union over all error exponent pairs over for which:
where and such that for .
where the equality in (45) follows from (136) and taking (to use all the power). Figure 7 shows this error exponent region. Observe that this region can be viewed as time-sharing two one-way schemes, each characterized by the exponent in Theorem 2, in which each direction operates for a fraction of time aided by the other terminal. The axis-crossing points are obtained when the whole block length and all power is dedicated to the transmission of one direction only.
Theorem 5 concludes the statement of our main results. The remainder of the paper consists of the proofs and numerical evaluations of these regions, so that their performance can be visually compared.
Iv Achievability scheme for the One-Way AWGN channel for under AS power constraint and active feedback: Proof of Theorem 1
In this section, the proof of Theorem 1, Equation (21) follows by generalizing the geometric technique used for messages introduced by Xiang and Kim  for passive feedback and modified towards the use of encoded feedback. We have reused most of their notation in order to highlight the differences when active feedback is used. We first derive the achievable error exponent expression for the transmission of messages, then generalize it to arbitrary but finite .
Some of the results presented here rely on the use of a simplex code denoted by for the transmission of symbols from the -th transmitter, using symbol energy and leading to codewords of length . Equation (46) shows the definition of the simplex code .
Iv-a Achievable error exponents for the transmission of three messages under the AS power constraint and active feedback
Consider transmitting one of three equally likely messages from over the one-way AWGN channel with active feedback of Figure 1 and assume the feedback link is strictly better than the forward channel, specifically: . The active feedback achievability scheme block diagram under the AS power constraint is shown in Figure 2 (right). Transmission of is performed in channel uses through three stages: transmission, active feedback and retransmission, and corresponds to a modified version of the scheme proposed in  that uses active feedback instead of passive. The three stages are analyzed next.
1. Transmission: The first stage occurs during channel uses, where message is transmitted without feedback as codeword using the simplex code defined in (46). At the receiver, the signal is decoded using protection regions , one for each transmitted codeword as shown in Figure 8 and defined in (IV-A), as in . All received signals inside region are immediately declared as the codeword (and no feedback is necessary). These regions are parametrized by following [20, Equation 6] for a parameter which is geometrically coupled to . Further reading regarding this idea can be found in .