https://orcid.org/0009-0002-7856-1804
Figures
Abstract
The Schalkwijk-Kailath (SK) scheme for the AWGN channel with noise-free feedback is well-known since its coding complexity grows linearly with the coding blocklength, and it is capacity-achieving and has extremely lower decoding error probability comparing with existing excellent codes such as LDPC and Polar codes. However, extension of SK scheme to more practical scenarios is challenging since it depends heavily on the feedback success. This paper aims to investigate how to design SK-type schemes for channels with intermittent feedback and establish the relationship between the coding blocklength, the desired decoding error probability, the achievable transmission rate and secrecy level of our proposed schemes. Specifically, first, for the additive white Gaussian noise (AWGN) channel with intermittent noise-free feedback, a variation of the well-known Schalkwijk-Kailath (SK) scheme for the AWGN channel with noise-free feedback is proposed, and the corresponding achievable rate is characterized for given coding blocklength and decoding error probability. Subsequently, the proposed scheme is extended to channels with noisy intermittent feedback. To quantitatively evaluate the secrecy performance, we adopt the eavesdropper’s normalized equivocation as the secrecy metric and analytically characterize the achievable secrecy level of the proposed schemes. In particular, we show that perfect weak secrecy, i.e., asymptotically vanishing information leakage rate, can be achieved under certain conditions. Numerical results show that for a given decoding error probability threshold, our proposed schemes require lower signal-to-noise ratio and significantly shorter coding blocklength comparing with LDPC code. The study of this paper may provide a way to construct efficient coding scheme for channels in the presence of intermittent feedback.
Citation: Yang J, Huang Q, Lin Y (2026) Coding for channels with intermittent feedback and its security analysis. PLoS One 21(4): e0347790. https://doi.org/10.1371/journal.pone.0347790
Editor: Neng Ye, Beijing Institute of Technology, CHINA
Received: October 31, 2025; Accepted: April 7, 2026; Published: April 29, 2026
Copyright: © 2026 Yang et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: This paper focuses on theoretic results, no data is used and all the detailed proof of our theorems are given in our manuscript. The simulation scripts utilized in this paper can be accessed at the following link: https://github.com/LightLion4/Codes.git.
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
Ultra-reliable and low-latency communications (URLLC) is becoming one of the critical services in 5G and future 6G wireless communications [1–3] since it aims to guarantee high reliability levels, and requires coding scheme with significantly short coding blocklength due to strict latency constraint in several practical scenarios, such as road safety information and autonomous driving.
One possible effective solution to URLLC is feedback control based coding scheme, which was first proposed in [4], known as the Schalkwijk-Kailath (SK) scheme. In [4], the additive white Gaussian noise (AWGN) channel with noise-free feedback was studied, where the channel feedback helps the transmitter to construct a highly efficient coding scheme. In this scheme, at the first time instant, the message is directly transmitted over the AWGN channel, and the receiver adopts a zero-forcing method to do his first estimation about the message. By noise free channel feedback, the transmitter knows the message’s first estimation by the receiver, and sends the estimation error (difference between the estimation and the real message) at the second time instant. Once receiving the signal, the receiver applies linear minimum mean square estimation (LMMSE) to the received signal and obtains a new estimation about the estimation error at the last time instant, and then he updates his estimation about the message by using this new estimation and the initial estimation. By iteration, it was shown that the receiver’s estimation error about the message vanishes with the increasing of the coding blocklength. Later, [5] showed that the SK scheme is in fact a feedback control based scheme, and re-presented this scheme from control-theoretic aspect. Note that at each time instant, the encoding-decoding procedure of the SK scheme consists of 5 linear operations, which indicates that the coding complexity of the SK scheme grows linearly with the coding blocklength N, and can be approximately denoted as . Besides this, for transmitting a bitstream with K bits, the SK scheme only requires at least 2 × K bits memory at both the transmitter and receiver. These indicate that the SK scheme has extremely low encoding-decoding complexity, which shows potential for channel coding in practical communication systems [6–10]. Besides this, in [4], it was shown that as the coding blocklength approaches infinity, the decoding error probability of the SK scheme declines doubly exponentially to zero, which indicates that to achieve a fixed decoding error probability, the coding blocklength of the SK scheme is significantly shorter than those of the well-known linear block codes, such as low-density parity-check (LDPC) codes and Polar codes.
Another interesting property of the SK scheme is that it satisfies the physical layer security (PLS) requirement by itself. Here recall that the PLS was first investigated by Shannon [11], and subsequently, Wyner [12] studied how to transmit a message over a noisy channel with perfect secrecy guaranteed. The secrecy capacity, which is the maximum transmission rate with perfect secrecy, was characterized. [13] showed that for the AWGN channel with noise-free feedback and an external eavesdropper, the SK scheme is the optimal secure scheme for such a model, which indicates that the SK scheme not only achieves the optimality, but also is self-secure. This self-secure property of the SK scheme has been extensively studied in literature, see [14–18].
Though the SK scheme performs excellent with low encoding-decoding complexity, its application to practical communication systems still has a long way to go since it is based on the noise-free feedback. In recent years, the SK-type schemes have been extensively studied for more practical feedback channel models, including quantized feedback channel [19], AWGN feedback channel [20], and feedback channel with arbitrary delay time [21]. However, note that in practical mobile communication scenarios, the feedback is often interrupted by obstacle and inter-cell handover etc., and this scenario is commonly modeled as intermittent feedback [22]. The optimal signaling strategies for the average-power-limited AWGN channel with intermittent feedback were investigated in [23,24], and [22,25] established bounds on the capacities of various channel models with intermittent feedback, which are based on random coding argument, and this implies that the coding schemes of [22,25] are non-constructive. Then it is natural to ask: is it possible to design efficient and low-complexity coding scheme for channels with intermittent feedback?
In this paper, we answer the aforementioned question by step-and-step extending the classical SK scheme to the following cases:
- We propose an SK-type schemes for the AWGN channel with noise-free intermittent feedback, and establish the relationship between the coding blocklength, the desired decoding error probability, and the achievable transmission rate of our proposed scheme.
- We further extend the above scheme to the noisy intermittent feedback cases.
- We analyze the secrecy levels of the above proposed schemes, and show that PLS can be achieved for some cases.
The remainder of this paper is organized as follows. Section 2 is about our results on the AWGN channel with noise-free/noisy intermittent feedback. Section 3 is about our results on the quasi-static fading channel with noise-free/noisy intermittent feedback. Security analysis of our proposed schemes is shown in Section 4. Section 5 concludes this paper and discusses future works.
2 The AWGN channel with intermittent feedback
2.1 Model formulation
Channel: Fig 1 shows the AWGN channel with intermittent feedback. For the feedfoward channel, the channel input-output is given by
For the noisy intermittent feedback, at time , the feedback channel input-output is given by
Here ,
are identical independent distributed (i.i.d.) Gaussian noises and they are independent of each other.
Intermittent feedback: We introduce the state sequence
which denotes the Bernoulli process, and it is distributed i.i.d. over all time instants. Here note that Si = 1 indicates that the transmitter perfectly obtains Yi, and Si = 0 indicates that the transmitter fails to receive the receiver’s feedback at time instant i. Define
for all , which is the probability that the transmitter fails to obtain the receiver’s feedback at time instant i.
Encoder:
- The input message W is uniformly distributed over the set
.
- For the noise-free feedback case, the encoder with output
satisfies the average power constraint.
where is an encoding function of the transmitter at time index i
,
and
. For simplification, define the feedforward signal-to-noise ratio as
.
- For noisy feedback case, the encoder with output
and
satisfy the average power constraints
where and
are encoding functions at time index i
,
,
, and
. For simplification, define the feedback signal-to-noise ratio as
.
Decoder:
- The output of the decoder is
, where
is the decoding function of the receiver.
- The average decoding error probability is given by
(8)
Achievable rate and capacity:
The -rate R is achievable for given blocklength N and decoding error probability
, there is a
-code such that
For the AWGN channel with noise-free intermittent feedback, the achievable rate is denoted by , while for the noisy case, denoted by
. Furthermore, the capacities
and
are the maximum among all achievable rates defined above.
2.2 Main results and numerical examples
Theorem 1: For given coding blocklength N and decoding error probability , the lower bound
on the capacity
of the channel with intermittent noiseless feedback is given by
where for sufficiently large N.
Proof: See subsection 2.3.
Theorem 2: For given coding blocklength N and decoding error probability , the lower bound
on the capacity
of the channel with intermittent noisy feedback is given by
where for sufficiently large N.
Proof: See subsection 2.3.
Numerical result:
Fig 2 compares the achievable rates of our schemes for noise-free and noisy feedback cases. From this figure, we conclude that the achievable rate for the noisy feedback case increases as the signal-to-noise ratio of the feedback channel increases, and it approaches its asymptotic value (the achievable rate for the noiseless feedback case) when the signal-to-noise ratio tends to infinity.
As shown in Figs 3 and 4, we see that to achieve a desired decoding error probability (10−7), the coding blocklength for the noise-free/noisy feedback case is about 50, which is significantly shorter than those of the LDPC and Polar codes. In addition, from Fig 4, we see that the feedback channel noise leads to the decreasing of the performance of our proposed scheme, e.g., for given N = 50, to achieve a desired decoding error probability (10−7), the transmission bits of the noisy feedback scheme is reduced to 75 percent of that of the noiseless feedback scheme.
Fig 5 shows the relationship between the achievable rates of the scheme for the Gaussian channel with intermittent feedback under different . When the coding blocklength is given, the achievable rate gradually decreases as
increases. Under the same
, the achievable rates increase with the increase of the coding blocklength.
We also provide performance comparison of our proposed schemes and LDPC code for the same model without using channel feedback. Specifically, Fig 6 shows that for and fixed transmission rate 1, to achieve a targeted decoding error probability, our proposed schemes require lower signal-to-noise ratio and significantly shorter coding blocklength comparing with LDPC code. Fig 7 shows that for
and fixed transmission rate 0.5, the advantage of our schemes still holds. Besides this, comparison between the SK code and the AttentionCode [26] is given (see Fig 8). Fig 8 compares the decoding error probabilities of the SK code and AttentionCode under various channel qualities, and it is easy to see that the SK code performs better than the AttentionCode for some cases.
2.3 Proofs
Proof of Theorem 1:
Coding procedure:
Since W is uniformly distributed in ,
is approximately uniformly distributed over the interval
and its variance is approximately equal to
, i.e.,
. At time instant 1, the transmitter sends
At time instant i (2 ≤ i ≤ N), if Si−1 = 0, the transmitter repeatedly sends
If Si−1 = 1, define , where
for
. For example, if Si−1 = (0, 1, 1, 0, 0),
.
If ai−1 = 1, the receiver gets
and sends it back to the transmitter, then computes his first estimation of by
where is the estimation error, and its variance is
. Thus, the transmitter sends the i-th time codeword Xi by
If ai−1 > 1, the receiver gets and sends it back to the transmitter. Then both the receiver and the transmitter calculate the estimation
by
where , and
is the Minimum Mean Squared Error (MMSE) estimation coefficient, which ensures that
is correctly estimated from Yi−1, and thus
where (a) follows from and
. The updated estimation error is
and . Then the transmitter sends the i-th time codeword Xi by
By calculation, it is not difficult to show that the general term of is given by
Decoding Error Probability Analysis: The receiver’s final estimation of is
, where aN = aN−1 + 1, and the decoding error occurring at time instant N is defined as
Thus, the decoding error probability Pe is given by
where (a) follows from Q(x) is the tail of unit Gaussian distribution evaluated at x, and denotes the variance of the receiver’s estimation error
at time instant N. From (25), we conclude that for obtaining the achievable rate Rf, the variance
should be determined first, which is obtained as follows.
Statistical Analysis: First, from the coding procedure in the proof of Theorem 1, we calculate the receiver’s estimation error at each time instant as follows:
By iteratively using (26), we have
which substituting into (25) and setting yields
The proof is completed.
Proof of Theorem 2:
Coding procedure:
In the following, we introduce a shared dither random i.i.d. sequence , which is perfectly known by both transmitter and receiver.
and
. The sequence
mutually independent of the noise sequences and the message, and it is used in the coding process. Then we describe the details of our coding scheme.
At time instant 1, the transmitter sends
At time instant i (2 ≤ i ≤ N), if Si−1 = 0, the transmitter repeatedly sends
If Si−1 = 1 and ai−1 = 1, the receiver obtains and calculates the first estimation of
by
where is the estimation error, and its variance is
.
Then the receiver sends the estimation of to transmitter over the feedback channel by
where is the modulation coefficient to avoid modulo-aliasing. To eliminate the impact of the feedback channel noise, we apply the modulo-lattice operation
.
The transmitter obtains , and then calculates a noisy version of the receiver’s estimation error by
In the case when , which means the modulo-aliasing errors do not occur, we obtain
then the transmitter sends the i-th time codeword Xi by
where is set to satisfy input power constraint P.
if ai−1 > 1, the receiver obtains , and updates the estimation of
by
where
the estimation error at time instant i − 1 is
and the variance of is denoted as
.
Then the receiver sends the estimation of to the transmitter by
the transmitter obtains , and then calculate a noisy version of estimation error of the receiver by
in the case when , we obtain
. The transmitter sends the i-th time codeword Xi by
Decoding Error Probability Analysis: The scheme is given in terms of the parameters ,
and
which dominate the transmission performance. The specific choices are described as follows.
where
and
are defined in (12). The proof of (42) and (43) and the derivation of the achievable rate of our scheme are given in Appendix A. The proof is completed.
3 The quasi-static fading channel with intermittent feedback
3.1 Model formulation
Channel: Fig 9 shows the quasi-static fading channel with intermittent feedback. For the feedfoward channel, the channel input-output is given by
For the noisy intermittent feedback, at time , the feedback channel input-output is given by
Here and
are i.i.d. circularly symmetric complex Gaussian noises and they are independent of each other.
The intermittent feedback is defined similar to that of the AWGN channel case, hence we omit its explanation here.
Encoder:
- The message W is uniformly distributed over the set
.
- For the scheme with noise-free feedback, the encoder with output
satisfies the average power constraint
(46)
whereis an encoding function of the transmitter at time index i
,
and
. For simplification, define the feedforward signal-to-noise ratio as
.
- For the scheme with noisy feedback, the encoder with output
and
satisfy the average power constraints
(47)
(48)
whereand
are encoding functions at time index i
,
,
, and
. For simplification, define the feedback signal-to-noise ratio as
.
Decoder:
- The output of the decoder is
, where
is the decoding function of the receiver.
- The average decoding error probability is given by
(49)
Achievable rate and capacity:
For the fading channel with intermittent feedback, the achievable rates are denoted by and
, and the capacities
and
are the maximum among all achievable rates.
3.2 Main results and numerical examples
Theorem 3: For given coding blocklength N and decoding error probability , the lower bound
on the capacity
of the channel with intermittent noiseless feedback is given by
where for sufficiently large N.
Proof: See subsection 3.3.
Theorem 4: For given coding blocklength N and decoding error probability , the lower bound
on the capacity
of the channel with intermittent noisy feedback is given by
where for sufficiently large N.
Proof: See subsection 3.3.
Numerical result:
Fig 10 compares the achievable rates of fading channel schemes for noise-free and noisy feedback cases. From this figure, we conclude that the achievable rate for the noisy feedback case increases as the signal-to-noise ratio of the feedback channel increases, and it approaches the achievable rate for the noiseless feedback case when the signal-to-noise ratio tends to infinity.
As shown in Fig 11, we see that to achieve a desired decoding error probability (10−7), the coding blocklength for the noise-free/noisy feedback case is about 50, which is significantly shorter than those of the LDPC and Polar codes. In addition, from Fig 12, we see that the feedback channel noise leads to the decreasing of the performance of our proposed scheme.
Fig 13 shows the relationship between the achievable rates of the scheme for fading channel with intermittent feedback under different . When the coding blocklength is given, the achievable rate gradually decreases as
increases. Under the same
, the achievable rates increase with the increase of the coding blocklength.
3.3 Proofs
Proof of Theorem 3:
Coding procedure: At time instant i, since the elements in (44) are complex numbers, (44) can be re-written as
where , YR,i = Re(Yi), YI,i = Im(Yi),
,
,
,
,
,
, where
and
denote the real and imaginary parts of a complex element, respectively. Here note that
and
, where
. Then convert (55) into real and imaginary parts
Then, we obtain
from (57), we define
thus (57) can be further expressed as
where and
. Hence, (59) is equivalent to (55), which indicates that the feedforward and feedback channels are divide into the two sub-channels.
Since W is divided two independent uniformly distributed sub-messages (WR, WI), where WR and WI take the value in the set and
, and we have
. Divide the interval
into
(
) equally spaced sub-intervals, and the center of each sub-interval is mapped to a message value in WR(WI). Let
(
) be the center of the sub-interval with respect to (w.r.t) the message WR(WI), and
.
At time instant 1, the transmitter sends
At time instant i (2 ≤ i ≤ N), if Si−1 = 0, the transmitter repeatedly sends
If Si−1 = 1, define , where
for
. For example, if Si−1 = (0, 1, 1, 0, 0),
.
If ai−1 = 1, the receiver gets , and calculates the first equivalent output signal
and sends it back to the transmitter, then computes his first estimation of and
by
then the transmitter can get the estimation error of the receiver
Thus, the transmitter sends the i-th time codeword by
If ai−1 > 1, the receiver gets and sends it back to the transmitter. Then both the receiver and the transmitter calculate the estimation by
where
the updated estimation error are
then the transmitter sends the i-th time codeword by
Statistical Analysis: By calculation, it is not difficult to show that the general term of is given by
Performance Analysis:
Since W is divided into two independent uniformly distributed sub-messages (WR, WI), we define the decoding error probability of (WR, WI) as Pe,R and Pe,I. The decoding error probability analysis process is the same for the real part and the imaginary part, and the corresponding decoding error probability of the real part is analyzed below. The receiver’s final estimation of is
, where
, and the decoding error probability
occurring at time instant N is defined as
For given coding blocklength N and decoding error probability , substituting (70) into (71), we can get
similarly, we can get
From (72) and (73), the achievable rate of the fading channel with intermittent noise-free feedback is given by
where for sufficiently large N. The proof is completed.
Proof of Theorem 4:
At time instant i, since the elements in (44) are complex numbers, (44) can be re-written as
where , YR,i = Re(Yi),
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, where
and
denote the real and imaginary parts of a complex element, respectively. Here note that
,
,
, and
where
and
. Then convert (75) into real and imaginary parts
where ,
,
, and
. Then, we obtain
from (77), we define
thus (77) can be further expressed as
where ,
,
and
. Hence, (79) is equivalent to (75), which indicates that the feedforward and feedback channels are divide into the two sub-channels.
Since W is divided two independent uniformly distributed sub-messages (WR, WI), where WR and WI take the value in the set and
, and we have
. The interval
divides into
(
) sub-intervals, and maps WR(WI) to the center of sub-interval. Since WR(WI) is uniformly distributed in
(
),
(
) is approximately uniformly distributed over the interval
and its variance is approximately equal to 1, i.e.,
.
In the following, we also introduce shared dither random i.i.d. sequence and
,
and
, where
,
. The sequence
and
mutually independent of the noise sequences and the message, and they are involved in the following iterative encoding procedure, which ensures that the encoded signals satisfy the transmission power constraints. Then we describe the details of our coding scheme.
At time instant 1, the transmitter sends
At time instant i (2 ≤ i ≤ N), if Si−1 = 0, the transmitter repeatedly sends
if Si−1 = 1 and ai−1 = 1, the receiver obtains and calculates the first equivalent output signal
and sends it back to the transmitter, then computes his first estimation of and
by
then the transmitter can get the estimation error of the receiver
where
Then the receiver sends the estimation of and
to transmitter over the feedback channel by
where and
are the modulation coefficient to avoid modulo-aliasing. To eliminate the impact of the feedback channel noise, we apply the modulo-lattice operation
.
The transmitter obtains
and then calculates a noisy version of the receiver’s estimation error by
In the case when and
, which means the modulo-aliasing errors do not occur, we obtain
then the transmitter sends the i-th time codeword by
where and
are set to satisfy input power constraint PR and PI.
If ai−1 > 1, the receiver obtains , and updates the estimation of
and
by
where
the estimation error at time instant i − 1 is
and the variance of and
is denoted as
and
.
Then the receiver sends the estimation of and
to the transmitter by
the transmitter obtains and
, and then calculate a noisy version of estimation error of the receiver by
in the case when and
. The transmitter sends the i-th time codeword by
Performance Analysis: In our proposed scheme, we define the parameters ,
and
as follows, and the transmission performance of our scheme is determined by these parameters.
where
and
are defined in (52) and (53). The proof of
,
,
and the derivation of the achievable rate of our scheme are given in Appendix B. The proof is completed.
4 Security analysis of our schemes
4.1 The fading wiretap channel with intermittent noise-free feedback
In this subsection, we analyze the secrecy level of our scheme for the fading wiretap channel with intermittent noise-free feedback, see Fig 14.
In Fig 14, the wiretap channel is given by
where ,
are the input and output of the feedforward channel at time instant i. The
,
and
are i.i.d as
,
and
, respectively.
are the fading coefficients of the eavesdropping channel.
Definition:
The uncertainty of the eavesdropper (also called the secrecy level) is defined as
where and
,
corresponds to perfect secrecy, which follows from the same definition in [27].
For the fading eavesdropping channel scheme with noise-free intermittent feedback, the codeword sent at the previous instant should be re-sent at the next instant due to the failure of feedback, hence if the feedback failure occurs, it has an impact on the secrecy level, to this end, we define until the time instant
the codeword containing the is successfully sent and feedback.
Theorem 5: For given coding blocklength N and decoding error probability , the lower bound on the security level of the fading channel with noise-free intermittent feedback is given by
where and
represent the information leaked through the feedforward channel and the feedback channel, respectively.
Proof. From (102), we can obtain
where (a) follows from , (b) follows from that
is a function of the channel noise, when the channel noise is known, Xi,
and
are all known, (c) follows from
, (d) follows from that
is a function of
, W and
are mutually independent, (e) follows from that
,
and
are mutually independent, (f) follows from
similarly, we can get , (g) follows from
. Therefore, the proof is completed.
4.2 The fading wiretap channel with intermittent noisy feedback
Fig 15 shows the fading wiretap channel with intermittent noisy feedback. In this model, the wiretap channel is given by
where ,
are the input of the feedforward channel and the feedback channel at time instant i. The
,
,
and
are i.i.d as
,
,
and
, respectively.
are the fading coefficients of the eavesdropping channel.
Definition:
The uncertainty of the eavesdropper (also called the secrecy level) is defined as
where and
,
corresponds to perfect secrecy. We also define until the instant
the codeword containing the is successfully sent and feedback.
Theorem 6: For given coding blocklength N and decoding error probability , the lower bound on the security level of the fading channel with intermittent noisy feedback is given by
where and
represent the information leaked through the feedforward channel and the feedback channel, respectively.
Proof. From (108), the is given by
where (a) follows from , (b) follows from that
is a function of the channel noise, (c) follows from
, W and
are mutually independent, (d) follows from that the elements in
are mutually independent, (e) follows from that
similarly, we can get . Substituting (111) into (108), we can obtain
therefore, the proof is completed.
4.3 Numerical examples
Fig 16 and 17 show the relationship between secrecy level, decoding error probability and coding blocklength. As the blocklength N and the decoding error probability Pe increase, the security level and
gradually rises. To achieve a desired decoding error probability (10−3) and the coding blocklength for the noiseless/noisy feedback case is about 60,
can achieve perfect security, but
can not guarantee perfect security.
Fig 18 illustrates the impact of the coding blocklength on security level under different power P. As the blocklength increases,
gradually improves and approaches perfect security. This is because, in the proposed scheme with k = 1, information leakage only occurs at initial transmission instant. With increasing blocklength, the average information leakage of the scheme asymptotically approaches zero. For given blocklength, the higher transmit power P leads to lower security level.
Fig 19 illustrates the impact of the coding blocklength on security level under different feedback power
. As the blocklength increases,
gradually improves. For given blocklength, as the feedback power
increases, the security level
correspondingly decreases. This occurs because the feedback information at each transmission instant is intercepted by the eavesdropper, and higher feedback power directly improves the eavesdropper’s reception quality, thereby increasing information leakage. Achieving perfect security requires the condition that
and sufficiently large blocklength.
As shown in Fig 20, we analyze how the instant k affects both and
under different feedback failure probability
. Under a fixed
,
and
decrease monotonically with increasing of instant k. For given instant k, higher
leads to degraded security levels
and
. These results indicate that ensuring system security requires minimizing both the
and the instant k.
5 Conclusion and future work
In this paper, we propose SK-type highly efficient coding schemes for channels with intermittent feedback, in both noise-free and noisy feedback cases. Numerical results show that to achieve a desired decoding error probability, the coding blocklength of our scheme is significantly shorter, and the encoding-decoding complexity is in linear proportion to the coding blocklength. Besides this, it was shown that when the coding blocklength is not long enough, our scheme almost approaches the PLS requirement, namely, weak secrecy. Though we have shown that our proposed schemes outperform existing ones in the literature from several aspects, some practical implementation challenges still exist, and they are given below.
- Synchronization: Following the frame synchronization schemes [28] for channels with feedback, we note that a short synchronization phase can be embedded at the beginning of each transmission frame to align the transmitter and receiver with respect to the effective operational state of the feedback-assisted SK-type scheme. Due to the state-forgetting property, the impact of initial synchronization mismatch becomes negligible after a finite transient period and may not affect the performance of following message transmissions.
- Dither Generation: The dither sequence is generated using a shared pseudo random seed, which can be established or refreshed during the synchronization phase or via feedback signaling. Occasional loss of dither alignment only affects a finite number of symbols and may not alter the asymptotic rate, reliability, or secrecy guarantees.
- Mitigation strategy for shared randomness assumptions: If no randomness is shared by the receiver and transmitter, one possible mitigation strategy is given below. At the first time instant, the transmitter sends
, the receiver estimates
by
, and he feeds back
to the transmitter. At time instant
, the transmitter obtains a noisy version of the previous feedforward channel output Yi−1, which is given by
. Hence, by
, the transmitter can calculate a noisy version of the receiver’s estimation error, and forwards it for helping the receiver update his estimation of
, namely, the transmitter’s codeword at time instant i is given by
(113)
From (113), it is easy to see that if setting , then
, and
which indicates that the codewords from time instant 2 to N are combinations of channel noises. Then along the lines of proof in [13], it is easy to prove that this strategy achieves perfect secrecy. However, this strategy may lead to significant degradation in transmission performance since it involves additional channel noise
in the feedforward transmission at each time instant.
- Imperfect CSI at the transceiver: If the transmitter does not know the perfect CSI, it will cause additional encoding-decoding error in the coding performance, and this kind of error will be accumulated, resulting in the decoding failure since the receiver’s decoding error probability does not vanish as coding blocklength tends to infinity.
Possible future work includes:
- Extension of our SK-type schemes to multi-user channel models is challenging, e.g., for the two-user Gaussian multiple-access channel with intermittent feedback, the SK-type scheme does not work since the two feedback channels may not be in feedback success at the same time, while this synchronization is the key to the success of original SK-type scheme used in this model.
- Here note that as shown in [20], the performance of the SK-type scheme for the noisy feedback case relies heavily on the sufficiently large feedback power. However, in covert communication [29,30], the encoding power should be sufficiently small, and this leads to the extension of our schemes to covert communication scenario becomes challenging.
- For the channels with feedback delay, how to design SK-type scheme is also interesting and challenging. Specifically, if the feedback delay is a constant, namely, at all time instants, the delay time is fixed. It is easy to extend to our schemes to such a case, for example, if the delay time is d (d ≥ 2, here note that d = 1 is exactly the same as the model of this paper), From time 1 to d − 1, the transmitter sends nothing, and at time d, he sends the initial message as SK scheme does, and from d + 1 to N, the iteration is the same as those given in this paper. Along the lines of performance analysis in this paper, it is not difficult to show that the asymptotic rate is the same as that of this paper, while the finite blocklength rate is related with d. On the other hand, if the feedback delay changes with time, for example, the feedback delay d is a random variable, which takes values in a finite set
, a trivial feedback coding scheme for such case is described as follows: Dividing the N coding blocklength into L blocks with each block consisting of dmax time instants, then using the first block to transmit the initial message and the following L − 1 blocks to transmit the receiver’s estimation errors as the SK scheme does, a feedback coding scheme for the variable feedback delay case is obtained. It is obvious that such scheme suffers from performance degradation as the number of iteration decreases in fact, and it is challenging to design superior feedback coding schemes for this case, which will be our future work.
Appendix
A Error probability analysis in the proof of theorem 2
The decoding error probability Pe of W is bounded as follows. Recall that the transmitter computes , via a modulo operation (34). For any time instant i (2 ≤ i ≤ N) and Si−1 = 1, define
as a modulo-aliasing error occurs during transmission, i.e.,
Furthermore, the final estimation of is
, define
to be the decoding error event at time instant N, i.e.,
Then the decoding error probability is upper bounded by
where ,
and Ec is the complement of the event E.
(i={2,...,N}) is denoted as the probability that modulo-aliasing error occurs and there is no such error occurs during all previous times, and
indicates the error probability of the final decoding without any modulo-aliasing error occurring before.
As mentioned before, the target error probability is set to , then let
, we have
Recalling the definition of the event in (114) and
, since
is Gaussian and
, so we obtain that
to simplify the calculation, parameter L is introduced, which is denoted as
using the definition of L, we can obtain that
from (120), the modulation coefficient of the transmitter is given by
Also note that, combining the (117), (118) and (120), we can obtain the following setting of L
Since XR,i should obey the power constraint
, we can easily obtain
from (120) and (B10) it follows that
Moreover, i* is defined, where and
, substituting
, (121) and (124) into (37), since
is independent of
and Xi−1, we obtained that
According to (37) and (38), we conclude that for any time instant i − 1 (2 ≤ i ≤ N)
substituting (121), (124) and (125) into (126), we obtain
where ,
. We also obtain
, which will be used in the decoding error probability analysis of receiver.
According to (115), we conclude that
and
substituting into (129), the achievable rate of the channel with intermittent noisy feedback is calculated by
which completes the proof.
B Error probability analysis in the proof of Theorem 4
Since W is divided into two independent uniformly distributed sub-messages (WR, WI), we define the decoding error probability of (WR, WI) as Pe,R and Pe,I. The decoding error probability analysis process is the same for the real part and the imaginary part, and the corresponding decoding error probability of the real part is analyzed below. Recall that the transmitter computes , via a modulo operation (88). For any time instant i (2 ≤ i ≤ N) and Si−1 = 1, define
as a modulo-aliasing error occurs during transmission, i.e.,
Furthermore, the final estimation of is
, define
to be the decoding error event at time instant N, i.e.,
Then the decoding error probability is upper bounded by
where (i={2,...,N}) is denoted as the probability that modulo-aliasing error occurs and there is no such error occurs during all previous times, and
indicates the error probability of the final decoding without any modulo-aliasing error occurring before.
As mentioned before, the target error probability is set to , then let
, we have
Recalling the definition of the event in (B1) and
, since
is Gaussian, we obtain that
to simplify the calculation, parameter L is introduced, which is denoted as
using the definition of L, we can obtain that
from (B7), the modulation coefficient of the transmitter is given by
Also note that, from (B5) and (B7), we can obtain the following setting of L
Since XR,i should obey the power constraint of the transmitter from (47), we can easily obtain
from (B7) and (B10) it follows that
Moreover, i* is defined, where and
, substituting
, (B8) and (B11) into (92), we obtained that
Substituting (B8), (B11) and (B12) into , we conclude that for any time instant i − 1 (2 ≤ i ≤ N)
where
According to (B2), we conclude that
and
substituting (B13) into (B16), the achievable rate is calculated by
similarly, we can get
From (B17) and (B18), the achievable rate of the fading channel with intermittent noisy feedback is calculated by
where for sufficiently large N. Therefore, the proof is completed.
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