A General Characterization of Sync Word for Asynchronous Communication
aa r X i v : . [ c s . I T ] M a y THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE. A General Characterization of Sync Word forAsynchronous Communication
Sundaram R M, Devendra Jalihal, Venkatesh RamaiyanIndian Institute of Technology Madras, Chennai 600036, India.Email: [email protected], { dj, rvenkat } @ee.iitm.ac.in Abstract —We study a problem of sequential frame synchro-nization for a frame transmitted uniformly in A slots. Fora discrete memoryless channel (DMC), Venkat Chandar et alshowed in [1] that the frame length N must scale with A as e Nα ( Q ) > A for the frame synchronization error to go to zero(asymptotically with A ). Here, Q denotes the transition proba-bilities of the DMC and α ( Q ) , defined as the synchronizationthreshold, characterizes the scaling needed of N for asymptoticerror free frame synchronization. We show that the asynchronouscommunication framework permits a natural tradeoff betweenthe sync frame length N and the channel (usually parameterisedby the input). For an AWGN channel, we study this tradeoffbetween the sync frame length N and the input symbol power P and characterise the scaling needed of the sync frame energy E = NP for optimal frame synchronisation. I. I
NTRODUCTION
Frame synchronization generally concerns the problem ofidentifying the sync word imbedded in a continuous streamof data (see e.g., [2]). The problem of detecting and decodingframes transmitted sporadically, possibly due to low informa-tion rate, is a subject of asynchronous communication. Theobjective of an asynchronous communication system could be,for example, to detect and decode a single frame transmittedat some random time and there may be no transmission beforeor after the frame (see e.g., [3]).The asynchronous communication setup has been discussedin earlier works such as [2] and [4], but the interest hasincreased in recent times with emerging applications in wire-less sensor networks and the Internet of Things (IoT). Inwireless sensor and actor networks (see e.g., [5] and [6]),the participating nodes would report a measurement or anevent to the fusion centre at random epochs. The nodes mayneed to transmit few bytes of data to the fusion centre overa relatively large time frame, e.g., a single packet possibly inan hour or even in a day. Also, in frameworks such as IoT[7], the nodes may report measurements sporadically leadingto an asynchronous communication framework. However, theconstraints on power may be less stringent in IoT than in wire-less sensor networks. Characterisation of the communicationoverheads (e.g., synchronisation overheads) needed in suchset-ups is crucial for optimal network design and operation.
Related Literature:
Earlier works on frame synchronization,such as [2] and [8], used the maximum-likelihood (ML)criteria for periodically occurring sync words. For aperiodicsync words, hypothesis testing (sequential frame sync) waspreferred in works such as [9], [10] and [11]. For the asyn-chronous set-up (one-shot frame sync), both ML criteria (e.g., [4]) and hypothesis testing (e.g., [12] and [13]) have been stud-ied. These works focus only on the design and performanceof receivers for a sync word designed independently.For the asynchronous set-up, Chandar et al [1] characterizedthe optimum system performance considering sync word andreceiver design jointly. They study a problem of sequentialframe synchronization for a frame transmitted randomly anduniformly in an interval of known size. For a discrete memory-less channel, they identified a synchronisation threshold thatcharacterises the sync frame length needed for asymptoticerror-free frame synchronisation. In [3], following [1], aframework for communication in an asynchronous set up wasproposed and achievable trade-off between reliable commu-nication and asynchronism was discussed. In our work, werestrict to frame synchronization but generalise the frameworkpresented in [1] to study a tradeoff between the sync framelength and the channel. For the AWGN channel, this tradeoffpermits us to characterise the scaling needed of the sync frameenergy (instead of the sync frame length considered in [1] and[3]) for optimal frame synchronisation.II. S
YSTEM S ET - UP The problem set-up is illustrated in Figure 1. We considerdiscrete-time communication between a transmitter and areceiver over a discrete memory-less channel. The discretememory-less channel is characterized by finite input andoutput alphabet sets X and Y respectively, and transitionprobabilities Q ( y | x ) defined for all x ∈ X and y ∈ Y .A sync packet s N = ( s , · · · , s N ) of length N sym-bols ( s i ∈ X for all i = 1 , · · · , N ) is transmitted atsome random time, v , distributed uniformly in { , , · · · , A } ,where A is assumed known. The transmission occupies slots { v, v + 1 , · · · , v + N − } as illustrated in Figure 1, i.e., x n = s n − v +1 for n ∈ { v, · · · , v + N − } , and, we assume thatthe channel input in slots other than { v, v + 1 , · · · , v + N − } is x (0) (where x (0) ∈ X and could represent zero input).The distribution of the channel output, { y n } , conditioned onthe random transmission time v and the sync sequence s N , is Q ( ·| s n − v +1 ) for n ∈ { v, v + 1 , · · · , v + N − } and Q ( ·| x (0)) otherwise.The receiver seeks to identify the location of the syncpacket v from the channel output { y n } . Let ˆ v be an estimateof v . Then, the error event is represented as { ˆ v = v } andthe associated probability of error in frame synchronizationwould be P ( { ˆ v = v } ) . We are interested in characterizing the HIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE. · · · · x (0) ’s As s v s · · s N s N · · · · · · · · · · · x (0) ’s Fig. 1. A discrete-time asynchronous communication model. A sync packet s N = ( s , · · · , s N ) is transmitted at some random time v ∼ U { , A } . Thechannel input in slots other than { v, · · · , v + N − } is assumed to be x (0) . sync sequence s N needed for error-free frame synchronizationas A tends to infinity. In this paper, we assume that thereceiver employs a sequential decoder to detect the sync frame.In particular, we assume that the decision ˆ v = t dependsonly on the output sequence up to time t + N − , i.e., { y , · · · , y t , · · · , y t + N − } .In [1], Chandar et al identify a synchronization thresholdthat characterizes the sync frame length needed for asyn-chronous optimal frame synchronisation. Definition 1 (from [1]) . Let A = e Nα denote the uncertaininterval length for a given sync frame length N and a constant α . An asynchronism exponent α is said to be achievable ifthere exists a sequence of pairs, sync pattern and sequentialdecoder ( s N , ˆ v ) , for all N ≥ , such that P ( { ˆ v = v } ) → as A → ∞ The synchronization threshold for the DMC, denoted as α ( Q ) ,is defined as the supremum of the set of achievable asynchro-nism exponents. In [1], the synchronization threshold for the discretememory-less channel was shown to be α ( Q ) = max x ∈X D ( Q ( ·| x ) k Q ( ·| x (0))) (1)where D ( Q ( ·| x ) k Q ( ·| x (0))) is the Kullback-Leibler distancebetween Q ( ·| x ) and Q ( ·| x (0)) . The authors also provide aconstruction of sync sequence s N entirely with two symbols, x (0) and x (1) , where x (1) := arg max x ∈X D ( Q ( ·| x ) k Q ( ·| x (0))) (2)and show asymptotic error-free frame synchronization witha sequential joint typicality decoder (see section IV or [1]for details). In our work, we generalise the above setup andstudy a tradeoff between the sync frame length and channelparameters. III. M OTIVATION
The synchronisation threshold for an AWGN channel withnoise power σ and input symbol power P can be shown tobe P σ (see [1]). Then, we know that the sync frame length N must scale as e N P σ > A for optimal frame synchronization.Note that this also implies a necessary scaling of the syncframe energy E = N P , i.e., e NP σ > A . This observationmotivates us to study the tradeoff between the sync framelength N and the channel (and input) parameters for optimalframe synchronisation. In Section IV, for a DMC, we firstpresent a general framework for asynchronous frame synchro-nisation and then study a tradeoff between N and α ( Q ) . In Section V, for the AWGN channel, we discuss the tradeoffbetween the sync frame length N and the input symbol power P and characterise the scaling needed of the sync frame energy E = N P for error-free frame synchronization.Chandar et al [1] studied the sequential frame synchronisa-tion problem for a fixed Q (and α ( Q ) ) and as a function of thesync frame length N only. Also, in [1], the setup and the proofbased on the joint typicality of input-output sequences requiresthe sync frame length N to scale to infinity. In our work, wegeneralise the framework and study a tradeoff between thesync frame length N and channel parameters and study thecase of finite sync frame length as well.IV. A G ENERAL F RAMEWORK FOR A SYNCHRONOUS F RAME D ETECTION
We now present a framework that permits a tradeoff betweenthe sync frame length N and the channel, represented by α ( Q ) , for the system setup described in Section II. Consider asequence of triples, channel, sync word and sequential decoder, ( {X A , Y A , Q A } , s N A , ˆ v ) parameterized by the asynchronousinterval length A . Define α ( Q A ) as α ( Q A ) = max x ∈X A D ( Q A ( ·| x ) k Q A ( ·| x A (0))) (3)and let x A (1) := arg max x ∈X A D ( Q ( ·| x ) k Q ( ·| x A (0))) . The fol-lowing theorem generalizes Theorem 1 in [1] and discussesthe necessary scaling needed of N A and α ( Q A ) for asymptoticerror-free frame synchronisation. Theorem 1.
Consider a sequence of triples, ( {X A , Y A , Q A } , s N A , ˆ v ) parameterized by the asynchronousinterval length A . Let N A → ∞ as A → ∞ . Let Q A ( ·| x A (1)) → Q ∗ ( · ) and Q A ( ·| x A (0)) → Q ∗ ( · ) suchthat α ( Q A ) → ∞ as A → ∞ . Then, the probability offrame detection error P ( { ˆ v = v } ) → as A → ∞ if e N A α ( Q A ) > A . Remarks IV.1.
1) Theorem 1 characterizes the rate at which N A and α ( Q A ) must scale with A for the frame synchronisationerror to tend to zero (asymptotically). In [1], the channelwas assumed to be the same independent of N or A . The generalisation proposed in Theorem 1 enablesus to study the tradeoff between N A and α ( Q A ) forsupporting asynchronism.2) For the AWGN channel, we know that α ( Q A ) = P A σ .Hence, N A × α ( Q A ) ∝ N A P A represents the energy ofthe sync packet. Thus, the above theorem also charac-terizes the necessary scaling needed of the energy of thesync packet for the frame synchronisation error to tendto zero. This observation is studied in detail in Section Vof this paper.Here, we have presented only the necessary outline ofthe proof for Theorem 1 as the argument is similar to thepresentation in [1]. Proof:Setup:
We consider the framework presented in Section IIfor every A . A sync packet s N A of length N A is transmitted HIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE. a. Error-less
Sync-likenoise b. Error E1
Sync-likeOverlap c. Error E2
Noise-likesync d. Error E3Fig. 2. Error events in sequential frame synchronization problem. at some random time v ∼ U { , A } . The discrete memory-lesschannel is characterised by finite input and output alphabet sets X A and Y A respectively, and transition probabilities Q A ( ·|· ) with synchronization threshold α ( Q A ) defined as in (3). Codeword:
Following [1], we consider a sync sequence s N A of length N A with the following properties.1) Fix some large K . Now, find a M A such that M A − − < N A K ≤ M A − for some M A = 1 , , · · · . Let s n = x A (1) for M A − < n ≤ N A . Consider amaximal-length shift register (MLSR) sequence { m n : n = 1 , , · · · , M A − } of length M A − and map itto { s n : n = 1 , , · · · , M A − } such that s n = x A (1) if m n = 0 and s n = x A (0) if m n = 1 .2) The sync sequence thus obtained, s N A , now has aHamming distance of Ω (cid:0) N A K (cid:1) with any of its shiftedsequences. Decoder:
We consider a simple version of the sequentialjoint typicality decoder for the problem setup. In [1], at everytime t + N A − , the decoder computes the empirical jointdistribution ˆ P of the sync word (the channel input of length N A ) and the output symbols in the previous N A slots, i.e., { y t , · · · , y t + N A − } . Whereas, we restrict our attention to thosepositions in the sync word where we transmit symbol x A (1) and only compute ˆP ( x A (1) , y ) = N ( x A (1) , y ) N A , for all y ∈ Y where, N A denotes the number of occurrences of x A (1) inthe sync word and N ( x A (1) , y ) denotes the number of jointoccurrences of ( x A (1) , y ) in the sync code word and thechannel output. We note that N A = Ω (cid:0) N A (cid:0) − K (cid:1)(cid:1) .If the empirical distribution is close enough to the expecteddistribution Q A ( ·| x A (1)) , i.e., if | ˆ P ( · ) − Q A ( ·| x A (1)) | < µ for some fixed µ > , then, the decoder declares ˆ v = t . Wehave assumed that Q A ( ·| x A (1)) → Q ∗ ( · ) and hence, we makea simplifying assumption and declare ˆ v = t only when | ˆ P − Q ∗ | < µ . Error event:
The failure to detect the exact instance ofsync word transmission, i.e., the error event { ˆ v = v } , canbe partitioned as given below and as shown in Figure 2 . • E : ˆ v ∈ { , · · · , v − N A } ∪ { v + 1 , · · · , A } . This cor-responds to the event that the output symbols generatedentirely by the zero input x A (0) is jointly typical. • E : ˆ v ∈ { v − N A + 1 , · · · , v − } . This corresponds tothe event that the output symbols generated partially by x A (0) and sync word is jointly typical. • E : ˆ v / ∈ { v } . This corresponds to the event that theoutput symbols generated by the sync word is not jointlytypical.In detection terminology, E and E both constitute falsealarm due to noise emulation of sync word and E is misseddetection. Performance Evaluation:
Using a union bound, we canupper bound the probability of error in frame synchronisationas P ( { ˆ v = v } ) ≤ P ( E ) + P ( E ) + P ( E ) Suppose that A = e ǫ · N A ( α ( Q A ) − ǫ ) for some < ǫ < and ǫ > , i.e., A < e N A α ( Q A ) . We will now show that P ( E ) , P ( E ) and P ( E ) tend to zero as A → ∞ .The proof follows the method of types (see [14] and [15]).A false alarm event of type E occurs at a time t , if an inputsequence composed entirely of x A (0) symbols generates anoutput type in the set Q ∗ = { Q ( · ) : | Q ( y ) − Q ∗ ( y ) | < µ, ∀ y ∈Y} . The probability of such an event is bounded as P ( E | t ) ≤ X Q ∈Q ∗ e − N A D ( Q k Q A ( ·| x A (0))) ≤ poly ( N A ) × e − N A ( α ( Q A ) − δ ) where δ is a function only of µ and is independent of A .The probability of false alarm of type E can now be upperbounded using a union bound (over t ) as follows. P ( E ) ≤ A × poly ( N A ) × e − N A ( α ( Q A ) − δ ) Substituting for A = e ǫ N A ( α ( Q A ) − ǫ ) and bounding N A , wehave, P ( E ) ≤ poly ( N A ) × e ǫ N A ( α ( Q A ) − ǫ ) × e − N A ( − K ) ( α ( Q A ) − δ ) (4)For large K and small δ (with an appropriate choice of µ ),we have, P ( E ) → as A → ∞ (i.e., as N A → ∞ or as α ( Q A ) → ∞ ).A false alarm event of type E occurs if an input sequencecomposed partially of x A (0) symbols and the sync word s N A generates an output type in the set Q ∗ . We note that, for everytransmission instant v , there are N A − possible positions thatcan lead to the error event. The MLSR sequence achieves aHamming distance of Ω (cid:0) N A K (cid:1) with any of its shifted versionsand, the Hamming distance corresponding to positions wherethe sync word is x A (1) is Ω (cid:0) N A K (cid:1) . Using similar argumentsas for E , the probability of false alarm of type E can nowbe upper bounded as P ( E ) ≤ poly ( N A ) × e − Ω (cid:16) NA K (cid:17) ( α − δ ) (5)Here again, P ( E ) → as A → ∞ (i.e., as N A → ∞ or as α ( Q A ) → ∞ ). HIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE. For the missed detection event E , we need to evaluate theprobability that an input sequence composed entirely of x A (1) symbols generates an output type outside the set Q ∗ . Clearly, P ( E ) ≤ X Q/ ∈Q ∗ e − N A D ( Q k Q A ( ·| x A (0)) ≤ poly ( N A ) × e − N A ( − K ) δ ′ (6)where δ ′ is a function only of µ and is independent of A .Now, P ( E ) → as A → ∞ (i.e., as N A → ∞ ). We notethat the scaling of α ( Q A ) does not take the error probability P ( E ) → .Thus, we have P ( { ˆ v = v } ) → if N A → ∞ and α ( Q A ) →∞ such that e N A α ( Q A ) > A .V. T RADE - OFF IN
AWGN C
HANNEL
In this section, we study the application of Theorem 1 to theadditive white Gaussian noise channel. We make the followingadditional assumptions to define a binary input binary outputDMC model for the AWGN channel.1) We consider a binary input alphabet set with X A = { x A (0) = 0 , x A (1) = √ P A } for every A . P A couldcorrespond to the symbol power constraint and P A σ would then be the SNR. We note that it is sufficientto consider the binary input alphabet set for the framesynchronisation problem (see Section II or [1] for de-tails).2) The received signal at time n is assumed to be x n + w n ,where w n is WGN with variance σ .3) We consider a binary alphabet set Y A for the outputchannel, i.e., Y A = { y A (0) , y A (1) } for every A . Inparticular, we consider the following map for the AWGNchannel: the output is y A (1) if x n + w n > τ A = a √ P A for some < a < and the output is y A (0) if x n + w n ≤ τ A . The binary input and binary output DMCmodel for the AWGN channel is illustrated in Figure 3where ǫ f and ǫ m denote the transition probabilities. Weshow in Section V-A that the two alphabet approxima-tion for the output channel is appropriate in the contextof asynchronous frame synchronisation. A. Binary Output DMC Model for AWGN Channel
The synchronisation threshold for the AWGN channel withnoise power σ and input symbol power P was shown to be P σ (see [1]). The following lemma shows that the binaryinput binary output model for the AWGN channel can achievea synchronisation threshold arbitrarily close to P σ . Lemma 1.
Consider the binary input binary output model forthe AWGN channel shown in Figure 3 . The synchronizationthreshold of the DMC tends to P σ for a ≈ and as P → ∞ .Proof: The channel transition probabilities for the DMCare ǫ f = P ( y A (1) | x A (0)) = P ( n > a √ P ) ≃ e − a P σ ǫ m = P ( y A (0) | x A (1)) = P ( n > (1 − a ) √ P ) ≃ e − (1 − a )2 P σ x A (1) x A (0) y A (1) y A (0)1 − ǫ m ǫ m ǫ f − ǫ f Fig. 3. A binary input binary output model for AWGN channel withtransition probabilities ǫ f = P ( y A (1) | x A (0)) = P ( n > a √ P ) and ǫ m = P ( y A (0) | x A (1)) = P ( n > (1 − a ) √ P ) . The synchronization threshold for the binary DMC is givenby α = (1 − ǫ m ) log 1 − ǫ m ǫ f + ǫ m log ǫ m − ǫ f Clearly, ǫ f , ǫ m → as P → ∞ . Hence, α → P →∞ − log ǫ f + ǫ m log ǫ m ≃ a P σ + (1 − a ) P σ e − (1 − a )2 P σ ≃ a P σ Thus, for large P and a close to , the synchronizationthreshold of the binary input binary output tends to thesynchronisation threshold of the AWGN channel.The above lemma permits us to apply the results of theSection IV for the AWGN channel. B. Tradeoff for the AWGN Channel
The following corollary discusses an application of Theo-rem 1 for the AWGN channel.
Corollary 1.
Consider an AWGN channel with noise variance σ . Let N A and P A denote the sync word length and the inputsymbol power parameterized by the asynchronous intervallength A . Let N A , P A → ∞ as A → ∞ . Then, the probabilityof frame detection error P ( { ˆ v = v } ) → if e N A P A σ > A .Proof: We know that α ( Q A ) → a P A σ ≃ P A σ for thebinary input binary output model for the AWGN channel as P A → ∞ . Also, as P A → ∞ , we see that Q A ( ·| x A (1)) → (0 , and Q A ( ·| x A (0)) → (1 , satisfying the assumptions.Hence, e N A α ( Q A ) → e N A PA σ as A → ∞ . From Theorem 1,we then have P ( { ˆ v = v } ) → as A → ∞ if e N A PA σ > A . Remarks V.1.
1) Define E A = N A P A as the energy of the sync packet.Then, the above corollary characterises the scaling nec-essary of the energy of the sync packet (when both N A and P A are adapted) for asymptotic error-free framesynchronisation.The following lemma extends the results of Corollary 1 fora sync word of finite length. HIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. COPYRIGHT MAY BE TRANSFERRED WITHOUT NOTICE, AFTER WHICH THIS VERSION MAY NO LONGER BE ACCESSIBLE. Lemma 2.
Consider an AWGN channel with noise variance σ . Let N A and P A denote the sync word length and the inputsymbol power parameterised by the asynchronous intervallength A . Let N A = N for all A and let P A → ∞ as A → ∞ .Then, the probability of frame detection error P ( { ˆ v = v } ) → if e NP A σ > A .Proof: The proof follows similar arguments as in The-orem 1. Here again, we seek to show that P ( { ˆ v = v } ) ≤ P ( E ) + P ( E ) + P ( E ) → under the suggested conditions. Codeword : Since N is finite, we will simplify the sync wordand let it consist only of x A (1) = √ P A in all the positions. Decoder : As the sync word comprises only of x A (1) , theentire length of the sync word is used for decoding. As P A →∞ , we see that Q A ( ·| x A (1)) → (0 ,
1) = Q ∗ . The decoder willdeclare ˆ v = t if | ˆ P − Q ∗ | < µ . For the finite N case, we willset µ = N . Then, for the choice of µ , we have Q ∗ = { Q ( · ) : | Q ( y ) − Q ∗ ( y ) | < N , ∀ y } = { (0 , } . This implies that thedecoder will declare the sync packet as received only whenall the previous N output symbols are decoded as y A (1) . Performance Evaluation : The probability of false alarm oftype E for the decoder can now be upper bounded as P ( E ) ≤ A × ǫ Nf ≤ A × e − N a PA σ If A = e ǫ N PA σ < e N PA σ for some < ǫ < , then we have P ( E ) → as P A → ∞ for a suitable choice of a .The probability of false alarm of type E can be upperbounded by considering the worst case overlap with the syncword and using a union bound as given below. P ( E ) ≤ ( N − ǫ f ≤ ( N − e − a PA σ Clearly, P ( E ) → as P A → ∞ .The missed detection occurs even if one of the symbols is inerror, since µ = N . Thus, using a union bound, the probabilityof missed detection is upper bounded as P ( E ) ≤ N ǫ m ≤ N e − (1 − a )2 PA σ P ( E ) → as P A → ∞ . Hence, P ( { ˆ v = v } ) → as P A →∞ . Remarks V.2.
1) Let N = 1 . The above lemma suggests that wecan achieve arbitrarily low packet detection error if e σ P A > A , even with a single length sync word.2) We note again that the proofs (in Section IV and inearlier references [1] and [3]) based on joint typicalityof input-output sequences require the sync frame length N to scale to infinity. In Lemma 2, we illustrate thatasynchronous frame synchronization over an AWGNchannel can be achieved with finite sync frame lengthas well.3) In the proof of Theorem 1, for a general DMC, we notedthat P ( E ) need not scale to zero as α ( Q A ) → ∞ .However, in the binary input binary output model forthe AWGN channel shown in Figure 3, P ( E ) → as α ( Q A ) → ∞ . This permits us to describe an asyn-chronous frame synchronisation framework for a finitelength sync word.Motivated by the results obtained so far for AWGN channel,1) P A = P , N A → ∞ as A → ∞ by Chandar [1]2) P A → ∞ , N A → ∞ as A → ∞ in Corollary 13) P A → ∞ , N A = N as A → ∞ in Lemma 2we can now define the synchronization threshold for theAWGN channel in terms of the sync packet energy. Lemma 3.
The synchronisation threshold for the AWGNchannel with respect to the sync packet energy is σ . VI. C
ONCLUSION
In this paper, we present a general framework for asyn-chronous frame synchronisation that permits a trade-off be-tween sync word length N and channel. The frameworkallowed us to characterise the synchronisation threshold forthe AWGN channel in terms of the sync frame energy (i.e., e E σ > A ) instead of the sync frame length. We alsoobserve that a finite sync word can achieve optimal framesynchronization for an AWGN channel. As future work, weseek to study this trade-off for wireless channel models.R EFERENCES[1] V. Chandar, A. Tchamkerten, and G. Wornell, “Optimal sequential framesynchronization,”
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