Protocol Sequences for Multiple-Packet Reception

0.24 MB·

by  Yijin Zhang

1
Protocol Sequences for Multiple-Packet
Reception
Yijin Zhang, Yuan-Hsun Lo, Feng Shu and Wing Shing Wong
4
1
0
2
Abstract

p
e
S Consider a time slotted communication channel shared by K active users and a single receiver. It

6 is assumed that the receiver has the ability of the multiple-packetreception (MPR) to correctly receive
2
at most γ (1 ≤ γ < K) simultaneously transmitted packets. Each user accesses the channel following

]
T a specific periodical binary sequence, called the protocol sequence, and transmits a packet within a
I
. channelslot if and only if the sequence value is equal to one. The fluctuation in throughputis incurred
s
c
[ by inevitable random relative shifts among the users due to the lack of feedback. A set of protocol

2 sequences is said to be throughput-invariant(TI) if it can be employed to produce invariantthroughput
v
for any relative shifts, i.e., maximize the worst-case throughput. It was shown in the literature that
0
6 the TI property without considering MPR (i.e., γ = 1) can be achieved by using shift-invariant (SI)
2
6 sequences, whose generalized Hamming cross-correlation is independent of relative shifts. This paper
.
1
investigatesTIsequencesforMPR;resultsobtainedincludeachievablethroughputvalue,alowerbound
0
4 on the sequence period, an optimal construction of TI sequences that achieves the lower bound on the
1
: sequence period, and intrinsic structure of TI sequences. In addition, we present a practical packet
v
i
X decoding mechanism for TI sequences that incorporates packet header, forward error-correcting code,
r and advanced physical layer blind signal separation techniques.
a
The material in this paper was presented in part at the IEEE International Symposium on Information Theory, Honolulu,
HI, July 2014. This work was partially supported by the National Natural Science Foundation of China (No. 61301107,
61174060, 61271230), theHongKongRGCEarmarkedGrantCUHK414012andtheShenzhenKnowledgeInnovationProgram
JCYJ20130401172046453.
Y.ZhangandF.ShuarewiththeSchoolofElectronicandOpticalEngineering,NanjingUniversityofScienceandTechnology,
Nanjing, China. E-mail: [email protected]; [email protected].
Y.-H. Lo is with the Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan. E-mail:
[email protected].
W.S.Wongiswiththe Department of Information Engineering, TheChinese Universityof Hong Kong, Shatin,N. T.,Hong
Kong. E-mail: [email protected].
September29,2014 DRAFT
2
Index Terms
Collision channel without feedback, protocol sequences, multiple-packet reception, throughput.
I. INTRODUCTION
The idea of using protocol sequences to define deterministic multiaccess protocols for a
collision channel without feedback was proposed by Massey and Mathys in [1], and recently has
attracted many research revisits [2]–[7] for different design criteria and applications. Compared
with time division multipleaccess (TDMA), ALOHA and carrier sense multiple access (CSMA),
protocol sequences do not require stringent synchronization, channel monitoring, backoff algo-
rithm and packet retransmissions. Such a simplicity is particularly desirable in ad hoc networks
and sensor networks, in which well-coordinated transmissions and time synchronization may be
difficult to achieve due to user mobility, time-varying propagation delays and energy constraints.
Moreover, in contrast to the random and contention based schemes, protocol sequences in [2]–[7]
can respectively provide a positive short-term throughput guarantee with probability one in the
worst case.
Previousstudies on multiaccessprotocols havetraditionallyassumed a collisionchannel model
of single-packet reception (SPR), in which a packet is received correctly only if it is not involved
in a collision, i.e., does not overlap with another. However, the assumption of SPR becomes
more and more unsuitable in practice, due to recent advances at reception techniques of the
physical (PHY) layer, such as antenna arrays, CDMA technique and beamforming algorithms,
which can be employed to ensure that the receiver has the ability of multiple-packet reception
(MPR) [8] to receive multiple packets simultaneously. In this paper, we restrict our attention
to protocol sequences for the γ-user MPR channel, in which a packet can be received error-
free only if at most γ − 1 other packets are being transmitted simultaneously. We refer to γ
as the MPR capability which has been commonly assumed in [9]–[13] for studying ALOHA
and CSMA. However, it is expected that protocol sequences will also behave differently from
what were reported in [2]–[7] for γ = 1 (i.e., SPR). Until only recently there has been no
research published on protocol sequences for MPR. Only one related result [14] stated that there
is no need of employing protocol sequences for a channel with a sufficiently large recovering
probability of collided packets, which is different from the MPR capability discussed here.
DRAFT September29,2014
3
The effective throughput of a user under the MPR capability γ is defined as the fraction
of packets it can be sent out without suffering any collision in which more than γ users are
involved. Due to a lack of feedback information, the relative shifts among users are unknown
to each other, and thus incur performance variation in throughput. As argued in [5], [6], our
main goal of designing protocol sequences is to maximize the worst-case individual throughput
for any relative shifts, i.e., minimize the variation in throughput. If the throughput of each user
is constant and independent of any relative shifts, then the assigned set of protocol sequences
is said to be throughput-invariant (TI). It was shown in [1], [5], [6] that this zero-variance on
throughput for γ = 1 can be achieved by using shift-invariant (SI) sequences, which enjoy
the special property that their generalized Hamming cross-correlation is independent of relative
shifts. The generalized Hamming cross-correlation here is a generalization of pairwise Hamming
cross-correlationand defined forallnonemptysubsetsofusers.However,thequestionofwhether
there exist TI sequences for γ = 1 which are shorter than SI sequences has not been answered.
Moreover, the impact of MPR capability on the throughput performance and sequence design of
TI sequences is not explored either. As such, this paper is a first attempt to study TI sequences
for MPR, and can be viewed as a generalization of results in [6].
The remainder of this paper is organized as follows. After setting up the channel model and
notation in Section II, we in Section III derive the throughput value of a system with MPR
capability. In Section IV, lower bounds on the period of TI sequences with MPR capability are
presented. In Section V, we use a known construction of SI sequences to design TI sequences
for any MPR capability, which are optimal in the sense that the sequence period achieves the
lower bound. Section VI proposes a mechanism of identification and decoding for TI sequences.
It is further shown in Section VII that TI sequences must be SI in many specific cases, which
indicates that the SI sequence set is the unique solution to the TI problem with MPR. Finally,
in Section VIII we close the discussion with some concluding remarks.
II. CHANNEL MODEL AND NOTATION
Consider a time slotted communication channel shared by K activeusers and a single receiver.
It is assumed that each of these users always has a fixed-length packet to send, knows the slot
boundaries and transmits its packet within a channel slot. However, these users do not know
the relative shifts of other users. Let τ , an integer measured in unit of slot duration, denote
i
September29,2014 DRAFT
4
the relative shift of user i for i = 1,2,...,K. Following [1], we define a deterministic binary
sequence, s := [s (0) s (1) ... s (L − 1)], called protocol sequence to schedule the packet
i i i i
transmission of user i for i = 1,2,...,K, where L is the common sequence period of all K
sequences. We consider that the channel is slot-synchronous so that there exists a system-wide
slot labeling, t, and user i transmits a packet at slot t if s (t + τ ) = 1, and keeps silent if
i i
s (t+τ ) = 0, in which the addition by τ is in modulo L arithmetic.
i i i
Following the assumption of the γ-user MPR channel, we focus on the non-trivial case in
which γ < K. A transmitted packet is correctly received if at most γ−1 other packets are being
transmitted at the same slot, and is considered lost otherwise. For practical considerations, the
users can employ forward error-correcting code across packets to recover data lost, as explained
in Section VI.
Some notation and definitions used in this paper are stated below.
Definition 1. For i = 1,...,K, define theduty factor f of a protocolsequence s , as the fraction
i i
of ones in s , namely,
i
L−1
1
f := s (t).
i i
L
Xt=0
The cyclic shift of s by τ is defined as
i i
s(τi) := [s (τ ) s (1+τ ) ... s (L−1+τ )],
i i i i i i i
where the addition is taken modulo L. Note that the t-th bit of s(τi) is denoted by s (t+τ ).
i i i
Definition 2. Let b ∈ {0,1} for j = 1,...,K. For a system with the MPR capability γ, the
j
throughput of user i with the assigned sequence s for given relative shifts τ ,...,τ is defined
i 1 K
as
1
R (τ ,τ ,...,τ ) = N(b ,...,b |s(τ1),...,s(τK)), (1)
i 1 2 K L 1 K 1 K
bXi=1,
qi≤γ−1
in which q = b and N(b ,...,b |s(τ1),...,s(τK)) denotes the number of time indices t,
i j6=i j 1 K 1 K
P
0 ≤ t < L, such that s (t+τ ) = b for all j. This computes the fraction of time slots in which
j j j
at most γ users including user i are transmitting. Note that the summation in (1) is taken over
(b ,...,b ) with b = 1,q ≤ γ −1.
1 K i i
A sequence set {s ,s ,...,s } is said to be TI with the MPR capability γ if R (τ ,τ ,...,τ )
1 1 K i 1 2 K
is a constant function of τ ,τ ,...,τ for any i. For simplicity, we sometimes use R to denote
1 2 K i
DRAFT September29,2014
5
the throughput of user i. To avoid the uninteresting cases, in this paper we only consider TI
sequences that ensure R is strictly bigger than zero for any i.
i
Definition 3. We identify the K users by means of the index set
K := {1,2,...,K}.
Let O be the set
K
K
{(i ,...,i ) ∈ Kn : i < i < ··· < i }.
1 n 1 2 n
n[=1
An element in O corresponds to an ordered tuple of users. For A = (i ,i ,...,i ) ∈ O , the
K 1 2 n K
generalized Hamming cross-correlation associated with A is defined as
L−1 n
H(τ ,...,τ ;A) := s (t+τ ).
i1 in ij ij
Xt=0 Yj=1
If |A| = 2, the generalized Hamming cross-correlation is reduced to the pairwise Hamming
cross-correlation function.
We further introduce the following definitions by means of the generalized Hamming cross-
correlation:
(i) Given an ordered tuple A ∈ O , then H(τ ,...,τ ;A) is said to be SI if it is a constant
K i1 in
for any τ ,...,τ .
i1 in
(ii) A sequence set is said to be SI [6] if H(τ ,...,τ ;A) is SI for every A in O .
i1 in K
(iii) A sequence set is said to be pairwise SI [15] if H(τ ,...,τ ;A) is SI for every A in O
i1 in K
with |A| = 2.
III. THROUGHPUT OF TI SEQUENCES
Let s ,s ,...,s be n binary sequences with a common period L. For b ,...,b ∈ {0,1},
1 2 n 1 n
Shum et al. [6] showed that
L−1 L−1 n
... N(b ,...,b |s(τ1),...,s(τn)) = L N(b |s ). (2)
1 n 1 n i i
τX1=0 τXn=0 Yi=1
The main result in this section is summarized in the following.
September29,2014 DRAFT
6
Theorem 4. Let s ,s ,...,s be K TI sequences with the MPR capability γ, 1 ≤ γ < K, and
1 2 K
duty factors f ,f ,...,f respectively. Then we have
1 2 K
R = f f (1−f ). (3)
i i j k
H⊆XK\{i} jY∈H k∈K\Y({i}∪H)
|H|<γ
Proof: Suppose that the relative shifts of s is τ , for i = 1,2,...,K. We can treat
i i
τ ,τ ,...,τ as independent and uniformly distributed random variables that are equally likely
1 2 K
to take on any of L values: 0,1,...,L − 1. After taking the expectation over (τ ,τ ,...,τ ),
1 2 K
we obtain the average throughput of user i as the following:
E R (τ ,τ ,...,τ )
i 1 2 K
(cid:16) (cid:17)
1
= E N(b ,...,b |s(τ1),...,s(τK)) (4)
L 1 K 1 K
(cid:16)qi≤γX−1,bi=1 (cid:17)
1
= E N(b ,...,b |s(τ1),...,s(τK))
L 1 K 1 K
qi≤γX−1,bi=1 (cid:16) (cid:17)
K
1 1
= N(b |s ) (5)
L LK−1 j j
qi≤γX−1,bi=1 Yj=1
K
1 N(1|s )
= i N(b |s )
j j
LK−1 L
qiX≤γ−1 j=Y1,j6=i
K
1
= f N(b |s )
LK−1 i j j
qiX≤γ−1j=Y1,j6=i
= f f (1−f ),
i j k
H⊆XK\{i} jY∈H k∈K\Y({i}∪H)
|H|<γ
where q = b . (4) directly follows from (1), while (5) is due to (2). Furthermore, since
i j6=i i
P
R (τ ,...,τ ) is a constant function of τ ,...,τ , it must be equal to its average value. This
i 1 K 1 K
completes the proof.
Note that Theorem 4 is a generalization of [6, Thm. 3], which focuses on the case of γ = 1.
For the symmetric case that each user has the same duty factor f, all users have the same
throughput:
γ−1
K −1
fj+1(1−f)K−1−j. (6)
(cid:18) j (cid:19)
Xj=0
DRAFT September29,2014
7
5
f=1/10,γ=1
4.5 f=1/10,γ=5
f=1/10,γ=10
4
f=1/20,γ=1
3.5 f=1/20,γ=5
put f=1/20,γ=10
h
ug 3
o
hr
m t 2.5
e
st
y 2
s
e
h
T 1.5
1
0.5
0
10 15 20 25 30 35 40 45 50
The number of users
Fig. 1. The symmetric system throughput for 10≤K ≤50.
The system throughput in the symmetric case is plotted in Fig. 1 for 10 ≤ K ≤ 50 with
γ = 1,5,10 and f = 1/10,1/20, respectively. We can see from (6) that there is an optimal
duty factor for maximizing the throughput of a given user number. For γ = 1 the optimal value
is 1/K, but for the other cases the closed-form expression is difficult to obtain. See numerical
results of optimal f in Fig. 2 for K = 20 and γ = 1,2,3,5,8,10,12,15.
IV. LOWER BOUND ON MINIMUM PERIOD
A long sequence period could result in large variation in throughput on a short-time scale.
With a weak assumption on duty factors, we derive lower bounds on the period of TI sequences
for any γ in this section. These are clearly constraints on constructing TI sequences for small
L values. Let gcd(x,y) denote the greatest common divisor of integers x and y.
Definition 5. Consider K binary sequences s ,s ,...,s with a common period L. Given an
1 2 K
ordered tuple A = (i ,...,i ) ∈ O and relative shifts τ ,...,τ for some M ≤ K, let
1 M K i1 iM
θ (τ ,...,τ ;A), for j = 0,1,...,M, denote the number of time indices t, 0 ≤ t < L, such
j i1 iM
September29,2014 DRAFT
8
12
γ=1
γ=2
10 γ=3
γ=5
γ=8
put 8 γ=10
h
ug γ=12
o
hr γ=15
m t 6
e
st
y
s
e
h 4
T
2
0
0 0.2 0.4 0.6 0.8 1
The symmetric duty factor
Fig. 2. The symmetric system throughput for K =20 and different γ.
that there are exactly j ‘1’s among s (t+τ ),...,s (t+τ ). Then, define
i1 i1 iM iM
j
θ (τ ,...,τ ;A) = θ (τ ,...,τ ;A).
≤j i1 iM k i1 iM
Xk=0
Note that θ (τ ,...,τ ;A) = H(τ ,...,τ ;A).
M i1 iM i1 iM
Proposition 6. Consider a set of K sequences which is TI with the MPR capability γ (1 ≤ γ <
K). If R > 0 for any i, then there are at most γ −1 all-one sequences.
i
Proof: Suppose there are γ all-one sequences. Then all packets from other K−γ sequences
cannot be received error-free. It implies R = 0 for some i, which contradicts the assumption.
i
Theorem 7. Let γ be any integer with 1 ≤ γ < K. If a set of K binary sequences is TI with
the MPR capability γ, then this sequence set is pairwise SI.
Proof: Denote by s ,...,s the K sequences. The proof of the case K = 2 is straightfor-
1 K
ward. For K > 2, we aim to show that θ (τ ,τ ;A) is a constant function of τ ,τ for any
2 i1 i2 i1 i2
A = (i ,i ) ∈ O . Without loss of generality, let A = (1,2), and let B = (3,...,K). That is,
1 2 K
we divide these K sequences into two parts: A = {s ,s } and B = {s ,...,s }.
1 2 3 K
DRAFT September29,2014
9
First, we fix the relative shifts τ∗,...,τ∗ of sequences in B such that θ (τ∗,...,τ∗;B) > 0.
3 K γ−1 3 K
This combination of relative shifts can always be found since there are at most γ − 1 all-one
sequences in B for the case 1 ≤ γ < K following Proposition 6. Suppose in B that there are
exactly h all-one sequences. As h ≤ γ −1 ≤ K −2, then we cyclically shift some γ −h−1
sequences which are not all-ones such that the first time slot of them are all equal to one, and
cyclically shift the remaining K −γ −1 sequences such that the first time slot of them are all
equal to zero. Hence there are exactly γ −1 ones in the first time slot.
For τ ,τ , we define
1 2
T (τ ,τ ) := N(b ,...,b |s(τ1),s(τ2),s(τ3∗),...,s(τK∗))
1 1 2 1 K 1 2 3 K
b1X+b2=1
b3+...+bK<γ
and
T (τ ,τ ) := N(b ,...,b |s(τ1),s(τ2),s(τ3∗),...,s(τK∗)).
2 1 2 1 K 1 2 3 K
b1X+b2=2
b3+...+bK<γ−1
We assume the relative shifts τ ,τ are uniformly distributed in 0,1,...,L−1. After taking the
1 2
expectation over τ ,τ , we have
1 2
E(LR +LR ) =E(T (τ ,τ )+2T (τ ,τ )) = E(T (τ ,τ ))+2E(T (τ ,τ )) (7)
1 2 1 1 2 2 1 2 1 1 2 2 1 2
2
1
= iθ (τ ,τ ;A)θ (τ∗,...,τ∗;B). (8)
L i 1 2 ≤γ−i 3 K
Xi=1
The proof of (8) is relegated to Appendix A.
We now change the pair of relative shifts from (τ ,τ ) to (τ′,τ′). Let
1 2 1 2
σ := θ (τ′,τ′;A)−θ (τ ,τ ;A). (9)
2 1 2 2 1 2
By the fact that
θ (τ ,τ ;A)+2θ (τ ,τ ;A) = Lf +Lf = θ (τ′,τ′;A)+2θ (τ′,τ′;A),
1 1 2 2 1 2 1 2 1 1 2 2 1 2
we have
θ (τ′,τ′;A)−θ (τ ,τ ;A) = −2σ. (10)
1 1 2 1 1 2
Since R +R has zero-variance, by (7) and (8), we have
1 2
2
iθ (τ ,τ ;A)θ (τ∗,...,τ∗;B)
i 1 2 ≤γ−i 3 K
Xi=1
2
= iθ (τ′,τ′;A)θ (τ∗,...,τ∗;B).
i 1 2 ≤γ−i 3 K
Xi=1
September29,2014 DRAFT
10
Then it follows from (9) and (10) that
σθ (τ∗,...,τ∗;B) = 0,
γ−1 3 K
which implies that σ = 0 because of θ (τ∗,...,τ∗;B) > 0 due to the choice of τ∗,...,τ∗.
γ−1 3 K 3 K
Thus, θ (τ ,τ ;A) is a constant function of τ ,τ .
2 1 2 1 2
Since θ (τ ,τ ;A) = H(τ ,τ ;A) and the choice of A is arbitrary, we conclude that a set of
2 1 2 1 2
these K sequences is pairwise SI for any γ < K.
Compared with the SI property which has been proved as a sufficient condition of a sequence
set being TI for γ = 1 [6], pairwise SI is conceptually a weaker requirement on sequence design.
However, they are known to enjoy the same lower bound on the minimum sequence period for
some special form of duty factors. Given any set of K pairwise SI sequences with duty factors
n /d , where gcd(n ,d ) = 1 for all i, Zhang et al. [15, Thm. 1] proved that its common period
i i i i
is divisible by d d ···d . By Theorem 7, we then have the following result.
1 2 K
Corollary 8. Let γ be an integer with 1 ≤ γ < K. If a set of K binary sequences with
duty factors n /d , where gcd(n ,d ) = 1 for all i, is TI with the MPR capability γ, then its
i i i i
common period is divisible by d d ···d . In particular, the minimum common period is at least
1 2 K
d d ···d .
1 2 K
With duty factors n /d where gcd(n ,d ) = 1 for all i, Corollary 8 further obtains that the
i i i i
lower bound on the period of TI sequences still grows exponentially with K for any γ, although
their combinatorial design requirement is different from that of pairwise SI and SI sequences.
V. AN OPTIMAL CONSTRUCTION
Shum et al. [6] showed that any SI sequence set is TI for the classical model (γ = 1). In this
section, we extend this property to general γ by means of the following result.
Theorem 9 ( [6, Thm. 1]). The sequence set s ,s ,...,s is SI if and only if for each choice
1 2 K
of b ,...,b , N(b ,...,b |s(τ1),...,s(τK)) is a constant function of τ ,...,τ .
1 K 1 K 1 K 1 K
Theorem 10. If a sequence set is SI, then it is TI for any MPR capability γ.
Proof:From (1), we obtain theresult that thethroughputR (τ ,τ ,...,τ ) can becomputed
i 1 2 K
only in terms of N(b ,...,b |s(τ1),...,s(τK)) for some particular choices of b ,...,b . By
1 K 1 K 1 K
DRAFT September29,2014