The performance of the Random Multiple Access (RMA) technique, applied to a direct terminal-to-terminal link with a large number of potential users, is determined. The average signal-to-noise ratio (SNR) is derived. Under Gaussian assumption, the approximation of the probability of error is given. The analysis shows that the system performance is affected by the sequence length, the number of simultaneous users, and the number of cochannel symbols, but is not sensitive to the thermal noise. The performance of using very small aperture antenna for both transmitting and receiving without a hub station is given

Phinainitisart, N.

Wu, W. W.

English

NASA Technical Reports Server

N92o.
Performance of Random Multiple Access
Transmission System
N. Phinainitisart
Chulalongkorn University
Bangkok, Thailand
W. W. Wu
Stanford Telecom
1761 Business Center Drive
Reston, VA 22090
Phone: 703-438-8110
FAX: 703-438-8112
ABSTRACT
This paper determines the performance of the
RMA technique applied to direct terminal-to-
terminal link with large number of potential
users. The average signal-to-noise (SNR) is
derived. Under Gaussian assumption, the
approximation of the probability of error is
given. The analysis shows that the system
performance is affected by the sequence length,
the number of simultaneous users, and the
number of cochannel symbols, but is not
sensitive to the thermal noise. The performance
of using very small aperture antenna for both
transmitting and receiving without a hub station
is given.
INTRODUCTION
Most satellite networks, at present, use earth
stations as shared nodes or involve a number of
small terminals in a star network interacting
with a central master station, or hub station.
Random Multiple Access (RMA) technique will
allow very small aperture terminals (VSAT) to
be installed directly on end-user premises (fixed
or mobile terminals) and have a mesh
configuration. The reason for mesh
configuration is to avoid double-hop of
messages through the gateway or hub stations,
thus, allowing small terminal to small terminal
interconnectivity.
Within a satellite antenna coverage area, all
terminals potentially can directly communicate
with any other terminal. For n terminals, the
number of possible terminal-to-terminal link is
n(n-1)/2. If the system consists of five thousand
terminals or more, then the potential number of
connection is over 12,497,500 different links.
Under such conditions, frequency division
multiple access (FDMA), or time division
multiple access (TDMA), or any other
ortbogonal access system are not likely to be
feasible. The direct sequence spread spectrum
(DS-SS) is also not shown for applications of
such large number of user links. Therefore, it
appears that RMA is an attractive solution to
provide access for large number of VSAT.
The basic transmission scheme of a RMA
system is the transformation of each binary bit
into a set of time-frequency (TF) symbols.
These symbols form a code sequence which is
uniquely assigned to a particular user. The
receiver of the system, then, performs the
inverse transformation from a sequence of
symbols back to the corresponding binary bit.
The procedures for RMA code sequence
generation, which utilizing Euclidean Geometry
(EG) Difference Set {De} and Projective
Geometry (PG) Difference Set {Dp}, were
presented in [1] [2] [4]. In this paper only the
system performance results based on the
derivation from {De } are presented.
International Mobile Satellite Conference, Ottawa, 1990
363
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Section Two describes the system model and
defines the notations. Section Three addresses
the issue of average signal to noise ratio.
Section Four relates signal to noise ratio to the
probability of error with sequence parameters.
Numerical examples are given in Section Five.
SYSTEM MODEL
Our model for a RMA transmission system
presented in Figure 1 is taken from [1]. The L
active users simultaneously transmit through the
satellite transponder to L receivers. Each RMA
transmitter contains a set of frequency
generators (FG), a set of delay units, biphase
modulators, and assembler. Assume that each
transmitter has the same power and there is only
one overlapping symbol between any two
sequences.
The #k user's data signal bk (t) is a sequence
of unit amplitude, positive and negative,
rectangular pulse of duration T. The #k user is
assigned a RMA code sequence, which contains
M symbols, to represent each of it's binary
information bit. Each FG generates a cartier
frequency with proper delay by the delay unit to
represent a symbol in the code sequence. The
biphase modulator phase modulates the data
signal b_ (t) of the #k user onto the carder. The
number of such FG and delay units depends on
the number of symbols/sequence (M). After
passing through the assembler, the corresponding
code sequence S_ (t) is formed for transmission.
The remaining (L-1) transmitters are identical to
the #k transmitter except that different FG and
delay units are used to generate different code
sequence for each of the transmitters.
A the receiving end, the corresponding FG
and delay units, and correlation detections are
used to reverse the encoding process from the
received signal W(t). The total output from all
M correlation receivers of any #k user (Ck) will
include the desired signal bk (t), channel noise
n(t), and multiple-access interference (MAI)
signals. The recovery of the data bit bk is done
by the threshold comparator which provides at T
interval a +1 or -1 output state, depending on
whether Ck is larger or smaller than zero.
The analysis of the interaction between RMA
signals in a multiple access transmission
environment is presented in [1] [3]. The
analysis shows that SNR depends on the number
of simultaneous users (c) the number of energy
per data bit, and the design of TF matrix.
Because the statistic of the simultaneous users
is time varying, the exact evaluations of SNR
and error rate Pe are difficult. Instead, an
approximation will be used, based on the
assumption that all noise is additive white
Gaussian noise and large number of users are
involved. The system is operating in the
environment where there are strong interfering
signals, hence, the system performance is not
sensitive to the variation in the thermal noise.
Consequently, allowing the use of very large
number of small aperture terminals for both
transmission and reception.
APPROXIMATION OF THE AVERAGE
SIGNAL-TO-NOISE RATIO
We have shown in [1] [3] that the output
from each correlation receiver includes the
desired signal, channel noise, and the multiple-
access interference (MAI). This MAI is caused
by all simultaneous users whose sequence
symbols are represented by the same carder
frequencies as the desired symbols. The total
number of cochannel symbols for each carrier is
simply the number of time-delay units in the TF
matrix, or T d.
The precise value of MAI can be calculated
only when the statistic of the simultaneous users
are known at a particular time for a particular
system determined by the corresponding TF
matrix. Therefore, the average value of multiple
access interference and consequently the signal-
to-noise ratio, which is useful for preliminary
system design where the statistic of each user is
not available but the sequence parameters and
TF matrix are known, will be determined. The
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International Mobile Satellite Conference, Ottawa, _990
SNR at the input of the threshold comparator
can be expressed as [1].
M ]x/2.._.___U+ No3M 2_ b
(i)
where-0- a + c_ - _ (2)
3M
The parameters _ and a are the part of MAI
of any user, which is caused by his own
transmitter, and any other simultaneous user
accordingly.
= average number of designated cochannel
symbols (0_<a<__M)
a = average number of cochannel symbols
between designated user and any other
user (0_<d<M)
c = total number of simultaneous users
(O_<c<L)
Both _ and d can be expressed in term of
sequence length, as followed:
= yM where 0<y<l (3)
_l- max. (U)-yM (4)
max. (c)
Parameter y is a user's cochannel symbols-to-
sequence symbols ratio, or
m
y=
average number of cochannel symbol l
m a sequence
total number of symbols in that]
sequence
When all users are activated simultaneously,
we have
max. (c) = L-1 (5)
where L = L_ = M(M+I) (6)
The maximum value of parameter U are given
by [1]
max. (U)= M[(M+I)T d - 1] (7)
Substituting (7) and (6) into (4)
d - M[(M+OTd - Z] - yM (8)
M(M+I) - 1
Since M>>I, (8) reduce to
d *' T d (9)
From (2), (3) and (9), we have
U = yM + cT d (I0)
m
Finally substituting U into (1), we have
. l. -- (11),%[yM+CTd] +-_a
In terms of energy per data bit, (11) can be
expressed as
SNR -
1 1
+
(g d N,,)uja (g flN ,,),h.
(12)
(E_o) energy -per-bit/channelwhere - thermal noise density ratio
th.
Eb ) 3Mtam yM+cTd
(13)
International Mobile Satellite Conference, Ottawa, 1990
365
or = energy-per-bit/multiple-
access-interference density
ratio
PROBABILITY OF ERROR
In this case of a communication system with
additive white Gaussion noise (AWGN), which
includes multiple access interference noise and
channel noise, a data bit error occurs at the
threshold comparator output when the integrated
amplitude of the noise is larger than the
integrated amplitude of the desired signal in the
opposite direction.
The threshold comparator provides, at T
intervals, a +1 or -1 output state, depending on
whether C is larger or smaller than zero,
respectively. Thus, the probability of error is
given by
Pe = P[C_>0/b i = -1] P [bi= -1] + P[Ci<0/b i =
+1 P[bi= +1] (14)
where P[C,>O/b,= -1] = probability of having
sampled C1 larger than zero, given that a
-1 (b_) is being transmitted
For equa-probable signaling, (14) becomes
or
Pe = 0.5 {P[C,>0/b_ =- 11 + P[C:0/b, = +1] }
Pe = 0.5 P[(total noise at the sampling
instant) > (total desired signals at the
sampling instant)]
The probability of error can be approximated
by
ee-QfSNRI (15)
From (11) and (15), we can conclude that the
probability of error of the RMA system depends
on the following parameters:
sequence length (M)
user's cochannel symbols-to-sequence
symbols ratio (y)
number of simultaneous users (c)
number of time-delay in the TF matrix or
number of cochannel symbols (Td)
energy-per-bit/channel-thermal-noise
density ratio (E_o)U,
Note that the parameters M, y and Td depend
on the structure of the sequence and the size of
the TF matrix, but the (E_No)_h can be
calculated from the link budget parameters, such
as transmitted carrier power, bit rate, etc. It is
of interest to know how the system will perform
when the number of simultaneous users are
varying, as this is the only parameter which can
not be predetermined or fixed due to the random
access nature of the system. Thus, the graphs in
Figures 2 - 5 are of Pe versus C for different
values of M, y, T_ and (Eb/No)th
Our objective is to keep the value of (Eb/No)u,
to a minimum; as the higher (E_No)_. may result
in higher uplink carrier power, or larger antenna
at the received station or a reduced data rate.
We assume that all stations have the same
power, thus, any increase in energy per data bit
from the desired user also means an increase in
the MAI. Therefore, (Eb/No)_. has little effect on
the Pe as we can observe in Figure 3. The Pe is
also not sensitive to the change of parameter y
as shown in the plot in Figure 4. On the
contrary Figures 2 and 5 demonstrate that the
sequence length and the number of cochannel
symbols in the TF matrix greatly affect the
probability of error, especially when the number
of simultaneous users are relatively very small
compared to the total number of users.
Plots in all figures are for sequences of length
59, 97, 107, 127, and 169 which can be
constructed from EG sets. The corresponding
number of users are calculated from (6) as
followed:
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International Mobile Satellite Conference, Ottawa. 1990
Table 1. Total Number of Users For Each
Sequence
Sequence Length (M) Total No. of Users (Le)
Table 2. Pe for Different Sequence Length
When Number of Simultaneous Users
is Fixed (c = 25), and T d = M
M Pe
59 3,450
97 9,560
107 11,556
127 16,256
169 28,730
289 83,810
For preliminary system design it is useful to
be able to carry out a tradeoff between the
parameters M, T d, and c as they all have
significant effect on Pe. Such a tradeoff can be
obtained from (11) and (15) as:
Pe - Q cTa No (16)
[y + --g] + --2E b
Given y = 0.2, E_No = 0.2 dB, (16) becomes
Pe-Q
cr,[0.2 + --_-1 + 0.477
(17)
The effects of these three parameters can be
summarized as followed:
1) For a fixed number of simultaneous users, we
have to increase the sequence length in order to
improve the system performance. An example
is shown in Table 2
289 5.95E-09
169 6.71E-06
127 8.00E-05
107 2.65E-04
97 4.72E-04
59 4.90E-03
2) If a certain percentage of users is expected to
transmit simultaneously, we'll find that smaller
size system will have a better performance than
the bigger one. This can be observed from the
Table 3.
Table 3 Pe for Different System Size When c
is a Fixed Percentage of Total Users
(c = 0.001% of L), and T d = M
c = 0.001% of
Total users Total users Pe
83,810 84 7.4E-04
28,730 29 2.5E-05
16,256 16 1.7E-06
11,556 12 6.3E-07
9,506 10 2.9E-07
3,540 4 1.0E-08
3) Since the noise in this RMA system is
dominated by cochannel interference, we can
improve the system performance by reducing the
number of cochannel symbols. In other words,
International Mobile Satellite Conference, Ottawa, 1990
367
we have to reduce Td while increasing F
accordingly, so that the total number of symbols
(V = Td*F) can remain constant.
Table 4 and Figure 5 show how we can
significantly reduce the probability of error by
increasing the carrier frequency. In this specific
example, the sequences are derived from a EG
set, hence, Td*F=M 2. When the TF matrix is a
square matrix, we have F = Td = M. If we
reduce the cochannel symbols by half, the "IT
matrix becomes a rectangular matrix and F =
2M.
Table 4. Pe for Different Cochannel Symbols
(T d) (M = 169)
C
(users) T a = M Td = 0.5M
14 4.48E-09 3.38E-15
16 3.45E-08 1.03E-13
18 1.64E-07 1.45E- 12
20 5.84E-07 1.29E- 11
22 1.68E-06 8.12E-11
24 4.12E-06 3.89E-10
26 8.87E-06 1.50E-09
28 1.72E-05 4.88E-09
30 3.0E-05 1.38E-08
4) If the system is designed to provide a fixed
nominal bit error rate performance, the number
of simultaneous users can not exceed a certain
limit. An example is shown in Table 5.
Table 5.
M
Maximum Number of Simultaneous
Users for Different Sequence Length
When Pe = 10 -7 and T d = M
Max. c as %
Maximum c of Total Users
289
169
127
107
97
59
31 0.037%
16 0.056%
13 0.08%
11 0.095%
10 0.1%
5 0.14%
EXAMPLES OF LINK BUDGET FOR
VERY SMALL APERTURE ANTENNAS
In this section some link budgets are presented
to illustrate that the proposed RMA technique is
of use to a system with very small aperture
terminals (VSAT).
We have demonstrated in the previous section
that the (EtfNo),_ has some contribution to the
system performance (Pe) but it will require a big
increase in (EdNo)tb ' to have any noticeable
effect on Pe. In addition, the carrier power
varies with (Ed_o)t_, thus, it is important to
keep the value of this parameter at some
minimum level. Since the lowest level of
(EdNo),_ ' assumed in this section is 2.0 dB, we
will use this same value for conveniently
relating the link budgets to the PE already
calculated. The link budgets are designed for
the system which the transmit and receive
stations are the same size.
14/11 GHz 6/4
Antenna Diameter 1.2
(m).
Antenna Gain (dB) 42.3
HPA (dB/carrier) -24.1
EIRP/carrier
(dBW) 18.2
Satellite G/T
(dB/K) 1.7
Up Link (C/T)
thermal (dBW/K)
Antenna Gain 1
ms (dB-m 2) 44.0
Saturation Flux
Density (dBW/m 1) -90.5
Input Backoff/
cartier (dB) -53.6
Output Backoff/
carrier (dB) -48.1
EIRP Downlink (dBW) -7.1
Earth Station G/I"
(ctB/K) 15.5
1 0.8 1
40.8 38.8 35.0
-21.6 -16.9 -1.4
19.2 21.9 33.6
1.7 1.7 -2.0
-188.1 -186.4 -184.4 -169.1
44.0 44.0 37.0
-90.5 -90.5 -81.0
-51.9 -49.9 -51.1
-46.4 -44.4 -46.6
-5.4 -3.4 -10.6
13.8 11.8 10.0
At Ku-band the EIRP is assumed as 41.0 dB,
368
International Mobile Satellite Conference, Ottawa, 1990
downlink (C/T) is -197.6 dBW/K which is
almost the same as the total (C/T). At C-band
the EIRP is 36.0 dB and the total (C/T) or
(C/No) is the same as that of Ku-band.
In order to increase the speed of information
bit, we have to increase the transmitted EIRP or
increase G/T of the earth terminal. Table 6
shows bit rate versus (EJNo)th..
Table 6. (C/No)th. Required for Different Bit
Rate
Rb Total (C/No)m
bit/sec, dB dBW/Hz
1200 30.8 31
2400 33.8 34
4800 36.8 37
9600 39.8 40
As an example, if we increase the information
bit rate of 1200 bit/sec, to 4800 bit/sec., we have
to increase the EIRP proportionally. That is to
raise the uplink EIRP level up to 6 dB. Another
alternative is to introduce error coding in order
to increase the G/T by 6 dB.
CONCLUSION
The expressions for approximation of the
signal-to-noise ratio and the probability of error
of the RMA system has been obtained, based on
the Gaussian interference assumption. It is
assumed that the system consists of a large
number of users, thus, long sequence are
required.
The evaluations of the error probability are
presented. The results show that the sequence
length, the number of cochannel symbols in the
TF matrix and the number of simultaneous
users, which all contribute to the multiple access
interference (MAI) parameters, affect the bit
error rate. On the contrary, the system is not
sensitive to the changes in the energy-per-
bit/channel thermal noise density ratio (E_qo)_,
thus, allowing the use of very small aperture
terminals (VSAT) for both transmission and
reception without a larger hub station. The
examples of link budget for VSAT are given for
illustration.
However, the system performance will
degrade rapidly as the number of simultaneous
users is increasing. Therefore, this access
technique is suitable for systems with large
number of users and light traffic. This
technique can be applied to emergency network,
military network, thin route commercial
network, and mobile satellite communication for
aeronautical, land or maritime.
REFERENCES
lo
1
*
e
a
.
N. Phinainitisart. An access technique for
very small aperture antenna. Thesis for the
Degree of Doctor of Engineering,
Chulalongkorn University, Bangkok, Thailand
(May, 1990).
N. Phinainitisart. W.W. Wu. Code
sequences for gatewayless transmission.
Proceeding IEEE GLOBECOM (Nov., 1989),
Vol. 3.
N. Phinainitisart. W.W. Wu. Analysis of
random multiple access transmission system.
International Journal of Satellite Commun. to
be published in 1990.
W.W. Wu, Elements of Digital Satellite
Communication. 2 Vols. Computer Science
Press, 1985.
K. Feher. Digital communications."
satellite/earth station engineering. Prentice
Hall Inc., 1983.
T.T. Ha. Digital satellite communications.
2nd ed. Mcgraw-Hill, 1990.
International Mobile Satellite Conference, Ottawa, 1990
369
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370
International Mobile Satellite Conference, Ottawa, 1990
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International Mobile Satellite Conference, Ottawa, 1990
371
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FIGURE 4: Pe AS A FUNCTION OF c FOR DIFFERENT y
(Eb/No)th" =0.2 dB, Td=M, M=169
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International Mobile Satellite Conference, Ottawa, 1990


Performance of random multiple access transmission system

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