Svetislav V. MariC, Edward L. Titlebaum and Zoran KostiC. Department of Electrical Engineering,. University of Rochester, Rochester, NY 14627. Abstract: In this ...
ADDRESS ASSIGNMENT FOR MULTIPLE-ACCESS SYSTEMS BASED UPON THE THEORY OF CONGRUENCE EQUATIONS* Svetislav V. MariC, Edward L. Titlebaum and Zoran KostiC Department of Electrical Engineering, University of Rochester, Rochester, NY 14627. Abstract: In this paper we discuss the address assignment via frequency hop patterns for n multiple-access spread spectrum communication system. In such a system the receiver simultaneously communicates with a large number of transmitters (users), distinguished by their assigned addresses. In order to insure that the receiver correctly ‘recognizes’ which users are active at the moment, it is necessary for the assigned addresses to possess minimum mutual inteflerence. Here we define an algebraic construction for frequency hop patterns based upon the theory of congruence equations. The fiequency hop patterns are constructed in such a way that they carry the address assignment and the message at the same time. The decoding scheme for the system is presented and we prove that in both synchronous and nonsynchronous case the mutual iniei-ference between the transmitted frequency hop patterns is minimal.
received there is no ambiguity about the sender and the message it carries [I]. In the paper we investigate the assignment of addresses to signals from different transmitters in order to solve the above mentioned problems in the multiple-access system. We implement the modified system proposed by Viterbi [2], in which each address is represented by a different frequency hop pattern, which besides the address carries also the transmitted message. After the frequency hop pattern is received, the decoder determines which user has sent the message and decodes it as well. We propose here an algebraic method for generating the addresses - frequency hop patterns that achieve minimum mutual interference. The method is based upon the theory of congruence equations and the system performance is equal to the one described in [3], where the address assignment was based on Reed-Solomon Codes [4]. The paper is organized as follows. In the next section we give the definitions necessary for the sequel. In the third section we give the description of the system, namely the address assignment and the decoding scheme as well as the algebraic construction of the frequency hop codes that are used in address and message definition. We then prove that the system indeed possess the minimal interference between the addresses. In the conclusion we give a short summary of the results and compare them with those in [3].
I Introduction This paper deals with the multiple-access spread spectrum communication system in which a large number of transmitters needs to simultaneously communicate with one receiver. For instance such a system is a satellite communication network in which a large number of earth stations needs to simultaneously communicate with one satellite receiver [I]. There are two main problems that such a multiple-access communication system must deal with: 1) it must receive the transmitted signal in such a way that it interferes as little as possible with the reception of signals from other users and 2) the signal must be coded so that once it is
I1 Preliminaries In order to transmit the addresses and messages the transmitter uses a frequency hop signals defined as follows:
*
This work was in part supported by the grant from SDIOflST and managed by the office of the naval research under contract N0014-8f.LK0511.
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0283
Definition 1: A frequency hop signal of length T is a train of N equal-length pulses with the kth pulse being frequency modulated with frequency f,,. The frequencies to be placed in appropriate time slots are determined via frequency hop coding, by the set of ordered integers, y , which is called the placement set'. Each element of this set is called the k-th placement operator and is denoted by: ~ ( k ) , k = O , l , ..., N - 1 .
The expression for the frequency fl; = y
B ( k ) ?;J
t7
5=I 3 Ck 3 2 1
(1)
0
is then:
fk
k = 0,1, ..., N
t
-b
time
0 -
1
(2)
1
2
3
4
Figure 2 Frequency Hop Pattern
where B is the signal bandwidth (note that the signals from all users occupy the same available bandwidth, B). The frequency hop signal is shown in figure 1. and can be written as:
We also assume that the system has both block and bit synchronization. Block synchronization means that the pulses from each signal are transmitted in the same relative time frame. Block synchronization reduces the error probabilities, but is not a necessary assumption for successful communication [6]. Bit synchronization means that there is no overlap from one signaling period to the next.
where
The interference effect in this type of the system comes from the simultaneous use of the same frequency channel. This event is referred to as a "coincidence" or a "hit". The receiver is not able to distinguish between the hit and the single use of the frequency channel, hence the greater the number of hits the greater the probability of error. Therefore, it is our goal to design frequency hop patterns with the smallest possible number of hits.
and
fc
is the carrier frequency.
I 0
TIN
iT/N
'N-4 T
time
Figure 1 Frequency Hop Signal
III Address and Message Assignment
In the sequel we assume that the number of different frequencies is equal to the number of time slots. Then a convenient way of representing a frequency hop signal is through a N x N matrix where N rows correspond to N frequency channels and N columns to N time slots. The element of the matrix, uk,l is equal to one if and only if the frequency fx is transmitted in the interval ti. Otherwise, ak,l is equal to zero [S].This representation is shown in figme 2. where the shaded squares correspond to the transmitted frequencies'.
In order to be identified by the receiver each of the Q users, u1,u2?...,U ; , ...?U Q , is assigned an address from the set A of frequency hop patterns
in such a way that the user ui is assigned the address yi. The users also transmit a message, m, from the set M:
M = {0,1, ..., R } .
In the paper we use the terms "placement set" and "frequency hop pznem" interchangeably. For simplicity we denote they axes in the figures by frequency although it is ihe placement set that is drawn. In order to obtain the real frequency the placerncnt should be multiplied by the constant
As noted above each frequency hop pattern, yz, consists of placement operators, yz( k ) , to wit yz = ( ~ (0) z 7 yz (1) ...> yz ( N - 1 ) ) .
5.
3
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(6)
We propose the following construction for the i-th frequency hop pattern - a d d r e ~ s : ~ :
the decoder subtracts from it (in modulo p arithmetic), placement operator by placement operator all the prespecified addresses. Only the correct address, yz, will completely fill the row m,indicating that the user, U , was active transmitting the message m. In order to prove that frequency hop patterns generated through (7) have at most one hit and therefore achieve minimum possible interference we need the following important theorem: Theorem 1: Every member of a complete residue system modulo a prime p is a solution to the congruence equation
ikn + m ( m o d p ) (7) k = 0 , 1 , ...,p - l , i = 1 , 2 ,..., Q, EM yz( k )
with p an odd prime such that p and n satisfy the additional constraint: d = ( $ ( p ) , n ) = 1.
(8)
d is the Euler-Phi function [7] and d = ( a , b) is the greatest common divisor, GCD, of a and b. Theorem 1. that follows explains the necessity for constraint (8). From (7) it is seen that for a prime p , maximum of Q = p - 1 users can simultaneously operate in the system with a possibility of transmitting a message, m, from the set 144 = { 0,1, ..., p - 1). Note that in a different context, congruence equations of the same type as (7), with 12 = 1 and n = 2 were used in [SI and [9] respectively. Also, for n = 1 these frequency hop patterns are equivalent to prime number codes as defined in [6], although they are generated differently. Some comments about (7) are in order. Since for any prime number
q I C n + c2 z (modp) k = ~1 = ~2 = 0,1, ...,p - 1 ,
if the greatest common divisor, d, of $ ( p ) and n is equal one. Proof: It is known from the classical number theory (see for instance [7]) that the equation an = b
(13)
(modp)
has a solution if and only if b*
(9)
d(P) = p - 1
(12)
1
(modp)
d = ( # ( p ) , n ).
(14)
Since in (12) the prime p defines a finite field [ 101 the element
it follows that for all odd primes the Euler-Phi function is even. Therefore, for n = 2s, s an
integer, equation (8) is never satisfied and in fact n must be of the form: n = 2s 1. Further, for n = 1 any prime satisfies (7), but for n > 1, a little attention has to be paid in prime selection. As an example for n = 3 or n = 5 the prime must not be of the form
+
exists and is unique. Now (12) can be written as
k”
zP--l
(11)
- 1 =
(16)
(modp)
and, since in the theorem d = 1 and z1 is a solution to (16) if :
or p=5s+1
= 21
4 (p)=p
(modp).
-
1,
(17)
By Fermat’s Little Theorem [7] this is true for all positive integers (except zero) less than p . Going back to z and noting that for c1 = c2 = 0, z = 0 it immediately follows that the values of z form a complete residue system. This finishes the proof. Note that as equations (7) and (12) have the same form, y ( I C ) also form a complete residue system modulo the prime p , meaning that in the pattern each frequency appears once and only once. We first consider the synchronous case (signals from different users are all aligned in time) and
respectively, since then (8) is not satisfied. An example of a placement set (generated through (7) with n = 3 and p = 5) for the user u1 transmitting a message m = 2 is shown in figure 2. Note that the arithmetic is done modulo five and that there are maximum four users. The receiver possess (stored in the memory or generated) all the possible placement sets - addresses. Once the frequency hop pattern is received “e”is to be read “is congruent to”
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prove that frequency hop patterns if generated by (7) have at most one hit. Suppose that two frequency hop patterns are
and
The hit occurs when
0 1 2 3 4 Figure 3 Receiver Mamx The decoder finds all occupied frequency channels and subtracts their values from the prespecified addresses. For instance, in order to determine if user one was active the decoder takes its address, (0 1 3 2 4),and first subtracts 0 from I and 2. Then it subtracts I from 0, 1, 2 and 3. Continuing the same procedure for every time pulse, at the end, one of the combinations will give 1 1 1 1 1 and the decoder will correctly conclude that user one was active transmitting message one. Repeating this procedure for the remaining addresses, it will decode all active users and their messages. Finally we comment on the performance of codes in the absence of synchronization. The effect of this is to randomly shift the frequency hop patterns, causing for instance the first time pulse of one message to arrive at the same time with the third time pulse of the other message. Now the hit occurs when
so, two hits occur when equation (20) is satisfied simultaneously for two different values of q:
Combining (21) and (7) two hits occur if and only if ( a - b ) j " = ( U - b)?.
(22)
Since a ; # a j and from the proof of theorem 1.
...,p - 1 )
and j # r , (23) equation (20) can never be satisfied and consequently frequency hop patterns can simultaneously form only one hit. jn#rn
j,rE{O,l,
In table 1. we present a system of four users with placement sets generated through (7) with n = 3. The assigned messages and the transmitted frequency hop patterns are also given. user
address
message
where r is a random time shift. Inserting equation (7) in (24) we have:
freq. hop pattern
U1
01324
1
12430
U2
02143
1
U3
03412
2
13204 20134
U4
04231
2
21403
By Lagrange's theorem [7] equation (23) can have at most n noncongruent solutions modulo prime p , allowing at most n simultaneous hits to occur. Therefore, when n = 1 the loss of synchronization does not affect the system performance. For other n the number of hits increases resulting in the increase of the error probability. This leads to the conclusion that when there is no synchronization in the system n = 1 should be used in equation (7) to assign the addresses .
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IV Conclusion
In Digital Satellite Communication Conference,
We have introduced a new algebraic construction of frequency hop patterns for address and message assignment in multiple-access systems, based on congruence equations. In the synchronous case, we have proved that for any address and message assignment frequency hop patterns generated by (7) overlap at most once and consequently the multipleaccess system has minimal possible interference. Further, we showed that in each pattern each frequency gets utilized once and only once. This kind of transmission causes minimum problems for the coherent receiver. Further, it is seen from equation (7), that when n = 1 these codes are equivalent to prime number codes as defined in [6], and that the perfomance of our frequency hop patterns is equal to the performance of the patterns generated in [3]. In the asynchronous case we have proved that the patterns possess at most n possible hits. Therefore in this case, the frequency hop patterns generated through (7) with n = 1 still have at most one hit and retain the minimal possible interference between the channels. Unlike in [3], the address and message assignment for congruence patterns is the same in both synchronous and asynchronous case. This results in an easier and faster decoding scheme, which makes the use of congruence frequency hop patterns more attractive.
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References [I] H.L. Van. Trees editor. Satellite Communications. John Wiley and Sons - IEEE Press, New York, NY, 1979. [2] A.J. Viterbi. A processing satellite transponder for multiple-access by low-rate mobile users.
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