1. Advanced Wireless Networks
Lecture 3: Main Parameters of Multiple Access Systems
Capacity of Multiple Access Systems
General Shannon's Formula:
1) In FDMA, each user is allocated a bandwidth
is a number of users. .
For one user capacity equals:
For K users capacity equals:
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

2. Capacity of FDMA Systems (Continuation)
The total capacity is equivalent to that of a single user with average power. It is interesting to note that for a fixed bandwidth, the total capacity limits to infinity as the number of users increases linearly with K. On the other hand, as K increases, each user is allocated a smaller bandwidth and, consequently, the capacity per user decreases.
If we now introduce a normalized capacity per user as and the energy of signal will change on the bit energy, we will obtain the following formula:
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

3. Capacity of FDMA Systems (Continuation)
The normalized capacity is presented in Figure versus for different parameter K from 1 (single user) to 10.
A more compact form of above formula is obtained
by defining the normalized total capacity
which is the total bit rate R for all K users per unit
of bandwidth:
or equivalently:
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

4. Capacity of FDMA Systems (Continuation)
The graph of versus is shown below.
We observe that increases as increases above the minimum
value of ln2.
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

5. 2) Capacity of TDMA Systems
In a TDMA system, each user transmits for 1/K of the time through the channel
of bandwidth
with average power KP.
Therefore the capacity per user is
which is identical to the capacity of an FDMA system.
However, it must be note that in TDMA it may not be possible to transmit power of KP when K is very large.
Therefore must be some threshold in power for the transmitter beyond which it cannot transmit as K is increased.
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

6. 3) Capacity of CDMA System
If we assume that each user's pseudo-random signal waveform is Gaussian and each user signal is corrupted by Gaussian interference power (K-1)P and additive Gaussian noise . Therefore, the capacity per user is
or, accounting bit energy, the normalized capacity is:
Figure illustrates the graph of normalized capacity
per user versus
, with K as a parameter.
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

7. Capacity of CDMA System (cont.)
For a large number of users, we can use approximation ln(1+x)<x, then
or for the total normalized capacity of all K users:
In this case, we observe that the total capacity does not increase with K
as in TDMA and FDMA.
The achievable K-dimensional rate region for the K users in an AWGN channel,
assuming equal power for each user, is given by the following inequalities:
(a)
(b)
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

8. Capacity of CDMA System (cont.)
In the special case when all rates are identical, the inequality (c) is dominant over the
other K-1 inequalities.
From the above discussions, we conclude that the sum of the rates of the K users goes to
infinity with K.
Therefore, with cooperative (correlative) synchronous users, the capacity of CDMA
has a form similar to that of FDMA and TDMA. Note that if all the rates in the CDMA
system are selected to be identical to R then (c) reduces to
which is identical to the rate constraint in FDMA and TDMA.
However, if the rates of the K users are selected to be unequal such that the inequalities
(a)-(c) are satisfied, then it is possible to find the points in the achievable rate region
such that the sum of the rates for the K users in CDMA exceeds the capacity of FDMA
and TDMA.
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

9. Capacity of CDMA System (cont.)
Example: Consider two users in a CDMA system that employs coded signals as described above. The rates of the two users must satisfy the inequalities:
Find: The capacity region for two-user CDMA system.
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

10. Capacity of CDMA System (cont.)
Solution:
The form of capacity region for two-user CDMA system can be presented schematically
as shown in Figure, where
are the capacities corresponding to two users with
We note if user 1 transmits at capacity
user 2 can transmit up to a maximum
rate equals:
which is illustrated in Figure as point A.
The same we have for user 1, if user 2 transmits at
which is illustrated in Figure as point B.
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

11. Some Aspects of CDMA System Performance
1) Signal-to-Noise Ratio in CDMA Channel
In the CDMA channel that is shared by K simultaneous users
is the energy per bit,
is the data block from the kth user,
if each transmission block has length N,
the cross-correlation function
Each user is assigned a signature waveform
, where .
is a pseudo-noise (PN) code sequence consisting L chips that take values
is a chip duration and
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

12. Some Aspects of CDMA System Performance (cont.)
The baseband transmitted waveform for k-th user can be presented as:
The total signal from K users is
If a channel is corrupted by AWGN, we can rewrite above at the output correlator as:
Then, the probability of bit error for the kth user at the output of the correlator is
The average probability of bit error is simply:
and
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

13. Some Aspects of CDMA System Performance (cont.)
Example: Consider two symbol-synchronous users with signal energies and
Find: The asymptotic efficiency of the conventional detector.
Solution
For this case from above formulas we get a probability of error
The asymptotic efficiency follows from its definition: .
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

14. Some Aspects of CDMA System Performance (cont.)
2) Performance of DS-SS Modulation
Let us consider a DS-SS system with K multiple access users, each has a PN sequence
with N chips (with duration
) per message symbol period T such that
If we assume that the average probability of the bit error can be described by the error
function Q related to Gaussian distribution function, we have for K-1 users, which are
served as identically distributed interferers, an expression for the average BER is: .
For a single user, K=1, this expression reduces to the BER expression for BPSK
modulation of the data signal. In the case of K-1 interferers for the interference
limited situation in the system, when thermal noise is not so significant and
the BER expression can be simplified to
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

15. Some Aspects of CDMA System Performance (cont.)
2) Performance of FH-SS Modulation
In FH-SS systems, several users independently hop their carrier frequency while using BFSK
modulation of data signal. If two users are not simultaneously utilizing the same frequency
band, the BER for BFSK can be expressed as
If there are K-1 interfering users in the system and M possible hopping slots, there is a 1/M
probability that a given interferer will be presented in the desired user’s slot and the overall
probability of BER can be modeled for fast FH as
For slow FH-SS system, the probability of BER is more complicated and described by the
following formula:
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

16. Some Aspects of CDMA System Performance (cont.)
Example: A FH-SS system uses 50 kHz channels over a continuous 20 MHz spectrum. Fast frequency hopping with 2 hopes per bit is used. For data signal modulation BFSK is used.
Find: a) the number of hops per second if each user transmits at 25kbps;
b) the probability of BER for single user operating at ;
c) the same probability, but for user operating at with 20 other
users, which are independently frequency hoped. .
Solution
a) For the number of hops per second is:
b) For single user we have for
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

17. Some Aspects of CDMA System Performance (cont.)
c) we first of all will find the number of hoped channels:
Then the probability that the desired user interferes with 20 other users
can be found as: .
which is of one order higher than the same probability for one user:
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.