Results for INR with JMUD

As a third, and last, case of MUD-based system, we analyze an INR with JMUD where the decoder either decodes all the users or asks to all the users to retransmit new redundancy. Since in principle this might not be the optimal INR strategy with INR, our result is a lower bound to \clever"

INR systems with MUD. We shall prove that this scheme actually achieves the ergodic rate-sum of the underlying block-fading channel, thus proving its optimality.

In a system characterized by user random activity, a given user is ac-tive on a sequence of slots that, in general, are not consecuac-tive in time and have dierent set of active users. In this scenario, it is not clear how to carry out joint decoding across the slots. In the two cases discussed above, joint decoding across the slots is not needed since an ALO protocol was con-sider. In general, for any protocol and any receiver structure the throughput cannot exceed the ergodic rate-sum of the underlaying MAC block-fading channel. For a system symmetric with respect to all the

K

users, the ergodic throughput is given by

^(ergodic) = E

2 4log

0

@1 +^XK

j=1

j

1 A 3 5

(a)= E[log(1 +

K

eq)]^jeq=_K^{1 P}_Kj⁼¹j (b)

log(1 +

K

_E[

]) (3.58)

where (a) shows that the

K

-user ergodic throughput is the single-user INR throughput with equivalent fading given by the arithmetic mean of the fad-ing powers of the dierent users; and (b) follows from Jensen inequality and shows that the single-user AWGN capacity is an upperbound to all system without power control at the transmitters. Since

^(ergodic) is increasing in

K

, the single-user unfaded performance is achieved by letting

K

^!1. The convergence to the RHS of (3.58) follows from the central limit theorem.

Any system with user random activity has throughput that satises

^(ergodic), hence also INR with JMUD (assuming we were able to dene what INR with JMUD is!). A lower bound to this INR with JMUD can be obtained by imposing the (possibly) sub-optimal decoding scheme \all or none", i.e. either the decoder is able to decode all the users or asks to all

3.4 Unspread system with joint decoding 95

the users to transmit a new \chunk" of their codeword. We can write

1 +^P

K R

¹_m=1

p

(

m

)

E where

p

(

m

) is the probability that successful joint decoding does not occur after

m

received blocks. From the denition of our decoding strategy we have that, for every

m

1,

p

(

m

) is given by all the users, i.e., the probabilities in (3.61) depend only on the cardinality of

S

and not on

S

itself, it follows

p

(

m

) Pr fading multiple-access channel with

K

symmetric users. It easy to see that

(

K

) is increasing in

K

while

(

K

)

=K

is decreasing in

K

. At this point, we can bound the probabilities in (3.62) as follows

Pr^"1 Large deviation Theorem[88, Sec.5.11]. With the bounds in (3.64), the sum of the

p

(

m

) over

m

1 is bounded by

and by taking the limit for

R

^!1, and by recalling (3.59), we arrive at the desired results This proves that an optimal system \forces" the users to transmit all the time, i.e.

G

=

K

and for every

K

, and decodes either all of them or no one.

While for SUD-based systems with innite population, optimal

G

^{! 1} in

E

n

=N

0²[(

E

n

=N

0)min

;

(

E

n

=N

0)th] and

G

^!1 in

E

n

=N

01, for this scheme

G

=

K

^!1 for all

E

n

=N

0.

3.5 Conclusions 97

0 2 4 6 8 10 12 14 16 18 20

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

E_n/N₀ [dB]

η [nat/dim]

Upper bound: AWGN single user JMUD SIC−SUD Single user Unspread Hybrid−ARQ MMSE SUMF

Figure 3.7: Throughput versus

E

n

=N

0 for ALO with Rayleigh fading.

3.5 Conclusions

To conclude the chapter, we put on the same chart the throughput of all the systems analyzed so far, in order to compare them and draw some conclu-sions. We claim that our comparison based on equal received energy per bit is fair. However, it is worth reminding that from a practical point of view there are other quantities of great interest, average delay and probability of packet loss for example, that were not considered in this work. Furthermore, comparison is made on total throughput versus

E

n

=N

0, the choice of other performance measures, like per-user rate or delay, etc, may lead to dierent conclusions from the one we are going to present here.

In the following we refer to Fig. 3.7,which shows the throughput curves versus

E

n

=N

0 of all the analyzed systems with ALO protocol in Rayleigh fading (we did not reported the cases without fading to make the picture more readable) and to Fig. 3.8, which shows the throughput curves versus

E

n

=N

0 of all the analyzed systems with INR without fading (again, the curves referring to the Rayleigh fading case were not added to make the picture more readable).

Minimum

^En=N0

.

We have identied two values of (

E

n

=N

0)min under which the throughput is zero. One is (

E

n

=N

0)min= e = 4

:

3429 dB for ALO with Rayleigh fading, the other is (

E

n

=N

0)min= 1 = 0 dB for ALO, RTD, INR without fading and RTD and INR with Rayleigh fading. This shows

0 2 4 6 8 10 12 14 16 18 20 0

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

E_n/N₀ [dB]

η [nat/dim]

Single user Unspread Hybrid−ARQ MMSE SUMF

Figure 3.8: Throughput versus

E

n

=N

0 for INR without fading.

the benet of packet combining over discarding previous transmission in a faded environment.

Low

^En=N0

regime.

For low

E

n

=N

0 all the protocols, regardless the fading statistic, have the same throughput which corresponds to vanishing information rate

R

and innite channel load

G

. This means that users must be active all the time, in the unspread case, or be innitely more than the spreading gain, in the CDMA system. Fading does not impair the system performance, compared to the unfaded case, because the available power is so little that fading uctuations do not matter. In this regime the simplest system can be implemented without loosing in throughput, in fact CDMA with SUMF does not loose in performance with respect to more complicated schemes as CDMA with MMSE. For the ALO with Rayleigh fading, the curves of Fig. 3.7 show that JMUD is superior to CDMA and unspread system. In this regime SIC-SUD is equivalent to JMUD and again the simpler systems have the same performance of the more complex ones.

High

^En=N0

regime.

The dierences in performance among systems and protocols are visible at high

E

n

=N

0. For all SUD-based systems but CDMA with SUMF, the optimal information rate

R

grows with

E

n

=N

0, the channel load

G

^!1 and the throughput grows linearly with log(

E

n

=N

0). This means that when the available power is high, the best strategy consists of encoding

3.5 Conclusions 99

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