MAC-DMT for General MIMO-MAC with Individual ML Decoding

In the previous section we investigated the MAC-DMT for a general MIMO-MAC with joint de-coding at the receiver end. We also observed in Example 2 that certain DMT performance loss could result from the use of joint decoder. However, such loss can be safely avoided by the use of individual ML decoder.

Recall that for the ith user, the truly optimal decoder, though having extremely high computa-tional complexity, is the individual ML decoder that seeks optimal ML estimate ˆS_iby

Sˆ_i = arg max

· · · × S_K−1. Clearly (4.7) outperforms (4.1) in error performance, but at a cost of much higher computational complexity.

Without loss of generality, below we focus on the error performance of the individual ML de-coding for the ith user. To distinguish the DMT performances of decoders (4.1) and (4.7), we shall call the DMT of the latter the individual MAC-DMT and will denote it by d^(i)∗_{n

0,··· ,nK−1},nr(r₀, · · · , r_K−1).

To characterize the DMT performance of the individual ML decoder, we only need to consider the outage events O (I) (cf. (4.5)) in which the ith user is a member of I. Event O(I) with i 6∈ I is not counted as an outage for the ith user for obvious reasons. Thus, along similar lines as in the proof of Theorem 8 we can show the following.

Theorem 9 (General individual MAC-DMT). Let K, n_i,r_iandn_rbe defined as before. If individ-ual ML decoding is performed at receiver end for the ith user, the optimal individual MAC-DMT is given by

Proof. For brevity we only outline the proof. Let Oi denote the outage event of the ith user; then following from the above discussion it can be seen that

O_i = [

I⊆{0,1,··· ,K−1}

i∈I

O (I) ,

since if i 6∈ I, the ith user is not in outage. Now let Pe⁽ⁱ⁾(r₀, · · · , r_K−1) denote the error probability of the individual decoder for the ith user; then it can be shown that

P_e⁽ⁱ⁾(r₀, · · · , r_K−1) ≥ Pr {O_i}

where the first inequality follows from [9, Lemma 5]. To show the converse, let E (I) denote the error event that the signal matrices of the users in I are erroneously decoded under joint decoding.

Then simply note that the error probability of an individual ML decoder is upper bounded by that of a joint ML decoder, i.e.,

P_e⁽ⁱ⁾(r₀, · · · , r_K−1) ≤ Pr

where the right-hand-side gives the probability of a joint ML decoder when the signal of the ith user is erroneously decoded. Now using the union bound argument and along similar lines as in the proof of Theorem 8 it can be shown that

P_e⁽ⁱ⁾(r₀, · · · , r_K−1) ≤ X

This completes the proof. 

0 0.5 1 1.5 2 0

0.5 1 1.5 2 2.5 3 3.5 4

Multiplexing Gain r

Diversity gain d

d_{1,2},2^* (r/4,r) d_{1,2},2^(1)* (r/4,r)

Figure 4.3: Comparison between the joint MAC-DMT and the individual MAC-DMT of the second user when r₁ = 4r₀ = r.

0 0.5 1 1.5 2

0 0.5 1 1.5 2 2.5 3 3.5 4

Multiplexing gain r

Diversity gain d

d{1,2},2

* (r,r) d{1,2},2

(0)* (r,r) d{1,2},2

(1)* (r,r) d2,2

* (r) d1,2

* (r)

Figure 4.4: Comparison between the joint MAC-DMT and the individual MAC-DMT when r₁ = r₀ = r.

With the above result, we now come back to Example 2 to investigate the individual MAC-DMT of the second user.

Example 3 (Continued from Example 2). In Example 2 we have considered the specific case of K = 2, n₀ = 1, n₁ = 2, n_r = 2 and r₀ = 0. Assuming the second user transmits at multiplexing gainr₁, from Theorem 9 the individual MAC-DMT of the second user is

d^(1)∗_{1,2},2(0, r₁) = min{d^∗_2,2(r₁), d^∗_3,2(r₁)} = d^∗_2,2(r₁).

Hence we see that the single-user performance of the second user is recovered by the use of an individual ML decoder. To illustrate further the difference in MAC-DMT between(4.1) and (4.7), in Fig. 4.3 we compare the MAC-DMT performances of joint and individual decoders at r₁ = 4r₀ = r. It can be clearly seen that the individual ML decoder outperforms significantly the joint ML decoder at low-multiplexing-gain regime.

Another comparison between the DMT performances of both decoders atr₁ = r₀ = r is given in Fig. 4.4. It shows that the joint ML decoder (given by d^∗_{1,2},2(r, r)) is not optimal for the second user. The truly optimal individual ML decoder for the second user has DMT performance d^(1)∗_{1,2},2(r, r). Furthermore, the individual ML decoder for the second user achieves the single-user DMT performanced^∗_2,2(r) as long as r ≤ 0.4. On the other hand, for the first user who has lesser number of transmit antennas, the DMT performances of the joint and individual decoders are the same and are actually equal to his single-user performanced^∗_1,2(r). 

Next we could apply Theorem 9 to the case of symmetric MIMO-MAC to see how the error probabilities of joint and individual ML decoders compare. The comparison is given in the fol-lowing corollary. It shows that in the symmetric MIMO-MAC there is no difference in terms of MAC-DMT performance between the joint and individual ML decoders.

Corollary 10. For symmetric MIMO-MAC with K users, each having nt transmit antennas and transmitting at multiplexing gain r_i = r, let P_e(r) denote the error probability of the joint ML decoder andPe⁽ⁱ⁾(r) denote the error probability of the ith individual ML decoder. Then

P_e(r) .

= P_e⁽ⁱ⁾(r) and in terms of DMT we have

d^(i)∗_{n

t,··· ,nt},nr(r, · · · , r) = d^∗_{{nt,··· ,nt},nr}(r, · · · , r) for alli = 0, 1, · · · , K − 1.

Proof. It suffices to show only the equality in DMT. First, from Theorem 9 we have d^(i)∗_{n

t,··· ,nt},nr(r, · · · , r) = min

1≤m≤Kd^∗_mn

t,nr(mr)

and the proof is complete after noting that the right-hand-side of the above is the same as the

MAC-DMT given in Theorem 1. 

Before concluding the section we have the following remarks. First, while the individual ML decoder could achieve a much higher DMT performance as seen in Examples 2 and 3, the com-putational complexity required by (4.7) is often extremely high. Thus, the individual ML decoder has widely been considered as being impractical in multiuser detections. The reason for including this receiver is only to clarify the unexpected DMT performance loss of the joint ML decoder in Example 2.

As the individual ML decoder is rarely used, below we will not consider this receiver anymore.

We will regard the joint MAC-DMT given in Theorem 8 as the optimal MAC-DMT in practice, although it is now clear that it is not the best one can actually achieve.