In this exercise, we further study the comparison between orthogo- nal multiple access and SDMA with multiple receive antennas at the base-station. While orthogonal multiple access is simple to implement, SDMA is the capacity achieving scheme and outperforms orthogonal multiple access in certain scenarios (cf. Exercise 10.4) but requires complex joint decoding of the users at the base-station.

Consider the following access mechanism, which is a cross between purely orthog- onal multiple access (where all the users' signals are orthogonal) and purely SDMA (where all the K users share the bandwidth and time simultaneously). Divide the K users into groups of approximately nr users each. We provide orthogonal resource allocation (time, frequency or a combination) to each of the groups but within each group the users (approximately nr of them) operate in an SDMA mode.

We would like to compare this intermediate scheme with orthogonal multiple access and SDMA. Let us use the largest symmetric rate achievable with each scheme as the performance criterion. The uplink model (same as the one in Exercise 10.5) is the following: receiver CSI with i.i.d.

Rayleigh fast fading. Each user has the same average transmit power constraint P, and SNR denotes the ratio of P to the background complex Gaussian noise power N0.

1. Write an expression for the symmetric rate with the intermediate access scheme (the expression for the symmetric rate with SDMA is in (10.90)).

2. Show that the intermediate access scheme has performance comparable to both orthogonal multiple access and SDMA at low SNR, in the sense that the ratio of the performances goes to 1 as SNR → 0.

3. Show that the intermediate access scheme has performance comparable to SDMA at high SNR, in the sense that the ratio of the performances goes to 1 as SNR → ∞.

4. Fix the number of users K (to, say, 30) and the number of receive antennas nr (to, say, 5). Plot the symmetric rate with SDMA, orthogonal multiple access and the intermediate access scheme as a function of SNR (0 dB to 30 dB). How does the intermediate access scheme compare with SDMA and orthogonal multiple access for the intermediate SNR values?