Design and Performance of a Noncoherent SIMO MAC System Alexandros Manolakos, Mainak Chowdhury, Andrea Goldsmith Electrical Engineering, Stanford University November 26, 2013 AM, MC, AG (EE, Stanford) Noncoherent massive mimo November 26, 2013

Motivation for large number of antennas The good Q High carrier frequencies make larger number of antennas possible More the number of antennas n, more the number of spatial streams The bad Q Channel estimation overhead is huge (even at receiver) Our question How well do we do with no CSIR (channel state info at receiver) ? We show Positive rate achievable schemes with vanishing BER and no CSIR Low encoding and decoding complexity Scaling law (with n) of the symmetric rate supported is same as that of the coherent case for 2 users AM, MC, AG (EE, Stanford) Noncoherent massive mimo November 26, 2013

System model Multiuser model with 1 n-antenna receiver and 2 transmitters y = Hx + n, y ∈ Rn, H ∈ Rn×m, Hi ,j ∼ N (0, 1), ni is i.i.d. N (0, σ2), x1 ∈ C1, x2 ∈ C2 Idea We don’t know the channel, let’s look at the received energy ! Simplified system model ||y||2 ||h1x1 + h2x2 + n||2 = n n = x 2 + x 2 + σ2 + ν for n large 1 2 AM, MC, AG (EE, Stanford) Noncoherent massive mimo November 26, 2013

dmin = BER performance Some facts about ν Is non zero even if σ2 is zero By large deviation bound, the following holds P(|ν| > c) ≤ e−nI (c), ∼ For large n, BER in the system dominated by I (c) for c = “minimum distance” where I (c) is the rate function, and I (c) ∝ c2 for small c dmin = min |c2 + c2 − c2 − c2 | 1,i 2,j 1,k 2,l i ,j ,k,l :(i ,j )I=(k,l ) Objective of “constellation” design for small BER Maximize dmin Non convex in general, symmetry critical to solving it efficiently AM, MC, AG (EE, Stanford) Noncoherent massive mimo November 26, 2013

Comparison with perfect CSI Perfect CSIR With CSIR known perfectly, an upper bound on the symmetric capacity of the system is given by Csym = log(I + HH H) ≈ log n for large n Scaling law optimality for 2 users If optimal dmin can be shown to be decaying polynomially with |C1|, |C2|, i.e. dmin = 1 , the BER is dominated by |C1|a |C2|b BER � e−n/(|C1|2a |C2|2b ), for a > 0, b > 0 AM, MC, AG (EE, Stanford) Noncoherent massive mimo November 26, 2013 5 / 7

Comparison with perfect CSI Perfect CSIR With CSIR known perfectly, an upper bound on the symmetric capacity of the system is given by Csym = log(I + HH H) ≈ log n for large n Scaling law optimality for 2 users If optimal dmin can be shown to be decaying polynomially with |C1|, |C2|, i.e. dmin = 1 , the BER is dominated by |C1|a |C2|b BER � e−n/(|C1|2a |C2|2b ), for a > 0, b > 0 Symmetric rate with vanishing probability of error scales like log n AM, MC, AG (EE, Stanford) Noncoherent massive mimo November 26, 2013 5 / 7

Numerical results: minimum distance Achievable Fitted Curve log(minimum distance) −0.5 Upper Bound −1 n1=4 −1.5 −2 −2.5 0 10 n1=16 20 30 n2 Figure : Minimum distance with |C1| and |C2|. Fit gives dmin ≥ 3.7 (|C1 ||C2 |)1.09 AM, MC, AG (EE, Stanford) Noncoherent massive mimo November 26, 2013

Numerical results: BER −1 nr = 100 nr = 200 nr = 500 nr = 1000 nr = 2000 nr = 5000 −2 log(BER) −3 −4 −5 6 − 2 3 4 5 6 Number of Constellation points per user 7 8 Figure : BER performance with |C1| and |C2|. AM, MC, AG (EE, Stanford) Noncoherent massive mimo November 26, 2013