EE359 Discussion Session 8 Beamforming, Diversity-multiplexing tradeoff, MIMO receiver design, Multicarrier modulation

Transcription

1 EE359 Discussion Session 8 Beamforming, Diversity-multiplexing tradeoff, MIMO receiver design, Multicarrier modulation November 29, 2017 EE359 Discussion 8 November 29, / 33

2 Outline 1 MIMO concepts Beamforming Diversity multiplexing tradeoff for point to point MIMO 2 MIMO Decoding Linear Decoders Sphere Decoding 3 Multicarrier modulation EE359 Discussion 8 November 29, / 33

3 Outline 1 MIMO concepts Beamforming Diversity multiplexing tradeoff for point to point MIMO 2 MIMO Decoding Linear Decoders Sphere Decoding 3 Multicarrier modulation EE359 Discussion 8 November 29, / 33

4 Brief recap of the notation for a point to point MIMO system y = Hx + n ỹ = Σ x + ñ where H = UΣV H = i σ iu i v H i and x = V x, ỹ = U H y N t transmit antennas and N r receive antennas Decomposition into parallel channels with perfect CSIT and CSIR σ 1 > σ 2 >... EE359 Discussion 8 November 29, / 33

5 Outline 1 MIMO concepts Beamforming Diversity multiplexing tradeoff for point to point MIMO 2 MIMO Decoding Linear Decoders Sphere Decoding 3 Multicarrier modulation EE359 Discussion 8 November 29, / 33

6 Beamforming Idea If CSIT available, simply transmit along vector with the largest singular value, i.e. make x C scalar - one value Some points Equivalent scalar channel ỹ = u H Hv x + ñ 1 Maximizes SNR if u and v are first singular vectors Optimal only if other parallel channels are weak (low SNR) Why is maximum SNR scheme not always optimal? Any choice of u and v other than u 1 and v 1 is suboptimal EE359 Discussion 8 November 29, / 33

7 Homework 7 Problem 1 In part (a), it can be assumed that the magnitude of path loss is the same in all transmissions. Show that optimal (i.e., SNR maximizing) choice for u and v are u 1 and v 1 respectively Problem 2 Problem 2 explores linear algebra inequality between 2-norm and Frobenius norm. As well as the connection between MIMO communication and Maximal Ratio Combining in Rayleigh fading. EE359 Discussion 8 November 29, / 33

8 Outline 1 MIMO concepts Beamforming Diversity multiplexing tradeoff for point to point MIMO 2 MIMO Decoding Linear Decoders Sphere Decoding 3 Multicarrier modulation EE359 Discussion 8 November 29, / 33

9 The tradeoff Setting CSI (channel state info) known at receiver but is unknown at transmitter, finite blocklengths. Intuition Antennas can be used for higher reliability (diversity) or rate (multiplexing) Fineprint We assume i.i.d. complex normal entries for H High SNR concept: Multiplexing gain r = R(SNR) limsnr log 2 (SNR) Diversity gain d = log P limsnr e log SNR EE359 Discussion 8 November 29, / 33

10 The tradeoff Diversity d Multiplexing r Figure: Blue curve for N t = 3, N r = 3, green for N t = 2, N r = 2 Blue dot corresponds to low rate, high reliability transmission Red dot corresponds to high rate, low reliability transmission Achievability Any point on this tradeoff curve may be achieved in general by a suitable space time code EE359 Discussion 8 November 29, / 33

11 Homework 7 Problem 3 Explores the diversity multiplexing tradeoff for a simple system Problem 4 Evaluate capacity using waterfilling (CSIT) or uniform allocation (only CSIR). Also evaluate bit error rates and total throughput for some achievable schemes. EE359 Discussion 8 November 29, / 33

12 Outline 1 MIMO concepts Beamforming Diversity multiplexing tradeoff for point to point MIMO 2 MIMO Decoding Linear Decoders Sphere Decoding 3 Multicarrier modulation EE359 Discussion 8 November 29, / 33

13 The optimal receiver Idea Maximum likelihood criterion: ˆx = argmax x X p(x H, y) Some more details about ML decoder For i.i.d. Gaussian noise statistics and uniformly random MQAM signalling, p(x H, y) e c y Hx 2, so ˆx = argmax y Hx 2 x X Problem An NP-hard combinatorial optimization problem. EE359 Discussion 8 November 29, / 33

14 Outline 1 MIMO concepts Beamforming Diversity multiplexing tradeoff for point to point MIMO 2 MIMO Decoding Linear Decoders Sphere Decoding 3 Multicarrier modulation EE359 Discussion 8 November 29, / 33

15 Simple approximations: Zero forcing Idea Use matrix inversion Math ˆx = H y where H = (H H H) 1 H H if H is tall Some features Requires O ( Nt 3 ) operations. (Can t expect to do better than this unless H is sparse) Nearly optimal when condition number is close to 1. Poor performance for ill-conditioned channels. EE359 Discussion 8 November 29, / 33

16 Zero Forcing in Pictures H H -1 EE359 Discussion 8 November 29, / 33

17 Zero Forcing in Pictures H H -1 EE359 Discussion 8 November 29, / 33

18 Zero Forcing in Pictures H H -1 EE359 Discussion 8 November 29, / 33

19 Zero Forcing in Pictures H H -1-1 EE359 Discussion 8 November 29, / 33

20 Simple approximations: Linear MMSE decoding Idea Write estimate as an affine function of y. Minimize expected squared error by choosing right affine function. Regular MMSE: Assume x to be i.i.d. multivariate Gaussian and compute optimal decoder (minimum expected mean squared error (MSE)) Math (assuming SNR = 1/σ 2 ) ˆx = (H H H + σ 2 I) 1 H H y Some features Good complexity (similar to zero forcing) Less sensitive to ill-conditioned matrices In practice x is not Gaussian EE359 Discussion 8 November 29, / 33

21 Outline 1 MIMO concepts Beamforming Diversity multiplexing tradeoff for point to point MIMO 2 MIMO Decoding Linear Decoders Sphere Decoding 3 Multicarrier modulation EE359 Discussion 8 November 29, / 33

22 An ML Algorithm: Enumeration ML Decoding: Find closest point in l 2 norm. How to search for close points? Naïve approach Check all M n points, return closest EE359 Discussion 8 November 29, / 33

23 Enumeration BPSK, N t = 2 x Nt x Nt x Nt 1 x Nt 1 x Nt 1 x Nt There is no reason two adjacent nodes are close! Questions Is there are smart way to traverse this graph? What is our stopping criteria? Can we prune nodes? EE359 Discussion 8 November 29, / 33

24 Better Enumeration: QR Decomposition H = QR = Q r 1,1 r. 1, r 1,n r n,n New Basis x = Rx, y Hx 2 = Q H y x 2 Notice: l 2 norm in this basis can be considered element-wise: x n,..., x 1 Partial objective : s m = m i=1 ( q n i, y x n i ) EE359 Discussion 8 November 29, / 33

25 Sphere decoder Prune branches below m if s m > r s 1 > r x Nt x Nt x Nt 1 x Nt 1 x Nt 1 x Nt 1 EE359 Discussion 8 November 29, / 33

26 Sphere decoder Prune branches below m if s m > r s 1 > r x Nt x Nt x Nt 1 x Nt 1 x Nt 1 x Nt 1 EE359 Discussion 8 November 29, / 33

27 Sphere decoder Prune branches below m if s m > r s 1 < r x Nt x Nt x Nt 1 x Nt 1 x Nt 1 x Nt 1 EE359 Discussion 8 November 29, / 33

28 Sphere decoder Prune branches below m if s m > r s 1 < r x Nt x Nt x Nt 1 x Nt 1 x Nt 1 x Nt 1 EE359 Discussion 8 November 29, / 33

29 Sphere decoder Prune branches below m if s m > r s 1 < r x Nt x Nt x Nt 1 x Nt 1 x Nt 1 x Nt 1 Return minimum value of objective function at last depth. EE359 Discussion 8 November 29, / 33

30 Further notes If r is large enough, gives ML estimate If correct solution is pruned, declare error (erased symbol) Reducing r reduces complexity. Complexity also based on channel condition number and signal to noise ratio Further techniques exist improve enumeration (e.g. LLL algorithm) EE359 Discussion 8 November 29, / 33

31 Homework 7 Problem 5 Apply ML, Zero Forcing and MMSE decoder. Naïve implementation of ML is fine. Problem 6 Simple exploration of sphere decoding. No implementation needed! EE359 Discussion 8 November 29, / 33

32 Outline 1 MIMO concepts Beamforming Diversity multiplexing tradeoff for point to point MIMO 2 MIMO Decoding Linear Decoders Sphere Decoding 3 Multicarrier modulation EE359 Discussion 8 November 29, / 33

33 Intersymbol interference Problem Coherence bandwidth of channel is small, thus channel spreads wideband signal in time Some common remedies Equalization/deconvolution/channel inversion Multicarrier modulation Spread spectrum EE359 Discussion 8 November 29, / 33

34 Multicarrier modulation Idea Split wideband (B) into N narrowband chunks each of bandwidth B/N, such that B n = B/N B c Common incarnations Frequency division multiplexing (FDM) Orthogonal FDM (OFDM) Uses 4G LTE, Wifi use OFDM 2G standards (GSM) used FDM heavily EE359 Discussion 8 November 29, / 33

35 FDM Idea Pack a bunch of orthonormal basis functions in frequency domain, thereby creating parallel channels Implementation issues Minimum carrier frequency separation with signal duration T N is 1/T N Usually need a rolloff factor β and guard bands ɛ, thus effective occupancy B n = N(1 + β + ɛ)/t N Need separate receiver hardware/modulation schemes at each carrier frequency EE359 Discussion 8 November 29, / 33

36 Homework 7 Problem 7 Two signals s i (t) and s j (t) over time T N are orthogonal if TN t=0 s i(t)s j (t)dt = 0 EE359 Discussion 8 November 29, / 33