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Multiple-input multiple-output (MIMO) communication systems

Multiple-input multiple-output (MIMO) communication systems. System model. N T transmit antennas. N R receive antennas. k : time index (k = 1, ..., K). (coded) data symbols a n (k), E[|a n (k)| 2 ] = E s. System model. (m = 1, ..., N R ; k = 1, ..., K).

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Multiple-input multiple-output (MIMO) communication systems

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  1. Multiple-input multiple-output (MIMO) communication systems Advanced Modulation and Coding : MIMO Communication Systems

  2. System model NT transmit antennas NR receive antennas k : time index (k = 1, ..., K) (coded) data symbols an(k), E[|an(k)|2] = Es Advanced Modulation and Coding : MIMO Communication Systems

  3. System model (m = 1, ..., NR; k = 1, ..., K) wm(k) : complex i.i.d. AWGN, E[|wm(k)|2] = N0 hm,n : complex Gaussian zero-mean i.i.d. channel gains (Rayleigh fading), E[|hm,n|2] = 1 r(k) = Ha(k) + w(k) R = HA + W Advanced Modulation and Coding : MIMO Communication Systems

  4. System model Rb : information bitrate Rs : symbol rate per transmit antenna Eb : energy per infobit Es : energy per symbol (code rate) Es = EbrMIMOlog2(M) Rb = RsNTrMIMOlog2(M) Advanced Modulation and Coding : MIMO Communication Systems

  5. System model Spectrum transmitted bandpass signal BRF Rs(necessary condition toavoid ISI between successive symbols) All NT transmitted signals are simultaneously in the same frequency interval Bandwidth efficiency : Rb/Rs = NTrMIMOlog2(M) (bit/s/Hz) Advanced Modulation and Coding : MIMO Communication Systems

  6. ML detection Log-likelihood function ln(p(R |A, H) : (minimization over all symbol matrices that obey the encoding rule) ML detection : Frobenius norm of X : Trace of square matrix M : Properties : Tr[PQ] = Tr[QP], Tr[M] = Tr[MT] • XH denotes Hermitian transpose (= complex conjugate transpose (X*)T) Advanced Modulation and Coding : MIMO Communication Systems

  7. Bit error rate (BER) symbol matrix A represents NTKrMIMOlog2(M) infobits : # infobits in which A(i) and A(i) are different Advanced Modulation and Coding : MIMO Communication Systems

  8. Bit error rate (BER) (averaging over statistics of H) PEP(i,0)(H) = Pr[||R - HA(i)||2 < ||R - HA(0)||2| A = A(0), H] PEP(i,0)(H) : pairwise error probability, conditioned on H PEP(i,0) = EH[PEP(i,0)(H)] : pairwise error probability, averaged over H Advanced Modulation and Coding : MIMO Communication Systems

  9. Bit error rate (BER) Define error matrix : D(i,0) = A(0) - A(i) Averaging over Rayleigh fading channel statistics : set of eigenvalues of NTxNT matrix D(i,0)(D(i,0))H(these eigenvalues are real-valued non-negative) Advanced Modulation and Coding : MIMO Communication Systems

  10. To be further considered Single-input single-output (SISO) systems : NT = NR = 1 Single-input multiple-output (SIMO) systems : NT = 1, NR 1 Multiple-input multiple-output (MIMO) systems : NT 1, NR 1 • Spatial multiplexing : ML detection, ZF detection • Space-time coding : trellis coding, delay diversity, block coding Advanced Modulation and Coding : MIMO Communication Systems

  11. Single-input single-output (SISO) systems Advanced Modulation and Coding : MIMO Communication Systems

  12. SISO : observation model Observation model : r(k) = ha(k) + w(k) k = 1, ..., K Bandwidth efficiency : Rb/Rs = rSISOlog2(M) (bit/s/Hz) Advanced Modulation and Coding : MIMO Communication Systems

  13. # codewords = SISO : ML detection ML detector looks for codeword that is closest (in Euclidean distance) to {u(k)} In general : decoding complexity exponential in K (trellis codes, convolutional codes : complexity linear in K by using Viterbi algorithm) Advanced Modulation and Coding : MIMO Communication Systems

  14. SISO : PEP computation Conditional PEP : squared Euclidean distance between codewords {a(0)(k)} and {a(i)(k)} decreases exponentially with Eb/N0 Average PEP, AWGN (|h| = 1) : inversely proportional to Eb/N0 Average PEP, Rayleigh fading : (occasionally, |h| becomes very small  large PEP) Advanced Modulation and Coding : MIMO Communication Systems

  15. SISO : numerical results BER fading channel >> BER AWGN channel Advanced Modulation and Coding : MIMO Communication Systems

  16. SISO : numerical results coding gain on fading channel << coding gain on AWGN channel Advanced Modulation and Coding : MIMO Communication Systems

  17. Single-input multiple-output (SIMO) systems Advanced Modulation and Coding : MIMO Communication Systems

  18. SIMO : observation model Same transmitter as for SISO; NR antennas at receiver Observation model : rm(k) = hma(k) + wm(k) m = 1, ..., NR; k = 1, ..., K Bandwidth efficiency : Rb/Rs = rSIMOlog2(M) (bit/s/Hz) Advanced Modulation and Coding : MIMO Communication Systems

  19. SIMO : ML detection ML detector looks for codeword that is closest to {u(k)} Same decoding complexity as for SISO Advanced Modulation and Coding : MIMO Communication Systems

  20. SIMO : PEP computation Conditional PEP : decreases exponentially with Eb/N0 PEP, AWGN (|hm| = 1) : total received energy per symbol increases by factor NR as compared to SISO  same PEP as with SISO, but with Eb replaced by NREb (“array gain” of 10log(NR) dB) inversely proportional to (Eb/N0)^NR PEP, Rayleigh fading : PEP is smaller than in SISO setup with Eb replaced by NREb - “array gain” of 10log(NR) dB - additional “diversity gain” (diversity order NR) : probability that (|h|2)avg is small, decreases with NR Advanced Modulation and Coding : MIMO Communication Systems

  21. SISO : statistical properties of <|h|2> Advanced Modulation and Coding : MIMO Communication Systems

  22. SISO : statistical properties of <|h|2> Advanced Modulation and Coding : MIMO Communication Systems

  23. SIMO : numerical results AWGN channel : array gain of 10log(NR) dB Advanced Modulation and Coding : MIMO Communication Systems

  24. arraygain diversity gain SIMO : numerical results fading channel : array gain + diversity gain Advanced Modulation and Coding : MIMO Communication Systems

  25. SIMO : numerical results fading channel : array gain + diversity gain Advanced Modulation and Coding : MIMO Communication Systems

  26. SIMO : numerical results fading channel : array gain + diversity gain Advanced Modulation and Coding : MIMO Communication Systems

  27. SIMO : numerical results fading channel : coding gain increases with NR Advanced Modulation and Coding : MIMO Communication Systems

  28. SIMO : numerical results fading channel : coding gain increases with NR Advanced Modulation and Coding : MIMO Communication Systems

  29. SIMO : numerical results fading channel : coding gain increases with NR Advanced Modulation and Coding : MIMO Communication Systems

  30. Multiple-input multiple-output (MIMO) systems Advanced Modulation and Coding : MIMO Communication Systems

  31. MIMO : observation model (m = 1, ..., NR; k = 1, ..., K) at any receive antenna, symbols from different transmit antennas interfere R = HA + W Bandwidth efficiency : Rb/Rs = NTrMIMOlog2(M) (bit/s/Hz) Advanced Modulation and Coding : MIMO Communication Systems

  32. # different symbol matrices A = MIMO : ML detection ML detector :  detector complexity in general increases exponentially with NTrMIMO and K Advanced Modulation and Coding : MIMO Communication Systems

  33. MIMO : PEP computation PEP, Rayleigh fading : # nonzero eigenvalues equals rank(D(i,0)), with 1  rank(D(i,0))  NT PEP(i,0) inversely proportional to (Eb/N0)^(NRrank(D(i,0))) PEPs that dominate BER are those for which D(i,0) = A(0)-A(i) has minimum rank.  Advanced Modulation and Coding : MIMO Communication Systems

  34. MIMO : spatial multiplexing Advanced Modulation and Coding : MIMO Communication Systems

  35. Spatial multiplexing codewords transmitted by different antennas are statistically independent codeword transmitted by antenna #n Aim : to increase bandwidth efficiency by a factor NT as compared to SISO/SIMO Bandwidth efficiency : Rb/Rs = NTrlog2(M) Advanced Modulation and Coding : MIMO Communication Systems

  36. # different symbol matrices A = Spatial multiplexing :ML detection ML detector : (un)T : n-th row of HHR As HHH is nondiagonal, codewords on different rows must be detected jointly instead of individually  ML detector complexity increases exponentially with NT as compared to SISO/SIMO Advanced Modulation and Coding : MIMO Communication Systems

  37. Spatial multiplexing : PEP computation D(i,0) nonzero  rank(D(i,0))  1 rank(D(i,0)) = 1 when all rows of D(i,0) are proportional to a common row vector dT = (d(1), ..., d(k), ..., d(K)). Example : D(i,0) has only one nonzero row; this corresponds to a detection error in only one row of A  Denoting the n-th row of a rank-1 error matrix D(i,0) by an dT, the bound on the corresponding PEP is dominating PEPs are inversely proportional to (Eb/N0)^NR  diversity order NR (same as for SIMO) Advanced Modulation and Coding : MIMO Communication Systems

  38. Spatial multiplexing : zero-forcing detection Complexity of ML detection is exponential in NT Simpler sub-optimum detection : zero-forcing (ZF) detection linear combination of received antenna signals rm(k) to eliminate mutual interference between symbols an(k) from differenttransmit antennas and to minimize resulting noise variance Condition for ZF solution to exist : rank(H) = NT (this requires NR NT) rank(H) = NT (HHH)-1 exists r(k) = Ha(k) + w(k) z(k) = (HHH)-1HHr(k) = a(k) + n(k) n(k) = (HHH)-1HHw(k) z(k) : no interference between symbols from different transmit antennas E[n(k)nH(k)] = N0(HHH)-1 : ni(k) and nj(k) correlated when ij Advanced Modulation and Coding : MIMO Communication Systems

  39. Spatial multiplexing : zero-forcing detection Sequences {an(k)} are detected independently instead of jointly, ignoring the correlation of the noise on different outputs : Complexity of ZF detector is linear (instead of exponential) in NT Performance penalty : diversity equals NR-NT+1 (instead of NT) Advanced Modulation and Coding : MIMO Communication Systems

  40. Spatial multiplexing : numerical results ML detection : penalty w.r.t. SIMO decreases with increasing NR Advanced Modulation and Coding : MIMO Communication Systems

  41. Spatial multiplexing : numerical results ZF detection : same BER as SIMO with NR-NT+1 receive antennas Advanced Modulation and Coding : MIMO Communication Systems

  42. MIMO : space-time coding Advanced Modulation and Coding : MIMO Communication Systems

  43. Space-time coding Aim : to achieve higher diversity order than with SIMO any D(i,0) must have rank larger than 1 (max. rank is NT, max. diversity order is NTNR)  coded symbols must be distributed over different transmit antennas and different symbol intervals (“space-time” coding) Property : rMIMO  1/NT is necessary condition to have rank NT for all D(i,0)  at max. diversity, Rb/Rs  log2(M) Rb/Rs limited by bandwidth efficiency of uncoded SIMO/SISO Advanced Modulation and Coding : MIMO Communication Systems

  44. Three classes of space-time coding : • space-time trellis codes (STTrC) • delay diversity • space-time block codes (STBC) Advanced Modulation and Coding : MIMO Communication Systems

  45. STTrC : trellis representation L input bits at beginning of k-th symbol interval S(k) : state at beginning of k-th symbol interval state transitions : (S(k+1), a(k)) determined by (S(k), b(k)) trellis diagram : (L=1) 2L branches leaving from each state2L branches entering each state Advanced Modulation and Coding : MIMO Communication Systems

  46. #states STTrC : bandwidth efficiency Bandwidth efficiency : rMIMO = L/(NTlog2(M))  Rb/Rs = L (bit/s/Hz) rMIMO 1/NT  L  log2(M) at max. diversity : (exponential in NT) ML decoding by means of Viterbi algorithm : decoding complexity linear (instead of exponential) in K but still exponential in NT (at max. diversity) Advanced Modulation and Coding : MIMO Communication Systems

  47. STTrC : example Example for NT = 2, 4 states, 4-PSK, L = 2 infobits per coded symbol pair entryi,j : 2 data symbols corresponding to transition from state i to state j any D(i,0) has columns (0, d1)T, (d2, 0)Twith nonzero d1 and d2  rank = 2, i.e., max. rank max. diversity (NTNR = 2NR), max. bandwidth efficiency (L = log2(M)), corresponding to max. diversity Advanced Modulation and Coding : MIMO Communication Systems

  48. Delay diversity : transmitter Example : NT = 3 maximum possible diversity order !  rank(D(i,0)) = NT  diversity order = NTNR Advanced Modulation and Coding : MIMO Communication Systems

  49. Delay diversity : bandwidth efficiency Equivalent configuration : Bandwidth efficiency (for K >> 1) : Rb/Rs = NTrMIMOlog2(M) = rlog2(M)  same bandwidth efficiency as SIMO/SISO with code rate r Max. bandwidth efficiency (at max. diversity) of log2(M) achieved for r = 1 (uncoded delay diversity) Advanced Modulation and Coding : MIMO Communication Systems

  50. #states = Delay diversity : trellis representation Uncoded delay diversity (r = 1) can be interpreted as space-time trellis code : state S(k) = (a(k-1), a(k-2), ..., a(k-NT+1)) input at time k : a(k) output during k-th symbolinterval : (a(k), a(k-1), ..., a(k-NT+1)) decoding complexity : linear in K, exponential in NT Advanced Modulation and Coding : MIMO Communication Systems

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