1. Diversity Receivers MIMO-Space Time Block Codes
Αναπλ. Καθηγητής Γεώργιος Ευθύμογλου
October 9, 2017
Module Title

2. Εισαγωγή BER performance in fading channels Diversity Gain
Multiple Input Multiple Output (MIMO) systems
Space Time Block Codes (STBCs)
Alamouti code
Code transmission matrices and rates
Spatial Multiplexing
Diversity performance
MRC
Selection combining
Alamouti scheme
Module Title

3. Average received SNR at distance d
Received power of desired signal at distance d based on free space path-loss only Simplified path-loss model with depending on propagation environment (Average) Received SNR at distance d

4. Received signal in a flat fading channel
In a flat fading channel (or narrowband system), the CIR (channel impulse response) reduces to a single impulse (just one path from tx to receiver) scaled by a time-varying complex coefficient.
The received (equivalent lowpass) signal is of the form
α(t) is the magnitude of h(t) (usually α(t) has Rayleigh pdf)
h(t) is constant for many symbol intervals
Phase θ(t) varies slowly and can be tracked

5. Instantaneous SNR for non-diversity system
System model Signal power Noise power Instantaneous SNR: (signal power / noise power)

6. BER vs. Average SNR (cont.)
Fading h varies with time  SNR γ varies with time
Let us define instantaneous SNR and average SNR:

7. BER vs. Average SNR (cont.)
Since
using
we get
Rayleigh distribution
Exponential distribution

8. BER vs. Average SNR (cont.)
Rayleigh pdf Exponential pdf
Pdf of γb = R2 Eb/N0 , όπου R έχει Rayleigh κατανομή

9. BER vs. Average SNR (cont.)
The average bit error probability is
where the bit error probability (BEP) for a fixed value of α is
We thus get the BEP for a given average SNR
Important formula for obtaining statistical average
2-PSK

10. BER vs. Average SNR (cont.)
Approximation for large values of average SNR is obtained in the following way. First, we write
Then, we use
which leads to

11. BER vs. Average SNR (cont.)
Frequency-selective channel (equalization or Rake receiver)
BER
Frequency-selective channel (no equalization)
“BER floor”
AWGN channel (no fading)
Flat fading channel
SNR
means a straight line in log/log scale

12. Better performance through diversity
Diversity  the receiver is provided with L copies of the transmitted signal. The multiple signal copies should experience uncorrelated fading in the channel.
In this case the probability that all signal copies fade simultaneously is reduced dramatically with respect to the probability that a single copy experiences a fade.
As a rough rule:
Diversity of L:th order
BER
Average SNR

13. BER vs. Average SNR (diversity effect)
Flat fading channel,
Rayleigh fading,
L = 1
AWGN channel (no fading)
SNR
L = 4
L = 3
L = 2

14. Different kinds of diversity methods
Common receive diversity techniques
–Selection diversity
–Maximal Ratio Combining (MRC)
–Equal Gain (EG) diversity
• Common transmit diversity techniques
–Transmit diversity with channel state information (CSI) feedback
Transmit Antenna Selection (TAS)
Beamforming
–Space time block code (STBC) without CSI at transmitter
• MIMO
– Spatial Multiplexing (SM)
– Space time block code
– Space time trellis code

15. Coding and Diversity Gains
BER asymptotic expressions (at high SNR regime)

16. Maximum likelihood (ML) decision rule
Received signal with fading h and AWGN n ML multiplies all possible symbols with h, and selects the one symbol that is ‘closer’ (minimum Euclidean distance) to the received signal y where is the estimated symbol. ML: Find which symbol when multiplied with channel weight h is “closer” to the received signal y.

17. Maximum likelihood (ML) decision rule
ML decision rule: choose signal if and only if όπου είναι η Ευκλείδεια απόσταση μεταξύ των σημάτων x και y.

18. SIMO - MISO - MIMO
Receive diversity – Transmit diversity – MIMO (max )

19. Space diversity: Receive diversity
1 Tx and N receive antennas - Single Input Multiple Output (SIMO)

20. Space diversity: Selection diversity
Out of the N received signals, at each symbol period, choose the one with the maximum instantaneous SNR. • Demodulation and decoding is performed for the signals with the maximal SNR. SNR of the k-th Rx antenna SNR after selection diversity

21. Space diversity: Maximal Ratio Combining (MRC)
The signals from all Rx antennas are combined together – The output on the k-th branch – The output of the MRC receiver

22. Maximum Ratio Combining (MRC)
Λαμβάνουμε τα παρακάτω σήματα:
Ο συνδυαμός (combining) των παραπάνω σημάτων δίνει
Στην επόμενη διαφάνεια θα δούμε ότι το σύστημα MRC με δύο κεραίες λήψης μπορεί να επιτευχθεί με δύο κεραίες μετάδοσης με το σύστημα Alamouti, αρκεί τα channel weights να είναι σταθερά για 2 διαδοχικά σύμβολα εκμπομπής!!!

23. Maximum Ratio Combining (MRC)
Λαμβάνουμε τα παρακάτω σήματα:
Θεωρούμε ότι έχουν Γκαουσιανή κατανομή, ο κανόνας απόφασης με μέγιστη πιθανοφάνεια στο δέκτη είναι να επιλέξουμε το σήμα αν και μόνο αν
όπου
είναι η Ευκλείδεια απόσταση μεταξύ των σημάτων y και x.

24. Maximum Ratio Combining (MRC)

25. Maximum Ratio Combining (MRC)
ML decision rule For PSK signals The ML decision rule simplifies to

26. Alamouti Space Time Code (STC) 2 Tx × 1 Rx
At time t1, transmit x1 from antenna 1 and x2 from antenna 2 At time t2, transmit –x2* from antenna 1 and x1* from antenna 2

27. Alamouti STC: 2 Tx × 1 Rx
System equation Conjugate transpose or Hermitian transpose Receiver

28. Alamouti STC: 2 Tx × 1 Rx
System equation Conjugate transpose or Hermitian transpose Receiver

29. Alamouti STC: 2 Tx × 1 Rx FINAL Receiver with normalized channel power
Results in
The term guarantees the 2nd order diversity !
Conclusion: diversity order 2 can be accomplished
1 transmit antenna and 2 receive (Receive diversity)
2 transmit antennas and 1 receive (Transmit diversity) using Alamouti space time code.

30. Maximum Likelihood (ML) detection of STC
ML receiver solves:
where is the estimated symbol vector.
ML: Find which symbols when multiplied with channel matrix H are “closer” to the receive vector y. The minimization is performed over all possible transmit vector symbols.
For example, for QPSK modulation, the alphabet size is 4 and assuming a system 2x1, the ML decoder needs to search over a total of 42=16 vector symbols.
It follows that the decoding complexity of this receiver grows exponentially with the number of transmit antennas.

31. Alamouti code
Alamouti code = 2 × 1 MISO. Note a little different notation: r instead of y s instead of x

32. ML detection of Alamouti code
The received signal expressions at time 1 and 2 are The combiner combines the received signal as follows and sends them to the maximum likelihood detector, which minimizes the following metric (why? See r1 = h1s1+h2s2+n1, r2=…) over all possible s1 and s2

33. ML detection of Alamouti code
Expanding and deleting terms, the above minimization reduces to separately minimizing for detecting s1, and for detecting s2

34. ML detection of Alamouti code
If we use the notation The decision rule for each combined signal becomes: Pick iff For MPSK symbols, it reduces to SAME as MRC decision rule !!!

35. Alamouti 2 Tx × 1 Rx: Zero Forcing decoder
Zero Forcing (ZF) decoder: the decoder complexity of the ML decoder is reduced using a linear filter to separate the transmitted data streams, and then independently decode each stream. Values r1 and r2 are sent to Maximum Likelihood detector to determine which symbols x1 and x2 are closer to r1 and r2.

36. Alamouti 2 Tx × 1 Rx: Zero Forcing decoder
Zero Forcing decoder is a linear receiver based on channel matrix inversion.

37. Maximum Likelihood (ML) detection of STC
Assuming that the transmitted symbols belong to a set S For example, for QPSK: S = [1+j, -1+j, 1-j, -1-j] Maximum likelihood detector selects the 2 symbols that have the minimum Euclidean distance from the received vector [r1, r2]

38. Maximum Likelihood demodulation of STC
Assume QPSK symbol set: S = [1+j, -1+j, 1-j, -1-j] For example, if the received vector is [0.5-j*0.3, j*0.6 ], we have And detect with maximum likelihood as received symbol … ? (1-j)

39. Alamouti code 2 Tx × N Rx: Diversity order 2N
- At time t1, transmit x1 from antenna 1 and x2 from antenna 2
System equation for k-th antenna
- At time t2, transmit –x2* from antenna 1 and x1* from antenna 2
- System equation for k-th antenna
Conjugate transpose or Hermitian transpose
Receiver

40. Alamouti STC 2 Tx × N Rx
System equation for all Rx antennas MRC receiver: performance equivalent to 1 Tx – 2N Rx antennas

41. Special case of interest 2 x 2 MIMO
Rx antenna1
Rx antenna2
Tx antenna1 h1 h3
Tx antenna2 h2 h4

42. Alamouti code with 2 Receivers
Alamouti code = 2 × 2 MIMO. Note a little different notation: r instead of y s instead of x

43. ML detection of Alamouti code with 2 Receivers
The combiner combines the received signal as follows Substituting the appropriate equations These combined signals are then sent to ML decoder:

44. ML detection of Alamouti code with 2 Receivers
Using the previous combined signals Choose iff (Find s1 using and s2 using ) For MPSK symbols, it reduces to SAME as MRC decision rule for 1 Tx and 4 Rx antennas !!!

45. Special case of interest 2 x 2 MIMO
Received vector at first time slot Received vector at second time slot

46. Special case of interest 2 x 2 MIMO
Zero Forcing Decoder

47. Special case of interest 2 x 2 MIMO
Zero Forcing Decoder is a diagonal matrix

48. Special case of interest 2 x 2 MIMO
ZF-Decoder

49. Special case of interest 2 x 2 MIMO
Given Maximum likelihood detector selects the 2 symbols that have the minimum Euclidean distance from the received vector [r1, r2]

50. Spatial Multiplexing (SM)
Spatially multiplexed MIMO (SM-MIMO): increase the data rate using antenna diversity techniques. Each antenna transmits a different symbol.

51. SM with 2Tx and 1Rx
No diversity gain but double the transmit bit rate with 2Tx and 1Rx antenna Non-diagonal means that there is intersymbol interference. Interference cancellation techniques can be applied to improve performance.

52. SM with 2Tx and 2Rx
Diversity gain of only 2 but double bit rate compared to 2x2 Alamouti space time coding scheme.

53. SM-MIMO 2x2
With algebra, it can be proved that the ML detector searches over all possible values of the transmitted symbols and decides in favor of (x1, x2) which minimizes the Euclidean distance: Using the notation and making combiner

54. SM-MIMO 2x2
Expanding the previous result, we obtain for decoding x1 and x2 based on and , respectively, that is: j=1,2 For M-PSK this technique obtains full receive diversity with low complexity decoders. No transmit diversity achieved but data rate is twice than using STC.

55. Alamouti code: space time block codes
The Alamouti codeword X is a complex orthogonal matrix

56. Generalization of space time block coding
Two main objectives of orthogonal space time code design are
to achieve the diversity order of NTNR and
to implement computationally efficient per-symbol detection at the receiver that achieves the ML performance.

57. Higher order Orthogonal STBCs
Tarokh et al. discovered Orthogonal STBC that are quite simple. In order to facilitate computationally-efficient ML detection at the receiver, the following property is required:
They proved that for more than 2 transmit antennas and for complex symbols there is no symbol mapper to achieve coding rate of 1.
They showed that for complex symbols, coding rate of ¾ can be achieved with 3 or 4 transmit antennas
a rate of ½ is achievable for any number of transmit antennas
On the other hand, for real symbols, coding rate of 1 is achievable for any number of transmit antennas.

58. STBC for real symbol constellations
For real constellations, like BPSK, it is possible to achieve full rate since matrices are orthogonal and diversity order of NT .
We consider square space-time block codes with real entries with N=T= NT , that is, R=N/T=1.

59. STBC for real symbol constellations
It is feasible to construct non-square STBCs with R=1 based on another rule (Tarokh, V., Jafarkhani H., and A.R. Calderbank, “Space Time Codes from Orthogonal Designs,” IEEE Trans. Inform. Theory, Vol 45, No. 5, July 1999, pp ) , for number of transmit antennas 3, 5, 6 and 7, for real constellations.
These matrices are as follows:

60. STBC for complex constellations QPSK, 8PSK, 16QAM.
Complex transmission matrices with nT=3 and nT=4 with coding rate r=1/2.
For matrix G3, there are s1, s2, s3 και s4 along with their complex conjugates, so that k=4, with eight symbol intervals, T=8.
The achieved coding rate is r = k/T = 4/8 =1/2.
Same with G4, we have r=1/2 with diversity gain nT=4.

61. STBC for complex constellations QPSK, 8PSK, 16QAM.
Other complex matrices that achieve higher coding rates, for example r = 3/4, but require substantial complex algebra manipulation are

62. Decoding STBC X4
Using similar formulation to the 2x1 and 2x2 Alamouti codes:

63. Decoding STBC
Using similar formulation to the 2x1 and 2x2 Alamouti codes:

64. Decoding STBC
Diversity gain achieved is nT=3

65. Channel estimation
Suppose we transmit a known QPSK (pilot) symbol 1+1j
After multiplication with the channel weight
the received signal is (assume no noise)
Therefore provided we obtain at the output of the correlation receiver the receive vector we can compute the channel weight

66. Channel estimation
Now if we transmit any QPSK symbol a+jb, the received signal is
Now, if we multiple the received vector with the conjugate of the channel weight and divide by the channel power

67. Maximal Ratio Combining (MRC)
To SNR στην έξοδο του MRC είναι
r1(t)=
r2(t)
rL(t)
SUM
MRC

68. Rayleigh fading => SNR in i:th diversity branch is
MRC performance
Rayleigh fading => SNR in i:th diversity branch is
Gaussian distributed quadrature components
Rayleigh distributed magnitude
In case of L uncorrelated branches with same fading statistics, the MRC output SNR is

69. MRC performance (cont.)
The pdf of follows the chi-square distribution with 2L degrees of freedom
Reduces to exponential pdf when L = 1
Gamma function
Factorial
For 2-PSK, the average BER is

70. MRC performance (cont.)
For large values of average SNR this expression can be approximated by
which is according to the general rule

71. Selection Combining (SC)
Selection Combining (SC), where the combiner selects the replica with the highest SNR
or equivalently, in the case of equal noise power among replicas, the combiner selects the replica with the highest channel amplitude

72. Selection diversity performance
(a) uncorrelated fading in diversity branches
(b) fading in i:th branch is Rayleigh distributed
(c) => SNR is exponentially distributedd:
PDF
Probability that SNR in branch i is less than threshold y :
CDF

73. Selection diversity (cont.)
Probability that SNR in every branch (i.e. all L branches) is less than threshold y :
This cdf is also the cdf of
Differentiating the cdf with respect to y gives the pdf

74. Selection diversity (cont.)
which can be inserted into the expression for average bit error probability
… but as a general rule, for large average SNR it can be shown that

75. PDF of Selection signal
Η θεωρία για επιλογή μεταξύ ανεξάρτητων μεταβλητών xi, i=1,…,L, με πυκνότητα πιθανότητας για τη μεταβλητή y = max(x1 , x2 , …, xL), είναι
όπου είναι η αθροιστική κατανομή πιθανότητας της κάθε τυχαίας μεταβλητής x. Για Rayleigh πλάτη έχουμε

76. PDF of Selection signal (cont.)
clear; N=100000; bins=100; % normalized power Ω = 1 για κάθε πλάτος καναλιού % σ2 = 1 για κάθε randn, για αυτό διαιρούμε το πλάτος rv με sqrt(2) L=3; rv1=(1/sqrt(2))*abs(randn(1,N)+i*randn(1,N)); rv2=(1/sqrt(2))*abs(randn(1,N)+i*randn(1,N)); rv3=(1/sqrt(2))*abs(randn(1,N)+i*randn(1,N)); for j=1:N y(j) = max([rv1(j), rv2(j), rv3(j)]); end

77. PDF of Selection signal (cont.)
[n xout]=hist (y, bins); bar(xout, (n/ N)* (bins/max(xout) ) ) % ! ! ! ! ! Total probability = 1 axis ([ ] ) hold on x=0 : 0.1 : 5 ; omega = 1; L=3; p = L.*(2.*x./omega).*(1-exp(-x.^2/omega)).^(L-1).*exp(-x.^2./omega); plot (x, p, 'ro' ) legend('simulation', 'theory') title('y = max(x1, x2, x3)') ylabel('PDF of y') xlabel('values of y')

78. PDF of Selection signal (cont.)
PDF of maximum of 3 random variables

79. BER vs. SNR (diversity effect)
Flat fading channel,
Rayleigh fading,
L = 1
AWGN channel (no fading)
SNR
L = 4
L = 3
L = 2

80. Single Transmit Antenna Selection (TAS)
(Lt, 1; Lr) scheme : select one transmit diversity based on channel state information (CSI) at the transmitter side

81. Generalized TAS
Select L out of N antennas to transmit

82. Generalized TAS (cont.)
(Lt, 2; Lr) scheme: select 2 tx antennas based on

83. Δημιουργία προσομοίωσης για STBC
Δημιουργία bit-stream
Διαμόρφωση για παραγωγή συμβόλων (BPSK, QPSK, 16-QAM)
STBC κωδικοποίηση σύμφωνα με κάποιο πίνακα μετάδοσης
Έξοδος καναλιού (multiply with fading weight and add AWGN)
ΜL detection και σύγκριση tx και rx (bits, symbols or packets)
Data Bit Stream
Modulation
STBC Coder
MIMO Channel
STBC decoder
(Maximum Likelihood Detector)
Receive Data Bit Stream
BER/SER/FER
Calculation

84. Modulator function
function [mod_symbols, sym_table, M] = modulator(bitseq,b) N_bits=length(bitseq); sym_table=exp(j*pi/4*[ ]); sym_table=sym_table([ ]+1); inp=reshape(bitseq,b,N_bits/b); mod_symbols=sym_table([2 1]*inp+1); M=4;

85. Alamouti_scheme.m (2 x 1 Alamouti)
N_frame=130; N_packet=4000; NT=2; NR=1; b=2; %QPSK
SNRdBs=[0:2:30];
for i_SNR = 1:length(SNRdBs)
SNRdB = SNRdBs(i_SNR);
sigma=sqrt(0.5/(10^(SNRdB/10)));
for i_packet = 1:N_packet
msg_symbol=randint(N_frame*b, NT);
tx_bits=msg_symbol.'; tmp=[]; tmp1=[];
for i=1:NT
[tmp1, sym_tab, P] = modulator(tx_bits(i,:), b); tmp=[tmp; tmp1];
end
Εύρος SNR per bit σε dB
δημιουργούμε την τιμή του σ2 ανά διάσταση (Ι & Q)
Δημιουργούμε Ν διαφορετικές στήλες συμβόλων για Ν κεραίες μετάδοσης

86. Alamouti_scheme.m (2 x 1 Alamouti)
X = tmp. ';
X1=X;
X2 = [-conj(X(:,2)) conj(X(:,1))];
Hr(:,:) =(randn(N_frame, NT) + j* randn(N_frame, NT) )/sqrt(2);
H=reshape(Hr, N_frame, NT);
Habs=sum(abs(H).^2, 2);
Δημιουργία πίνακα μετάδοσης για πίνακα Alamouti
Δημιουργία των συντελεστών του καναλιού

87. Alamouti_scheme.m (2 x 1 Alamouti)
% received signal per receive antenna
for m=1:NR
r1(:,m) = sum(H.*X1, 2)/sqrt(NT) +… sigma*(randn(N_frame,1)+j*randn(N_frame,1));
r2(:,m) = sum(H.*X2, 2)/sqrt(NT) + ...
sigma*(randn(N_frame,1)+j*randn(N_frame,1));
% create the combined signals
z1(:,m)=r1(:,m).*conj(H(:,1))+conj(r2(:,m)).*H(:,2);
z2(:,m)=r1(:,m).*conj(H(:,2))-conj(r2(:,m)).*H(:,1);
end
Δημιουργούμε τα σήματα στο δέκτη
Δημιουργούμε τα σήματα στον αποδιαμορφωτή

88. Alamouti_scheme.m (2 x 1 Alamouti)
% form estimates
for q=1:P % P = # bits / symbol
tmp=(-1+sum(Habs,2))*abs(sym_tab(q)^2;
d1(:,q)=abs(sum(z1,2)-sym_tab(q)).^2 + tmp;
d2(:,q)=abs(sum(z2,2)-sym_tab(q)).^2 + tmp;
end
για κάθε σύμβολο εκπομπής j=1,2 δημιουργούμε τις στατιστικές που θα σταλούν στον ML detector
όπου η Ευκλείδεια απόσταση d:

89. Alamouti_scheme.m (2 x 1 Alamouti)
% determine the minimum of estimates
% decision for detecting s1
[y1,i1]=min(d1, [ ], 2);
s1d=sym_tab(i1).';
clear d1
% decision for detecting s2
[y2,i2]=min(d2, [ ], 2);
s2d=sym_tab(i2).';
clear d2
το κριτήριο απόφασης είναι να επιλέξουμε το σύμβολο που ελαχιστοποιεί dj,q
Eπέλεξε xi iff:
Στη θεωρία για j =1 βρίσκαμε το s1 και για j=2 βρίσκαμε το s2

90. Alamouti_scheme.m (2 x 1 Alamouti)
% form received symbol stream
Xd=[s1d s2d];
tmp1=X>0;
tmp2=Xd>0;
noes_p(i_packet) = sum(sum(tmp1~=tmp2));
end % end of FOR loop for "packet_count"
% calculate BER
BER(i_SNR)=sum(noes_p)/(N_packet*N_frame*b)
end % end of FOR loop for i_SNR

91. Alamouti_scheme.m ( 2 x 1 Alamouti)
semilogy(SNRdBs, BER)
axis([SNRdBs([1 end]) 1e-6 1e0])
grid on
xlabel('SNR (dB)')
ylabel (‘BER')

92. ΒΙΒΛΙΟΓΡΑΦΙΑ
[1] Alamouti, S. M., “A Simple Transmit Diversity Technique for Wireless Communications,” IEEE Journal Select. Areas Commun., Vol. 16, No. 8, October 1998, pp
[2] Tarokh, V., Jafarkhani H., and A.R. Calderbank, “Space-Time Block Coding for Wireless Communication: Performance Results,” IEEE Journal on Selected Areas in Communications, Vol 17, No. 3, March 1999, pp
[3] Tarokh, V., Jafarkhani H., and A.R. Calderbank, “Space Time Codes from Orthogonal Designs,” IEEE Trans. Inform. Theory, Vol 45, No. 5, July 1999, pp
[4] Yong Soo Cho, et al., MIMO-OFDM Wireless Communications with Matlab, John Wiley & Sons, 2010.
[5] David Tse and Pramod Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005.