MIMO Wireless Systems

Transcription

1 MIMO Wireless Systems Andreas Constantinides Assaf Shacham May 14, Introduction Communication in a slow flat Rayleigh fading channel with AWGN is not reliable as the channel frequently enters into deep fades, (i.e., the channel attenuation is large). More specifically, as seen in class, Rayleigh fading converts an exponential depency of bit error probability on the signal-to-noise ratio (SNR) into ( an inverse relationship. For BPSK, the probability of bit error in an AWGN channel is P e = Q 2Eb N 0 ), where E b /N 0 is the SNR per bit. However, with Rayleigh fading the average probability of bit error is P e = 1 2 ( 1 β 1 + β ), where β = E b N 0 E(α 2 ), and α is Rayleigh distributed Figure 1 shows clearly this severe degradation in the probability of bit error for BPSK in a Rayleigh fading channel for E(α 2 ) = Probability of bit error for BPSK vs SNR per bit AWGN channel Rayleigh fading channel 10 1 Probability of bit error SNR per bit (db) Figure 1: Effect of Rayleigh fading on the probability of bit error for BPSK In this report, we will be mostly talking about QAM. So for completeness, in Fig. 2, we plot MAT- LAB simulation results for the probability of symbol error of 16-QAM. As it can be seen, the performance degradation is as severe as in the BPSK case. 1

2 QAM SISO Probability of Symbol Error AWGN channel Rayleigh fading channel 10 1 Probability of a Symbol Error SNR per bit (db) Figure 2: Effect of Rayleigh fading on the probability of bit error for 16-QAM Diversity and coding are two well known techniques for combating fading. Stuber [8] points out that the basic idea of diversity systems is to provide the receiver with multiple replicas of the same information bearing signal, where the replicas are affected by uncorrelated fading. In the first part of this report, we will concentrate on antenna diversity techniques, giving both analytical as well as simulation results for the performance of the different techniques. The second part of our report will concentrate on a recent important extension of antenna diversity, and more specifically the Multiple-Input-Multiple-Output (MIMO) wireless systems idea. The seminal works of Foschini [2, 3] and Telatar [9] on the capacity of MIMO channels, helped transformed the view that fading should be considered as an enemy. MIMO systems can be defined simply as having multiple transmitting and receiving antennas, and one of their key feature is the ability to turn multipath propagation,traditionally a pitfall in wireless transmission, into a benefit for the user [4]. We will look at the benefits derived by using MIMO, and in particular we will look at a specific implementation, V-BLAST [5]. Again here, we will include extensive MATLAB simulation results that give the performance of V-BLAST. Finally, we will compare and contrast the benefits gained from using antenna diversity or MIMO. 2 Antenna Diversity As mentioned above, diversity is one of the most effective ways to combat deep fades. Assume that the receiver is provided with multiple replicas of the same information bearing signal, and denote by p the probability that the instantaneous SNR is below the receiver threshold on a single diversity branch (p denotes the probability of outage for that specific threshold in this case). If the receiver is provided with L replicas that fade indepently, then the probability that all the branches are at, or below the threshold at the same time is equal to p L. Proakis [7] identifies that the most important diversity techniques are: Frequency Diversity: The same information bearing signal is transmitted on L carriers, where the 2

3 separation between successive carriers is equal to or greater than the coherence bandwidth of the channel. Time Diversity: The same information bearing signal is transmitted in L different time slots, where the separation between successive time slots is equal to or greater than the coherence time of the channel. Antenna Diversity: A single transmitting antenna and L receiving antennas are used. The receiving antennas are spaced sufficiently apart to achieve indepence between the received signals. Antenna diversity, which will be our main topic of discussion in this section, can be achieved via these arrangements [8]: (i) Spatial, which is the most common method, is achieved by using multiple transmit and/or receive antennas. The spatial separation between isotropic antennas has to be at least half-wavelength in order to experience indepent fading. (ii) Angle (or direction) diversity requires a number of directional antennas that select waves arriving from a narrow angle of arrival in order to achieve indepent fading. (iii) Polarization diversity uses the property that scattering ts to de-polarize the signal and uses vertically and horizontally polarized receive antennas. In this report, we will concentrate on spatial diversity. Next we describe the methods for combining the signals received on the different diversity branches. 2.1 Diversity Combining Techniques There are many methods for combining the different diversity branches at the receiver, the most important of which and most widely used are: Maximal Ratio Combining (MRC), Equal Gain Combining (EGC) and Selective Combining (SC) [8]. Let us assume at this point that we have L receivers (diversity branches), and let us denote by γ i, i = 1,..., L, the instantaneous received symbol energy-to-noise ratio on the i-th diversity branch. As we saw in class, with Rayleigh fading γ i has an exponential pdf. where γ c is the average received branch symbol energy-to-noise ratio Selective Combining f γi (x) = 1 γ c e x/ γ c (1) With selective combining, the diversity branch yielding the highest SNR is always selected. Thus, the output of the selective combiner is γ sc s = max{γ 1,..., γ L } (2) If the branches experience indepent fading, it can be shown that the cdf function of γ sc s is [8]: F γ sc s (x) = P r[γ 1 x,..., γ L x] = [1 e x/ γc ] L (3) The probability of outage that we discussed in class is in fact equal to F γ sc(x). Figure 3 shows the s probability of outage for selective combining. Note that the largest diversity gain is obtained when going from L = 1 to L = 2, and diminishing returns are obtained with increasing L. 3

4 10 0 L=1 L=2 L=3 L= cdf of γ s sc γ s sc (db) Figure 3: Cdf of γ sc s (probability of outage) for selective combining 10 0 L=1 L=2 L=3 L= cdf of γ s mrc mrc γ s (db) Figure 4: Cdf of γ mrc s (probability of outage) for maximal ratio combining 4

5 SC is impractical for systems that use continuous transmission because it requires monitoring of all diversity branches. If such monitoring is performed, then it is better to use MRC which is optimal Maximal Ratio Combining With maximal ratio combining, the diversity branches are weighted by their respective complex fading gains and combined. MRC gives an optimal performance. As we have seen in class, the output of the MRC combiner, γs mrc has the following cdf: F γ mrc s (x) = 1 e x/ γc L i=0 ( ) 1 x i (4) i! γ c Figure 4 shows the cdf of the output of the maximal ratio combiner. Note, the similarities with Figure 3 in terms of diminishing returns with increasing L, and also that MRC performs better with SC in terms of probability of outage Equal Gain Combining Equal gain combining is similar with MRC with the only difference that the diversity branches are not weighted. As it is noted in [8], the cdf and pdf of the output of the EGC combiner, γs egc cannot be obtained in closed form for L > 2. EGC is useful for modulation techniques having equal energy symbols such as M-PSK. Besides the plots for the cdf of the combiners for MRC and SC, we also performed extensive MATLAB simulations to see the performance of antenna diversity systems when 16-QAM is used. Figure 5 shows the performance of dual diversity systems in terms of probability of symbol error. As we can see, MRC has superior performance and is closely followed by EGC. As expected, SC has the worst performance. Then, in Figures 6 and 7 we calculate the probability of symbol errors for selective combining and equal gain combining for different number of antennas (2, 4, 6, 8). Note again the diminishing returns with increasing L. 5

6 10 0 Dual Antenna Diversity (16 QAM): slow flat Rayleigh AWGN Channel 1 Antenna Selective Combining Equal Gain Combining Maximal Ratio Combining Probability of Symbol Error SNR per bit (db) Figure 5: Dual antenna diversity for 16-QAM 10 0 Antenna Diversity Selective Combining (16 QAM): slow flat Rayleigh AWGN Channel Antenna 2 Antennas 4 Antennas 6 Antennas 8 Antennas Probability of Symbol Error SNR per bit (db) Figure 6: Selective Combining with different number of antennas for 16-QAM 6

7 10 0 Antenna Diversity Equal Gain Combining (16 QAM): slow flat Rayleigh AWGN Channel Antenna 2 Antennas 4 Antennas 6 Antennas 8 Antennas Probability of Symbol Error SNR per bit (db) 3 MIMO Figure 7: Equal Gain Combining with different number of antennas for 16-QAM As we have seen in the previous section, there are huge performance gains in terms of symbol error probabilities when antenna diversity is used. In this section, we will briefly look at the Multiple-Input-Multiple- Output (MIMO) idea which was was inspired by the seminal works of Foschini [2, 3] and Telatar [9] on the capacity of MIMO channels. As mentioned above, the ideas behind MIMO helped transformed the view that fading should be considered as an enemy. MIMO systems can be defined simply as having multiple transmitting and receiving antennas. Let us assume that the number of transmitting antennas is M, and the number of receiving antennas is N. We will first look at the capacity of different antenna systems in order to see the dramatic increases in capacity obtained by using MIMO systems. 3.1 Capacity of conventional multiple antenna systems The Shannon capacity of different multiple antenna systems will be now presented. Our expressions are approximate, but they give an intuition for the derived benefits in terms of channel capacity when using multiple antennas. Our analysis is based on the book of Durgin [1] Single-Input, Single-Output (SISO) This is the conventional system that is used everywhere. Assume that for a given channel, whose bandwidth is B, and a given transmitter power of P the signal at the receiver has an average signal-to-noise ratio of SNR 0. Then, an estimate for the Shannon limit on channel capacity, C, is C B log 2 (1 + SNR 0 ) (5) 7

8 3.1.2 Single-Input, Multiple-Output (SIMO) For the SIMO system, we have N antennas at the receiver. If the signals received on these antennas have on average the same amplitude, then they can be added coherently to produce an N 2 increase in the signal power. On the other hand, there are N sets of noise that are added incoherently and result in an N-fold increase in the noise power. Hence, there is an overall increase in the SNR SNR N 2 (signal power) N (noise) = N SNR 0 (6) Thus, the channel capacity for this channel is approximately equal to Multiple-Input, Single-Output (MISO) C B log 2 (1 + N SNR 0 ) (7) In the MISO system, we have M transmitting antennas. The total transmitted power is divided up into the M transmitter branches. Following a similar argument as for the SIMO case, if the signals add coherently at the receiving antenna we get approximately an M-fold increase in the SNR as compared to the SISO case. Note here, that because there is only one receiving antenna the noise level is the same as in the SISO case. Thus, the overall increase in SNR is approximately SNR M 2 (signal power/m) noise Thus, the channel capacity for this channel is approximately equal to = M SNR 0 (8) C B log 2 (1 + M SNR 0 ) (9) Multiple-Input, Multiple-Output (MIMO)- Same signal transmitted by each antenna The MIMO system can be viewed in effect as a combination of the MISO and SIMO channels. In this case, it is possible to get approximately an MN-fold increase in the SNR yielding a channel capacity equal to C B log 2 (1 + MN SNR 0 ) (10) Thus, we can see that the channel capacity for the MIMO system is higher than that of MISO or SIMO. However, we should note here that in all four cases the relationship between the channel capacity and the SNR is logarithmic. This means that trying to increase the data rate by simply transmitting more power is extremely costly [6] Multiple-Input, Multiple-Output (MIMO)- Different signal transmitted by each antenna Our assumption here is that N M, so that all the transmitted signals can be decoded at the receiver. The big idea in MIMO is that we can s different signals using the same bandwidth and still be able to decode correctly at the receiver. Thus, it is like we are creating a channel for each one of the transmitters. The capacity of each one of these channels is roughly equal to C single B log 2 (1 + N M SNR 0) (11) 8

9 But, since we have M of these channels (M transmitting antennas), the total capacity of the system is C M B log 2 (1 + N M SNR 0) (12) Thus, as we can see from (12), we get a linear increase in capacity with respect to the number of transmitting antennas. So, the key principle at work here, is that it is more beneficial to transmit data using many different low-powered channels than using one single, high-powered channel [1]. In the next section, we will briefly describe a specific implementation of the MIMO idea, the V-BLAST algorithm [5, 10], and analyze its performance using a MATLAB simulation. 4 MIMO Implementation: V-BLAST Since the V-BLAST architecture was extensively discussed in class, we will only briefly describe it here. We assume that we have M transmitting and N receiving antennas, where M N, and also that there is a rich scattering environment. Each transmitter is an ordinary QAM transmitter operating co-channel with synchronized symbol timing. In effect, the collection of transmitters comprises a vector valued transmitter. We assume that the same constellation, (16-QAM in our MATLAB simulation), is used by each transmitter. The total transmitted power is constant, and thus the power transmitted by each transmitted is proportional to 1/M. The main idea behind the V-BLAST algorithm is to use symbol cancellation as well as linear nulling to perform detection in a recursive way. The full details of how the recursive V-BLAST algorithm is implemented can be found in [10, 5], and are not reproduced here. In our MATLAB simulation, we used the V-BLAST algorithm in 3 different architectures: 4x8, 8x12, and 12x16. The details of the MATLAB code and implementation are given in the appix. Here, we just present plots of symbol error probabilities vs SNR for the 3 different simulated architectures BLAST: M=4, N=8, 16 QAM Probability of Symbol Error SNR per bit (db) Figure 8: V-BLAST: 4 Transmitters, 8 Receivers 9

10 10 0 BLAST: M=8, N=12, 16 QAM Probability of Symbol Error SNR per bit (db) Figure 9: V-BLAST: 8 Transmitters, 12 Receivers 10 0 BLAST: M=12, N=16, 16 QAM Probability of Symbol Error SNR per bit (db) Figure 10: V-BLAST: 12 Transmitters, 16 Receivers 10

11 5 Conclusion As it can be seen from Figures 8,9,10 our simulated results for V-BLAST resemble quantitatively the experimental results of [5]. These results hint at the great capacity gains that can be achieved by using MIMO systems. So far however, only indoor experiments were achieved very successfully with V-BLAST architectures. It remains to be seen if the capacity gains promised by the seminal works of Foschini [2] and Telatar [9] can be also obtained in other environments. APPENDIX SIMULATION DOCUMENTATION In this research project we have made 4 groups of simulations: Single Input Single Output Single Input Dual Output - comparing combining techniques. Single Input Multiple Outputs - showing the effect of additional receivers. Multiple Input Multiple Outputs - BLAST SISO - Single Input Single Output (File: SISO.m) In this simulation we used a single 16-QAM transmitter and a single receiver and simulated two channel between the two: and additive white Gaussian noise channel (AWGN) and a Rayleigh fading channel. For the Rayleigh fading channels we used the Rayleigh distribution parameter r=0.24. This parameter was picked after simulating numerous Rayleigh fading channels and picking the channel that gave a -30 db fading 2We compared the channels, sweeping across a wide range of Eb/No and calculating the symbol error rates for 100 blocks of 256 symbols, per each energy level. SIDO - Single Input Dual Output (File: SIDO.m) Based on the SISO simulation, we added another receiving antenna and simulated the three combining methods. The comparison is done against the mean symbol error rate of the two antennas. SIMO - Single Input Multiple Output (File: SIMO.m) Based on the SISO simulation, we tested two combining (selection combining and equal gain combining) methods with multiple antennas. The combined signals symbol error rates are compared to that of a single antenna. MIMO - Multiple Input Multiple Output - V-BLAST (File: VBLAST.m) Using the V-BLAST algorithm from [5] we simulated a multiple inputs multiple outputs environment. The results for 4 transmitter, 8 receiver setup are shown along with the results for 8x12 and 12x16 setups. 11

12 References [1] Gregory D. Durgin. Space-Time Wireless Channels. Prentice Hall, New Jersey, [2] G. J. Foschini. Layered space-time architecture for wireless communication in a fading environment when using multiple antennas. Bell Labs Technical Journal, 1(2):41 59, Autumn [3] G.J. Foschini and M.J. Gans. On limits of wireless communication in a fading environment when using multiple antennas. Wireless Personal Communications, 6(3): , [4] D. Gesbert, M. Shafi, D. S. Shiu, P. Smith, and A. Naguib. From theory to practice: An overview of mimo space-time coded wireless systems. IEEE Journal on Selected Areas in Communications, 21(3): , April [5] G. Golden, G. Foschini, R. Valenzuela, and P. Wolniasky. Detection algorithm and initial laboratory results using the v-blast space-time communication architecture. Electronics Letters, 35(1):14 15, January [6] Angel Lozano, Farrokh R. Farrokhi, and Reinaldo A. Valenzuela. Lifting the limits on high-speed wireless data access using antenna arrays. IEEE Communications Magazine, pages , September [7] John G. Proakis. Digital Communications. McGraw Hill, New York, 4th edition, [8] Gordon L. Stuber. Principles of Mobile Communication. Kluwer Academic Publishers, Boston, 2nd edition, [9] I. Emre Telatar. Capacity of multi-antenna gaussian channels. Technical Memorandum, Bell Laboratories, Lucent Technologies, October [10] P. Wolniansky, G. Foschini, G. Golden, and R. Valenzuela. V-blast: an architecture for realizing very high data rates over the rich-scattering wireless channel,

13 MATLAB CODE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % SISO.m % Single Input Single Output 16-QAM Simulation % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clear; close all; SignalEnergyMin = 2; SignalEnergyMax = 42; SignalEnergy_d = 1; BlocksPerEnergy = 100; BlockSize = 2^8; SamplingRate = 6; WGNPower = 1; RayleighParam =.4; Antennas = 2; M=16; Gvec1 = randn(1,blocksperenergy); Gvec2 = randn(1,blocksperenergy); RayleighVec(aa,:) = sqrt((rayleighparam^2.*gvec1.^2)+(rayleighparam^2.*gvec2.^2)); RayleighVec(1,:) = ones(1,size(rayleighvec,2)); EnergiesdB = SignalEnergyMin:SignalEnergy_d:SignalEnergyMax; Energies = 10.^(EnergiesdB./10); for ee = 1:size(Energies,2) for bb = 1:BlocksPerEnergy A = randint(1,blocksize,16); % Generate Data Stream B = mod16qam (A,1,SamplingRate,Energies(ee))'; % 16-QAM Modulation disp(['eb = ',int2str(energies(ee)),'; Block ',int2str(bb),'; Antenna ',int2str(aa)]); C(aa,:) = RayleighVec(aa,bb).*B; % Rayleigh fading D(aa,:) = AWGN(C(aa,:),WGNPower); % AWGN [E(aa,:),Eb(aa,bb)] = demod16qam (D(aa,:),1,SamplingRate); % Demodulate [TT,SER(aa,bb)] = symerr(a,e(aa,:)); % Measure BER per antenna ESER(aa,ee) = mean(ser(aa,:),2); Eaxis = 10*log10((SignalEnergyMin:SignalEnergy_d:SignalEnergyMax)./(WGNPower^2)); figure; semilogy(energiesdb,eser,'+'); title ('\fontsize{12}\bf16-qam SISO Bit Error Rate in a slow flat Rayleigh AWGN Channel'); xlabel ('Eb/N_0'); ylabel ('SER'); leg('no fading','rayleigh fading (r=0.24)'); %% EOF %% EOF %% EOF %% EOF %% EOF %% EOF %% EOF %% EOF %% EOF %% EOF 13

14 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%% % % SISO.m % Single Input Dual Output 16-QAM Simulation % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%% clear; close all; SignalEnergyMin = 4; SignalEnergyMax = 52; SignalEnergy_d = 1; BlocksPerEnergy = 200; BlockSize = 2^8; SamplingRate = 1; WGNPower = 1; RayleighParam =.24; Antennas = 2; M=16; Gvec1 = randn(1,blocksperenergy); Gvec2 = randn(1,blocksperenergy); RayleighVec(aa,:) = sqrt((rayleighparam^2.*gvec1.^2)+(rayleighparam^2.*gvec2.^2)); EnergiesdB = SignalEnergyMin:SignalEnergy_d:SignalEnergyMax; Energies = 10.^(EnergiesdB./10); for ee = 1:size(Energies,2) for bb = 1:BlocksPerEnergy A = randint(1,blocksize,16); % Generate Data Stream B = mod16qam (A,1,SamplingRate,Energies(ee))'; % 16-QAM Modulation disp(['eb = ',int2str(energies(ee)),'; Block ',int2str(bb),'; Antenna ',int2str(aa)]); C(aa,:) = RayleighVec(aa,bb).*B; % Rayleigh fading D(aa,:) = AWGN(C(aa,:),WGNPower); % AWGN [E(aa,:),Es(aa,bb)] = demod16qam (D(aa,:),1,SamplingRate); % Demodulate [TT,SER(aa,bb)] = symerr(a,e(aa,:)); % Measure SER per antenna % Selective Combining stronger = (find (Es(:,bb) == max (Es(:,bb)))); [SC,E_sc(1,bb)] = demod16qam (D(stronger,:),1,SamplingRate); [TT,SER_sc(bb)] = symerr(a,sc); % Measure SER per antenna % Equal Gain Combining [EGC,E_egc(1,bb)] = demod16qam (mean(d,1),1,samplingrate); [TT,SER_egc(bb)] = symerr(a,egc); % Measure SER per antenna % Maximal Ratio Combining r(aa) = sqrt(es(aa,bb)/energies(ee)); a(aa) = r(aa)/sum(sqrt(r)); 14

15 [MRC,E_mrc(1,bb)] = demod16qam (r(1)*d(1,:)+r(2)*d(2,:),1,samplingrate); [TT,SER_mrc(bb)] = symerr(a,mrc); % Measure SER per antenna ESER(aa,ee) = mean(ser(aa,:),2); ESER_sc(ee) = mean(ser_sc,2); ESER_egc(ee) = mean(ser_egc,2); ESER_mrc(ee) = mean(ser_mrc,2); figure; semilogy(energiesdb,mean (ESER,1),'+',EnergiesdB,ESER_sc(1,:),'d',EnergiesdB,ESER_egc(1,:),'d',EnergiesdB,ESER_mrc(1,:),'d'); title ('\fontsize{10}\bfantenna Diversity (16-QAM): \rm2 Received signals with close power levels (slow flat Rayleigh AWGN Channel)'); xlabel ('Eb/N_0'); ylabel ('SER'); leg('antenna 1+2','Selective Combining','Equal Gain Combining','Maximum Ratio Combining'); %% EOF %% EOF %% EOF %% EOF %% EOF %% EOF %% EOF %% EOF %% EOF %% EOF 15

16 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % SIMO.m % Single Input Multiple Outputs 16-QAM Simulation % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clear; close all; SignalEnergyMin = 4; SignalEnergyMax = 52; SignalEnergy_d = 1; BlocksPerEnergy = 100; BlockSize = 2^8; SamplingRate = 1; WGNPower = 1; RayleighParam =.24; Antennas = 8; M=16; Gvec1 = randn(1,blocksperenergy); Gvec2 = randn(1,blocksperenergy); RayleighVec(aa,:) = sqrt((rayleighparam^2.*gvec1.^2)+(rayleighparam^2.*gvec2.^2)); EnergiesdB = SignalEnergyMin:SignalEnergy_d:SignalEnergyMax; Energies = 10.^(EnergiesdB./10); for ee = 1:size(Energies,2) for bb = 1:BlocksPerEnergy A = randint(1,blocksize,16); % Generate Data Stream B = mod16qam (A,1,SamplingRate,Energies(ee))'; % 16-QAM Modulation disp(['eb = ',int2str(energies(ee)),'; Block ',int2str(bb),'; Antenna ',int2str(aa)]); C(aa,:) = RayleighVec(aa,bb).*B; % Rayleigh fading D(aa,:) = AWGN(C(aa,:),WGNPower); % AWGN [E(aa,:),Es(aa,bb)] = demod16qam (D(aa,:),1,SamplingRate); % Demodulate [TT,SER(aa,bb)] = symerr(a,e(aa,:)); % Measure SER per antenna for aa=2:2:antennas TempEs = Es(1:aa,bb); % Selective Combining stronger = (find (TempEs == max (TempEs))); [SC,E_sc(1,bb)] = demod16qam (D(stronger,:),1,SamplingRate); [TT,SER_sc(aa/2,bb)] = symerr(a,sc); % Measure SER per antenna % Equal Gain Combining TempD = D(1:aa,:); [EGC,E_egc(1,bb)] = demod16qam (mean(tempd,1),1,samplingrate); [TT,SER_egc(aa/2,bb)] = symerr(a,egc); ESER(aa,ee) = mean(ser(aa,:),2); 16

17 ESER_sc(:,ee) = mean(ser_sc,2); ESER_egc(:,ee) = mean(ser_egc,2); % ESER_mrc(ee) = mean(ser_mrc,2); figure; semilogy(energiesdb,mean (ESER,1),'+',EnergiesdB,ESER_sc,'d-'); title ('\fontsize{10}\bfantenna Diversity - Selective Combining (16-QAM): \rmslow flat Rayleigh AWGN Channel'); xlabel ('Eb/N_0'); ylabel ('SER'); leg('antenna 1+2','2 Antennas','4 Antennas','6 Antennas','8 Antennas'); figure; semilogy(energiesdb,mean (ESER,1),'+',EnergiesdB,ESER_egc,'x-'); title ('\fontsize{10}\bfantenna Diversity - Equal Gain Combining (16-QAM): \rmslow flat Rayleigh AWGN Channel'); xlabel ('Eb/N_0'); ylabel ('SER'); leg('antenna 1+2','Equal Gain Combining 2','Equal Gain Combining 4','Equal Gain Combining 6','Equal Gain Combining 8'); %% EOF %% EOF %% EOF %% EOF %% EOF %% EOF %% EOF %% EOF %% EOF %% EOF 17

18 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % VBLAST.m % Multiple Inputs Multiple Outputs (V-BALST algorithm) 16-QAM Simulation % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clear; close all; SignalEnergyMin = 2; SignalEnergyMax = 13; SignalEnergy_d = 0.5; BlockSize = 2^8; BlocksPerEnergy = 20; SamplingRate = 1; WGNPower = 1; RayleighParam = 1; Transmitters = 4; Receivers = 8; EnergiesdB = SignalEnergyMin:SignalEnergy_d:SignalEnergyMax; Energies = 10.^(EnergiesdB./10); for ee = 1:size(Energies,2) Eb = Energies(ee)/Transmitters; for bb=1:blocksperenergy disp(['eb = ',int2str(energies(ee)),'; Block ',int2str(bb)]); RayleighMat = RayleighParam^2*randn (Receivers,Transmitters) + j*rayleighparam^2*randn (Receivers,Transmitters); %RayleighMat = RayleighMat*.1; A = randint(transmitters,blocksize,16); % Generate Data Stream A(rows = xmitters, cols = symbols) B = mod16qam (A,1,SamplingRate,Eb)'; % 16-QAM Modulation noise = (WGNPower).*randn(size(RayleighMat*B)) + j*(wgnpower).*randn(size(rayleighmat*b)); C = RayleighMat*B + noise; k = []; H = RayleighMat; r=c; for tt=1:transmitters % find best signal G = pinv(h); normg = sum(abs(g).^2,2); normg(k) = inf; k(tt) = find(normg == min(normg)); w = G(k(tt),:) ; y = w*r; [E(k(tt),:),Eb_rec(k(tt))] = demod16qam (y,1,samplingrate); % Demodulate % nulling starts here r = r - H(:,k(tt))*(mod16qam (E(k(tt),:),1,SamplingRate,Eb_rec(k(tt))))'; 18

19 H(:,k(tt)) = zeros(size(h(:,k(tt)))); SER(:,bb) = 1-sum(A==E,2)/size(A,2); NC_ESER(ee)=mean(mean(SER,2),1); % Measure BER per T-R pair. figure; semilogy(energiesdb,nc_eser,'b+'); title (['\fontsize{12}\bfblast: \rmm=',int2str(transmitters),', N=',int2str(Receivers),', 16-QAM']); xlabel ('Eb/N_0'); ylabel ('SER'); %% EOF %% EOF %% EOF %% EOF %% EOF %% EOF %% EOF %% EOF %% EOF %% EOF 19

20 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % mod16qam.m %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function Y = mod16qam(x,fd,fs,eb); % 16-QAM modulation % % Y = mod16qam(x,fd,fs,eb) % % X - data stream % Fd - data sampling rate % Fs - Modulation signal samling rate (Fs/Fd integer) % Eb - Average bit energy % M = 16; half_d = sqrt(sqrt(0.4*eb)); Y = half_d*dmodce(x,fd,fs,'qask',m)'; % QAM modulation %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % demod16qam.m %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [Y,Eb] = demod16qam(r,fd,fs); % 16-QAM modulation % % [Y,Eb] = demod16qam(r,fd,fs) % % R - received signal (row vector) % Fd - data sampling rate % Fs - Modulation signal samling rate (Fs/Fd integer) % Y - Output signal % Eb - Average energy per received symbol M = 16; n = size(r,2); Es = sum(abs(r).^2)/n; Eb = Es/4; half_d = sqrt(0.4*eb); % scatterplot(ynoisy,5,0,'b.'); % scatter plot of signal+noise Y = ddemodce(r/half_d,fd,fs,'qask',m); % demodulated signal 20