Quadrature Amplitude Modulation (QAM) is an important modulation scheme as it allows for higher data rates and spectral efficiencies. The bit error rate (BER) of QAM can be calculated through Monte Carlo simulations. However this becomes quite complex as the constellation size of the modulation schemes increases. Therefore a theoretical approach is sometimes preferred. The BER for Gray coded QAM, for even number of bits per symbol, is shown below.
BER of 4-QAM, 16-QAM, 64-QAM, 256-QAM in AWGN
Gray coding ensures that a symbol error results in a single bit error. The code for calculating the theoretical QAM BER for k even (even number of bits per symbol) is given below. The formula for calculating the BER for k odd is different, however, the formula given below can be used a first estimate.
The Shannon Capacity of a channel is the data rate that can be achieved over a given bandwidth (BW) and at a particular signal to noise ratio (SNR) with diminishing bit error rate (BER). This has been discussed in an earlier post for the case of SISO channel and additive white Gaussian noise (AWGN). For a MIMO fading channel the capacity with channel not known to the transmitter is given as (both sides have been normalized by the bandwidth [1]):
Shannon Capacity of a MIMO Channel
where NT is the number of transmit antennas, NR is the number of receive antennas, γ is the signal to interference plus noise ratio (SINR), INR is the NRxNR identity matrix and H is the NRxNT channel matrix. Furthermore, hij, an element of the matrix H defines the complex channel coefficient between the ith receive antenna and jth transmit antenna. It is quite obvious that the channel capacity (in bits/sec/Hz) is highly dependent on the structure of matrix H. Let us explore the effect of H on the channel capacity.
Let us first consider a 4×4 case (NT=4, NR=4) where the channel is a simple AWGN channel and there is no fading. For this case hij=1 for all values of i and j. It is found that channel capacity of this simple channel for an SINR of 10 dB is 5.36bits/sec/Hz. It is further observed that the channel capacity does not change with number of transmit antennas and increases logarithmically with increase in number of receive antennas. Thus it can be concluded that in an AWGN channel no multiplexing gain is obtained by increasing the number of transmit antennas.
We next consider a more realistic scenario where the channel coefficients hij are complex with real and imaginary parts having a Gaussian distribution with zero mean and variance 0.5. Since the channel H is random the capacity is also a random variable with a certain distribution. An important metric to quantify the capacity of such a channel is the Complimentary Cumulative Distribution Function (CCDF). This curve basically gives the probability that the MIMO capacity is above a certain threshold.
Complimentary Cumulative Distribution Function of Capacity
It is obvious (see figure above) that there is a very high probability that the capacity obtained for the MIMO channel is significantly higher than that obtained for an AWGN channel e.g. for an SINR of 9 dB there is 90% probability that the capacity is greater than 8 bps/Hz. Similarly for an SINR of 12 dB there is a 90% probability that the capacity is greater than 11 bps/Hz. For a stricter threshold of 99% the above capacities are reduced to 7.2 bps/Hz and 9.6 bps/Hz.
In a practical system the channel coefficients hij would have some correlation which would depend upon the antenna spacing. Lower the antenna spacing higher would be the antenna correlation and lower would be the MIMO system capacity. This would be discussed in a future post.
The MATLAB code for calculating the CCDF of channel capacity of a MIMO channel is given below.
clear all
close all
Nr=4;
Nt=4;
I=eye(Nr);
g=15.8489;
for n=1:10000
H=sqrt(1/2)*randn(Nr,Nt)+j*sqrt(1/2)*randn(Nr,Nt);
C(n)=log2(det(I+(g/Nt)*(H*H')));
end
[a,b]=hist(real(C),100);
a=a/sum(a);
plot(b,1-cumsum(a));
xlabel('Capacity (bps/Hz)')
ylabel('Probability (Capacity > Abcissa)')
grid on
[1] G. J. Foschini and M. J. Gans,”On limits of Wireless Communications in a Fading Environment when Using Multiple Antennas”, Wireless Personal Communications 6, pp 311-335, 1998.
Variable system bandwidth to accommodate users with different data rates, 1.25, 2.50, 5.00, 10.00, 15.00 and 20.00 MHz, actual transmission bandwidth is a bit lower than this
2. Frequency-selective scheduling
Not possible
A key advantage of OFDMA, although it requires accurate real-time feedback of channel conditions from receiver to transmitter
3. Symbol period
Very short—inverse of the system bandwidth
Very long—defined by subcarrier spacing and independent of system bandwidth
4. Equalization
Complicated time domain equalization
Simple frequency domain equalization
5. Resistance to mulitpath
Rake receiver can combine various multipath components
Highly resistant to multipath due to insertion of cyclic prefix (CP)
6. Suitability for MIMO
MIMO is not suited to a wideband frequency selective channel
MIMO is suited to the independent narrowband flat fading channels that the subcarriers provide
7. Resistance to narrowband interference
Resistant to narrow band interference
Some subcarriers to be affected by narrowband interference
8. Separation of users
Scrambling and orthogonal spreading codes
Frequency and time although scrambling and spreading can be added as well
Reference: Agilent 3GPP Long Term Evolution System Overview, Product Development and Test Challenges Application Note.
Peak to Average Power Ratio (PAPR) as the name suggests is the ratio of peak signal power to the average signal power and has received considerable attention in the context of multicarrier signals like OFDM which exhibit a high PAPR. The down side of this high PAPR is that the power amplifier in the transmitter is operated at a relatively lower power level so that the peaks in the signal are not distorted by the saturating amplifier. This is called the amplifier backoff and it plays an important part in wireless system design.
Power Amplifier Input Output Behavior
The reason for this high PAPR is that when multiple sinusoids are added together in a multicarrier transmission the resulting signal exhibits constructive and destructive behavior. The higher the number of these sinusoids higher is the PAPR. The following example illustrates this behavior.
Constructive and Destructive Behavior of Multiple Carriers
It is observed that the PAPR of a signal composed of two sinusoids is greater than that of a single sinusoid. Similarly the PAPR of a signal composed of three carriers is even higher. The PAPR for the case of a single tone, two tones and three tones is 2.00, 3.10 and 4.15 respectively. Or on a logarithmic scale 3.01 dB, 4.91 dB and 6.18 dB respectively. So if the power amplifier in a wireless system starts saturating at 24 dBm then the average signal power of three tone system must not exceed 24-6.18=17.82 dBm. In practical systems techniques are adopted that decrease the PAPR so that the power amplifier can operate close to its maximum limit. One simple technique clips the peaks in the signal, while another adds tones at the unused frequencies such that the total PAPR is reduced.
Note:
1. The three tones in the above example are at ‘f’, ‘2f’ and ‘3f’ Hz.
2. Another metric that we will discuss in a future post is the cubic metric (CM).
We had previously presented a method of generating a temporally correlated Rayleigh fading sequence. This was based on Smith’s fading simulator which was based on Clark and Gan’s fading model. We now present a highly efficient method of generating a correlated Rayleigh fading sequence, which has been adapted from Young and Beaulieu’s technique [1]. The architecture of this fading simulator is shown below.
Modified Young’s Fading Simulator
This method essentially involves five steps.
1. Generate two Gaussian random sequences of length N each.
2. Multiply these sequences by the square root of Doppler Spectrum S=1.5./(pi*fm*sqrt(1-(f/fm).^2).
3. Add the two sequences in quadrature with each other to generate a length N complex sequence (we have added the two sequences before multiplying with the square root of Doppler Spectrum in our simulation).
4. Take the M point complex inverse DFT where M=(fs/Δf)+1.
5. The absolute value of the resulting sequence defines the envelope of the Rayleigh faded signal with the desired temporal correlation (based upon the Doppler frequency fm).
A point to be noted here is that although the Doppler spectrum is defined from -fm to +fm the IDFT has to be taken from -fs/2 to +fs/2. This is achieved by stuffing zeros in the vacant frequency bins from -fs/2 to +fs/2. The MATLAB code for this simulator is given below.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% RAYLEIGH FADING SIMULATOR BASED UPON YOUNG'S METHOD
% N is the number of paths
% M is the total number of points in the frequency domain
% fm is the Doppler frequency in Hz
% fs is the sampling frequency in Hz
% df is the step size in the frequency domain
% Copyright RAYmaps (www.raymaps.com)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all;
close all;
N=64;
fm=70;
df=(2*fm)/(N-1);
fs=7.68e6;
M=round(fs/df);
T=1/df;
Ts=1/fs;
% Generate 2xN IID zero mean Gaussian variates
g1=randn(1,N);
g2=randn(1,N);
g=g1-j*g2;
% Generate Doppler Spectrum
f=-fm:df:fm;
S=1.5./(pi*fm*sqrt(1-(f/fm).^2));
S(1)=2*S(2)-S(3);
S(end)=2*S(end-1)-S(end-2);
% Multiply square root of Doppler Spectrum with Gaussian random sequence
X=g.*sqrt(S);
% Take IFFT
F_zero=zeros(1, round((M-N)/2));
X=[F_zero, X, F_zero];
x=ifft(X,M);
r=abs(x);
r=r/mean(r);
% Plot the Rayleigh envelope
t=0:Ts:T-Ts;
plot(t,10*log10(r))
xlabel('Time(sec)')
ylabel('Signal Amplitude (dB)')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The question now is that how do we verify that the generated Rayleigh fading sequence has the desired statistical properties. This can be verified by looking at the level crossing rate (LCR) and average fade duration (AFD) of the generated sequence as well as the PDF and Autocorrelation function. The LCR and AFD calculated for N=64 and fm=70 Hz and threshold of -10 dB (relative to the average signal power) is given below.
LCR
Simulation: 15.55
Theoretical: 15.46
AFD
Simulation: 453 usec
Theoretical: 508 usec
It is observed that the theoretical and simulation results for the LCR and AFD match reasonably well. We next examine the distribution of the envelope and phase of the resulting sequence x. It is found that the envelope of x is Rayleigh distributed while the phase is uniformly distributed from -pi to pi. This is shown in the figure below. So we are reasonably satisfied that our generated sequence has the desired statistical properties.
Envelope and Phase Distribution for fm=70Hz
Rayleigh Fading Simulator Based on Young’s Filter
In the above simulation of Rayleigh fading sequence we reduced the computation load of Smith’s simulator by reducing the IFFT operations on two branches to a single IFFT operation. However, we still used the Doppler spectrum proposed by Smith. Now we use the filter with spectrum Fk defined by Young in [1]. The code for this is given below.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% RAYLEIGH FADING SIMULATOR BASED UPON YOUNG'S METHOD
% N is the number of points in the frequency domain
% fm is the Doppler frequency in Hz
% fs is the sampling frequency in Hz
% Copyright RAYmaps (www.raymaps.com)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all;
close all;
N=2^20;
fm=300;
fs=7.68e6;
Ts=1/fs;
% Generate 2xN IID zero mean Gaussian variates
g1=randn(1,N);
g2=randn(1,N);
g=g1-j*g2;
% Generate filter F
F = zeros(1,N);
dopplerRatio = fm/fs;
km=floor(dopplerRatio*N);
for k=1:N
if k==1,
F(k)=0;
elseif k>=2 && k<=km,
F(k)=sqrt(1/(2*sqrt(1-((k-1)/(N*dopplerRatio))^2)));
elseif k==km+1,
F(k)=sqrt(km/2*(pi/2-atan((km-1)/sqrt(2*km-1))));
elseif k>=km+2 && k<=N-km,
F(k) = 0;
elseif k==N-km+1,
F(k)=sqrt(km/2*(pi/2-atan((km-1)/sqrt(2*km-1))));
else
F(k)=sqrt(1/(2*sqrt(1-((N-(k-1))/(N*dopplerRatio))^2)));
end
end
% Multiply F with Gaussian random sequence
X=g.*F;
% Take IFFT
x=ifft(X,N);
r=abs(x);
r=r/mean(r);
% Plot the Rayleigh envelope
T=length(r)*Ts;
t=0:Ts:T-Ts;
plot(t,10*log10(r))
xlabel('Time(sec)')
ylabel('Signal Amplitude (dB)')
axis([0 0.05 -15 5])
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The above code was used to generate Rayleigh sequences of varying lengths with Doppler frequencies of 5 Hz, 70 Hz and 300 Hz. The sampling frequency was fixed at 7.68 MHz (corresponding to a BW of 5 MHz). It must be noted that in this simulation the length of the Gaussian sequence is equal to the filter length in the frequency domain. It was found that to generate a Rayleigh sequence of reasonable length the length of the Gaussian sequence has to be quite large (2^20 in the above example). As before we calculated the distribution of envelope and phase of the generated sequence as well as the LCR and AFD. These were found to be within reasonable margins.
Envelope Phase Distribution at fm=300 Hz
Note:
1. Level crossing rate (LCR) is defined as number of times per second the signal envelope crosses a given threshold. This could be either in the positive direction or negative direction.
2. Average fade duration (AFD) is the average duration that the signal envelope remains below a given threshold once it crosses that threshold. Simply it is the average duration of a fading event.
3. The LCR and AFD are interconnected and the product of these two quantities is a constant.
[1] David J. Young and Norman C. Beaulieu, "The Generation of Correlated Rayleigh Random Variates by Inverse Discrete Fourier Transform", IEEE Transactions on Communications vol. 48 no. 7 July 2000.
Just to recap, building an LTE fading simulator with the desired temporal and spatial correlation is a three step procedure.
1. Generate Rayleigh fading sequences using Smith’s method which is based on Clarke and Gan’s fading model.
2. Introduce spatial correlation based upon the spatial correlation matrices defined in 3GPP 36.101.
3. Use these spatially and temporally correlated sequences as the filter taps for the LTE channel models.
We have already discussed step 1 and 3 in our previous posts. We now focus on step 2, generating spatially correlated channels coefficients.
3GPP has defined spatial correlation matrices for the Node-B and the UE. These are defined for 1,2 and 4 transmit and receive antennas. These are reproduced below.
Spatial Correlation Matrix
The parameters ‘alpha’ and ‘beta’ are defined as:
Low Correlation
alpha=0, beta=0
Medium Correlation
alpha=0.3, beta=0.9
High Correlation
alpha=0.9, beta=0.9
The combined effect of antenna correlation at the transmitter and receiver is obtained by taking the Kronecker product of individual correlation matrices e.g. for a 2×2 case the correlation matrix is given as:
Correlation Matrix for 2×2 MIMO
Multiplying the square root of the correlation matrix with the vector of channel coefficients is equivalent to taking a weighted average e.g. for the channel between transmit antenna 1 and receive antenna 1 the correlated channel coefficient h11corr is given as:
h11corr=w1*h11+w2*h12+w3*h21+w4*h22
where w1=1 and w2, w3 and w4 are less than one and greater than zero. For the high correlation case described above the channel coefficient is calculated as:
From a practical point of antenna correlation is dependent on the antenna separation. Greater the antenna spacing lower is the antenna correlation and better the system performance. However, a base station requires much higher antenna spacing than a UE to achieve the same level of antenna correlation. This is due to the fact the base station antennas are placed much higher than a UE. Therefore the signals arriving at the base station are usually confined to smaller angles and experience similar fading. A UE on the other hand has a lot of obstacles in the surrounding areas which results in higher angle spread and uncorrelated fading between the different paths.
