Monthly Archives: November 2011

MIMO Capacity in a Fading Environment

November 25, 2011Capacity, LTE, WiMAX4G, Channel, LTE, MIMO, Multipath, Rayleigh, Rayleigh Fading, Shannon Capacity, SINR, SNRYasir Ahmed (aka John)

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 N_T is the number of transmit antennas, N_R is the number of receive antennas, γ is the signal to interference plus noise ratio (SINR), I_NR is the N_RxN_R identity matrix and H is the N_RxN_T channel matrix. Furthermore, h_ij, 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 (N_T=4, N_R=4) where the channel is a simple AWGN channel and there is no fading. For this case h_ij=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 h_ij 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 h_ij 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.

CDMA vs OFDMA

November 13, 2011LTE, WCDMA, WiMAX4G, FDMA, LTE, OFDM, WCDMA, WiMAXYasir Ahmed (aka John)

Property	CDMA	OFDMA
1. Channel bandwidth	Full system bandwidth	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)

November 7, 2011LTE4G, Amplifier, LTE, PAPRYasir Ahmed (aka John)

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).

Computationally Efficient Rayleigh Fading Simulator

November 1, 2011Channel Modeling4G, Channel, LTE, Rayleigh, Rayleigh Fading, WiMAXYasir Ahmed (aka John)

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 — 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 F_k 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.