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 h11_{corr} is given as:

h11_{corr}=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.

Since the wireless channel is time varying the channel taps c(0) c(1)…..c(L-1) are also time varying with either Rayleigh or Rician distribution. It is quite easy to generate Rayleigh random variables with the desired power and distribution, however, when these Rayleigh random variables are required to have temporal correlation the process becomes a bit complicated. Temporal correlation of these variables depends upon the Doppler frequency which is turn depends upon the speed of the mobile device. The Doppler frequency is defined as:

fd=v*cos(theta)/lambda

where

fd is the Doppler Frequency in Hz
v is the receiver velocity in m/sec
lambda is the wavelength in m
and theta is the angle between the direction of arrival of the signal and the direction of motion

A simple method for generating Rayleigh random variables with the desired temporal correlation was devised by Smith [1]. His method was based on Clark and Gans fading model and has been widely used in simulation of wireless communication systems.

The method for generating the Rayleigh fading envelope with the desired temporal correlation is given below (modified from Theodore S. Rappaport Text).

1. Define N the number of Gaussian RVs to be generated, fm the Doppler frequency in Hz, fs the sampling frequency in Hz, df the frequency spacing which is calculated as df=(2*fm)/(N-1) and M total number of samples in frequency domain which is calculated as M=(fs/df).

2. Generate two sequences of N/2 complex Gaussian random variables. These correspond to the frequency bins up to fm. Take the complex conjugate of these sequences to generate the N/2 complex Gaussian random variables for the negative frequency bins up to -fm.

3. Multiply the above complex Gaussian sequences g1 and g2 with square root of the Doppler Spectrum S generated from -fm to fm. Calculate the spectrum at -fm and +fm by using linear extrapolation.

4. Extend the above generated spectra from -fs/2 to +fs/2 by stuffing zeros from -fs/2 to -fm and fm to fs/2. Take the IFFT of the resulting spectra X and Y resulting in time domain signals x and y.

5. Add the absolute values of the resulting signals x and y in quadrature. Take the absolute value of this complex signal. This is the desired Rayleigh distributed envelope with the required temporal correlation.

The Matlab code for generating Rayleigh random sequence with a Doppler frequency of fm Hz is given below.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% RAYLEIGH FADING SIMULATOR BASED UPON SMITH'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=10;
fm=300;
df=(2*fm)/(N-1);
fs=7.68e6;
M=round(fs/df);
T=1/df;
Ts=1/fs;
% Generate two sequences of N complex Gaussian random variables
g=randn(1,N/2)+j*randn(1,N/2);
gc=conj(g);
g1=[fliplr(gc), g];
g=randn(1,N/2)+j*randn(1,N/2);
gc=conj(g);
g2=[fliplr(gc), g];
% Generate Doppler Spectrum S
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 the complex sequences with the Doppler Spectrum S, take IFFT
X=g1.*sqrt(S);
X=[zeros(1,round((M-N)/2)), X, zeros(1,round((M-N)/2))];
x=abs(ifft(X,M));
Y=g2.*sqrt(S);
Y=[zeros(1,round((M-N)/2)), Y, zeros(1,round((M-N)/2))];
y=abs(ifft(Y,M));
% Find the resulting Rayleigh faded envelope
z=x+j*y;
r=abs(z);
t=0:Ts:T-Ts;
plot(t,10*log10(r/mean(r)),'r')
xlabel('Time(sec)')
ylabel('Envelope (dB)')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The above code generates a Rayleigh random sequence with samples spaced at 0.1302 usec. This corresponds to a sampling frequency of 7.68 MHz which is the standard sampling frequency for a bandwidth of 5MHz. Similarly, the sampling rate for 10 MHz and 20 MHz is 15.36 MHz and 30.72 MHz respectively. The Doppler frequency can also be changed according to the scenario. LTE standard defines 3 channel models EPA, EVA and ETU with Doppler frequencies of 5 Hz, 70 Hz and 300 Hz respectively. These are also known as Low Doppler, Medium Doppler and High Doppler respectively.

The above code generated Rayleigh sequences of varying lengths for the three cases. But in all the cases it is in excess of 10 msec and can be used as the fading sequence for an LTE frame. Just to recall an LTE frame is of 10 msec duration with 20 time slots of 0.5 msec each. If each slot contains 7 OFDM symbols the total length of a fading sequence is 140 symbols.

It is seen that the fluctuation in the channel increases with Doppler frequency. The channel is almost static for a Doppler frequency of 5 Hz and varies quite rapidly for a Doppler frequency of 300 Hz. It is also shown above that the envelope of z is Rayleigh distributed and phase of z is Uniformly distributed. However, the range of phase is from 0 to pi/2. This needs to be further investigated. The level crossing rate and average fade duration can also be measured.

This is the process for generating one Rayleigh distributed channel tap. This step would have to be repeated for the number of taps in the channel model which could be 7 or 9 for the LTE channel models. A Ricean distributed channel tap can be generated in a similar fashion. MIMO channel taps can also be generated using the above described method, however, we would need to understand the concept of antenna correlation before we do that.

Level Crossing Rate and Average Fade Duration

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. 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. The LCR and AFD are interconnected and the product of these two quantities is a constant. The program given below calculates the LCR and AFD of the above generated envelope r.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% PROGRAM TO CALCULATE THE LEVEL CROSSING RATE AND AVERAGE FADE DURATION
% Rth: Level to calculate the LCR and AFD
% Rrms: RMS level of the signal r
% rho: Ratio of defined threshold and RMS level
% Copyright RAYmaps (www.raymaps.com)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Rth=0.30;
Rrms=sqrt(mean(r.^2));
rho=Rth/Rrms;
count1=0;
count2=0;
r=r/mean(r);
for n=1:length(r)-1
if r(n)<Rth && r(n+1)>Rth
count1=count1+1;
end
if r(n) < Rth
count2=count2+1;
end
end
LCR=count1/(T)
AFD=((count2*Ts)/T)/LCR
LCR_num=sqrt(2*pi)*fm*rho*exp(-(rho^2))
AFD_num=(exp(rho^2)-1)/(rho*fm*sqrt(2*pi))
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The program calculates both the simulated and theoretical values of LCR and AFD e.g. for a threshold level of 0.3 (-5.22 dB) the LCR and AFD values calculated for fm=70 Hz and N=32 are:

LCR simulation = 45.16

LCR theoretical = 43.41

AFD simulation = 0.0015 sec

AFD theoretical = 0.0016 sec

It can be seen that the theoretical and simulation results match quite well. This gives us confidence that are generated envelope has the desired statistical characteristics.

Note:

1. According to Wireless Communications Principles and Practice by Ted Rappaport “Perform an IFFT on the resulting frequency domain signals from the in-phase and quadrature arms to get two N-length time series, and add the squares of each signal point in time to create an N-point time series. Note that each quadrature arm should be a real signal after IFFT”. Now this point about the signal being real after IFFT is not always satisfied by the above program. The condition can be satisfied by playing around with the value of N a bit.

2. Also, we take the absolute value of both the time series after IFFT operation to make sure that we get a real valued sequence. However, taking the absolute value of both the in-phase and quadrature terms makes z fall in the first quadrant and the phase of z to vary from 0 to pi/2. A better approach might be to use the ‘real’ function instead of ‘abs’ function so that the phase can vary from 0 to 2pi.

3. A computationally efficient method of generating Rayleigh fading sequence is given here.

[1] John I. Smith, “A Computer Generated Multipath Fading Simulation for Mobile Radio”, IEEE Transactions on Vehicular Technology, vol VT-24, No. 3, August 1975.