Category Archives: Channel Modeling

Rayleigh fading channel and its various implementations.

LTE Multipath Channel Models

Channel Modeling, LTE, WiMAX, , , ,

When a wireless signal travels from a transmitter to a receiver it follows multiple paths. The signal may travel directly following the line of sight between the transmitter and receiver, it may bounce off the ground and reach the receiver or it may be reflected by multiple buildings on the way to the receiver. When these copies of the same signal arrive at the receiver they are delayed and attenuated based upon the path length that they have followed and various other factors.

A well known technique to model such a wireless channel is to model it as an FIR (Finite Impulse Response) filter. The wireless channel thus performs the convolution operation on the transmitted signal. The multipath profile of three well known LTE channel models is shown below.

LTE Channel Models
LTE Channel Models

The channel profile quantifies the delays and relative powers of the multipath components. It can be observed that the EPA model has 7 multipath components whereas the EVA and ETU models have 9 multipath components each. However there is a small caveat here. The multipath components described in the above table are not uniformly spaced in the time domain. So if an FIR filter has to perform convolution operation on a signal uniformly sampled at 100 MHz (Ts=10 nsec) the number of filter taps would be much larger. To be exact the FIR filters corresponding to the above channel models would have 42, 252 and 501 filter taps respectively. Most of these taps would have no power so the FIR filter can be efficiently implemented in hardware.

Also, if the channel is time-varying as most wireless channels are, each filter tap can be modeled to have a Rayleigh or Ricean distribution with a mean value described in the table above. Lastly, the variation in the value of a channel tap from one sample to the next depends upon the Doppler frequency which in turn depends upon the speed of the mobile unit. Higher the velocity of a mobile unit higher would be the Doppler frequency and greater would be the variations in the channel. The Doppler frequency is defined as:

fd=v*cos(theta)/lambda

where

‘fd’ is the Doppler Frequency
‘v’ is the receiver velocity
‘lambda’ is the wavelength
and ‘theta’ is the angle between the direction of arrival of the signal and the direction of motion

The exact method of generating Rayleigh distributed channel co-efficients with the desired temporal correlation requires some more explanation and would be the subject of a future post.

Note:
1. The above multipath channel models have a maximum delay of 410 nsec, 2510 nsec and 5000 nsec which is well within the range of a long cyclic prefix of length 16.67 usec (16670 nsec). So the Intersymbol Interference (ISI) would not adversely effect the system performance.
2. If you are doing simulation of a system that operates on individual symbols then temporal correlation between channel co-efficients is not that important. But if the system operates on blocks of symbols or bits (as an interleaver or convolutional encoder does) then temporal correlation plays an important part in determining the system performance.

WiMAX Path Loss Calculation

Channel Modeling, WiMAX, ,

Calculation of the path loss is fundamental to Wireless System Design. There are many models available for calculating the path loss such as Okumura Model, Hata Model, COST-231 Model and more recently the SUI (Stanford University Interim) Model. The SUI Model has been specifically proposed for Broadband Wireless Access Systems such as WiMAX. It defines three types of environments namely A, B and C which are equivalent to the urban, suburban and rural environments defined in the earlier models. According to this model the path loss can be calculated as:

PL=A+10*n*log10(d/do)+Xf+Xh+s

where

n=a-(b*hb)+(c/hb)
A=20*log10(4*pi*do/lambda)
Xf=6.0*log10(f/2000)
Xh=-10.8*log10(hr/2) for A&B
Xh=-20.0*log10(hr/2) for C

and

frequency of operation = f = >2000MHz
transmit receive separation = d = 100 m to 8000 m
reference distance = do = 100 m
base station antenna height = hb = 10 m to 80 m
receive antenna height = h = 2 m to 10 m
shadowing factor with lognormal distribution = s = 8.2 dB to 10.6 dB

The values for the parameters a,b and c for the three environment are given in the table below.

SUI Parameters
SUI Parameters

Doing a quick calculation for f=2500 MHz, hb=30 m, hr=2 m, s=8.2, do=100 m and d=1000 m gives us a path loss 137.13 dB for Type-A channel. Increasing the frequency to 3500 MHz (another WiMAX band) increases the path loss to 140.93 dB i.e. there is a 3.8 dB increase in the path loss.

So to recap the path loss given by the SUI model is composed of 5 elements:

SUI Path Loss Equation
SUI Path Loss Equation

1. The free space path loss (A) up to the reference distance of ‘do’.
2. Additional path loss for distance ‘d’ with path loss exponent ‘n’.
3. Additional path loss (Xf) for frequencies above 2000 MHz.
4. Path gain (Xh) for receive antenna heights greater than 2 m.
5. Shadowing factor (s).

Simulating a SISO Ring Model

Channel Modeling,

We simulate the SISO Ring Model described previously by varying the transmit receive separation from 50m to 500m. Keeping the ring radius fixed at 20m the angular spread of the channel decreases as the receiver moves away from the transmitter.

Ring Model

It is observed that the power level of the received signal fluctuates as the distance ‘d’ is varied. However, after a certain critical distance (around 250 m) the power vs distance curve approaches a straight line.

Simulating a SISO Ring Model

Channel Modeling,

A Ring Model is a well known spatial channel model. It models the propagation channel as an unobstructed transmitter and a receiver surrounded by a ring of reflectors. The distance between the transmitter and receiver is usually much larger than the radius of the ring. The reflectors are distributed uniformly around the ring. This model is useful for modeling a scenario where a base station is located at sufficient altitude and is unobstructed whereas the mobile station is at ground level and is surrounded by a bunch of reflectors.

Ring Model

Given below is the MATLAB code for a SISO Ring Model consisting of eight reflectors distributed uniformly around the ring.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [Et]=ring_mod0(d,r,B)
Et=0;
for phi=0:pi/4:(2*pi)-(pi/4);
R=sqrt((d-r*cos(phi))^2+(r*sin(phi))^2);
E=exp(-i*B*(r+R))/(r+R);
Et=Et+E;
end
return
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Simulating a MIMO Ring Model

Channel Modeling,
A Ring Model is a well known spatial channel model. It models the propagation channel as an unobstructed transmitter and a receiver surrounded by a ring of reflectors. The distance between the transmitter and receiver is usually much larger than the radius of the ring. The reflectors are distributed uniformly around the ring. This model is useful for modeling a scenario where a base station is located at sufficient altitude and is unobstructed whereas the mobile station is at ground level and is surrounded by a bunch of reflectors.
Ring Model

Given below is the MATLAB code that calculates the composite signal (from nth Tx element to mth Rx element) at the receiver for a MIMO channel composed of eight reflectors.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [Et]=ring_mod2(d,r,psi,l,B,alpha,a)
Et=0;
for phi=0:pi/4:(2*pi)-(pi/4);
R_=sqrt((d-l*cos(psi)-r*cos(phi))^2+(r*sin(phi)-l*sin(psi))^2);
r_=sqrt((r*cos(phi)-a*cos(alpha))^2+(r*sin(phi)-a*sin(alpha))^2);
E=exp(-i*B(r_+R_))/(r_+R_);
Et=Et+E;
end
return
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%