Let us study a faded signal in an omnidirectional satellite receiver sent by an earth transmitter. We consider that the transmitter (TX) is sending a constant QAM symbol, i.e., the TX baseband signal is given by

(1) |

(2) |

Given this model, we are interested in the time scale during which varies, i.e., the coherence time. The coherence time depends on the Doppler shift, and the satellite's orbital speed is given by

(3) |

s | (4) |

The following figures illustrate the RX baseband signal changing in time while the satellite moves. As expected, the variance in the signal strength is larger in the Rayleigh faded case.

Here is the Octave code:

function signals = satellite_baseband_simulation() pkg load statistics close all; clear all; clear imread; clear imwrite; h = 600*1000; #Altitude of satellites. K = 0; #Rician parameter. t = 0; #Initial time. [refs bbsignals] = scatteredsignals(K); #Random baseband signals ”bbsignals' and the scatterer-obstacle's locations in 'refs'. signal = RXbaseband(refs, bbsignals); N = 200; history =[]; filename = 'basebandgif.gif'; if(exist(filename)) delete(filename); end for iii = 1 : N ##Observe the progress. if(mod(iii,10) == 0) iii end refs = rotateearth(refs); #Rotate Earth. signal = RXbaseband(refs, bbsignals); #New received baseband signal history = [history [real(signal); imag(signal)]]; #History of the signals. ##Write GIF. quiver(0,0,real(signal), imag(signal)); hold on; plot(history(1,:), history(2,:)) axis([[-0.1, 0.1], [-0.1, 0.1]]); t = t + 1/(4*8*100); #Time hop. t = 1/(8*300) is the initial coherence time of the example. text(0.03, 0.03, mat2str(t, 3), 'fontsize',25); text(0.075, 0.03, 's', 'fontsize',25); frame = getframe(); imwrite(frame.cdata, filename,'gif','writemode','append','DelayTime',0.05, 'Compression','lzw') hold off; end end ##Returns a table of 101 (1 LOS signal and 100 signals from the scattered paths) randomly phased complex baseband signal symbols each corresponding to one of a obstacle location given in 'refs'. 'K' is the Rician parameter. function [refs, signals] = scatteredsignals(K) ##First, generate the Poisson distributed random obstacle locations. refs = [0; 0]; #LOS component. yMin = 1-0.0000000001; yMax = 1; xMin = -pi; xMax = pi; xDelta = xMax - xMin; yDelta = yMax - yMin; #Rectangle dimensions numbPoints =100; #Number of points. x = xDelta*(rand(numbPoints,1)) + xMin; #Pick points from uniform distribution y = yDelta*(rand(numbPoints,1)) + yMin; refs = [refs [pi/2-asin(y)' ; x'] ]; #Map referencepoints to spherical coordinates A = 30000; #Amplitude. signals = [A*sqrt(K/(K+1)).*exp(-rand(1,1).*i*2*pi)]; signals = [signals A/(sqrt(100)*sqrt(1+K)).*exp(-rand(1,length(refs(1,:))-1).*i*2*pi)]; end ##Derives the received baseband signal. function bb = RXbaseband(refs, signals) R = 6378*1000; #Radius of Earth in m. h = 600*1000; #Altitude of the satellite. d = @(gamma) sqrt((cos(gamma).*(R+h)-R).^2+(sin(gamma).*(R+h)).^2); ##Distance to the satellites in m. c = 299792458; ##Speed of light. if(!isempty(refs)) a = 1./d(refs(1,:)); tau = d(refs(1,:))/c; else a = 0; tau = 0; end fm = 12*10^9; ##Modulation frequency. ab = a.*exp(-i*2*pi*tau.*fm); ##Received individual signals. bb = sum(ab.*signals); ##Aggregate received signal. end ##Rotates the positions in 'refs' as the satellite "moves". In this model, the satellite stays still at (0,0) but the Earth moves. function refs = rotateearth(refs) t = 1/(4*8*100); #Time hop in seconds. h= 600*1000; #Altitude of the satellite. R = 6378*1000; #Radius of Earth. GM = 3.986*10^14; #Gravitational constant. orbitalspeed = sqrt(GM/(h + R)); #Satellite speed in m/s. angularspeed = orbitalspeed/(h + R); #Angular speed of the satellite. rotation = angularspeed*t; #Rotation of Earth. eucpos = pol2euc(refs); #Transform the polar coordinates to euclidean coordinates. newpos = [[1 0 0]; [0 cos(rotation) -sin(rotation)]; [0 sin(rotation) cos(rotation)]]*eucpos; #Rotation about x-axis. refs = euc2pol(newpos); #Back to the polaroordinates. end function p = euc2pol(e) R = 6378*1000; #Radius of Earth. p = [acos(e(3,:)./R); (e(2,:) >= 0).*atan2(e(2,:),e(1,:)) + ... (e(2,:) < 0).*(atan2(e(2,:),e(1,:)) + 2*pi)]; #Polar coordinates from the given Euclidean coordinates in 'e'.. end function e = pol2euc(p) R= 6378*1000; #Radius of Earth. e = [R*cos(p(2,:)).*sin(p(1,:));... R*sin(p(2,:)).*sin(p(1,:));... R*cos(p(1,:))]; #Euclidean coordinates from the given polar coordinates in 'p'. endReferences:

- Tse, David., and Pramod. Viswanath. Fundamentals of Wireless Communication. Cambridge, U.K. ;: Cambridge University Press„ 2005.