Free Space Path Loss

Support for Modeling Propagation in Free Space

Propagation environments have significant effects on the amplitude, phase, and shape of propagating space-time wavefields. In some cases, you may want to simulate a system that propagates narrowband signals through free space. If so, you can use the phased.FreeSpace System object™ to model the range-dependent time delay, phase shift, Doppler shift, and gain effects.

Consider this object as a point-to-point propagation channel. By setting object properties, you can customize certain characteristics of the environment and the signals propagating through it, including:

• Propagation speed and sampling rate of the signal you are propagating

• Signal carrier frequency

• Whether the object models one-way or two-way propagation

Each time you call a phased.FreeSpace object, you specify not only the signal to propagate, but also the location and velocity of the signal origin and destination.

You can use fspl to determine the free space path loss, in decibels, for a given distance and wavelength.

Free Space Path Loss in dB

Assume a transmitter is located at (1000,250,10) in the global coordinate system. Assume a target is located at (3000,750,20). The transmitter operates at 1 GHz. Determine the free space path loss in decibels for a narrowband signal propagating to and from the target.

[tgtrng,~] = rangeangle([3000; 750; 20],[1000; 250; 10]);

Determine the wavelength for 1 GHz.

lambda = physconst('LightSpeed')/1e9;

Solve for the two-way loss in dB using fspl. The two-way loss is twice the one-way loss.

L = 2*fspl(tgtrng,lambda)

L = 
197.4635

The free space path loss in decibels is approximately 197.5 dB.

Alternatively, you can compute the loss directly from

Loss = 2*pow2db((4*pi*tgtrng/lambda)^2)

Loss = 
197.4635

Propagate Linear FM Pulse Waveform to Target and Back

Construct a linear FM pulse waveform of 50 ms duration with a bandwidth of 100 kHz. Model the range-dependent time delay and amplitude loss incurred during two-way propagation. The pulse propagates between the transmitter located at (1000,250,10) and a target located at (3000,750,20). The signals propagate at the speed of light.

waveform = phased.LinearFMWaveform('SweepBandwidth',1e5,...
    'PulseWidth',5e-5,'OutputFormat','Pulses',...
    'NumPulses',1,'SampleRate',1e6,'PRF',1e4);
signal = waveform();
channel = phased.FreeSpace('SampleRate',1e6,...
    'TwoWayPropagation',true,'OperatingFrequency',1e9);
y = channel(signal,[1000; 250; 10],[3000; 750; 20],[0;0;0],[0;0;0]);

Plot the magnitude of the transmitted and received pulse to show the amplitude loss and time delay.

t = unigrid(0,1/waveform.SampleRate,1/waveform.PRF,'[)');
subplot(2,1,1)
plot(t.*1e6,abs(signal))
title('Magnitude of Transmitted Pulse')
xlabel('Time (microseconds)')
ylabel('Magnitude')
subplot(2,1,2)
plot(t.*1e6,abs(y))
title('Magnitude of Received Pulse')
xlabel('Time (microseconds)')
ylabel('Magnitude')

The delay in the received pulse is approximately 14 μs, the expected value for a distance of 4.123 km.

One-Way and Two-Way Propagation

The TwoWayPropagation property of the phased.FreeSpace System object™ lets you simulate either one- or two-way propagation. The following example demonstrates how to use this property for a single linear FM pulse propagating to a target and back. The sensor is a single isotropic radiating antenna operating at 1 GHz located at (1000,250,10). The target is located at (3000,750,20) and has a nonfluctuating RCS of 1 square meter.

The following code constructs the required System objects and calculates the range and angle from the antenna to the target.

waveform = phased.LinearFMWaveform('SweepBandwidth',1e5,...
    'PulseWidth',5e-5,'OutputFormat','Pulses',...
    'NumPulses',1,'SampleRate',1e6);
antenna = phased.IsotropicAntennaElement('FrequencyRange',[500e6 1.5e9]);
transmitter = phased.Transmitter('PeakPower',1e3,'Gain',20);
radiator = phased.Radiator('Sensor',antenna,'OperatingFrequency',1e9);
channel = phased.FreeSpace('SampleRate',1e6,...
    'TwoWayPropagation',true,'OperatingFrequency',1e9);
target = phased.RadarTarget('MeanRCS',1,'Model','Nonfluctuating');
collector = phased.Collector('Sensor',antenna,'OperatingFrequency',1e9);
sensorpos = [3000;750;20];
tgtpos = [1000;250;10];
[tgtrng,tgtang] = rangeangle(sensorpos,tgtpos);

Because the TwoWayPropagation property is set to true, you compute the total propagation only once.

Compute the radiated signal.

pulse = waveform();
pulse = transmitter(pulse);
pulse = radiator(pulse,tgtang);

Propagate the pulse to the target and back.

pulse = channel(pulse,sensorpos,tgtpos,[0;0;0],[0;0;0]);
pulse = target(pulse);
sig = collector(pulse,tgtang);

Alternatively, you can break up the two-way propagation into two separate one-way propagation paths. You do so by setting the TwoWayPropagation property to false.

channel1 = phased.FreeSpace('SampleRate',1e9,...
    'TwoWayPropagation',false,'OperatingFrequency',1e6);

Radiate the pulse.

pulse = waveform();
pulse = transmitter(pulse);
pulse = radiator(pulse,tgtang);

Propagate the pulse from the antenna to the target.

pulse = channel1(pulse,sensorpos,tgtpos,[0;0;0],[0;0;0]);
pulse = target(pulse);

Propagate the reflected pulse from the target to the antenna.

pulse = channel(pulse,tgtpos,sensorpos,[0;0;0],[0;0;0]);
sig = collector(pulse,tgtang);

Propagate Signal from Stationary Radar to Moving Target

This example shows how to propagate a signal in free space from a stationary radar to a moving target.

Define the signal sample rate, propagation speed, and carrier frequency. Define the signal as a sinusoid of frequency 150 Hz. Set the sample rate to 1 kHz and the carrier frequency to 300 MHz. The propagation speed is the speed of light.

fs = 1.0e3;
c = physconst('Lightspeed');
fc = 300e3;
f = 150.0;
N = 1024;
t = (0:N-1)'/fs;
x = exp(1i*2*pi*f*t);

Assume the target is approaching the radar at 300.0 m/s, and the radar is stationary. Find the Doppler shift that corresponds to this relative speed.

v = 1000.0;
dop = speed2dop(v,c/fc)

dop = 
1.0007

From the formula, the one-way Doppler shift is 1 Hz.

Create a phased.FreeSpace System object™, and use it to propagate the signal from the radar to the target. Assume the radar is at (0, 0, 0) and the target is at (100, 0, 0).

channel = phased.FreeSpace('SampleRate',fs,...
   'PropagationSpeed',c,'OperatingFrequency',fc);
origin_pos = [0;0;0];
dest_pos = [100;0;0];
origin_vel = [0;0;0];
dest_vel = [-v;0;0];
y = channel(x,origin_pos,dest_pos,origin_vel,dest_vel);

Plot the spectrum of the transmitted signal. The peak at 150 Hz reflects the frequency of the signal.

window = 64;
ovlp = 32;
[Pxx,F] = pwelch(x,window,ovlp,N,fs);
plot(F,10*log10(Pxx))
xlabel('Frequency (Hz)')
ylabel('Power/Frequency (dB/Hz)')
title('Transmitted Signal')

Plot the spectrum of the propagated signal. The peak at 250 Hz reflects the frequency of the signal plus the Doppler shift of 100 Hz.

window = 64;
ovlp = 32;
[Pyy,F] = pwelch(y,window,ovlp,N,fs);
plot(F,10*log10(Pyy))
grid
xlabel('Frequency (Hz)')
ylabel('Power/Frequency (dB/Hz)')
title('Propagated Signal')

The Doppler shift is too small to see. Overlay the two spectra in the region of 150 Hz.

figure
idx = find(F>=130 & F<=170);
plot(F(idx),10*log10(Pxx(idx)),'b')
grid
hold on
plot(F(idx),10*log10(Pyy(idx)),'r')
hold off
xlabel('Frequency (Hz)')
ylabel('Power/Frequency (dB/Hz)')
title('Transmitted and Propagated Signals')
legend('Transmitted','Propagated')

The peak shift appears to be approximately 1 Hz.