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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 step on 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 Decibels

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]);
% Multiply range by two for two-way propagation
tgtrng = 2*tgtrng;
% Determine the wavelength for 1 GHz
lambda = physconst('LightSpeed')/1e9;
L = fspl(tgtrng,lambda)```

The free space path loss in decibels is approximately 105 dB. You can express this value as:

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

which is a direct implementation of the equation for free space path loss.

Propagation of a Linear FM Pulse Waveform to and from a Target

Construct a linear FM pulse waveform 50 ms in 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 location of [3000; 750; 20].

```hwav = phased.LinearFMWaveform('SweepBandwidth',1e5,...
'PulseWidth',5e-5,'OutputFormat','Pulses',...
'NumPulses',1,'SampleRate',1e6,'PRF',1e4);
wf = step(hwav);
hpath = phased.FreeSpace('SampleRate',1e6,...
'TwoWayPropagation',true,'OperatingFrequency',1e9);
y = step(hpath,wf,[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. Scale the time axis in microseconds.

```t = unigrid(0,1/hwav.SampleRate,1/hwav.PRF,'[)');
subplot(2,1,1)
plot(t.*1e6,abs(wf)); title('Magnitude of Transmitted Pulse');
xlabel('Microseconds'); ylabel('Magnitude');
subplot(2,1,2);
xlabel('Microseconds'); ylabel('Magnitude');```

The delay in the received pulse is approximately 14 μs, which is exactly what you expect for a distance of 4.123 km at the speed of light.

One-Way and Two-Way Propagation

The TwoWayPropagation property of the phased.FreeSpace object enables you to use the step method for one- or two-way propagation. The following example demonstrates how to use this property for a single linear FM pulse propagated to and from a target. 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 objects and calculates the range and angle from the antenna to the target.

```hwav = phased.LinearFMWaveform('SweepBandwidth',1e5,...
'PulseWidth',5e-5,'OutputFormat','Pulses',...
'NumPulses',1,'SampleRate',1e6);
hant = phased.IsotropicAntennaElement(...
'FrequencyRange',[500e6 1.5e9]);
htx = phased.Transmitter('PeakPower',1e3,'Gain',20);
hpath = phased.FreeSpace('SampleRate',1e6,...
'TwoWayPropagation',true,'OperatingFrequency',1e9);
hcol = phased.Collector('Sensor',hant,'OperatingFrequency',1e9);
sensorpos = [3000; 750; 20];
tgtpos = [1000; 250; 10];
[tgtrng,tgtang] = rangeangle(sensorpos,tgtpos);```

Because the TwoWayPropagation property is set to true, you call the step method for the phased.FreeSpace object only once. The following code calls the step after the pulse is radiated from the antenna and before the pulse is reflected from the target.

```pulse = step(hwav); % Generate pulse
pulse = step(htx,pulse); % Transmit pulse
% Propagate pulse to and from target
pulse = step(hpath,pulse,sensorpos,tgtpos,[0;0;0],[0;0;0]);
pulse = step(htgt,pulse); % Reflect pulse
sig = step(hcol,pulse,tgtang); % Collect pulse```

Alternatively, if you prefer to break up the two-way propagation into two separate calls to the step method, you can do so by setting the TwoWayPropagation property to false.

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

pulse = step(hwav); % Generate pulse
pulse = step(htx,pulse); % Transmit pulse
% Propagate pulse from the antenna to the target
pulse = step(hpath,pulse,sensorpos,tgtpos,[0;0;0],[0;0;0]);
pulse = step(htgt,pulse); % Reflect pulse
% Propagate pulse from the target to the antenna
pulse = step(hpath,pulse,tgtpos,sensorpos,[0;0;0],[0;0;0]);
sig = step(hcol,pulse,tgtang); % Collect pulse```

Propagation 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's sample rate, propagation speed, and carrier frequency. Define the signal as a sinusoid of frequency 150 Hz.

```fs = 1000;
c = 1500;
fc = 300e3;
N = 1024;
t = (0:N-1)'/fs;
x = exp(1i*2*pi*150*t);```

Assume the target is approaching the radar at 0.5 m/s, and the radar is stationary. Find the Doppler shift that corresponds to this relative speed.

```v = 0.5;
dop = speed2dop(v,c/fc)```
```dop =

100```

Create a phased.FreeSpace 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).

```hpath = 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 = step(hpath,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.

```figure;
window = 64;
ovlp = 32;
[Pxx,F] = pwelch(x,window,ovlp,N,fs);
plot(F,10*log10(Pxx));
grid;
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.

```figure;
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');```