# phased.FreeSpace System object

Package: phased

Free space environment

## Description

The phased.FreeSpace System object™ models narrowband signal propagation from one point to another in a free-space environment. The object applies range-dependent time delay, gain and phase shift to the input signal. The object accounts for doppler shift when either the source or destination is moving. A free-space environment is a boundaryless medium with a speed of signal propagation independent of position and direction. The signal propagates along a straight line from source to destination. For example, you can use this object to model the propagation of a signal from a radar to a target and back to the radar.

For non-polarized signals, the FreeSpace System object lets you propagate signals from a single point to multiple points or from multiple points to a single point. Multiple-point to multiple-point propagation is not supported.

To compute the propagated signal in free space:

1. Define and set up your free space environment. See Construction.

2. Call step to propagate the signal through a free space environment according to the properties of phased.FreeSpace. The behavior of step is specific to each object in the toolbox.

When propagating a signal in free-space to an object and back, you can either using a single FreeSpace System object to compute a two-way free space propagation delay or two FreeSpace System objects to perform one-way propagation delays in each direction. Because the free-space propagation delay is not necessarily an integer multiple of the sampling interval, it may turn out that the total round trip delay in samples when you use a two-way propagation phased.FreeSpace System object differs from the delay in samples when you use two one-way phased.FreeSpace System objects. For this reason, it is recommended that, when possible, you use a single two-way phased.FreeSpace System object.

## Construction

H = phased.FreeSpace creates a free space environment System object, H.

H = phased.FreeSpace(Name,Value) creates a free space environment object, H, with each specified property Name set to the specified Value. You can specify additional name-value pair arguments in any order as (Name1,Value1,...,NameN,ValueN).

## Properties

 PropagationSpeed Signal propagation speed Specify signal wave propagation speed in free space as a real positive scalar. Units are meters per second. Default: Speed of light OperatingFrequency Signal carrier frequency A scalar containing the carrier frequency of the narrowband signal. Units are hertz. Default: 3e8 TwoWayPropagation Perform two-way propagation Set this property to true to perform round-trip propagation between the origin and destination that you specify in the step command. Set this property to false to perform one-way propagation from the origin to the destination. Default: false SampleRate Sample rate A scalar containing the sample rate. Units of sample rate are hertz. The algorithm uses this value to determine the propagation delay in number of samples. Default: 1e6

## Methods

 clone Create free space object with same property values getNumInputs Number of expected inputs to step method getNumOutputs Number of outputs from step method isLocked Locked status for input attributes and nontunable properties release Allow property value and input characteristics changes reset Reset internal states of propagation channel step Propagate signal from one location to another

## Examples

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### Signal Propagation from Stationary Radar to Stationary Target

Calculate the amplitude of a signal propagating in free-space from a radar at (1000,0,0) to a target at (300,200,50). Assume both the radar and the target are stationary. The sample rate is 8000 Hz while the operating frequency of the radar is 300 MHz. Transmit five samples of a unit amplitude signal. The signal propagation speed takes the default value of the speed of light. Examine the amplitude of the signal at the target.

fs = 8e3; fop = 3e8; henv = phased.FreeSpace('SampleRate',fs,... 'OperatingFrequency',fop); pos1 = [1000;0;0]; pos2 = [300;200;50]; vel1 = [0;0;0]; vel2 = [0;0;0]; 

Compute the received signal at the target.

x = ones(5,1); y = step(henv,x,... pos1,... pos2,... vel1,... vel2); disp(y) 
 1.0e-03 * 0.0000 + 0.0000i 0.0129 - 0.1082i 0.0129 - 0.1082i 0.0129 - 0.1082i 0.0129 - 0.1082i 

The first sample is zero because the signal has not yet reached the target.

Manually compute the loss using the formula

R = sqrt( (pos1-pos2)'*(pos1-pos2)); lambda = fop/physconst('Lightspeed'); L = (4*pi*R/lambda)^2 
L = 8.3973e+07 

Because the transmitted amplitude is unity, the square of the signal at the target equals the inverse of the loss.

disp(1/abs(y(2))^2) 
 8.4205e+07 

### Signal Propagation from Moving Radar to Moving Target

Calculate the result of propagating a signal in free space from a radar at (1000,0,0) to a target at (300,200,50). Assume the radar moves at 10 m/s along the x-axis, while the target moves at 15 m/s along the y-axis. The sample rate is 8000 Hz while the operating frequency of the radar is 300 MHz. The signal propagation speed takes the default value of the speed of light. Transmit five samples of a unit amplitude signal and examine the amplitude of the signal at the target.

fs = 8000; fop = 3e8; sProp = phased.FreeSpace('SampleRate',fs,... 'OperatingFrequency',fop); pos1 = [1000;0;0]; pos2 = [300;200;50]; vel1 = [10;0;0]; vel2 = [0;15;0]; y = step(sProp,ones(5,1),... pos1,... pos2,... vel1,... vel2); disp(y) 
 1.0e-03 * 0.0000 + 0.0000i 0.0129 - 0.1082i 0.0117 - 0.1083i 0.0105 - 0.1085i 0.0093 - 0.1086i 

Because the transmitted amplitude is unity, the square of the signal at the target equals the inverse of the loss.

disp(1/abs(y(2))^2) 
 8.4205e+07 

## Algorithms

When the origin and destination are stationary relative to each other, the output Y of step can be written as Y(t) = x(t-τ)/L. The quantity τ is the signal delay and L is the free-space path loss. The delay τ is given by R/c, where R is the propagation distance and c is the propagation speed. The free space path loss is given by

$L=\frac{{\left(4\pi R\right)}^{2}}{{\lambda }^{2}}$

where λ is the signal wavelength.

This formula assumes that the target is in the far-field of the transmitting element or array. In the near-field, the free-space path loss formula is not valid and can result in a loss less than one, equivalent to a signal gain. For this reason, the loss is set to unity for range values, R ≤ λ/4π.

When there is relative motion between the origin and destination, the processing also introduces a frequency shift. This shift corresponds to the Doppler shift between the origin and destination. The frequency shift is v/λ for one-way propagation and 2v/λ for two-way propagation. The parameter v is the relative speed of the destination with respect to the origin.

For further details, see [2].

## References

[1] Proakis, J. Digital Communications. New York: McGraw-Hill, 2001.

[2] Skolnik, M. Introduction to Radar Systems, 3rd Ed. New York: McGraw-Hill, 2001.