Why does power loss computed via phased.FreeSpace appear to be non-linear with respect to frequency?

Using the Phased Array System Toolbox, I am modeling signal propagation with the phased.FMCWWaveform and phased.FreeSpace system objects. The signal is first created using the phased.FMCWaveform system object with a symmetric sweep interval and and upward sweep direction. The sweep bandwidth is set to 1 GHz.

The signal is then propagated with the phased.FreeSpace system object at an operating frequency of 76.5 GHz, and a distanced traveled by the signal being 1 meter.

Since the signal created with the phased.FMCWWaveform system object has a symmetric sweep interval, it is implied that it sweeps from f-B/2 to f+B/2, where f is the operating frequency. For the FMCWWaveform created above, the effective frequency sweep would be from 76 to 77 GHz.

It is expected that the amplitude of the propagated signal will vary linearly with frequency, as per the free space path loss formula. But, if the power loss of the signal is plotted in dB over frequency, it appears that the signal power attenuation is curved in shape. This is assuming that the frequency sweep of the propagated signal occurs over the duration of the sweep time. (i.e. at t = 0, f = 76 GHz and at t = SweepTime, f = 77 GHz)

If the free-space path loss is computed directly using the "fspl" function and plotted, as expected, the power attenuation is linear with increasing frequency.

Refer to the attached M-File, "Phased_FSPL_Example.m", for a clearer demonstration of this effect.

Why does the power loss computed using the phased.FreeSpace system object appear to be curve and not linear with frequency, whereas a similar computation using "fspl" is?

Answer 1

[MathWorks Support Team]

There are two concepts involved here which are affecting the results:

1. The phased.FreeSpace system object is a narrowband propagation model, therefore, even though the waveform occupies the sweep bandwidth, it is still considered as a narrowband signal compared to the carrier. The loss is computed against the carrier frequency for the entire signal. So, the loss should actually be constant over the entire duration of the signal, as it is calculated only against the carrier/operating frequency.

2. The reason why the result for the FreeSpace propagation model is not linear is due to the fact that the delayed signal does not start on an exact sampled time instant. The resulting signal is passed through a fractional delay filter causing some distortion in the signal, which in turn presents the nonlinear, curved behavior in power that is seen when plotted.

To show this effect in the attached code, try replacing the "Range" variable assignment on line 30 with

Range = 7*physconst('lightspeed')/(2*OverSampling*SweepBandwidth);

If the script is run with this variable assignment, the signal is sampled without distortion and the loss is a constant for the entire duration of the signal. The apparent non-linear behavior seen previously is actually a limitation and an artifact of the sampling setting. This effect can be mitigated by increasing the oversampling ratio, which should flatten out the power loss curve.

By implementing the "Range" assignment above in the attached code, the power loss computed with the FreeSpace method is actually shown to be constant, with its value computed based on the setting for 'OperatingFrequency'.

In order to prove that the power loss computed by the FreeSpace method does change linearly with operating frequency, the 'OperatingFrequency' setting must be updated for each frequency. This can be done by placing the signal propagation calculation in a FOR loop while updating the operating frequency used by the FreeSpace system object.