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The doppler effect exists regardless of whether we have a conventional or doppler VOR but why are we not interested in the doppler effect of the aircraft?

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Short answer

The Doppler effect is an apparent change in a signal frequency. When the source is moving relatively to the observer the distance between the source and the observer changes, thus the signal reaches the observer with a phase constantly changing. For the observer, this change is undistinguishable from a frequency change.

The apparent frequency shift value is given by $fv/c$, where $f$ is the signal frequency and $v$ the relative velocity. VOR have a carrier frequency around 110 MHz. For an aircraft approaching a VOR at a groundspeed of 500 kt, this gives a significant shift of 100 Hz. For a Doppler VOR this effect increases or decreases the station own Doppler shift by 20%.

A VOR station provides eventually two pieces of information to the receiver under the form of two 30 Hz sinusoids: Reference and variable signals. The variable signal sinusoid appears phase-shifted relatively to the reference according to the receiver bearing. At a given position and time, if the receiver gets phase values of 150° for the reference and 30° for the variable signal, then it's current bearing from the VOR is 150-30=120°. This is true both for conventional and Doppler VOR.

Thus your question boils down to whether the aircraft-induced Doppler shift affects the phase of the reference and the phase of the variable signal equally , or differently. The answer is the effect is the same on both signals, thus the difference is unaffected, for the conventional VOR as well as for the Doppler VOR.


Variable signal construction

From the two sinusoids used by the receiver, the reference is sent by the VOR station, the variable signal is the result of a collaborative work between the VOR station and the receiver. The receiver is the same for all VOR, the way the variable signal is constructed varies between the conventional VOR (CVOR) and the Doppler VOR (DVOR). Thus the aircraft-induced Doppler shift doesn't affect the variable signal the same way in both families.

CVOR

In the CVOR, the station radiates two high frequency signals: The carrier modulated by a 30 Hz reference sinusoidal signal using an omnidirectional pattern, and the same unmodulated carrier using a directional pattern. The receiver perceives this directed unmodulated signal with a variable strength, its amplitude varying according to a sinusoid at the rate of rotation of the pattern, also 30 Hz.

From the receiver standpoint, it takes the time of a full round (clockwise) for the phase of this variable signal to varies from 0° to 360°. When the beam is pointing in the direction of the receiver, the receiver sees this phase as 90°, because from a strength standpoint the curve is reaching the top of the ascending portion of the sinusoid.

The phase of the reference varies at the same rate and is synchronized so that its value is 90° when the rotating beam faces north. Thus if the receiver is on a north bearing, the difference is always 0°, if the receiver is on an east bearing, the difference is always 90°, etc.

The Doppler shift from aircraft motion affects omnidirectional and directional high-frequency signals the same way, introducing the same phase change in the two low frequency signals, thus this has no effect on bearing calculation.

DVOR principle

In the DVOR, the station radiates three high frequency signals: The carrier modulated by the reference from a fixed antenna and according to an omnidirectional pattern, and two sidebands, representing the variable signal, using two moving antennas and according to an omnidirectional pattern too. This design is a bit more complex and cumbersome.

Tens of antennas are set along a circle with a diameter of 14 m. Antennas are selected two at a time, at opposite locations on the circle, and all pairs are used successively within 1/30 s. Thus in one second, the active pair of antennas has covered 30 rounds, like the directional pattern of the CVOR. The DVOR has no directionality, so rotating the active antennas is not to radiate in some privileged direction, but to give the signal source an apparent linear velocity of 1300 m/s.

Each antenna of the active pair radiates a different unmodulated frequency, one radiates the lower sideband, the other the upper sideband. Sidebands are normal by-products of amplitude modulation, but in this case fake sidebands are deliberately created ex-nihilo in order to simulate AM. For the receiver, fixed sidebands belong to a carrier modulated in amplitude by a constant signal which frequency is the frequency difference with the carrier.

In this case the DVOR transmits sidebands at a distance of 9.96 kHz of the carrier. This frequency corresponds to the subcarrier used in the CVOR (to transmit the reference). Scanning the array at 1300 m/s creates a Doppler effect which actually changes the apparent frequency of the sidebands, by 480 Hz for a sideband frequency of 110 MHz, adding 480 Hz to the frequency of the advancing antenna, subtracting 480 Hz for the retreating antenna. This shift varies with the direction of the antenna motion, maximum when the receiver is at right angle of the antenna pair baseline, null when aligned with the baseline. Thus both fake sidebands move either closer or away from the carrier frequency at the rate of 30 Hz created by scanning the array. For the receiver this motion of the sidebands is a 30 Hz signal modulating the 9.96 kHz subcarrier in FM (the 30 Hz signal modulates the 9.96 kHz subcarrier in FM, and the subcarrier modulates the 110 MHz carrier in AM). As the phase of the signal depends on the bearing of the receiver, this signal can be compared with the reference. The DVOR station takes care of synchronizing the circular scanning and the reference phase.

The 9.96 kHz subcarrier and the variable signal it carries are affected by the velocity of the aircraft, the phase of the variable signal is changed, but by exactly the same amount than the reference signal, their difference remains constant, the bearing calculation is not affected.

Conclusion

The velocity of the aircraft influences the phase of both the reference and the variable signal, including in the DVOR and for the same reasons than in the CVOR. As only their difference is meaningful this doesn't change the resulting bearing calculation.


If by chance you wanted to dig deeper into VOR: There is more.

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