Pressure Gradient Microphones, 2nd Order Microphones, Proximity Effect

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Pressure gradient microphones measure the difference of air pressure between the two sides of the diaphragm. Depending on the angle of incidence, sound from a source will arrive at the two sides at different times. Because it is the same diaphragm, there will be an interference. Due to the delay between the two signals, we will see a heavy combfilter , resulting in several dips in the frequency response and a loss of bass response by -6 dB per octave. Two main goals of making a good pressure gradient microphone is to move the first dip of the combfilter to the frequency range that can't be heard by human ear and to (internally) equalize the response to balance the bass loss.
The pure pressure gradient microphone shows a polar pattern of the figure eight. With an additional delay of one of the electrical signals it is possible to achieve cardioidal patterns (super card., card., wide card. etc.)

It is possible to simulate this with two omni directional (pressure) microphones, that are mounted slightly shifted. The shift makes the physical delay. And the signal of one of the microphones can be delayed, phase inverted and then mixed with the other. That way it is possible to create any directional pattern in post.

Also this virtually created pressure gradient microphone will show a proximity effect

The resulting virtual mic is one order higher than the involved mics. Two omni-directional microphones, being zero-order microphones, create a microphone with a 1st order pattern. Replacing the omnis with 1st order microphones will result in a virtual mic with a 2nd order pattern.

There is one limitation. This works only in the horizontal plane of the microphones. This construction will not help to attenuate diffuse (room) sound.

Model/Pattern of the two mics:
Shift of 2nd mic :
Delay of mic . Granularity: 0.1mm Sample delay @3MHz Sample delay @1536kHz Sample delay @768kHz Sample delay @384kHz Sample delay @192kHz Sample delay @176.4kHz Sample delay @96kHz Sample delay @88.2kHz Sample delay @48kHz Sample delay @44.1kHz
Distance of sound :
Presets	Wide Cardioid Cardioid Super Cardioid Fig. 8 2nd Order Cardioid A 2nd Order Cardioid B 2nd Order Super Cardioid A 2nd Order Super Cardioid B

Signal has to be amplified by .