Notices by Don Romano (alt) (thor@noagendasocial.com), page 44

Okay, so I now *think* I know how to use transfer functions, i.e. those previously so mysterious "H(s) = foo / bar" things you always find on Wikipedia instead of actual explanations when you're looking for information about audio equalisation filters.

I knew that s was the frequency, but not much else.

Solution:

s can be substituted with jω where j is the imaginary unit and ω is angular frequency.

The output format is a phasor, from which you can derive an amplitude and a phase.

Bode plots and polar patterns for simulation of 2-element array of omnidirectional microphones, with their outputs subtracted and delayed to create a single cardioid microphone, with a low-shelf filter added to linearise the frequency response.

It looks like you want to keep them really close together to move the comb filter out of the audible range. The drawback of doing this is the need for even more amplification on the low end.

Polar patterns at various frequencies for an array of 2 omnidirectional microphones, 2 cm apart, subtracted and time-delayed to behave like a cardioid pattern microphone.

Cardioids are typically bass-boosted to give them a flat frequency response. This explains why they rumble so much when handled! I haven't done that in this case, so it doesn't start flattening out until about 4500 kHz.

Screenshots are from my JavaScript microphone array simulator.

I wonder if your average cardioid pattern microphone boosts the bass frequencies to get a flat response as much as I'm doing now. A +24 dB low-shelf filter is pretty extreme. That's the same as amplifying the signal 251 times. I could make it less extreme, but that comes at the cost of the low-end response. I should take a look at the low-end response of some professional cardioid mics for reference.

Bode plots and polar patterns for simulation of 2-element array of omnidirectional microphones, with their outputs subtracted and delayed to create a single cardioid microphone, with a low-shelf filter (+24 dB @ 20 Hz) added to linearise the frequency response.

It looks like you want to keep them really close together to move the comb filter out of the audible range. The drawback of doing this is the need for even more amplification on the low end.

I'm trying to understand how to apply a H(s) transfer function to phasors in order to find the filtered phasors.

I have understood that s is the complex frequency, and can be replaced with jω, where j is the imaginary unit and ω is the angular frequency, but when confronted with a function like...

H(s) = 1 + (G f) / (s + f)

...do you just plug in jω like...

H(jω) = 1 + (G f) / (jω + f)

...and perform a complex multiplication with the phasor, and ta-da, you have the response?

Something took a walk on this car hood.

@monerica Which, I suspect, is exactly what they do for large diaphragm mics. I'd then have a rather small and cheap digital microphone array that would behave like one of those. The mics I'm looking at are only 10 dB worse than a Neumann U87 in terms of their SNR and are quite flat below 10 kHz, and could be made pretty flat above 10 khz too with further filtering.

@monerica My current mic matrix has no filtering, just time delays. The natural response of a figure-eight microphone falls at lower frequencies. Mine does the same. I have turned it into a cardioid with a phase flip and a fixed time delay, but it still falls off. I suppose I could make the inverted/delayed mic fall off at a dB/Hz slope that matches that, and then you'd have a mic that's fairly flat across the spectrum, at the cost of LF dirctionality.

@monerica If you gradually gain down the 2nd (rear) microphone of the array as you reach lower frequencies, you'd get a warmer sound. You'd then have a microphone that has a cardioid pattern at higher frequencies and gradually gets omnidirectional at lower frequencies.

@monerica The frequency response of a large diaphragm microphone could probably be emulated. As far as the literature tells me, the reason they sound warm is that they become more omnidirectional at lower frequencies. It sounds to me like you could emulate this with a bit of filtering.

@monerica Well, at least in the case of a free space diaphragm, not in the case of an enclosed diaphragm, which is what I'm modeling.

@monerica More specifically, I'm attempting to model MEMS microphones the size of a grain of rice. A bigger diaphragm would probably have the effect of delaying the sound wave more before it pushes against the opposite side of the diaphragm.

@monerica In this case, the mics themselves are modeled as points, not diaphragms. Most omni mics have enclosed diaphragms with a small aperture (hole) that essentially make them measure the sound pressure at the point in space where the aperture is, without caring about where the sound wave is coming from.

Mom thinks cardioid microphone polar patterns look more like butts than hearts. Maybe we should've called them butt patterns.

Polar patterns at various frequencies for an array of 2 omnidirectional microphones, 2 cm apart, subtracted and time-delayed to behave like a cardioid pattern microphone.

Cardioids are typically bass-boosted to give them a flat frequency response. This explains why they rumble so much when handled! I haven't done that in this case, so it doesn't start flattening out until about 4500 kHz.

Screenshots are from my JavaScript microphone array simulator.

I wrote a little microphone array simulator in JavaScript.

I wanted to create an array with a cardioid response. One text I found said I should get that if I combine omni and figure-eight microphones, but that didn't work.

Another text said you needed to make a figure-eight microphone with a delayed rear port.

This is the 20 - 20000 Hz polar pattern of a two-microphone array, with the elements spaced 2 cm apart, with one element inverted and time-delayed by 2 cm.

I wrote a little microphone array simulator in JavaScript.

I wanted to create an array with a cardioid response. One text I found said I should get that if I combine omni and figure-eight microphones, but that didn't work.

Another text said you needed to make a figure-eight microphone with a delayed rear port.

This is the 20 - 20000 Hz polar pattern of a two-microphone array, with the elements spaced 2 cm apart, with one element inverted and delayed by 58.3 us.

Math equations as you typically find them in texts are a lot like optimised code. Like optimised code, they evaluate faster. Like optimised code, they're less intuitive and pedagogical than a naive implementation.

For example, the matrix for rotating a 3D vector around the origin makes a lot more sense if you write it out as a set of three equations, one per axis, because that way, you can actually work out how the sines and cosines act on each other and the point as the rotation angles change.