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Pointing a microphone to a connected loudspeaker creates a feedback path and usually results in acoustic resonance, which manifests in a sustained oscillation: https://youtu.be/_XTtjZ8aZbc. A similar effect can be observed with guitar feedback, where the guitar coils pick up the sound coming out of the amplifier.

This survey paper provides a great insight into the phenomenon and its mitigation approaches: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.231.9808. It refers to the Nyquist stability criterion on p.5 (292) and explains the effect from the usual closed-loop system stability analysis prespective.

My question is: what are actual the physical factors that define the frequency of resonant oscillation in microphone feedback?

My inexperienced guesses include:

  • frequency responses and resonant frequencies of both the microphone and the loudspeaker, which in turn depend on their internal circuitry
  • distance between microphone and loudspeaker

P.S. Please let me know if my question can be reformulated in a better way using the correct terminology. I am a non-English EE undergrad without formal control theory education (yet) and I form my assumptions based on my personal research in this exciting field.

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    $\begingroup$ I wrote an article on the subject a few years ago. I’ll try to dig that up somehow. $\endgroup$
    – Jdip
    Aug 8, 2023 at 14:13

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Let's look at a simple block diagram

enter image description here

The microphone receives the input sound but also reproduced sound from the loudspeaker. There are two transfer functions in place. One, $A(\omega)$ from the microphone to the loudspeaker which consists of pre-amp, mixer, effects, amplification, etc. The second one, $R(\omega)$ is from the loudspeaker back to the microphone. This is a room transfer function and most factors are acoustic: room modes, reverb, reflections, directivities of loudspeaker and microphone, etc.

The product of those $L(\omega) = A(\omega) \cdot R(\omega)$ is called the open loop gain. If the open loop gain at any frequency is larger than 1, than you get feedback simply because you ended up with more sound than you started with and any go around through the loop will amplify the sound further until something clips or tops out.

Mathematically the closed loop transfer function is

$$H(\omega) = \frac{1}{1+A(\omega) \cdot R(\omega)} = \frac{1}{1+ L(\omega)}$$

When the Loop transfer function becomes -1, the denominator becomes zeros and you have "divide by zero" problem which is exactly what feedback is.

Technically the criteria for feedback includes both an amplitude and a phase condition. However, if a room transfer function is involved, the phase rotates very quickly with frequency and you will always get feedback at the peak frequency of the open loop transfer function.

The forward transfer function $A(\omega)$ is typically pretty straight forward and well behaved. However the room transfer function is very complicated. The room has 1000s of resonance which can be very close together in frequency. They change a lot when you move and the transfer function depends a lot on how the directivities of speaker and microphone couples in the room modes (node vs. antinodes).

Both microphone and speaker can also have pronounced peaks and dips on their frequency response.

what are actual the physical factors that define the frequency of resonant oscillation in microphone feedback?

  1. Room acoustics, shape, materials, etc
  2. Location of mic and speaker in the room
  3. Directivity and orientation of mic and speaker
  4. Acoustic transfer functions of mic and speaker
  5. Blocking, absorbing, and diffracting objects in the room
  6. Mechanical vibrations and resonances: stage floor, microphone stand, etc,
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  • $\begingroup$ Thank you for your detailed response @Hilmar! The thing is that this effect is also present soundproof rooms and on volumes where acoustic reflections are negligible. In this case, I feel like the room's characteristic transfer function doesn't play a significant role in the phenomenon. I would like to know more about how it is related to the distance between the mic and the loudspeaker, if it is at all. $\endgroup$
    – Theodote
    Aug 8, 2023 at 20:42
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    $\begingroup$ @Theodote the room response does play a significant role! Hilmar's math explains every linear feedback system you can model, so even the scenario you describe in your comment. Even if your channel was perfectly "flat" in spectrum (which it isn't, every speaker and every microphone are frequency-selective), then you'd still incur a phase delay, which fully defines the feedback phenomena you're getting. Remember how the closed-loop transfer function becomes -1 for division by zero, i.e. feedback? exactly! -1, and inversion, is mathematically a pure phase shift by 180°. $\endgroup$ Aug 8, 2023 at 20:52
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    $\begingroup$ You get that in your anechoic chamber when the distance between microphone and speaker (including the phase shift that all the electronics introduce) amounts to half a wavelength. Wavelength is inversely proportional to frequency, so you get exactly one frequency that has a phase shift of 180°. So, yes, Hilmar's model fits your scenario perfectly. $\endgroup$ Aug 8, 2023 at 20:55
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    $\begingroup$ @Theodote: rooms where acoustic reflections are negligible are very rare owing to the fact that they are extremely expensive (don't ask how I know :-) ). Even in a very dry room, you still have the directivity of the microphone and the speaker. In fact, one of the main remedies is to use a cardioid microphone and orient it so that the null points towards the speakers. Distance does indeed play a role and in general further is better but the exact relationship is quite complicated and depends on the type of speaker (cylindrical is better than spherical) $\endgroup$
    – Hilmar
    Aug 8, 2023 at 21:54
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    $\begingroup$ Re 2) yep! And that's also why the technician at a venue adjusts the filter curve of the processing happening between mic sand speaker when they combat the occurrence of feedback. $\endgroup$ Aug 9, 2023 at 8:46

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