The first circuit looks to be a small speaker or headphone driver (something with a coil) so I hope it's not a limiter at the mic's normal input levels.

I don't know what it's driving, but the last 3 stages look like limiters to me. With 4 stages, and a gain of 50-100 per stage, the overall gain will be well over 120 dB. I think that most microphones have outputs well above a few microvolts.

 

Series negative feedback is used in a bootstrap circuit and parallel negative feedback is used in the OP circuit. So it can be used either way to increase or decrease impedance. My bad for getting them reversed. I stand corrected.

Actually, the bootstrapping makes the input impedance increase because it is positive feedback, not negative feedback.

 

Granted you might assume that but my old professor would want a proof.

If you do small signal analysis you will get the same result.

Full input impedance is equal to

\(Zin = \frac{re(Rc+Rf)(\beta+1)}{Rf+(Rc+re)(\beta+1)}\)

But we but usually prefer this simplify version

\(Zin = \frac{Rf}{Av+1}||(\beta+1)re\)

Where
Av - voltage gain
re - dVbe/dIe ≈ 26mV/Ic

I use in LTspice this circuit:

The Hfe ≈ 210 and Ic ≈ 1mA. So the voltage gain is equal

Av ≈ Rc/re ≈ 4.2KΩ/26Ω = 161.5V/V

And the input impedance is equal :

Zin = Rf/(Av+1) ||Hie = (1MΩ/162)||( 211 * 26Ω) =6.17KΩ||5.46KΩ = 2.89KΩ

And result show by simulation look like this

Zin = 2.86KΩ

As you can see this simply equation gives "correct" result in audio range frequency.