I'm trying to understand how poor I am in analog design, to improve, and I wanted to make a sample project, all the way (on paper), of a transimpedance amplifier, which takes the input signal from a photodiode illuminated by the 'external. The specifications I want to respect are:
1. power of the incident radiation, when there is a signal of -30 dBm,
2. diode responsiveness = 0.8 A / W,
3. peak output voltage of at least 5 mV,
4. 50 Ohm output impedance,
5. input impedance < 50 Ohm,
6. band > 100 MHz.
The mental path I take immediately is: I need a transimpedance that earns at least 5mV/(0.8A/W*1uW) = 6250

, it must accept a current signal (so I will use a common base at the beginning to obtain a low Z_in which I can then convert to voltage and amplify (therefore a common emitter stage also). I also think of a possible cascode on the common emitter in the case that, after having feedbacked it, the gain and bandwidth are not sufficient to cover the specifications. Finally, being the required output impedance low enough, I think a common collector buffer would be useful.
Hoping I didn't say stupid things and I didn't miss fundamental considerations, I could start thinking about a circuit for the small signal, which I will see later on how to polarize well (*):

I'd like to understand if I have made heavy mistakes so far that I don't notice, so I can correct myself and try to continue the project.
The feedback that I thought of using, at a first cause-effect-cause-effect analysis, is actually a negative feedback.
I would also like to know how to make a few more considerations on the possible choice of pnp transistors instead of npn, which I have not really considered now.
Please don't give me ready circuits, I would like to get there on my own, albeit slowly.
Thanks a lot to anyone who wants to help me.
(*) surely the polarization will insert components that will then influence the signal circuit again, but in any case starting from the latter and then worrying about polarization seems to me the best way to proceed, at least I hope.
Thank you.
PS: is there a way to insert formulas (latex)?