How to choose NTC thermistor?
Obviously to me is your DCDC convert is unable to meet the inrush current and the NTC is too high impedance.

To solve the problem you need to define some design specs for step load regulation ESR and current limit based on voltage and impedance.  There may be a better bandaid NTC solution but likely inadequate. 


perhaps a reed relay bypass of NTC is best. with a closure time of 20 ms where the DCDC has recovered from the step load.


The only siren specs I could find with my english corrections are:   

Rated voltage: 12V DC
Sound index: 120dB
Sound frequency: 3.8KH2+10%
Transmit speed: 3Hz increase or decrease 10% (me) this is frequency modulated +/-10%


Siren DC resistance   


Measure it then model the impedance as a RC//R to match test results  with 20ms voltage recovery time of  22ms with USB connection.








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I presume this is a magnetic speaker style siren not piezo. 
If it were like a tweeter speaker the DC resistance would be 3/8 of the rated impedance 


or 3 Ohms DC for an 8 Ohm speaker.  

If it were like a linear motor the DC resistance would be 8 ~ 10% of resonant frequency impedance meaning like a DC motor inrush is 10 ~ 12x the operating current. 


Suggestion

If it were me , I would use a tiny 12V battery with ESR &lt; 100 mOhm charged by "UPS" with a float charge.   This gives the >>> 1 Farad needed with low ESR needed to start the siren.  A Buck then Boost DCDC will never hack it.  Not enough stored energy in a small package and too high ESR.


fix up all power and signal interfaces to suppress 5V Oscillations from Boost regulator. 


Start with a transistor active load test to simulate Alarm step load 400mA@12V= 5W . 
For example disconnect Boost regulator and test with NPN or switch and 0.5 to 2 Ohm resistor with a loop smaller than a C with your fingers. 
Use 20MHz filter because your probes will have step resonance >20MHz with a long ground lead.  
Use better EMI techniques for RF interference, Like twisted pairs only. You have an EMI problem easily fixed with experience.









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