# Does 46 dB gain of a filter for a frequency imply 200 times more amplitude?

Consider we have a IIR filter or order 2 and it has gain of say 46.02 dB for frequency $$f_0$$. Now if we apply a time domain signal of frequency $$f_0$$ to the filter for a long time, can we expect to get 200 times higher amplitude in the output of the filter comapre to the input of the filter? The 46.02 db and 200 comes from the fact that $$20\log_{10}(200)= 46.0206$$.

## 1 Answer

can we expect to get 200 times higher amplitude in the output of the filter comapre to the input of the filter

Yes.

For a sine wave input the amplitude of the steady state output is simply the amplitude of the input multiplied with the magnitude of the transfer function at that frequency.

• Is there a proof or equation to show this please? – Creator Feb 25 '20 at 22:52
• Proof of what exactly ? That the output of an LTI system for a sine wave is also a sine wave of the same frequency ? See en.wikipedia.org/wiki/Linear_time-invariant_system – Hilmar Feb 25 '20 at 23:19
• yes, got it, thanks. – Creator Feb 25 '20 at 23:22