# Parallel LTI system

Imagine I know that a system comprises two LTI subsystems $$H_a$$ and $$H_b$$ connected in parallel. Also, suppose that I can measure the impulse response of the whole system, $$H_a \| H_b$$, and the impulse response of $$H_a$$. Is there any way to find the impulse response of $$H_b$$?

I may be tired, but I think is not possible.

• Yes you can calculate it! It all comes from the LTI property too. – Engineer Nov 19 '19 at 15:12
• thank you Engineer!, I don’t think you can! You only know the composite response and that of one of the subsystems, how can you use that info to find out the other? You maybe right, but I don’t see how and that’s why I asked... – julovi Nov 19 '19 at 15:16
• could you elaborate on “connected in parallel”. A resistor and capacitor are ideally LTI components and can be connected in parallel but that usage relates to circuit topology. One can also consider that parallel can describe where the composite impulse response is a sum of component impulse responses – user28715 Nov 19 '19 at 15:32
• Thanks Stanley, I’m not measuring electric components but acoustic IRS... – julovi Nov 19 '19 at 15:46
• @julovi there are acoustic analogs to circuit components. – user28715 Nov 19 '19 at 15:49

For a parallel system, the individual subsystems just add, so you simply have $$H_{||} = H_A + H_B$$ . That goes for both the transfer function and the impulse response, so you can simply do $$h_b = h_{||} - h_a$$