Interpolation filter gain

If somebody could clarify why everywhere is written that the interpolation filter should have gain $L$ where $L$ is upsampling factor.

1. What this gain actually means?



2. Does it compensates for the amplitude scaling inherent to upsampling?


3. If I would hypothetically have an "interpolation" filter which is not attenuating spectral replicas (let's call it unity filter) would this filter still have gain $L$, or in other words is gain $L$ something what is by the definition within the interpolation filter and then the filter coefficients which define how well the replicas will be attenuated is something which depends on the design/quality of the filter?

Answer

1. What this gain actually means?

The factor between in- and output amplitudes for a signal in the passband

2. Does it compensates for the amplitude scaling inherent to upsampling?

Exactly!

3. If I would hypothetically have an "interpolation" filter which is not attenuating spectral replicas (let's call it unity filter) would this filter still have gain L, or in other words is gain L something what is by the definition within the interpolation filter and then the filter coefficients which define how well the replicas will be attenuated is something which depends on the design/quality of the filter?

You design a filter to have a certain gain; setting gain is easy, just multiply your filter with a scalar.