I would like to clarify some confusion I have about linear phase FIR filters of which do not seem to have symmetric impulse responses.
Starting with a simple case, a delay, $ h[n] =\delta(n-n_0) $ does not seem to have have a symmetric impulse response. For example,
$$h[n] = [0, 0 ,0,1,0] $$ is not a symmetric impulse response. How is it that the property of symmetry for a linear phase FIR filter holds true, in this case?
For another example, say I have a filter defined as
$$h[n] = [0,0,0,0,1,0,0,0,0,0,1]$$
I believe this filter is linear phase too. But it's impulse response is not symmetric.
How does the symmetric property hold true? Does it have to do with zero padding? But even if we don't zeropad the filter, it is still linear phase? If we zeropad it by a large factor, is it also linear phase?
For one more example, say I have this filter:
$$ h[n] = [1, 2, 3, 4, 3, 2, 1]$$
This is obviously linear phase. When I zeropad it by a factor N, the impulse response is no longer symmetric, but it stays linear phase for all values of N. Why is this the case?