# Linear phase FIR filter for impulse responses that don't appear symmetric

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?

• All your example filters have (generalized) linear phase, since all have a symmetrical impulse response. How are you calculating group delay (or phase response)? Zero padding should not affect the frequency response. – Juancho Oct 14 '19 at 18:09
• All of your examples are symmetric around some point in $n$; they're just not necessarily symmetric around the center point of the vector. – TimWescott Oct 14 '19 at 18:51

If you consider the definition of linear phase FIR filter and the associated symmetry conditions on their impulse responses, then you can arrive the conclusion that the first two cases

$$h_1[n] = [0,0,0,1,0]$$

and

$$h_2[n] = [0,0,0,0,1,0,0,0,0,0,1]$$

are non-symmetric. However, as you use zeros and ones in those impulse responses, it can be seen that the following two new augmented impulse responses are equivalent to those non-symmetric looking ones:

$$h_3[n] = [0,0,0,1,0,0,0]$$

and

$$h_4[n] = [0,0,0,0,1,0,0,0,0,0,1,0,0,0,0]$$

And they are symmetric and linear phase according to the definition as well. This happens because of the zeros and use of ones in a specific way.

However, the following impulse response

$$h_5[n] = [0,0,1,0,0,0,0,1,0,1,0]$$

is not symmetric, is not linear phase and cannot be made linear phase by augmenting with zeros...