How to determine the filter type of a discrete frequency spectrum I have the discrete impulse response of a filter and i want to determine which kind of filter it is by using the DFT. 

The impulse response is:

h0 = [0,1,2,1,0]


which I zero-pad for a better result

N0 = 10;
h1 = [zeros(1,N0),h0, zeros(1, N0)];


Then i created the N-Point DFT of it using FFT

H1 = fft(h1);


The picture shows the plot of abs(H1)


Now i know that k1 = 0 represents the steady component and at first I thought that this is the spectrum of a band-stop filter but now that I read a little bit more about the DFT I'm not quite sure because the DFT is always mirrored for real signals.

So my question is: Do I have to use only one half of the DFT spectrum to determine the filter characteristics? (Which would mean that this is a low-pass filter?)
The $k_1$ indexes you get from the FFT correspond to frequencies 0 to $2\pi$ (actually, 1 sample short of $2\pi$). Note that frequencies in the range $\pi$ to $2\pi$ also correspond to the negative half of the specturm ($-\pi$ to 0).

You can get a nice visualisation with frequency 0 in the middle by applying fftshift to the result (Octave or Matlab assumed).

You can also check freqz, which plots a nice frequency and phase repsonse (try freqz([1 2 1], [1])). Or try plot(abs(freqz([1 2 1],[1]))) to get a linear plot rather than log.

What you are getting is a low-pass filter (gain 4 at low frequency, 0 at high frequency).

In VERY general terms: You can know beforehand it is a LPF because the impulse response is a single bump. A high-pass filter will have some sort of antisymetric shape (like [1 2 -2 -1]), and a band-pass or band-reject will have some oscillations in the impulse response.