DFT frequency resolution exercise I have a discrete signal that goes as follows:
$$x[n]=[-1,4,-1,0]$$
I have already done the DFT for the signal, with the following result:
$$X[0] = 2, X[1]=-4i,X[2]=-6,X[3]=4i$$
But for some reason, I seem to have trouble getting the frequency resolution for these samples(?).
My sampling interval (T) is 0.03125 seconds, and with some logical deducting I came up with these following frequencies:
$$f_0=0,f_1=8,f_2=16,f_3=8$$
However, I do not understand how I came to this conclusion, but it seems correct. I tried to use the following formula:
$$\Delta f = \frac{1}{T_{0}} = \frac{1}{NT}$$
Could someone explain how I ended up with these frequencies?
Carry the units and it becomes understandable.

T =  0.03125 seconds per sample

Thus your sampling rate is

1/T = 32 samples per second

Each bin corresponds to the cycles per frame.  Your frame has four samples.  Suppose the bin index is k.

k (cycles per frame) * 32 (samples per second) / 4 (samples per frame) = 8k cycles per second = 8k Hz

So the frequencies for your bins, in Hz, become


k   f_k
0   0
1   8 
2   16
3   24


However, the DFT wraps around, and choice of the principal frequency among many aliases is a matter of convention.  It is mathematically sounder to consider the upper half of the DFT as negative frequencies.  For real valued signals, the upper half is just the complex conjugate of the lower half and thus provides no new information so it is usually ignored.  So, if you want to consider f_3 as 24 or -8, it doesn't really matter.  For a real valued signal, a negative frequency is the same as a positive frequency.