Sign on results of fft

Question

I am attempting to calculate the MTF from a test target. I calculate the spread function easily enough, but the FFT results do not quite make sense to me. To summarize,the values seem to alternate giving me a reflection of what I would expect. To test, I used a simple square wave and numpy:

from numpy import fft

data = []
for x in range (0, 20):
    data.append(0)

data[9] = 10
data[10] = 10
data[11] = 10

dataFFT = fft.fft(data)

The results look correct, with the exception of the sign... I am seeing the following for the first 4 values as an example:

30.00000000 +0.00000000e+00j

-29.02113033 +7.10542736e-15j

26.18033989 -1.24344979e-14j

-21.75570505 +1.24344979e-14j

So my question is why positive->negative->positive->negative in the real plane? This is not what I would expect... It I plot it, it almost appears that the correct function is mirrored around the x axis.

Note: I was expecting the following as an example:

This is what I am getting:

Answer 1

Your pulse is symmetric and positioned in the center of your FFT window (around N/2). Symmetric real data corresponds to only the cosine or "real" components of an FFT result. Note that the cosine function alternates between being -1 and 1 at the center of the FFT window, depending on the frequency bin index (representing cosine periods per FFT width). So the correlation of these FFT basis functions with a positive going pulse will also alternate as long as the pulse is narrower than half the cosine period.

If you want the largest FFT coefficients to be mostly positive, try centering your narrow rectangular pulse around time 0 (or circularly, time N), where the cosine function is always 1 for any frequency.

Answer 2

It works if you shift the data around 0 instead of half your array, with:

dataFFT = fft.fft(np.fftshift(data))

Answer 3

This isn't all that unexpected. If you want to check against conventional plots, make sure you convert that info to magnitude and phase before coming to any conclusions.

I did a quick check using your code and numpy.abs for mag, numpy,angle for phase. It sure looks like a sinc() function to me, which is what would be expected if the time-domain is a square pulse. If you do this, you'll find a pretty wide sinc, as would be expeceted for a short duration pulse on so few samples.

Answer 4

1. you forget to specify if your data is Real or Complex

   not everyone code in python/numpy (including me) and if you do not know this then you probably handle data to/from FFT the wrong way.

   • FFT input can be both real or complex domain
   • FFT output is complex domain

   so check the docs for your FFT implementation and specify it and also repair your data handling accordingly. Complex domain usually have first value Re and Second Im but that depends on FFT implementation/configuration.

2. signal

   here is an example of impulse response from FFT first is input Real domain signal (Im=0) single finite nonzero width pulse and second is the Re part of FFT output. The third is the Im part of FFT output. If you zoom it a bit then you will see amplitude range of y axis of each signal (on left).

   Do not forget that different FFT implementations can have different normalization constants which will change the amplitude of signal. If you want magnitude and phase convert it like this:

   mag=sqrt(Re*Re+Im*Im); // power
   ang=atanxy(Re,Im); // phase angle
   

   atanxy(dx,dy) is 4 quadrant arctan also called atan2 but be careful to get the operand order the same as your atanxy/atan2 implementation needs. Also can use mine C++ atanxy implementation

[Notes]

if your input signal is Real domain then FFT output is symmetric. Both Re and Im signals will be like:

{ a0,a1,a2,a3,...,a(n-1),a(n-1)...,a3,a2,a1,a0 }

exactly like on the image above. On the left are low frequencies and in the middle is the top frequency. If your input signal is Complex domain then the output can be anything.