How to properly take fft in MATLAB plus take real and imaginary components?

I wish to calculate the real and imaginary parts of the fft of a function f(t) that I can determine with arbitary time spacing. How do I do this, and how can I adjust the sampling frequency of the resulting fft? When I change dt for the time vector, I no longer sample that function as precisely, but then df for the fft (I think) is more finely spaced. I guess I am misunderstanding something. Also, when I take the absolute value to obtain the one-sided fft, I no longer have any imaginary component. I think I misunderstanding some basic aspects of the fft that I hope somebody can walk me through.

t = linspace(0.01,100,10000); % arbitrary time spacing here
nfft = 2^nextpow2(numel(t)); % not sure why this must be the case, but I've seen this in examples
dt = t(2) - t(1); % linear spacing
df = 1/dt;
Freq = (df/2)*linspace(0,1,nfft/2+1); % I guess df/2 is the Nyquist criterion as is the nfft/2 ?
f_w = abs(fft(f_t,nfft)); % Take one-sided of f_t
A = real(f_w);
B = imag(f_w); % doesn't return anything

Answer 1

[KSSV]

You have used abs at the fft, how do you expect aa complex out put? You should not take abs if you want the complex output.

t = linspace(0.01,100,10000); % arbitrary time spacing here
nfft = 2^nextpow2(numel(t)); % not sure why this must be the case, but I've seen this in examples
dt = t(2) - t(1); % linear spacing
df = 1/dt;
Freq = (df/2)*linspace(0,1,nfft/2+1); % I guess df/2 is the Nyquist criterion as is the nfft/2 ?
f_w = fft(f_t,nfft); % Take one-sided of f_t
A = real(f_w);
B = imag(f_w); % doesn't return anything

[L'O.G.]

Shouldn't I take the one-sided fft here? If so, how?

[KSSV]

You may take the one-sided fft, but to have real and complex parts; you need not to do abs.

Answer 2

[Walter Roberson]

Arbitrary time values means your times might not be uniform intervals. You should use nufft

https://www.mathworks.com/help/matlab/ref/double.nufft.html

[L'O.G.]

By arbitrary, I just meant that I can calculate the function f(t) with any time values that I want.