How should I tell if the analyzed signal is real or complex?

[giorgi3092]

One of the test questions had these problems:

So, I need to tell whether the signal real or complex from its DFT magnitude and phase spectra.
DFT formula is:

How should I even think about this problem? I did some experiments in Matlab and got even more confused. Any hints?

 

[kaz1]

One of the test questions had these problems:

View attachment 173781
View attachment 173785

So, I need to tell whether the signal real or complex from its DFT magnitude and phase spectra.
DFT formula is:
View attachment 173782

How should I even think about this problem? I did some experiments in Matlab and got even more confused. Any hints?

One of the test questions had these problems:

View attachment 173781
View attachment 173785

So, I need to tell whether the signal real or complex from its DFT magnitude and phase spectra.
DFT formula is:
View attachment 173782

How should I even think about this problem? I did some experiments in Matlab and got even more confused. Any hints?

Try this, change the vector test as you like: (look how amplitude/phase show symmetry/antisymmetry)

xr=4*cos(2*pi*(0:19)*0);
xi=4*sin(2*pi*(0:19)*0);

%noise
nr = randn(1,20);
ni = randn(1,20);

test = complex(xr,xi) + complex(nr,ni);
stem(abs(fft(test)));
stem(angle(fft(test)));

 

[giorgi3092]

Thanks for the reply.

I'll answer my own question:
The function f is real-valued iff the FT of f is Hermitian (conjugate symmetric).
f is Hermitian iff:
a) real part of f is even.
b) imaginary part of f is odd.

Summary: for a function f to be real:
a) The amplitude spectrum must have even symmetry.
b) The phase spectrum must have odd symmetry.

source: https://en.wikipedia.org/wiki/Hermitian_function

Answers to the problem:
1) real
2) real
3) complex