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

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#### giorgi3092

##### Junior Member level 2
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?

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Solution

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

1) real
2) real
3) complex
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)));

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