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I have one question about the real part of a real-valued and causal system with the imaginary part of its Fourier transform given by

$$\textrm{Im}\big\{X(e^{j\omega})\big\}=3\sin(2\omega)-2\sin(3\omega)$$

How can I obtain the real part of the Fourier transform?

i have simulated this by octave here:

>> w=[1:100];
>> Im_X=sin(2.*w)+sin(3.*w);
>> plot (Im_X);

enter image description here

Is it possible to solve in by the Octave/Matlab?

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I suppose you've learned about the Hilbert transform relationship between the real part and the imaginary part of the Fourier transform of a real-valued and causal signal. So you can obtain the real part from the imaginary part using this relationship. Since the imaginary part is just composed of sinusoids, the real part can be obtained very easily (note that the Hilbert transform of a sine is a negative cosine).

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