# Discrete Fourier Transform in Signal Processing - Interpreting graphs of transformed signals

Given above are the real parts of the signals I to IV. Which of the following statements are correct?

(i): Signal III is the result of the discrete Fourier transform of signal I. The associated imaginary part is 0.

(ii): Signal III is the frequency spectrum of an oscillation with a frequency.

(iii): Signal IV is the frequency spectrum of an oscillation with 2 different frequencies.

(iv): Signal IV is the result of the discrete Fourier transform of Signal III. The associated imaginary part is 0.

(iii) seems be to correct. Spectrum of two frequencies.

The other choices can be verified,

(i) Fourier transform of $$f(t=0)=1$$ is 1. So it cannot be graph (iii)

(ii) Signal (iii) is sine wave and there is no spectrum associated with that.

(iv) It's Fourier transform of sine wave (iii) which is

$$f(t)=0.5\cdot \sin(2\pi Ft)$$

and it's transform is

$$\frac{\pi}{i}\left[\delta\left(\omega-2\pi F\right)-\delta\left(\omega+2\pi F\right)\right]$$

Imaginary part is not zero