Discrete Fourier Transform real and imaginary parts

Question

I am trying to find the discrete Fourier transform of a waveform with known x, y coordinates. I know that the Fourier transform produces transformed vectors in the form of sin and cosine statements. I use a Matlab plugin to generate the first 3 coefficients of the waveform, for example:

For the first coefficient: Cos:0.64, Sin:1.16

However, I use a different implementation of the Fourier transform that generates sets of real and imaginary parts for this type of transform:

For the first coefficient: Real part:3.07, Imaginary part:-45.18

I don't understand the relationship between sin and cosine statements and the real and imaginary parts of the first coefficient (How can I map output 1 to output 2). I need to compare the result of these two methods and see which one is more similar to the original form. Is there an online service that I can use to display each output in terms of sins and cosins and real and imaginary parts?

Answer 1

Take a look at the complex plane:

Clearly:

$$z=a+ib=r\cos (\alpha)+i\cdot r\sin(\alpha)$$

Where:

• $a$ is $z$'s real part.
• $b$ is $z$'s imaginary part.
• $r=\sqrt{a^2+b^2}$ (since $r$ is the hypotenuse in the triangle that is formed)
• $\alpha=\arctan{(\frac ba)}$.

Thus, you can rewrite the relationship between the two parts of the equation as:

$$ \left\{ \begin{array}{c} a=r\cos(\alpha)\\ b=r\sin(\alpha) \end{array} \right. $$