Fourier Transform of Complex and Real Functions

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Fourier Transform

For a continuous-time function x(t), the Fourier transform of x(t) can be defined as,

$$\mathrm{X(\omega)\:=\:\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt}$$

And the inverse Fourier transform is defined as,

$$\mathrm{x(t)\:=\:\frac{1}{2\pi }\int_{-\infty}^{\infty}X(\omega)e^{j\omega t}d\omega}$$

Fourier Transform of Complex Functions

Consider a complex function x(t) that is represented as −

$$\mathrm{x(t)\:=\:x_{r}(t)\:+\:jx_{i}(t)}$$

Where, x_r (t) and x_i (t) are the real and imaginary parts of the function respectively.

Now, the Fourier transform of function x(t) is given by,

$$\mathrm{F\left [ x\left ( t \right ) \right ]\:=\:X\left ( \omega \right )\:=\:\int_{-\infty}^{\infty}x\left (t\right )e^{-j\omega t}dt\:=\:\int_{-\infty}^{\infty}\left [ x_{r}\left ( t \right )\:+\:jx_{i}\left ( t \right ) \right ]e^{-j\omega t}dt}$$

$$\mathrm{\Rightarrow\: X\left ( \omega \right )\:=\:\int_{-\infty}^{\infty}\left [ x_{r}\left ( t \right )\:+\:jx_{i} \left ( t \right ) \right ]\left [ \cos \omega t\:-\:j\sin\: \omega t \right ]dt}$$

$$\mathrm{\Rightarrow X\left ( \omega \right )\:=\:\int_{-\infty}^{\infty}\left [ x_{r}\left ( t \right )\cos \omega t\:+\:x_{i}\left ( t \right )\sin \omega t \right ]dt\:+\:j\int_{-\infty}^{\infty}\left [ x_{i}\left ( t \right )\cos \omega t\:-\:x_{r}\left ( t \right )\sin \omega t \right ]dt}$$

Therefore, the Fourier transform of complex function is,

$$\mathrm{X(\omega )\:=\:X_{r}\left ( \omega \right )\:+\:jX_{i}\left ( \omega \right )}$$

Where,

$$\mathrm{X_{r}(\omega )\:=\:\int_{-\infty}^{\infty}\left [ x_{r}\left ( t \right )\cos \omega t\:+\:x_{i}\left ( t \right )\sin \omega t \right ]dt}$$

And

$$\mathrm{X_{i}(\omega )\:=\:\int_{-\infty}^{\infty}\left [ x_{i}\left ( t \right )\cos \omega t\:-\:x_{r}\left ( t \right )\sin \omega t \right ]dt} $$

Inverse Fourier Transform of Complex Functions

From the definition of inverse Fourier transform, we have,

$$\mathrm{x\left ( t \right )\:=\:\frac{1}{2\pi }\int_{-\infty }^{\infty}X\left ( \omega \right )e^{j\omega t}d\omega}$$

$$\mathrm{\:=\:\frac{1}{2\pi }\int_{-\infty }^{\infty}\left [ X_{r}\left ( \omega \right )\:+\:jX_{i}\left ( \omega \right ) \right ]\left [ \cos \omega t\:+\:j\sin \omega t \right ]d\omega} $$

$$\mathrm{\Rightarrow\: x\left ( t \right )\:=\:\frac{1}{2\pi }\int_{-\infty }^{\infty}\left [ X_{r}\left ( \omega \right )\cos \omega t\:-\:X_{i}\left ( \omega \right )sin \omega t \right ]d\omega\:+\:j\frac{1}{2\pi }\int_{-\infty }^{\infty}\left [ X_{r}\left ( \omega \right )\sin \omega t\:+\:X_{i}\left ( \omega \right )cos \omega t \right ]d\omega}$$

Therefore,

$$\mathrm{x\left ( t \right )\:=\:x_{r}\left ( t \right )\:+\:jx_{i}(t)}$$

Where,

$$\mathrm{x_{r}\left ( t \right )\:=\:\frac{1}{2\pi }\int_{-\infty }^{\infty}\left [ X_{r}\left ( \omega \right )\cos \omega t\:-\:X_{i}\left ( \omega \right )sin \omega t \right ]d\omega}$$

And

$$\mathrm{ x_{i}\left ( t \right )\:=\:\frac{1}{2\pi }\int_{-\infty }^{\infty}\left [ X_{r}\left ( \omega \right )\sin \omega t\:+\:X_{i}\left ( \omega \right )cos \omega t \right ]d\omega}$$

Fourier Transform of Real Functions

Case I – When x(t) is a real function,

$$\mathrm{x_{i}\left ( t \right )\:=\:0\:\: and\:\: X\left ( -\omega \right )\:=\:X^{\ast }\left ( \omega \right )}$$

Hence, the Fourier transform of the real and imaginary parts of the function is,

$$\mathrm{X_{r}\left ( \omega \right )\:=\:\int_{-\infty }^{\infty}x\left ( t \right )\cos \omega t\: dt} $$

$$\mathrm{X_{i}\left ( \omega \right )\:=\:-\int_{-\infty }^{\infty}x\left ( t \right )\sin \omega t\: dt}$$

$$\mathrm{\therefore\: X\left ( \omega \right )\:=\:\int_{-\infty }^{\infty}x\left ( t \right )\cos \omega t\: dt\:-\:j\int_{-\infty }^{\infty}x\left ( t \right )\sin \omega t\: dt}$$

Case II – When x(t) is real and even,

$$\mathrm{X_{r}\left ( \omega \right )\:=\:\int_{-\infty }^{\infty}x\left ( t \right )\cos \omega t\: dt\:=\:2\int_{0 }^{\infty}x\left ( t \right )\cos \omega t\: dt} $$

$$\mathrm{X_{i}\left ( \omega \right )\:=\:0}$$

$$\mathrm{\therefore\: X\left ( \omega \right )\:=\:2\int_{0 }^{\infty}x\left ( t \right )\cos \omega t\: dt}$$

Case III – When x(t) is real and odd,

$$\mathrm{X_{r}\left ( \omega \right )\:=\:0}$$

$$\mathrm{X_{i}\left ( \omega \right )\:=\:jX\left ( \omega \right )\:=\:-j\int_{-\infty }^{\infty}x\left ( t \right )\sin \omega t\: dt}$$

$$\mathrm{\Rightarrow\: X_{i}\left ( \omega \right )\:=\:-j2\int_{0 }^{\infty}x\left ( t \right )\sin \omega t\: dt}$$

$$\mathrm{\therefore\: X\left ( \omega \right )\:=\:-j2\int_{0 }^{\infty}\:x\left ( t \right )\sin \omega t\: dt}$$

If x_e (t) and x_o (t) are the even and odd parts of the function x(t), then for a non-symmetric function, we have,

$$\mathrm{F\left [ x\left ( t \right ) \right ]\:=\:X\left ( \omega \right )\:=\:X_{r}\left ( \omega \right )\:+\:jX_{i} \left ( \omega \right )} $$

$$\mathrm{\Rightarrow X\left ( \omega \right )\:=\:\int_{-\infty }^{\infty}x_{e}\left ( t \right )\cos \omega t\: dt\:-\:j\int_{-\infty }^{\infty}x_{0}\left ( t \right )\sin \omega t\: dt\:=\:X_{e}\left ( \omega \right )\:+\:X_{0}\left ( \omega \right )}$$