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Fourier Transform Linearity Property Derivation
January 5, 2022

The Fourier transform \mathcal{F} \{ \cdot \} is linear such that

(1)   \begin{equation*}\begin{split}\mathcal{F} \{ x(t) + y(t) \} & = \mathcal{F} \{ x(t) \} + \mathcal{F} \{ y(t) \} \\& = X(f) + Y(f).\end{split}\end{equation*}

The Fourier transform of x(t) + y(t) is

(2)   \begin{equation*}\mathcal{F} \{ x(t) + y(t) \} = \int_{-\infty}^{\infty} \left( x(t) + y(t) \right)e^{-j2\pi ft} ~ dt.\end{equation*}

As the integral is a linear operator the Fourier transform of the summation can be written as the summation of two Fourier transforms,

(3)   \begin{equation*}\mathcal{F} \{ x(t) + y(t) \} = \int_{-\infty}^{\infty} x(t) \right)e^{-j2\pi ft} ~ dt + \int_{-\infty}^{\infty} y(t) \right)e^{-j2\pi ft} ~ dt,\end{equation*}

therefore

(4)   \begin{equation*}\mathcal{F} \{ x(t) + y(t) \} = X(f) + Y(f).\end{equation*}

If you liked this post please check out the List of Important Math for DSP!

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