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DSP Algorithms for RF Systems

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DSP for Beginners cover
Euler's formula
Negative frequency in complex sinusoids
Complex sinusoids on the complex plane
Aliasing
Phase relationship between sine and cosine
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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