DFT Linearity

Linearity implies that the DFT of the sum of input signals is the sum of the DFTs of the individual input signals. Let’s find out how.

Derivation

Assume that two input signals $x_1[n]$ and $x_2[n]$ have DFTs given by $X_1[k]$ and $X_2[k]$, respectively. If that’s the case, the DFT of their sum is equal to the sum of their individual DFTs.

$$\begin{align*} \text{DFT }x_1[n]\quad \rightarrow\quad X_1[k]\\ \text{DFT }x_2[n]\quad \rightarrow\quad X_2[k] \end{align*}$$

This implies that:

$$\text{DFT }\Big\{y[n]=x_1[n]+x_2[n]\Big\}\quad \rightarrow\quad Y[k]=X_1[k]+X_2[k]$$

This is because:

$$\begin{align*} Y[k]&=\sum_{n=0}^{N-1}\Big\{x_1[n]+x_2[n]\Big\}e^{-j2\pi\frac{k}{N}n}\\ &=\sum_{n=0}^{N-1}x_1[n]e^{-j2\pi\frac{k}{N}n}+\sum_{n=0}^{N-1}x_2[n]e^{-j2\pi\frac{k}{N}n}\\ &=X_1[k]+X_2[k] \end{align*}$$

This helps determine the DFT of a signal without explicit calculations.

Example