We will now construct a signal, take its DFT, and interpret its results by looking at amplitude and phase plots.

Analog signal

Consider a signal made up of two sinusoids as follows:

\begin{align*} x(t) &= x_1(t) + x_2(t) \\ &= 2\cos(2\pi 500 t+65^\circ) + 3\sin(2\pi 1000 t + 15 ^\circ) \end{align*}

This is drawn in the figure below:

This is a real signal with no quadrature component.

Sampled signal

Let’s sample this signal at a rate of $f_s=1/T_s=10$ kHz.

\begin{align*} x[n] &= 2\cos(2\pi 500 nT_s+65^\circ) + 3\sin(2\pi 1000 nT_s + 15^\circ) \nonumber \\ &= 2\cos\left(2\pi \frac{1}{20}n+65^\circ\right) + 3\sin\left(2\pi \frac{2}{20}n + 15^\circ \right) \end{align*}