Spectrum

• When sinusoids of different frequencies are added together, the resulting signal is no longer sinusoidal.

• The spectrum of a signal is a graphical representation of the complex amplitudes, or phasors $Ae^{j\phi}$, of its frequency components (obtained using the DFT).

• The DFT of $y(\cdot)$ at frequency $\omega_k$ is a measure of the the amplitude and phase of the complex DFT sinusoid $e^{j\omega_k nT}$ which is present in $y(\cdot)$ at frequency $\omega_k$:
  $$Y(\omega_k) = \sum_{n=0}^{N-1}y(n)e^{-j\omega_k nT}$$

• Projecting a signal $y(\cdot)$ onto a DFT sinusoid provides a measure of ``how much that DFT sinusoid is in $y(\cdot)$''.