Spectrum Representation

• Recall that summing sinusoids of the same frequency but arbitrary amplitudes and phases produces a new single sinusoid of the same frequency.

• Summing several sinusoids of different frequencies will produce a waveform that is no longer purely sinusoidal.

• The spectrum of a signal is a graphical representation of the frequency components it contains and their complex amplitudes.
• Consider a signal that is the sum of $N$ sinusoids of arbitrary amplitudes, phases, AND frequencies:
  $$x(t)= A_0+\sum_{k=1}^{N}A_k\cos(\omega_kt+\phi_k)$$