Harmonic Sum of Sinusoids

• Harmonic frequencies are integer multiples of a fundamental frequency
• Signals that are a sum of sinusoids are periodic as multiples of the fundamental period
  • $T_0 = \frac{2\pi}{\omega_0}= \frac{1}{f_0}$
• General form:

• The fundamental frequency is the greatest common denominator of the frequency components of the signal
  • $f_0 = \gcd(f_1, f_2, \dots, f_N)$ (measured in Hz, without the $2\pi$)
  • $\omega_0 = \gcd(\omega_1, \omega_2, \dots, \omega_N)$ (measured in rad/s, with the $2\pi$)

Fourier Series

• Allows representation of non-sinusoids as harmonic sums of sinusoids

  • Fourier analysis: finding spectrum from signal
  • Fourier synthesis: finding signal from spectrum

• What signals have a Fourier series representation?

  • Must be periodic
  • Must meet Dirichlet conditions

• Fourier series representations:

  • $$x(t) = \sum_{k=0}^\infty A_{k} \cos(\omega_{0}kt)$$
  • $$x(t) = \sum_{k=0}^\infty B_{k} \sin(\omega_{0}kt)$$
  • $$x(t) = \sum_{k=-\infty}^\infty a_{k}e^{j\omega_{0}kt}$$
  • $$x(t) = \sum_{k=0}^\infty A_{k} \cos(\omega_{0}kt) + \sum_{k=0}^\infty B_{k} \sin(\omega_{0}kt)$$

Fourier Analysis

• To get the Fourier series representation of a signal:
  1. Determine $T_0$ and $\omega_0 = \frac{2\pi}{T_0}$
  2. Determine the coefficient of the DC (constant) term
  3. Determine the coefficients of the remaining harmonics
• For the Sum of complex exponentials Fourier series:
  • $k$ is the harmonic number
  • $k\omega_0$ is the frequency of the harmonic
  • $a_k$ is the complex amplitude of the harmonic
  • $(k\omega_0, a_k)$ pairs correspond to the two-sided spectrum
• Complex amplitude formula
  • The DC amplitude lacks an exponential term inside the integrand

Fourier Synthesis

• To synthesize a signal from its spectrum, perform a running loop of each term in the Fourier series
• Should already have complex amplitude and frequency arrays
  • $a_k$ should be halved if summing over complex exponentials
• Make sure to initialize the following:
  • $t$ array over the proper time interval
  • $x(t)$ array with all zeros
  • sum variable for running sum of the Fourier series

Complex exponentials

1. Loop over $k \in [-N, N]$
2. Compute each harmonic $x_k(t) = a_k e^{j\omega_0 kt}$
3. Add each computed harmonic to the running sum $x(t)$

Sum of cosines

1. Loop over $k \in [0, N]$
2. Compute each harmonic $x_k(t) = A_{k} \cos(\omega_{0}kt)$
3. Add each computed harmonic to the running sum $x(t)$

References

• Fourier series (course handout)