Hilbert Transform Filters

• For more complicated signals (which are the sum of sinusoids), the Hilbert Transform may be used to shift each sinusoidal component by a quarter cycle.

• When a real signal $x(t)$ and its Hilbert transform $y(t) = \mathcal{H}_t\{x\}$ are used to form a new complex signal
  $$z(t) = x(t) + jy(t),$$
  the signal $z(t)$ is the (complex) analytic signal corresponding to the real signal $x(t)$.

• Problem: Given the modulated signal
  $$x(t) = A(t)\cos(\omega t).$$
  How do you obtain $A(t)$ without knowing $\omega$?

  Answer: Use the Hilbert tranform to generate the analytic signal

  $$z(t) \approx A(t)e^{j\omega t},$$
  and then take the absolute value
  $$A(t) = \vert z(t)\vert.$$