Bandpass and Hilbert

$$x(t)=\cos 2\pi f_0 t$$

$$z(t)=e^{j2\pi f_0 t}$$

*******FIG.8************

• In general:

  Z(f)=2U(f)X(f) and x(t) is assumed real usually. So,

  
  

  $$z(t)=2(\frac{1}{2}\delta(t)+\frac{j}{2\pi t})\ast x(t)=x(t)+j\hat{x}(t)$$
  
  

  where

  
  

  $$\underbrace{\hat{x}(t)=x(t)\ast \frac{1}{\pi t}}_{\mbox{Hilbert Xform}}$$
  
  

  
  

  $$\hat{X}(f)=-j sgn(-f)X(f)=-j sgn(f)X(f)$$
  
  

  In examples above

  
  

  $$x(t)=\cos 2\pi f_0 t$$
  
  

  
  

  $$\hat{x}(t)=\sin 2\pi f_0 t$$
  
  

  Hilbert Xform gives a 90^o phase shift.

  
  

  $$e^{j2\pi f_0 t}=z(t)=x(t)+j\hat{x}(t)=\underbrace{\cos 2\pi f... ...in phase}}+ j \underbrace{\sin 2\pi f_0 t}_{\mbox{quadrature}}$$
  
  

• Hilbert neatos

  1.

  x(t) even, $\hat{x}(t)$ odd
  x(t) odd, $\hat{x}(t)$ even
  

  2.

  $\hat{\hat{x}}(t)=-x(t)$

  3.

  $E_x=E_{\hat{x}}$

  4.

  $x(t)\perp\hat{x}(t)$ (if x(t) real)
  

  This is worth a proof.

  
  

  $$\begin{eqnarray*}<x(t),\hat{x}(t)>& = &\int_{-\infty}^{\infty}x(t) \left(\int_{-... ...\\ & + & \int^0_{-\infty} \frac{j Im[R_x(\tau)]}{\pi\tau}d\tau \end{eqnarray*}$$

  
  

  and if x(t) real (or X(f)=X^*(f), i.e. hermitian),
  $Im[R_x(\tau)]=0$ QED