Segnali a banda stretta

Nel caso in cui un segnale modulato x $\left(\vphantom{ t}\right.$ t $\left.\vphantom{ t}\right)$ presenti una occupazione di frequenza molto piccola rispetto alla frequenza portate, si assume spesso che H $\left(\vphantom{ f}\right.$ f $\left.\vphantom{ f}\right)$ nella banda di segnale non vari di molto, ossia presenti sia modulo che fase pressoché costanti e pari al valore assunto per f = f₀, ossia H $\left(\vphantom{ f}\right.$ f $\left.\vphantom{ f}\right)$ $\simeq$ H $\left(\vphantom{ f_{0}}\right.$ f₀ $\left.\vphantom{ f_{0}}\right)$ = G₀e^j. Questa approssimazione permette di trascurare l'effetto delle distorsioni lineari, che in questo caso equivalgono ad una semplice rotazione degli assi dell'inviluppo complesso, reversibile scegliendo opportunamente la fase della portante di demodulazione.

Nelle ipotesi poste, risulta infatti $\underline{H}$ $\left(\vphantom{ f}\right.$ f $\left.\vphantom{ f}\right)$ = 2H⁺ $\left(\vphantom{ f+f_{0}}\right.$ f + f₀ $\left.\vphantom{ f+f_{0}}\right)$ = 2G₀e^j e quindi $\underline{h}$ $\left(\vphantom{ t}\right.$ t $\left.\vphantom{ t}\right)$ = $\mathcal {F}$ ^-1 $\left\{\vphantom{ \underline{H}\left( f\right) }\right.$ $\underline{H}$ $\left(\vphantom{ f}\right.$ f $\left.\vphantom{ f}\right)$ $\left.\vphantom{ \underline{H}\left( f\right) }\right\}$ = 2G₀e^j $\delta$ $\left(\vphantom{ f}\right.$ f $\left.\vphantom{ f}\right)$ ; pertanto l'inviluppo complesso di h $\left(\vphantom{ t}\right.$ t $\left.\vphantom{ t}\right)$ è un impulso di area complessa 2G₀e^j = 2G₀ $\left(\vphantom{ \cos \phi _{0}+j\sin \phi _{0}}\right.$ cos $\phi_{0}^{}$ + jsin $\phi_{0}^{}$ $\left.\vphantom{ \cos \phi _{0}+j\sin \phi _{0}}\right)$ . All'uscita del canale H $\left(\vphantom{ f}\right.$ f $\left.\vphantom{ f}\right)$ troviamo quindi

$\displaystyle \underline{y}$ $\displaystyle \left(\vphantom{ t}\right.$ t $\displaystyle \left.\vphantom{ t}\right)$	=	$\displaystyle {\textstyle\frac{1}{2}}$ ^. $\displaystyle \underline{x}$ $\displaystyle \left(\vphantom{ t}\right.$ t $\displaystyle \left.\vphantom{ t}\right)$ * $\displaystyle \underline{h}$ $\displaystyle \left(\vphantom{ t}\right.$ t $\displaystyle \left.\vphantom{ t}\right)$ = $\displaystyle \left(\vphantom{ x_{c}\left( t\right) +jx_{s}\left( t\right) }\right.$ x_c $\displaystyle \left(\vphantom{ t}\right.$ t $\displaystyle \left.\vphantom{ t}\right)$ + jx_s $\displaystyle \left(\vphantom{ t}\right.$ t $\displaystyle \left.\vphantom{ t}\right)$ $\displaystyle \left.\vphantom{ x_{c}\left( t\right) +jx_{s}\left( t\right) }\right)$ *G₀ $\displaystyle \left(\vphantom{ \cos \phi _{0}+j\sin \phi _{0}}\right.$ cos $\displaystyle \phi_{0}^{}$ + jsin $\displaystyle \phi_{0}^{}$ $\displaystyle \left.\vphantom{ \cos \phi _{0}+j\sin \phi _{0}}\right)$ $\displaystyle \delta$ $\displaystyle \left(\vphantom{ f}\right.$ f $\displaystyle \left.\vphantom{ f}\right)$ =
	=	G₀ $\displaystyle \left[\vphantom{ \left( x_{c}\left( t\right) \cos \phi _{0}-x_{s}... ...ft( t\right) \sin \phi _{0}+x_{s}\left( t\right) \cos \phi _{0}\right) }\right.$ $\displaystyle \left(\vphantom{ x_{c}\left( t\right) \cos \phi _{0}-x_{s}\left( t\right) \sin \phi _{0}}\right.$ x_c $\displaystyle \left(\vphantom{ t}\right.$ t $\displaystyle \left.\vphantom{ t}\right)$ cos $\displaystyle \phi_{0}^{}$ - x_s $\displaystyle \left(\vphantom{ t}\right.$ t $\displaystyle \left.\vphantom{ t}\right)$ sin $\displaystyle \phi_{0}^{}$ $\displaystyle \left.\vphantom{ x_{c}\left( t\right) \cos \phi _{0}-x_{s}\left( t\right) \sin \phi _{0}}\right)$ + j $\displaystyle \left(\vphantom{ x_{c}\left( t\right) \sin \phi _{0}+x_{s}\left( t\right) \cos \phi _{0}}\right.$ x_c $\displaystyle \left(\vphantom{ t}\right.$ t $\displaystyle \left.\vphantom{ t}\right)$ sin $\displaystyle \phi_{0}^{}$ + x_s $\displaystyle \left(\vphantom{ t}\right.$ t $\displaystyle \left.\vphantom{ t}\right)$ cos $\displaystyle \phi_{0}^{}$ $\displaystyle \left.\vphantom{ x_{c}\left( t\right) \sin \phi _{0}+x_{s}\left( t\right) \cos \phi _{0}}\right)$ $\displaystyle \left.\vphantom{ \left( x_{c}\left( t\right) \cos \phi _{0}-x_{s}... ...ft( t\right) \sin \phi _{0}+x_{s}\left( t\right) \cos \phi _{0}\right) }\right]$

$\displaystyle \left[\vphantom{ \begin{array}{c} y_{c}\left( t\right) \\ y_{s}\left( t\right) \end{array}}\right.$ $\displaystyle \begin{array}{c} y_{c}\left( t\right) \\ y_{s}\left( t\right) \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{c} y_{c}\left( t\right) \\ y_{s}\left( t\right) \end{array}}\right]$ = $\displaystyle \left[\vphantom{ \begin{array}{rr} \cos \phi _{0} & -\sin \phi _{0}\\ \sin \phi _{0} & \cos \phi _{0} \end{array}}\right.$ $\displaystyle \begin{array}{rr} \cos \phi _{0} & -\sin \phi _{0}\\ \sin \phi _{0} & \cos \phi _{0} \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{rr} \cos \phi _{0} & -\sin \phi _{0}\\ \sin \phi _{0} & \cos \phi _{0} \end{array}}\right]$ $\displaystyle \left[\vphantom{ \begin{array}{c} x_{c}\left( t\right) \\ x_{s}\left( t\right) \end{array}}\right.$ $\displaystyle \begin{array}{c} x_{c}\left( t\right) \\ x_{s}\left( t\right) \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{c} x_{c}\left( t\right) \\ x_{s}\left( t\right) \end{array}}\right]$