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6.20  Chapter Six

                       (SSB-AM). The x Q (t) that results in this case is important enough to get a name;
                       the Hilbert transform [ZTF89]. The transfer function of the Hilbert transformer
                       is

                                                   H Q (f ) =− jsgn(f )                   (6.22)

                       where

                                                           1     f > 0
                                                sgn(f ) =                                 (6.23)
                                                           −1    f < 0
                         Note, because of the sharp transition in the transfer function at DC it is
                       only possible to use SSB-AM with message signals that do not have significant
                       spectral content near DC. It should be noted that in analog video signals the DC
                       value is important in a simple way to synchronize the scanning of the picture so
                       SSB-AM cannot be used with video signals. The transmitted signal for SSB-AM
                       is

                                                x z (t) = A c (m(t) + jm h (t))           (6.24)

                       where m h (t) is the Hilbert transform of m(t).



                       EXAMPLE 6.9
                       Single-sideband modulation with a message signal

                                                   m(t) = β sin(2π f m t)

                       has an in-phase signal of
                                                             A c β          A c β
                               x I (t) = A c β sin(2π f m t)  X I (f ) =  δ( f − f m ) −  δ( f + f m )
                                                              2 j           2 j
                       Applying the Hilbert transform, H Q (f ) =− j sgn(f ), to x I (t) produces a quadrature
                       signal with
                                                        −A c β          A c β
                                   X Q (f ) = H Q (f )X I (f ) =  δ( f − f m ) −  δ( f + f m )
                                                         2              2
                       or

                                                 x Q (t) =−A c β cos(2π f m t)

                       This results in
                                 x z (t) = A c β(sin(2π f m t) − j cos(2π f m t)) =− jA c β exp( j 2π f m t)
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