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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)