Digital Signal Processing Reference
In-Depth Information
h
θ
LPF
(
)
H
{
x
}
−
1
x
′
=
A
cos[tan
−
θ
]
x
x
HTF
g
θ
Figure 7.19
Introducing an arbitrary phase shift θ into a real signal.
sets of coefficients to form the analytical phase-shifted signal; one pair for the real
channel and another pair for the imaginary (see Figure 7.19 for the real channel).
Let
h
θ
represent the low-pass component of the phase-shifting filter with the same
order as
g
θ
,
the Hilbert phase-shifting filter. We can write the final filter
coefficients for the real component in (7.24) and for arbitrary phase shift
θ
as
h
=
h
cos
θ
(7.27a)
θ
,
k
k
g
=
g
sin
θ
(7.27b)
θ
,
k
k
where
k
= 1,2,…
L
/2+1, and
g
k
are the original Hilbert transforming filter given in
(7.3). The corresponding coefficients for the imaginary component are given by
h
=
−
h
sin
θ
(7.28a)
θ
,
k
k
g
=
g
cos
θ
(7.28b)
θ
,
k
k
The filter implementation follows directly from (7.8), so that
L
+
1
1
2
∑
′
x
=
g
(
x
−
x
)
+
h
(
x
+
x
)
(7.29)
j
+
k
−
1
L
−
j
+
k
+
1
j
+
k
−
1
L
−
j
+
k
+
1
L
+
k
θ
,
j
θ
,
j
K
2
j
=
1
The Hilbert transform of (7.29) is found by making the phase shift . Figure
7.20 shows typical results for a sinusoidal signal and a modulated signal using
(7.29). The carrier frequency for the sinusoidal signal was 0.04. However, while
the errors for the sinusoidal signal were relatively insignificant, for a modulated
signal with low carrier frequencies the errors may be substantial. These are shown
as failure points in Figure 7.20(b). In fact, this is responsible for the demodulation
phase errors seen in both Schemes 1 and 2. The reasons for this are believed to be
due to the loss in amplitude when the carrier frequency gets too close to dc, and is
linked to the transition edge of the frequency response profile of the Hilbert
transform at dc.
If the carrier frequency is increased better results are obtained, provided the
modulation amplitude is not too severe so that the signal is well represented by a
π
±
θ
2
