Digital Signal Processing Reference
In-Depth Information
The goal of such demodulation schemes is to determine either the frequency
component
ϕ
t
. If the phase is required and the
frequency is a carrier only of such information, then we would need a reference
signal with identical frequency but uncorrupted phase information. The latter
signal usually drives the whole frequency modulation (FM) process forward and,
as is often the case in electrical or optical communications, forms part of the
transmission system [4]. The aim of this chapter is to present the digital Hilbert
transform within the context of digital demodulation, emphasizing typical errors
inherent in utilizing the transform. Two techniques will be described.
ω
or the time-dependent phase
Terminology
The terms
Hilbert transforming filter
or
Hilbert transformer
will be used to
describe the said filter. In general, we treat these as one type of unity gain filter.
7.2 HILBERT TRANSFORM REALIZATION
The digital Hilbert transformer can be realized either as a finite impulse response
(FIR) or infinite impulse response (IIR) system. We have chosen the FIR
approach for its stability, phase linearity, and general simplicity. Furthermore, we
will use the window technique, utilizing the Gaussian window for this purpose.
One major reason for this approach is that in some applications involving the
Hilbert transforming filter, it is useful to ensure that all filters are compatible in
terms of their pass-band ripple, phase, and attenuation. The noncausal digital
Hilbert transformer
g
in the time domain has the following form [1]
[
]
2
π
L
sin
F
(
+
1
−
k
)
2
c
2
2
g
=
W
K
k
=
1
2
Λ
,
L
k
k
2
π
L
(7.3)
(
k
−
+
1
2
g
=
0
g
=
−
g
L
L
−
k
+
2
k
+
1
2
where
L
is the order of the filter, and is even for the implementation given in this
chapter;
K
= 2
9
in the said filters. As such, all Hilbert transforming filters given
here are odd in length. The Gaussian window function
W
k
has been given in
Chapter 2. Figure 7.1 shows a 99-point Hilbert filter and its frequency response
for
F
c
= 0.5. Interestingly, this is a half-band filter but could only be used to
reduce the coefficient count by 25% since every fourth coefficient is zero.
The Hilbert transformer in (7.3) is linear in phase; however, it is offset by
90
o
. Thus, over the bandwidth of the filter the phase changes, and in principle
should be compensated for in the original real part of the signal by a filter of
similar length and phase response. In fact, the phase response
−
(
F
) for the Hilbert
filter in (7.3) is given for filter order
L
and normalized frequency
F
by
θ
π
θ
(
F
)
=
−
(
+
LF
)
(7.4)
2
