Digital Signal Processing Reference
In-Depth Information
s
LPF1
t
H
t
s
t
c
HTF
LPF
Q
H
t
{
s
c
}
LPF
−
1
t
ϕ
=
tan
c
t
t
{
s
c
}
LPF
t
t
I
LPF
LPF1
c
t
s
t
Figure 7.14
Schematic of a second phase/frequency demodulation process.
7.7 PHASE-FREQUENCY DEMODULATION SCHEME 2
A second and closely related phase/frequency demodulation technique is
described here. The schematic is shown in Figure 7.14. In this method, the input
signal
s
t
is low-pass filtered with LPF1 only, while the reference signal
c
t
is made
analytic via LPF1 and the Hilbert filter HTF. The resulting analytical signal is
multiplied by the filtered real signal
s
t
, then low-pass filtered a second time to
yield the in-phase
I
and quadrature component
Q
. This approach is based on the
fact that a real signal is half the sum of an analytical signal and its complex
conjugate. Thus, in multiplying the real signal
s
t
with the complex reference
c
t
, we
obtain a signal
2
at twice the carrier frequency and a dc phasor term that depends
only on the modulated phase
ϕ
t
. The high-frequency component at 2
ω
is readily
low-pass filtered by LPF, so that the demodulated phase
ϕ
t
is given by
H
t
{
s
c
}
LPF
−
1
t
ϕ
=
tan
(7.19)
t
{
s
c
}
LPF
t
t
The numerator in (7.19) corresponds to the imaginary part of the phasor whose
argument is
ϕ
t
, whereas the denominator is the real part of this low-pass filtered
analytical signal.
Note that replacing the analytical reference signal
c
t
with its sine and cosine
signals at the carrier frequency has no detrimental effect on performance of the
scheme, since with appropriate sign changes, sine and cosine functions are Hilbert
transforms of each other [8]. This could be useful in applications where an internal
digital sin/cos generator is available. However, there may still be a need for
correction of frequency drifts as discussed in Section 7.5.
2
LPF
j
(
ω
t
+
ϕ
)
−
j
(
ω
t
+
ϕ
)
−
j
ω
t
[
e
+
e
]
e
=
[cos(
2
ω
t
+
ϕ
)
+
cos
ϕ
]
+
j
[sin
ϕ
−
sin(
2
ω
t
+
ϕ
)]
→
cos
ϕ
+
j
sin
ϕ
t
t
t
t
t
t
t
t
