Digital Signal Processing Reference
In-Depth Information
0.4
0
(
a
)
(
b
)
0.3
-20
0.2
-40
0.1
-60
0
-80
-0.1
-100
-0.2
-120
-0.3
-140
-0.4
-160
0
50
100
0 .2 .4 .6 .81
Normalised Frequency,
F
Index,
k
Normalized Frequency,
F
Figure 7.1
(a) Hilbert transform filter coefficients. (b) Frequency response.
7.3 PHASE-FREQUENCY DEMODULATION SCHEME 1
In this section we will review a frequency demodulation scheme employing the
Hilbert transform. We mentioned earlier that a second reference signal is needed
to recover the phase
t
; we will show how this is employed. For all practical
purposes, this process will also demodulate FM signals as well as phase
modulated (PM) signals. Figure 7.2 shows the schematic of the first demodulation
process. The input real signal and its reference are all low-pass filtered and
Hilbert transformed, then multiplied and summed to give the in-phase
I
and
quadrature
Q
components. To see how this is achieved, let the initial signal and its
reference be
s
t
and c
t
, and their Hilbert transforms and, , respectively. This
creates two analytical signals
S
t
and
C
t
, which are then multiplied to form the
demodulated signal
D
t
. This could be written as
ϕ
H
t
s
c
H
t
∗
H
t
H
t
H
t
H
t
D
=
S
C
=
(
s
c
+
s
c
)
+
j
(
s
c
−
s
c
)
t
t
t
t
t
t
t
(7.5)
j
ϕ
=
G
e
t
t
where * means complex conjugate. The real part of
D
t
is the in-phase component
I
, whereas the quadrature component
Q
is the imaginary part, so that
H
t
H
t
I
=
s
c
+
s
c
t
t
t
(7.6)
H
t
H
t
Q
=
s
c
−
s
c
t
t
t
