Journal of Computer and Communications, 2014, 2, 87-92
Published Online July 2014 in SciRes. http://www.scirp.org/journal/jcc
http://dx.doi.org/10.4236/jcc.2014.29012
How to cite this paper: Xu, G.L., Wang, X.T., Xu, X.G. and Shao, L.M. (2014) Amplitude and Phase Analysis Based on Signed
Demodulation for AM-FM Signal. Journal of Computer and Communications, 2, 87-92.
http://dx.doi.org/10.4236/jcc.2014.29012
Amplitude and Phase Analysis Based on
Signed Demodulation for AM-FM Signal
Guanlei Xu1*, Xiaotong Wang2, Xiaogang Xu2, Limin Shao1
1Ocean Department, Dalian Navy Academy, Dalian, China
2Navigation Dep art men t, Dalian Navy Academy, Dalian, China
Received May 2014
Abstract
This paper proposes a new amplitude and phase demodulation schem e different from the tradi-
tional method for AM-FM signals. The traditional amplitude demodulation assumes that the am-
plitude should be non-negative, and the phase is obtained under the case of non-negative ampli-
tude, which approximates the true amplitude and phase but dis tor ts the true amplitude and phase
in some cases. In this paper we assume that the amplitude is signed (zero, positive or negative),
and the phase is obtained under the case of signed amplitude by optim iza ti on, as is called signed
demodulation. The main merit of the signed demodulation lies in the revelat ion of senseful p hysi-
cal meaning on phase and frequency. Experiments on the real-world data show the effici ency of
the me th od.
Keywords
Amplitude Demodulation, Phase, Hilbert Transform, Signed Demodulation
1. Introduction
In many signal processing fields such as communication, wireless navigation and machine systems, the modula-
tion and demodulation are often used to process the amplitude component and the phase component [1]-[10]. In
fact, the basic problem in processing AM-FM signals is d emodulation, i.e., estimation of the information stored
in the amplitude and phase signals while given the composite signal. For monocomponent AM-FM signals many
successful demodulation approaches have existed, ranging from standard methods such as Hilbert transform
demodulation [1] or phase-locked loops (PLL’s) to the recent energy separation algorithm (ESA) that tracks and
demodulates the energy of the source producing the AM-FM signal using instantaneous nonlinear differential
operators [2]-[18]. While each of these monocomponent algorithms may have its advantages and disadvantages,
they more or less offer a solution to the monocomponent AM-FM demodulation problem. However, these me-
thods shown in above only process the positive amplitudes, i.e., they think that the amplitudes should be the ab-
solute values (or non-negative). In other words, they fail to demodulate the signed amplitudes (or non-positive
AM component).
*
Corresponding author.

G. L. Xu et al.
88
In this paper, we derive a new scheme, different from the traditional method for AM-FM signals that can ob-
tain the signed amplitude and accordingly senseful physical meaning phase and frequency by optimization.
2. Signed Demodulation
2.1. Traditional Demodulation of Amplitude and Phase
For a real function, a direct and simple way to obtain the complex signal is via Hilbert transform [????]. A real
function
( )
xt
and its Hilbert transform
( )
xt
are related to each other in such a way that they together create
a so called strong analytic signal
( )( )( )
h
xtx tjxt= +
. The strong analytic signal can be written with the am-
plitude and the phase where the derivative of the phase can be identified as the instantaneous frequency. The
Hilbert transform defined in the time domain is a convolution between the Hilbert transformer
1t π
and the
real function
( )
xt
:
( )( ) ( ) ( )( )
11
*d
x
xtH xtxtP
tt
ττ
ππ τ
∞
−∞
===−
∫
, (1)
where the P in front of the integral denotes the Cauchy principal value which expanding the class of functions
for which the integral in (1) exist and “*” denotes the convolution operator and “
( )
H⋅
” denotes the Hilbert
transform operator. In frequency domain, we have the following relation:
( )( )( )( )
sgn
H
X uXuuXu= +
, (2)
where
( )
10
sgn 00
10
for u
ufor u
for u
>


= =

−<

and
( )
Xu
is the Fourier transform of the real function
( )
xt
. We also
see that
( )
xt
is the inverse Fourier transform of
( )
H
Xu
. The biggest advantage of Hilbert transform is that
one can directly obtain the amplitude and phase for AM-FM components, e.g., a real signal
( )( )( )
cosxt att φ =
,
one can obtain the follows via Hilbert transform
Amplitude:
( )( )() ()
atxtjH xt= +
, (3)
Phase:
( )( )( ) ( )
( )
( )
Im lntxtjH xt φ = +
, (4)
where “
Im( )
” denotes the imaginary part operator and “
ln()
” is the natural logarithm operator. After Hilbert
transform,
( )( )( )
cosxt att φ =
turns to a complex signal
( )( )
( )
jt
h
x tate
φ
=
, which is typical AM-FM signal
with the AM component
( )
at
and the FM component
( ) jt
e
φ
. In fact, the FM component
( ) jt
e
φ
is a special
AM-FM signal with the amplitude being 1. This process is called demodulation.
However, there is one question: 1) why the Hilbert transform of
( )
xt
is
( )( )
sinat t φ
but not
( ) ( ) ( )
cosHat t φ
? 2) why cannot we obtain the amplitude function
( )
at
rather than
( )
at
that nearly all the
demodulation methods obtain? For the first question, Bedrosian’s theorem [15] has yielded the answer. Now let
us review the Bedrosian’s theorem.
Bedro sian’s theorem [15]: For two real functions
( )
1
xt
and
( )
2
xt
, if
ua >
then
( )
1
0Xu=
, and if
ub<
then
( )
2
0Xu=
,
0ba≥≥
, then
( )( ) { } ( )( ) { }
12 12
Hxtx txtHx t= ⋅
, (5)
where
( )
1
Xu
and
( )
2
Xu
are the Fourier transform of
( )
1
xt
and
( )
2
xt
respectively. Bedrosian’s theorem
tells us that for two functio ns,
( )
at
and
( )
cos t φ
, of which
( )
at
is with low frequency and the other com-
ponent
( )
cos t φ
is with high frequency, then through Hilbert transform we have
( )( ) { } ( )( ) { } ( )( )
coscos sinHatt atHt att φ φφ = =
.
For the second question, we will answer it in the following few sections via the signed demodulation and
some optimizations.

G. L. Xu et al.
89
2.2. The Proposed Signed Demodulation Method
The first work is to o b tain the signed amplitude function out of the positive amplitude function via taking abso-
lute value of the complex signal
( )()
( )
h
jt
hh
x ta te
φ
=
. However, it is not hard to see that a function and its Hil-
bert transform are not absolutely orthogonal (even though they are orthogona l in principle) because of trunca-
tions in numerical calculations and the boundary effects. Therefore, for a real function
( )( )( ) ( )
cosxtatt φ =
with its complex function
( )()
( )
h
jt
hh
x ta te
φ
=
after Hilbert transform has no more than the following relations:
()( )
h
a tat ≅
and
( )( )
h
tt φφ ≅
. (6)
Therefore, taking absolute value of the complex signal
( )()
( )
h
jt
hh
x ta te
φ
=
, we have
( )()( )
hh
x ta tat= ≠
.
Thus, we only can obtain an approximation (i.e.
( )
at

) of
( )
at
even if we know the exact si gns of
( )
at
.
Hence, we have the following method.
The process of signed demodulation:
1) For the signal
( )( )( )
12
xtx txt=
with low-frequency component
( )
1
xt
and high-frequency component
( )
2
xt
, obtain the complex signal
( )( )( ) { }
12h
x tHxtx t=
via Hilbert transform, then find all the zero positions
{}
0,10,20,
, ,,
M
tt t
(indeed these positions make
( )
h
xt
be the local minima) in
( )
h
xt
, and M is the total num-
ber of zero positions;
2) Obtain the high-frequency signal by
( )( )( )
2h
x txtx t=

;
3) Estimate the amplitud e function
( )
1
xt

by
( )( )( )
( )( )
1
ifis evenodd
ifis oddeven
h
h
xt m
xt xt m


=−



, where
0,0, 1mm
t tt
+
≤<
and
1, 2,,1 mM= − 
;
4) Reconstruct the high-frequency signal
( )
2
xt

by
( )( )( )
( )
( )
2 12
sgn
h
xtxt xtxt= ⋅⋅
 
where
( )
10
sgn 00
10
s
ss
s
>

=

−<

.
3. Experiment and Discussion
Here we have
( )()
1
cos0.002xt t π =
,
( )()
2
cos0.015xtt π =
,
() ()()
12
xtx txt =
,
[ ]
0, 2000t∈
. Now we use
the traditional demodulation method and our signed demodulation to demodulate signal
()( )( )
12
xtx txt=
and
give the comparison (see Figure 1). The first row ((a) (b) (c)) is the composed two signals with low-frequency
and high-frequency respectively. The second row ((d) (e) (f)) is the demodulated amplitude, the high-frequency
signal and the phase respectively by traditional method. The third row ((g) (h) (i)) is the demodulated amplitude,
the high-fr equency signal and the phase respectively by our method.
(a) (b)

G. L. Xu et al.
90
(c) (d)
(e) (f)
(g) (h)

G. L. Xu et al.
91
(i)
Figure 1. The comparison of two methods for demodulation of amplitude and phase. (a) The low-frequency signal; (b) The
high -frequency signal; (c) The composed signal by (a) × (b); (d) Demodulated amplitude by traditional method; (e) Demo-
dulated high-frequency signal by traditional method; (f) The phase of (e); (g) Demodulated amplitude by our method; (h)
Demodulated high -frequency signal by our method; (i) The phase of (h).
Clearly, our demodulation method gives more rational physical sense. We allow our amplitude to be negative,
under such case we obtain the rational phase in (i) (compared with (f)).
4. Conclusion
This paper proposes a new amplitude and phase demodulation scheme different from the traditional method for
AM-FM signals. We assume that the amplitude is signed (zero, positive or negative), and the phase is obtained
under the case of signed amplitude by optimizatio n, as is called signed demodulation. The main merit of the
signed demodulation lies in the revelation of senseful physical meani ng on phase and frequency. Experiments on
the real-world data show the ef ficienc y of the method.
Acknowledgem ents
This work is sponsored by NSFCs (Grant No. 61002052, 61273262, 61250006).
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