AM-FM demodulation using zero crossings and
local peaks
K.V.S. Narayana and T.V. Sreenivas
Department of Electrical Communication Engineering
Indian Institute of Science, Bangalore, India 560012
Phone: +91 80 2360 2167, Fax: +91 80 2360 0683.
Email: narayana.kvs@ece.iisc.ernet.in, tvsree@ece.iisc.ernet.in
Abstract—We developed a new algorithm for estimating Instantaneous Amplitude(IA) and Instantaneous Frequency(IF) of a
band limited AM-FM signal, using zero crossings(ZC) intervals
and local peaks information(ZC-LP). In this method IA and IF are
estimated independently. For estimating IF, zero crossing interval
information is used in a K-Nearest Neighbor(K-NN) frame work
and for estimating IA local peaks of the given band limited
signal are used, also in a K-NN frame work[2]. Experimental
results shows that the proposed algorithm gives better results than
existing methods like Hilbert Transform[3], DESA1, DESA2[1].
I. I NTRODUCTION
All naturally occurring signals such as speech, biomedical
signals, music, etc., are non-stationary in nature. The nonstationarity of these signals are mainly due to their respective
production phenomena. The temporal variability of these signals can be modeled in general as amplitude variability and
frequency variability. In general, in communications AM is
used for modeling amplitude variability and FM is used for
modeling frequency variability of a signal. Hence to model
a non-stationary signal with both amplitude and frequency
variability, we can use AM and FM together. For a wideband
signal such as speech or music, there can be multiple modes of
frequency and amplitude variability, which can be represented
by a linear combination of AM-FM signals[4].
Let an AM-FM modulated signal be represented by
combination of AM-FM signals:
S(t) =
K
X
Ak (t)sin(Φk (t))
(2)
k=1
where K is number of band pass filters in the filter bank.
Given a bandpass AM-FM signal xk (t), there are many ways
of estimating AM and FM components. The basic method for
doing this is by using the Hilbert transform approach which
determines the analytic signal of a given signal and then
estimate AM and FM components separately. Other methods
are DESA1 and DESA2 which use teager energy operator[1].
In this paper we are presenting a new method(ZC-LP)
for separating AM and FM components of a given signal
independent of each other. FM decomposition is done using
zero crossing intervals and AM decomposition is done using
local peaks information. From the experiments it is shown that
AM estimation is not affected by the modulation index of the
FM and FM estimation is not effected by the modulation index
of the AM.
The earlier approaches for AM-FM decomposition from our
group has utilized ZC and level crossing(LC) information[2]
[13] directly whereas this paper proposes ZC intervals. Other
approaches in the literature use ZC interval statistics[12], but
not ZC intervals, which has certain advantages.
II. AM-FM D ECOMPOSITION
USING
ZC-LP
Let the AM-FM signal be
x(t) = A(t)sin(Φ(t))
(1)
where Φ(t) is the phase component and A(t) is the amplitude variation. The frequency of the signal is given by
1 d
f (t)= 2π
dt Φ(t).
To represent natural wideband signals such as speech, music,
or biomedical signals, it is convenient to decompose them into
narrow-band signals using a perfect reconstruction filter bank
and then represent each filter output as an AM-FM signal.
Let s(t) be a wideband which can be represented as the linear
x(t) = A(t)sin(Φ(t)),
where A(t) is the amplitude variation, generally given by
A(t) = (1 + µ ma (t)). Here µ is the modulation index
of AM and ma (t) modulating signal for AM. For all t,
|µma (t)| < 1 i.e. A(t) > 0. Similarly IF of the signal is given
1 dΦ(t)
by f (t) = 2π
dt = fc + β mf (t) where β is the modulation
index of FM and mf (t) is the FM modulation component.
Again, let |βmf (t)| < fc . Also, let ma (t) be slowly varying
compared to min [f (t)].
F = {f (t = τi ) =
1
2(ti+1 −ti ) , 0 ≤ i
(ti +ti+1 )
}.
2
≤ L − 1, where τi =
In continuous domain we can say that there is a zero crossing
when x(t) = 0, but in our case we are dealing with a discrete
signal x[nTs ] where Ts is the sampling period. In order to find
the ZC location we have used signal sign changes between
successive samples i.e. X[nTs ]X[(n + 1)Ts ] < 0. After that
th
we interpolated the signal between nth and (n + 1) samples
to a finer resolution using sinc interpolation for finding the ZC
instant.
Now we have a discrete set of frequencies at some nonuniform instants of time. In order to find the IF at any instant t
we have to interpolate the non-uniform samples of f (t). To
interpolate the data we can use the least squares approach.
To retain the generality of different applications, we make a
smooth local polynomial approximation of f (t) of aP
fixed order
P
p .i.e., we can write f(t) at any instant t as f (t) = p=0 cp tp .
By minimizing the cost function
PL−1
2
S(c) = L1 i=0 kfi − CTi k
where F is set of all L frequencies fi at discrete time instants.
′
C = [c0 , c1 ....cP ] and Ti = [1, ti , t2i ...tP
i ] . Minimizing the
cost function S(c) with respect to C will give us optimum
¯ ¯
Estimated IF Signal
where L is the total number of ZC’s in a given interval 0 ≤ t ≤
T signal. Now from the above discussion of sampling of f (t),
we can choose the IF between two consecutive zero crossings
ti and ti+1 be approximated by fi := 2(ti+11 −ti ) [11]. This
frequency can be assigned anywhere in the half cycle interval
and we found it empirically that assigning it to the middle of
the interval gave the best approximation for the time varying
IF. When we assigned this frequency to the left ZC we got a
delay in estimated IF with respect to the original IF and when
we assigned it to the other end we found a delay in the other
direction. By computing the frequency values between all the
zero crossing intervals we have a set of non-uniform samples
of IF function corresponding to each ZC interval. Let F be the
set of IF samples:
0.22
0.2
0.18
0.05
0.04
0.03
0.02
0.01
0.16
0
100
200
300
400
0
0
500
100
Sample Index
200
300
400
500
400
500
Sample Index
−4
−5
x 10
4
5
Error in IF estimation
TZ = {ti , 0 ≤ i ≤ L, where x(ti ) = 0}
0.06
0.24
Estimated IF Signal
For the simple case of a sine wave sin(2πf t), where f is
the frequency of the sine wave, the time interval between two
1
consecutive zero crossings τ is given by τ = 2f
. With A.M
because the envelope A(t) is slowly varying and is positive, it
does not affect the ZC’s. Thus, each successive ZC interval can
be interpreted as due to only the IF f (t). The ZC’s provide a
sampled information of f(t).
For the given signal x(t) let TZ be the set of all successive
zero crossing instants, i.e.,
values of C ∗ given by C ∗ = T + F (T + is pseudo inverse).
and T is the matrix whose ith column is Ti . After obtaining
the optimum solution for coefficients of the polynomial we can
get the predicted value of fb(t) at any time instant t.
Error in IF estimation
A. FM Computation
0
−5
100
200
300
400
x 10
2
0
−2
−4
−6
−8
0
500
100
Sample Index
200
300
Sample Index
Fig. 1. The plot of the estimated IF signal and Error signals.In 1st case the
IF signal is a sin wave give by f (t) = 0.2 + 0.045sin(0.0565n + π4 ) and in
2nd case the IF signal is given by f (t) =
and N=512 in this case.
.7
2π
∗ (.25 + .25 ∗
(n−N/2+1)3
(N
)
2
3
)
We can use the least square solution for interpolating the IF
only when we know the prior information about the smoothness
of the modulating signal. If we do not know the prior information about the smoothness of the signal then in that case
we cannot use the least squares solution. In that case we have
to go for either K Nearest Neighbor(K-NN)[2] or splines as
a solution for interpolation. In K-NN we will take K nearest
points surrounding t, where we want to calculate IF and fit a
polynomial locally in that region. In our experiments we have
used K=11 and P=3.
B. AM computation
Given an amplitude modulated signal (AM), one way of getting back the modulating signal is to pass it through a low pass
filter[8], which will output the envelope of the signal. Envelope
of a signal can also be approximated by the curve passing
through all the local peaks of the modulated carrier wave (i.e.
AM signal). Similarly joining all negative peaks(valleys) of the
signal will also give an envelope which is exactly negative of
= 0|t=ti }
1.3
1.25
Estimated AM Signal
Estimated AM Signal
1.6
1.4
1.2
1
0.8
0.6
1.2
1.15
1.1
1.05
1
0.4
0.2
0
100
200
300
400
500
0.95
0
100
Sample Index
200
300
400
500
Sample Index
−3
−4
x 10
x 10
1.5
1
Error in IA estimation
TP = {ti , 0 ≤ i ≤ Lp , where
d
dx x(t)
(b)
(a)
Error in IA estimation
the modulating signal. We can consider the negative of the
local valleys as additional samples of the envelope function,
enhancing the estimation accuracy and transient nature of the
AM function.
In the ZC-LP method, we calculate the local peaks of the
given AM signal. For finding local peaks of the signal, we
are finding the first difference of the signal and checking for
the sign changes to determines ZC’s of the first derivative.
If there is a sign change at ith sample that means there is
th
th
a peak between (i − 1) and (i + 1) samples. After finding
all intervals where there is a peak, we interpolate the difference
signal to a finer resolution to determine the exact location of
the peak and the value of the peak.
0.5
0
−0.5
−1
5
0
−5
−1.5
where TP is set of time instant, where there is a local peak
and LP is the number of local peaks in the signal interval.
−2
0
100
200
300
400
500
0
Sample Index
100
200
300
400
500
Sample Index
XP = {x(ti ), 0 ≤ i ≤ Lp , ti ǫTP }
We have a non-uniform sampled version of the envelope XP
at TP . To find the IA at any instant t, we can use either KNN or splines. In our experimentation we have used the same
method which we used in the computation of IF and computed
b
A(t).
For the experimentation we have chosen K=11 and the
order of polynomial p=3. We can see from the results that the
estimation error in estimation of AM is quite low of the order
of 10−4 .
The usual method for envelope estimation is through coherent demodulation which utilizes the IF component of the signal
x(t). But, here we are not using any information about the IF
of the signal and computing the IA of the signal independently.
Thus, are computing both IA and IF independently from the
ZC’s of the signal and its derivative. In the next section we show
that the performance of IA estimation is not effected much by
the parameters of FM and estimation of IF is not effected by
the parameters of the AM, when compared to other techniques
in the literature.
III. PERFORMANCE COMPARISON WITH DESA
The performance of the new ZC-LP based AM-FM estimation is compared with the DESA1 which uses Teager Energy
Operator (TEO) for computing AM and FM components.
The AM-FM signal is given by
R
x(t) = (1 + µma (t))sin(fc t + β mf (t)dt)
For comparison we have taken an AM-FM signal with both
modulating signals ma and mf as sinusoidal signals. We studied the effect on performance by varying µ ,β , parameters of
ma (frequency of the modulating signal) and carrier frequency
fc . We estimated the signals A(t) and f (t) using both the
methods DESA1 and the ZC-LP method. While calculating
Fig. 2. The plot of estimated IA and error in estimation.In first case we have
used a sine wave as modulation signal give as A(t) = 1 + sin(0.0471n)
and in second case the envelope signal is given by A(t) = 1 +
N=512 in this case.
(n− N
)
2
N2
2
and
AM and FM using DESA1 if there are any imaginary term we
have replaced them using real value neglecting the imaginary
part.We have taken mean square error(MSE) as a measure of
performance. The MSE of AM estimation is calculated by :
ξAM
1
=
N − 2Q + 1
N −Q+1
X
n=Q+1
2
b
(A(n) − A(n))
(3)
In our simulation we have used Q=24. As error in the edges
will be high we have excluded those regions for computation
of the ξAM . A similar measure is defined for computing MSE
ξF M of FM also .
First we have studied the performance of AM and FM
estimation by varying the parameters of AM µ and fAM . From
Fig, 3 we can see that the performance of ZC-LP is consistently
better than the performance of DESA1 for all variations in µ
and fAM . We can see that estimation of FM is not affected
by the AM parameters, the error in estimation of FM ξF M
is almost constant in the ZC-LP method, whereas in DESA1
estimation of FM is effected by the AM parameters. The
estimation error in FM increased with increase in modulation
index of AM (µ) and frequency fAM of the modulating wave.
This is because for estimating FM in ZC-LP method we have
used ZC’s, which are not effected by AM of the signal as
discussed in the previous sections.
The performance of the estimation of AM and FM is studied
with variations in fc and β. As shown in the Fig 4 we can
ZC−LP
DESA−1
−55
−10
−40
−60
−20
−65
−30
−60
−75
−70
0.8
−80
−70
ω
AM
0.06
0.4
(rad/s)
µ
−60
ξFM
ξ
FM
−70
−80
−90
0.8
0.6
0.4
µ
0.2
0.02
0.04
ω
AM
0.06
(rad/s)
0.08
0.2
0.02
0.04
ω
AM
0.06
4
(rad/s)
β
−50
−50
−60
−60
−70
−90
−100
µ
0.2
0.02
0.04
ω
AM
0.06
infer that ZC-LP is consistently performing better than DESA1
through out the range of fc and β. At the same time we
can see that the performance of AM estimation is also doing
consistently better than DESA1. We can see as fc is decreasing
the performance of DESA1 in estimating FM is degrading
fast, whereas the performance of the ZC-LP method is hardly
affected.
ω
FM
(rad/s)
0.15
2 0.1
β
ω
FM
(rad/s)
−50
−60
−70
−80
6
0.2
4
(rad/s)
Fig. 3.
The performance comparison of ZC-LP and DESA1 in case
of estimation of IA and IF with respect to AM modulation index µ and
Frequency of modulating sinusoidal wave.The AM modulating wave is given
by ma = 1 + µsin(ωAM t)
2 0.1
6
0.08
0.2
4
−80
−90
0.8
0.4
6
0.15
−70
−80
0.6
−60
0.2
FM
0.2
0.04
−50
6
0.08
−40
ξ
µ
0.02
0.6
FM
0.4
0.08
ξ
0.6
β
0.2
4
0.15
2 0.1
ω
FM
(rad/s)
0.15
2 0.1
β
ω
FM
(rad/s)
Fig. 4.
The performance comparison of ZC-LP and DESA1 in case of
estimation of IF and IA with respect to FM modulation index β and Carrie
n
frequency fc .The phase of the am-fm wave is given as ωf m +βsin(π 100
+ π4 )
Bandpass Filtered speech signal and envelope
80
speech signal
60
Envelope
40
Signal value
0.8
−70
ξ
AM
ξ
ξAM
AM
ξ
−60
−50
AM
−30
−40
−50
DESA−1
ZC−LP
20
0
−20
−40
−60
−80
0
50
100
150
200
250
300
350
400
450
500
Sample Index
IV. APPLICATION OF NEW MODEL TO NATURAL
SIGNALS
V. C ONCLUSION
We propose a new zero crossings based IF estimation and
local peaks based IA estimation of a band limited signal.From
the experimental results, we can see that for a clean signal
the estimation error is of the order of -70 to -80 dB. This
ZC−LP
Hilbert Method
4000
IF value
We tried applying the ZC-LP method for estimation of IA
and IF of natural occurring signals like speech, and music. We
took a bandpass filtered(BPF) signal multiplied by a trapezoidal
window of length N=512. As we know when a speech wave is
passed through a BPF the resulting signal will be of AM-FM
structure[4], [2]. For this experimentations as we know that the
frequency variations are high we used k=5 and p=3 (for K-NN).
As we can see from the Fig 5, 6 small variations in frequency
are captured by the ZC-LP in great detail because ZC’s will
capture all variations in frequency very closely. Amplitude
variations are also captured in good detail.
Estimated IF Using ZC method and Hilbert
5000
3000
2000
1000
0
0
50
100
150
200
250
Sample Index
300
350
400
450
500
Fig. 5. New Algorithm when applied on a BPF speech signal,sampled at
16kHz. When passband is 0.25 to 0.45(normalized frequency).
performance is much better than the existing methods like
DESA1,DESA2. We have also seen that this method can
be applied to natural signals like speech and music signal
also. AM-FM decomposition can be used for analysis of and
synthesis of speech signals[10].
Bandpass Filtered flute signal and envelope
Flute signal
100
Envelope
Signal value
50
0
−50
−100
0
50
100
150
200
250
Sample Index
300
350
400
450
500
Estimated IF Using ZC method and Hilbert
6000
ZC−LP
Hilbert Method
IF value
5000
4000
3000
2000
1000
0
50
100
150
200
250
Sample Index
300
350
400
450
500
Fig. 6. New Algorithm when applied on a BPF Music signal,sampled at
16kHz. When passband is 0.3 to 0.45 (normalized frequency).
R EFERENCES
[1] J.F. Kaiser P.Maragos and T.F. Quatieri, Energy separation in
signal modulations with application to speech analysis, IEEE
Trans. Signal Proc., vol. 41, pp. 30243051, Oct 1993.
[2] S. Chandra Sekhar and T.V. Sreenivas, “Novel approch to amfm
decomposition with application to speech and music analysis,“
Proc. IEEE ICASSP-04, vol. 2, pp. 7536, may 2004.
[3] B. Boashash, “Estimating and interpreting the instantaneous frequency of a signal,” Proc. IEEE, vol. 80, pp. 519568, April 1992.
[4] J.F. Kaiser, P. Maragos and T.F. Quatieri, “Speech nonlinearities,modulations, and energy operators,“ Proc. IEEE ICASSP,vol.
1, pp. 421424, May 1991.
[5] Bishwarup Mondal and T.V. Sreenivas “Mixture gaussian envelope
chirp model for speech and audio” in Proc. IEEE ICASSP-01,Vol.
1, pp. 857-860, May 2001.
[6] Doh-Suk Kim Soo-Young Lee R.M.Kil “Auditory processing of
speech signals for robust speech recognition in real-world noisy
environments”IEEE Transaction on Speech and Audio Processing,vol. 7, no. 1, pp. 55-69, 1999.
[7] T.E. Hanna, T.F. Quatieri and G.C. OLeary, “Am-fm separation
using auditory-motivated filters,” IEEE Trans on Speech and
Audio Proc., vol. 5, pp. 465480, Sep 1997.
[8] George kennedy and R.W.Tinnell, , in Electronic communications
systems. McG-Kog, 1991, vol. 3rd edition.
[9] J.K Gupta, S. Chandra Sekhar and T.V. Sreenivas, “Performance
analysis of am-fm estimators,” TENCON, vol. 3, pp.954958, Oct
2003.
[10] A. Potamianos and P.Maragos, “speech analysis and synthesis
using an am-fm modulation model,” Speech Comm., vol. 28, pp.
195209, July 1999.
[11] Sharif, Z. Zainal, M.S. Sha’ameri, A.Z. Salleh, S.H.S. “Analysis
and classification of heart sounds and murmurs based onthe
instantaneous energy and frequency estimations” Proc. IEEE TENCON pp. 130-134 vol.2 2000.
[12] T. V. Sreenivas and R. J. Niederjohn, “Formant contour estimation in noise: comparison of a zero-crossing based measure
with other spectral estimates,” Proc. IEEE Int. Conf. Svst. Enara.,
Dayton, Ohio, pp. 399-402, August,1989.
[13] S. Chandra Sekhar and T. V. Sreenivas, Polynomial instantaneous frequency estimation using level-crossing information, in
Proc. IEEE-EURASIP Workshop on Nonlinear Signal and Image
Processing, Grado, Italy, June 2003.