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 In- stantaneous 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. INTRODUCTION
All naturally occurring signals such as speech, biomedical signals, music, etc., are non-stationary in nature. The non- stationarity of these signals are mainly due to their respective production phenomena. The temporal variability of these sig- nals 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
x(t) =A(t)sin(Φ(t)) (1)
where Φ(t) is the phase component and A(t) is the am- plitude variation. The frequency of the signal is given by f(t)=2π1 dtdΦ(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
combination of AM-FM signals:
S(t) = XK
k=1
Ak(t)sin(Φk(t)) (2) where K is number of band pass filters in the filter bank.
Given a bandpass AM-FM signalxk(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 DECOMPOSITION USINGZC-LP Let the AM-FM signal be
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 byf(t) = 2π1 dΦ(t)dt =fc+β mf(t)whereβ is the modulation index of FM and mf(t) is the FM modulation component.
Again, let |βmf(t)| < fc. Also, letma(t) be slowly varying compared to min [f(t)].
A. FM Computation
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 consecutive zero crossings τ is given by τ = 2f1. 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 signalx(t)letTZ be the set of all successive zero crossing instants, i.e.,
TZ={ti,0≤i≤L, where x(ti) = 0}
where L is the total number of ZC’s in a given interval0≤t≤ T signal. Now from the above discussion of sampling off(t), we can choose the IF between two consecutive zero crossings ti and ti+1 be approximated by fi := 2(t 1
i+1−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. LetF be the set of IF samples:
F={f(t=τi) = 2(ti+11−ti),0≤i≤L−1, where τi =
(ti+ti+1) 2 }.
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 signalx[nTs]whereTsis 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 we interpolated the signal betweennthand(n+ 1)thsamples to a finer resolution using sinc interpolation for finding the ZC instant.
Now we have a discrete set of frequencies at some non- uniform instants of time. In order to find the IF at any instantt 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 off(t)of a fixed order p.i.e., we can write f(t) at any instant t asf(t) =PP
p=0cptp. By minimizing the cost function
S(c) = L1PL−1
i=0 kfi−CTik2
where F is set of allL frequenciesfi at discrete time instants.
C = [c0, c1....cP] and Ti = [1, ti, t2i...tPi ]′. Minimizing the cost function S(c) with respect to C
¯ w
¯ill give us optimum
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 offb(t)at any time instant t.
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Estimated IF Signal
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Estimated IF Signal
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x 10−4
Sample Index
Error in IF estimation
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Error in IF estimation
Fig. 1. The plot of the estimated IF signal and Error signals.In1stcase the IF signal is a sin wave give byf(t) = 0.2 + 0.045sin(0.0565n+π4)and in 2ndcase the IF signal is given byf(t) = 2π.7 ∗(.25 +.25∗(n−N/2+1)
3 (N2)3 ) and N=512 in this case.
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 infor- mation 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 get- ting 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
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 a peak between(i−1)thand(i+ 1)th 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.
TP={ti,0≤i≤Lp, where dxd x(t) = 0|t=ti} where TP is set of time instant, where there is a local peak andLP is the number of local peaks in the signal interval.
XP={x(ti),0≤i≤Lp, tiǫTP}
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 K- NN or splines. In our experimentation we have used the same method which we used in the computation of IF and computed A(t). For the experimentation we have chosen K=11 and theb 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 of10−4.
The usual method for envelope estimation is through coher- ent 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 signalindependently.
Thus, are computing both IA and IFindependentlyfrom 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 estima- tion is compared with the DESA1 which uses Teager Energy Operator (TEO) for computing AM and FM components.
The AM-FM signal is given by
x(t) = (1 +µma(t))sin(fct+βR
mf(t)dt)
For comparison we have taken an AM-FM signal with both modulating signalsma andmf as sinusoidal signals. We stud- ied the effect on performance by varyingµ,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
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0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
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Estimated AM Signal
(a)
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0.95 1 1.05 1.1 1.15 1.2 1.25 1.3
Sample Index
Estimated AM Signal
(b)
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x 10−3
Sample Index
Error in IA estimation
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x 10−4
Sample Index
Error in IA estimation
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 byA(t) = 1 +(n−
N 2)2 N2 and N=512 in this case.
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
(A(n)−A(n))b 2 (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 theξAM. 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 AMµandfAM. 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 frequencyfAM 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
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ωAM (rad/s) ZC−LP
µ ξAM
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ξFM
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ωAM (rad/s) DESA−1
µ ξAM
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ωAM (rad/s) µ
ξFM
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 byma= 1 +µsin(ωAMt)
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 asfcis decreasing the performance of DESA1 in estimating FM is degrading fast, whereas the performance of the ZC-LP method is hardly affected.
IV. APPLICATION OF NEW MODEL TO NATURAL SIGNALS
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.
V. CONCLUSION
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
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ωFM (rad/s) ZC−LP
β ξAM
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ξFM
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β ξAM
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ωFM (rad/s) β
ξFM
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 frequencyfc.The phase of the am-fm wave is given asωf m+βsin(π100n +π4)
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Bandpass Filtered speech signal and envelope
speech signal Envelope
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IF value
Estimated IF Using ZC method and Hilbert
ZC−LP Hilbert Method
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].
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Signal value
Bandpass Filtered flute signal and envelope
Flute signal Envelope
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IF value
Estimated IF Using ZC method and Hilbert
ZC−LP Hilbert Method
Fig. 6. New Algorithm when applied on a BPF Music signal,sampled at 16kHz. When passband is 0.3 to 0.45 (normalized frequency).
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