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 52 Phone: +9 8 23 27, Fax: +9 8 23 83. 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], DESA, DESA2[]. I. INTRODUCTION 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[]. Let an AM-FM modulated signal be represented by x(t) = A(t)sin(Φ(t)) () where Φ(t) is the phase component and A(t) is the amplitude variation. The frequency of the signal is given by f(t)= d 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 combination of AM-FM signals: S(t) = K A k (t)sin(φ k (t)) (2) k= where K is number of band pass filters in the filter bank. Given a bandpass AM-FM signal x k (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 DESA and DESA2 which use teager energy operator[]. 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] [3] directly whereas this paper proposes ZC intervals. Other approaches in the literature use ZC interval statistics[2], but not ZC intervals, which has certain advantages. II. AM-FM DECOMPOSITION USING ZC-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) = ( + m a (t)). Here is the modulation index of AM and m a (t) modulating signal for AM. For all t, m a (t) < i.e. A(t) >. Similarly IF of the signal is given by f(t) = dφ(t) 2π dt = f c + m f (t) where is the modulation index of FM and m f (t) is the FM modulation component. Again, let m f (t) < f c. Also, let m a (t) be slowly varying compared to min [f(t)].
A. FM Computation For the simple case of a sine wave sin(2πft), where f is the frequency of the sine wave, the time interval between two 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 T Z be the set of all successive zero crossing instants, i.e., T Z = {t i, i L, where x(t i ) = } where L is the total number of ZC s in a given interval t T signal. Now from the above discussion of sampling of f(t), we can choose the IF between two consecutive zero crossings t i and t i+ be approximated by f i := 2(t i+ t i) []. 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: F = {f(t = τ i ) = 2(t i+ t i), i L, where τ i = (t i+t i+) 2 }. In continuous domain we can say that there is a zero crossing when x(t) =, but in our case we are dealing with a discrete signal x[nt s ] where T s is the sampling period. In order to find the ZC location we have used signal sign changes between successive samples i.e. X[nT s ]X[(n + )T s ] <. After that we interpolated the signal between n th and (n + ) th 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 a fixed order p.i.e., we can write f(t) at any instant t as f(t) = P p= c pt p. By minimizing the cost function S(c) = L L i= f i CT i 2 where F is set of all L frequencies f i at discrete time instants. C = [c, c...c P ] and T i = [, t i, t 2 i...tp i ]. Minimizing the cost function S(c) with respect to ill give us optimum C w values of C given by C = T + F (T + is pseudo inverse). and T is the matrix whose i th column is T i. After obtaining the optimum solution for coefficients of the polynomial we can get the predicted value of f(t) at any time instant t. Estimated IF Signal Error in IF estimation 2.8. 2 3 5 5 5 x 2 3 5 Estimated IF Signal Error in IF estimation..5..3.2. 2 3 5 2 2 x 5 8 2 3 5 Fig.. The plot of the estimated IF signal and Error signals.in st case the IF signal is a sin wave give by f(t) = +.5sin(.55n + π ) and in 2 nd case the IF signal is given by f(t) =.7 (n N/2+)3 (.25 +.25 ) 2π ( N 2 and N=52 in this case. )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= 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
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 i th sample that means there is a peak between (i ) th and (i + ) 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. T P = {t i, i L p, where d dx x(t) = t=t i } where T P is set of time instant, where there is a local peak and L P is the number of local peaks in the signal interval. X P = {x(t i ), i L p, t i ǫt P } We have a non-uniform sampled version of the envelope X P at T P. 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 Â(t). For the experimentation we have chosen K= 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. 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 DESA which uses Teager Energy Operator (TEO) for computing AM and FM components. The AM-FM signal is given by x(t) = ( + m a (t))sin(f c t + m f (t)dt) For comparison we have taken an AM-FM signal with both modulating signals m a and m f as sinusoidal signals. We studied the effect on performance by varying,, parameters of m a (frequency of the modulating signal) and carrier frequency f c. We estimated the signals A(t) and f(t) using both the methods DESA and the ZC-LP method. While calculating Estimated AM Signal Error in IA estimation...2.8.. (a) 2 3 5 x 3.5.5.5.5 2 2 3 5 Estimated AM Signal.3.25.2.5..5 (b).95 2 3 5 Error in IA estimation 5 5 x 2 3 5 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) = + sin(.7n) and in second case the envelope signal is given by A(t) = + (n N 2 )2 N 2 and N=52 in this case. AM and FM using DESA 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 : = N Q+ (A(n) N 2Q + Â(n))2 (3) n=q+ In our simulation we have used Q=2. As error in the edges will be high we have excluded those regions for computation of the. A similar measure is defined for computing MSE of FM also. First we have studied the performance of AM and FM estimation by varying the parameters of AM and f AM. From Fig, 3 we can see that the performance of ZC-LP is consistently better than the performance of DESA for all variations in and f AM. We can see that estimation of FM is not affected by the AM parameters, the error in estimation of FM is almost constant in the ZC-LP method, whereas in DESA estimation of FM is effected by the AM parameters. The estimation error in FM increased with increase in modulation index of AM () and frequency f AM 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 f c and. As shown in the Fig we can
DESA DESA 5 7.8.....2 ω AM.8 3 5 7.8...2...8 55 5 7 75 8 2..5 2 3 5 2..5 5 5 5 7 8 7 8 7 8 9 7 9.8...2...8 9.8...2...8 2..5 8 2..5 Fig. 3. The performance comparison of ZC-LP and DESA in case of estimation of IA and IF with respect to AM modulation index and Frequency of modulating sinusoidal wave.the AM modulating wave is given by m a = + sin( t) Fig.. The performance comparison of ZC-LP and DESA in case of estimation of IF and IA with respect to FM modulation index and Carrie frequency f c.the phase of the am-fm wave is given as ω fm +sin(π n + π ) infer that ZC-LP is consistently performing better than DESA through out the range of f c and. At the same time we can see that the performance of AM estimation is also doing consistently better than DESA. We can see as f c is decreasing the performance of DESA 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=52. As we know when a speech wave is passed through a BPF the resulting signal will be of AM-FM structure[], [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, 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 -7 to -8 db. This Signal value IF value 8 2 2 Bandpass Filtered speech signal and envelope speech signal Envelope 8 5 5 2 25 3 35 5 5 5 3 2 Estimated IF Using ZC method and Hilbert 5 5 2 25 3 35 5 5 Hilbert Method Fig. 5. New Algorithm when applied on a BPF speech signal,sampled at khz. When passband is 5 to.5(normalized frequency). performance is much better than the existing methods like DESA,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[].
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