Adaptive Window Zero-Crossing-Based Instantaneous Frequency Estimation

We address the problem of estimating instantaneous frequency (IF) of a real-valued constant amplitude time-varying sinusoid. Estimation of polynomial IF is formulated using the zero-crossings of the signal. We propose an algorithm to estimate nonpoly-nomial IF by local approximation using a low-order polynomial, over a short segment of the signal. This involves the choice of window length to minimize the mean square error (MSE). The optimal window length found by directly minimizing the MSE is a function of the higher-order derivatives of the IF which are not available a priori. However, an optimum solution is formulated using an adaptive window technique based on the concept of intersection of conﬁdence intervals . The adaptive algorithm enables minimum MSE-IF (MMSE-IF) estimation without requiring a priori information about the IF. Simulation results show that the adaptive window zero-crossing-based IF estimation method is superior to ﬁxed window methods and is also better than adaptive spectrogram and adaptive Wigner-Ville distribution (WVD)-based IF estimators for di ﬀ erent signal-to-noise ratio (SNR).


INTRODUCTION
Almost all information carrying signals are time-varying in nature. The adjective "time-varying" is used to describe an "attribute" of the signal that is changing/evolving in time [1]. For most signals such as speech, audio, biomedical, or video signals, it is the spectral content that changes with time. These signals contain time-varying spectral attributes which are a direct consequence of the signal generation process. For example, continuous movements of the articulators, activated by time-varying excitation, is the cause of the timevarying spectral content in speech signals [2,3]. In addition to these naturally occurring signals, man-made modulation signals, such as frequency-shift keyed (FSK) signals used for communication [4] carry information in their time-varying attributes. Estimating these attributes of a signal is important both for extracting their information content as well as synthesis in some applications.
Typical attributes of time-varying signals are amplitude modulation (AM), phase/frequency modulation (FM) of a sinusoid. Another time-varying signal model is the output of a linear system with time-varying impulse response. However, the simplest and fundamental signal processing model for time-varying signals is an AM-FM combination [5,6,7] of the type s(t) = A(t) sin(φ(t)). Further, if the amplitude does not vary with time, the signal is simplified to s(t) = A sin(φ(t)). Estimating the IF of such signals is a well-studied problem with limited performance for arbitrary IF laws and low SNR conditions [8,9].
In [9], a novel auditory motivated level-crossing approach has been developed for estimating instantaneous frequency (IF) of a polynomial nature, that is, the instantaneous phase (IP), φ(t) is of the form φ(t) = p k=0 a k t k and the IF is given by f (t) = (1/2π)(dφ(t)/dt). In this paper, we address the estimation of IF of nonpolynomial nature of monocomponent phase signals, in the presence of noise, using zero-crossings of s(t). We achieve this by performing local polynomial approximation to the IF using the zero-crossings (ZCs). This involves the choice of optimum window length to minimize the mean square error (MSE). The minimum MSE (MMSE) formulation gives rise to an optimum window length solution which requires a priori information about the IF. Also, the length of the window introduces a "bias-variance tradeoff " which is resolved using an adaptive approach [10,11,12] based on the intersection of confidence intervals of the zero-crossing-based IF (ZC-IF) estimator.
Fundamental contributions related to the ZCs of amplitude and frequency modulated signals were made in [13], wherein the factorization of an analytic signal in terms of real and complex time-domain zeros was proposed. A model based pole-zero product representation of an analytic signal was proposed in [14]. Recent contributions include the use of homomorphic signal processing techniques for factorization of real signals [15]. In contrast to these, we use the real ZCs of the signal s(t) that can be directly estimated from its samples. The use of zero-crossing (ZC) information is a nonlinear approach to estimating IF; this has been reported earlier using either ZC rate information [8,16] or ZC interval histogram information [17,18] (in the context of speech signals). These earlier approaches are quasistationary and are inherently limited to estimating only mild frequency variations. The new approach developed in this paper fits a local, nonstationary model for the IF and uses the ZC instant information [19] for IF estimation.
This paper is organized as follows. In Section 2, we formulate the problem. In Section 3, we discuss ZC-based polynomial IF estimation and the need for local polynomial approximation for nonpolynomial IF estimation. Bias and variance of the ZC-IF estimator are derived in Section 4. The problem of optimal window length selection is addressed in Section 5 and an adaptive algorithm is discussed in Section 6. Simulation results are presented in Section 7. Section 8 concludes the paper.

IF ESTIMATION PROBLEM
Let s(t) = A sin(φ(t)) be the phase signal with constant amplitude and IF 1 is given by f (t) = (1/2π)(dφ(t)/dt). Let the frequency variation be bounded, but arbitrary and unknown. The signal s(t) has strictly infinite bandwidth, but we assume that it is essentially bandlimited to [−Bπ, Bπ]. Let s(t) be corrupted additively with Gaussian noise, w(t), which has a flat power spectral density, S ww (ω) = σ 2 w for |ω| ≤ Bπ and zero elsewhere. w(t) is therefore bandlimited in nature. However, samples taken from this process at a rate of B samples/second are uncorrelated. Let the noisy signal be denoted by where N is the number of discrete-time observations. The noise w[n] is white Gaussian with a variance σ 2 w . The signal-tonoise ratio (SNR) is defined as SNR = A 2 /2σ 2 w . The problem is to estimate the IF of the signal s(t) using the samples y[n] and estimating the ZCs of the signal, y(t). Negative IF is only conceptual; naturally occurring IF is always positive and hence we confine our discussion to positive IF. 1 It must be noted that this definition of IF is different from that obtained using the Hilbert transform.

ZC-IF ALGORITHM
Let the ZCs of the noise-free signal, s(t) = A sin(φ(t)), t ∈ [0, T], be given by Z = {t j | s(t j ) = 0; j = 0, 1, 2, . . . , Z}, where Z + 1 is the number of ZCs of s(t) in [0, T]. Correspondingly, the values of the phase function φ(t) are given by P = { jπ; j = 0, 1, 2, . . . , Z}. The phase value corresponding to the first ZC over [0, N − 1] has been arbitrarily assigned to 0. This does not affect IF estimation because of the derivative operation. If the phase function φ(t) is a polynomial of order p, 0 < p < Z, of the form, φ(t) = p k=0 a k t k , then, up to an additive constant, it can be uniquely recovered from the set of ZC instants Z. This property of uniqueness eliminates the need for a Hilbert transform based definition of IF. Corresponding to each ZC instant, t j , we have an equation The set of (Z + 1) equations, in general, is more than the number of unknowns, p, and in the absence of ZC estimation errors, they are consistent. Due to arbitrary assignment of φ(t 0 ) to 0, the coefficient estimate of a 0 will be in error; however, this does not affect the IF estimate and the IF can be recovered uniquely.
In practice, since ZC instants of s(t) have to be estimated using s [n], there is a small, nonzero error. 2 In such a case, the coefficient vector, a = {a k , k = 0, 1, 2, . . . , p} can be estimated by minimizing the cost function C p (a) defined as where a = a k , k = 0, 1, 2, . . . , p , e j = 1 t j t 2 j · · · t p j T (2) (T stands for transpose operator). The optimum coefficient vector is obtained in a straightforward manner as where Φ is a column vector whose jth entry is jπ and H is a matrix whose jth row is e j T . a = a 0 a 1 a 2 · · · a p . At the sample instants, the IF is estimated as We refer to this as the ZC-IF estimator.

Performance of the ZC-IF estimator
To illustrate the performance of the ZC-IF algorithm, 256 samples of a quadratic IF signal were generated. The ZC instants were estimated using 10 iterations, each through the root-finding approach. The actual and the estimated IF corresponding to p = 3 are shown in Figures 1a, 1b, IF estimation error: It must be noted that C p ( a) can be computed using the ZC information, whereas J p ( a) can be computed only when the actual IF, f [n], is known. Also, while C p ( a) is a nonincreasing function of p, J p ( a) need not be. These error measures are plotted in Figures 1c, 1d, respectively. From Figure 1c, it is clear that beyond p = 3 (cubic phase fitting or equivalently, quadratic IF), the error reduction is not appreciable. Thus, a measure of saturation of the IP fitting error can be used for order selection. The algorithm works best when the actual IF and the as-sumed IF model are matched, that is, the underlying IF is a polynomial and the assumed IF model is also a polynomial of the same order. However, when there is a mismatch, that is, the underlying IF is not a polynomial but we approximate it using polynomials, the following problems arise.
(1) The choice of the order of the polynomial becomes crucial. A value of p that keeps the IP fitting error below a predetermined threshold does not necessarily yield the minimum IF estimation error. This problem occurs even when the underlying IF is a polynomial of unknown order as demonstrated in Figures 1c, 1d. (2) Not all kinds of IF variations can be approximated by finite-order polynomials to a desired degree of accuracy.
(3) Fast IF variations in a given interval require very high polynomial orders and hence large amounts of data. However, this can often lead to numerically unstable set of equations in solving for the coefficients of the polynomial yielding erroneous and practically useless IF estimates.
A natural modification of the ZC-IF algorithm is to perform local polynomial fitting, that is, use lower-order polynomial functions to locally estimate the IF rather than use one large order polynomial over the entire observation window. If we always use a fixed low order polynomial, say p = 3, we are still faced with the question of window length selection; that is, over what window length should a local polynomial approximation be performed? An algorithm that helps us choose the appropriate window length should have the following features: (1) require no a priori information about the IF, (2) yield an IF estimate with the MMSE for all values of SNR.
The objective of this paper is to develop such an algorithm. The relevant cost function is the MSE [22] of the estimate f , of the quantity f , defined as where E denotes the expectation operator. MSE can be rewritten as The first term is the squared bias and the second term is the variance of the ZC-IF estimator. In the following sections, we obtain the bias and variance of the ZC-IF estimator and develop the algorithm.
At the ZCs, the noisy signal Thus, under a high SNR assumption, one can approximate the effect of additive noise on the signal samples to have an additive phase noise effect [23].
Let t ∈ [0, T] be the point where the IF estimate is desired. The basic principle in the new approach to IF estimation is to fit a polynomial, locally, to the ZCs and IP values within an interval L about the point t. The IF is obtained by the derivative operation. We use a rectangular window symmetric about t, that is, choose the window function, as h(τ) = 1/L for τ ∈ [−L/2, +L/2] and zero elsewhere. The window function is normalized to have unit area. Define the set I t,L = {τ | t − L/2 ≤ τ ≤ t + L/2} which is the set of all points within the L-length window centered at τ = t.

Consider the quadratic cost function
The coefficients {a 0 , a 1 , a 2 , . . . , a p } are specific to the time instant t and can be obtained as the minimizers of the above quadratic cost function. The optimal coefficient estimates are denoted by { a 0 , a 1 , a 2 , . . . , a p } and defined as In other words, a is a solution to ∂C/∂a = 0 or, equivalently, ∂C/∂a | a = a = 0. We have The estimation error, ∆a = a − a , is due to the following: (1) error due to additive noise, δw, (2) error due to mismatch between the actual phase and the estimated phase using a local polynomial model (residual phase error), δ ∆φ .
The minimum of the cost function therefore is perturbed due to noise and residual phase effects. We can rewrite ∂C/∂a as follows: where | 0 indicates that the quantities are those corresponding to zero-phase error and absence of noise, that is, ∆φ = 0 andw(t) = 0. Unlike the results in [10,11,24], where the derivative of the time-frequency distribution (TFD) is nonquadratic and approximate linearization of the derivative around the peak is done, here, the cost function is quadratic and hence its derivative is linear in the parameters to be estimated. Therefore, the above linear equation is exact and not approximate. The terms ∂C/∂a | 0 δ ∆φ and ∂C/∂a | 0 δw indicate the perturbations in the derivative as a result of phase error and noise, respectively. Evaluation of these quantities, bias, and variance computation of the IF estimates is given in the appendix. The asymptotic expressions for bias, variance, and covariance (denoted by Bias(·), Var(·), and Cov(·), respectively) of the coefficient estimates are given by Using these, the bias and variance of the IF estimator can be obtained as Directly substituting the expressions for bias and covariance of the coefficient estimates gives rise to very complicated expressions for the bias and variance of the ZC-IF estimator. However, a considerable simplification can be achieved by using the idea of data centering about the origin, that is, without loss of generality, assume that the data is shifted to lie in Data centering is very useful in obtaining simplified expressions for the bias and variance of IF estimators [25]. It must be noted that data centering is an adjustment to yield simplified expressions and the IF estimate is unaffected in doing so because the estimates are computed using the centered data. This yields the following expressions for bias and variance of the coefficients: From the coefficient estimates, the expressions for bias and variance of the IF estimate at the center of the window (t = 0) are obtained as It may be noted that these are approximate asymptotic expressions for bias and variance of the ZC-IF estimator.

OPTIMUM WINDOW LENGTH SELECTION
Substituting the expressions for bias and variance obtained above, we can write the expression for MSE, MSE( f (0)) as follows: The MSE is a function of the window length L. In Figure 2, we illustrate the variation of bias, variance, and MSE, as a function of window length. Since the bias, variance, and MSE characterize an estimator, the y-axis is commonly labelled  as characteristic and plotted in decibel (dB) scale. From the figure, we infer that the MSE has a minimum with respect to window length. The optimal window length, L opt corresponding to MMSE is given as All the mathematically valid minimizers of the MSE are not practically meaningful. Only the real solution, L opt above, is relevant. The optimum window length is a function of the higher-order derivatives of the IF which are not known a priori, because the IF itself is not known and it has to be estimated. The above expression for the optimal window length is mainly of theoretical interest. The analysis, however, throws light on the issues and tradeoff involved in window length selection for MMSE-ZC-IF estimation. Unlike the expression for bias, the expression for variance does not require any a priori knowledge of the IF, but depends only on the SNR which can be estimated. The expression for variance can be used to devise an adaptive window algorithm to solve the bias-variance tradeoff for MMSE ZC-IF estimation.

Bias-variance tradeoff
The expressions for squared bias and variance can be restated as follows: Large bias, small variance where At L = L opt , From the expressions above, it is clear that the squared bias is directly proportional to L 2p and the variance is inversely proportional to L 3 , clearly indicating bias-variance tradeoff frequently encountered in devising estimators operating on windowed data [12,26]. The increased smoothing of the estimate for a long window decreases variance but increases bias; conversely, reduced smoothing with a short window increases variance but bias decreases. The asymptotic distribution of the estimator is shown in Figure 3.
We need to emphasize an important aspect specific to the ZC-IF estimator. Unlike regular sampling, in an irregular sampling scenario (ZC data belongs to this class), the distribution of data is not uniform. In the case of uniform sampling, as the window length is increased, in multiples of the sampling period, the window encompasses more data and hence the associated bias and variance change monotonically. However, in the irregular sampling case, as the window length is increased in multiples of the sampling period, the window may or may not encompass more data, depending on the data distribution. Thus, the associated bias and  variance do not vary smoothly. This is illustrated through an example.
A noise sequence, white and Gaussian distributed, 256 samples long, was lowpass filtered (filter's normalized cutoff frequency arbitrarily chosen as 0.05 Hz). The filtered signal was rescaled and adjusted to have amplitude excursions limited to [0, 0.45]. This was used as the IF to simulate a constant amplitude, frequency modulated sinusoid. Additive white Gaussian noise was added to achieve an SNR of 25 dB. Since this is a synthesized example, the underlying IF is known and hence bias can be computed directly. Using ZCs to perform a third-order polynomial phase fitting, the IF was estimated at the center of the observation window for different window lengths. The experiment was repeated 100 times and the bias and variance were computed and plotted in Figure 4. The figure clearly illustrates the bias-variance tradeoff for the ZC-IF estimator using noisy signal data.

ADAPTIVE WINDOW ZC-IF TECHNIQUE (AZC-IF)
Asymptotically, the IF estimate 3 f L (the subscript L denotes the window length) can be considered as a Gaussian random variable distributed around the actual value, f , with bias, b( f L ) and standard deviation, σ( f L ). Thus, we can write the following relation: for a given SNR. This inequality holds with probability ). In terms of the standard normal distribution, N (0, 1), this probability is given as P(κ) and tends to unity as κ tends to infinity. We can rewrite this inequality as which holds with probability we can rewrite the inequality as which holds with probability P(| f − f L | ≤ (∆κ + κ)σ( f L )). Therefore, we can define a confidence interval for the IF estimate (using window length L) as We define a set of discrete-window lengths, H = {L s | L s = a s L 0 , s = 0, 1, 2, . . . , s max ; a > 1}. 4 If a = 2, this set is dyadic in nature. Likewise, if a = 3, it is a triadic window set. We choose a = 2. At this point, we recall a theorem from [10] using which we can show that, for the present case, For a third-order fit, that is, p = 3, we have ∆κ = 3.6569 and κ < 43.2013. Together, we have κ + ∆κ < 46.8582. This is only an upper bound obtained using the approximate asymptotic analysis in Section 4. For simulations reported in this paper, a 5σ confidence interval, that is, κ + ∆κ = 2.5 was used. For this value of κ + ∆κ, the coverage probability is nearly 0.99. For a detailed discussion on the choice of κ + ∆κ, see [27]. We can also define H as H = {L s | L s = (s + 1)L 0 , s = 0, 1, 2, . . . ,s max }, that is, the window lengths are in arithmetic progression. The consequence of such a choice is studied in Section 7.2.

Algorithm
The algorithm for AZC-IF estimation at a point t is summarized as follows.
(1) Initialization. Choose H = {L s | L s = a s L 0 , s = 0, 1, 2, . . . , s max ; a > 1}, κ + ∆κ = 2.5. Set s = 0. L 0 is chosen as the window length encompassing p + 1 farthest ZCs. This ensures that at any stage of the algorithm, there is sufficient data to perform a pth-order fit. s max is chosen such that L smax+1 just exceeds the observation window length. The IF estimate f Ls is obtained using the window length L s , that is, the ZC data (after data centering) within the window is used to perform a pth-order fit to obtain the IP and the IF. Let f Ls be the corresponding AZC-IF estimate. 4 The choice of s max and L 0 is discussed in Sections 6.1 and 7.2.
(2) Confidence interval computation. The limits of the confidence interval are computed as follows: (3) Estimation. Obtain f Ls+1 using the next window length, L s+1 = 2L s , from the set H. Compute the confidence interval limits as follows: Q s+1 ] and ∅ denotes the empty set). In other words, the following condition is checked: The smallest value of s for which the condition is satisfied yields the optimum window length, that is, if s * is the smallest value of s for which the condition is satisfied, then L opt = L s * ; else s ← s + 1 and steps 3 and 4 are repeated.
Since the bias varies as L 2p , large values of p imply a fastvarying bias. This results in an MSE that is steep about the optimum. With a discretized search space of window lengths, small changes in the window length about the optimum can cause steep rise in the MSE. Also, large values of p can give rise to numerically unstable set of equations. On the other hand, small values of p, that is, p = 1, 2, correspond to a not-so-clearly defined minimum; p = 3 was found to be a satisfactory choice and is used in the simulations reported in this paper.
For implementing the algorithm, the computation of variance requires an estimate of the SNR. The SNR estimator suggested in [10,11] requires oversampling of the signal. Though robust at very low SNRs, in general, it was found to yield poor estimates of the SNR even with considerably large oversampling factors. Therefore, an alternative method of moments estimator is proposed for estimating SNR. A detailed study of its properties and improved adaptive TFDbased IF estimation is reported separately. For the sake of completeness, the SNR estimator is given below (the hat is used to denote an estimate):

SIMULATIONS
We present here simulation results evaluating the performance of the AZC-IF technique and also compare it with the fixed window approaches.

Fixed window versus adaptive window ZC-IF estimator
To illustrate the adaptation of window length, we consider the following IF laws. (1) Step IF.
For each of the IF above, the following experiments were conducted: (1) ZC-IF estimation using a fixed medium window (51samples long), (2) ZC-IF estimation using a fixed long window (129 samples long), (3) adaptive window ZC-IF estimation.  (2) In the vicinity of a fast change in IF (like the discontinuity in the case of step IF), the algorithm chooses shorter window length thereby reducing bias and hence capturing "events in time." This improves time resolution but at the expense of large variance. At the discontinuity, the local polynomial approximation does not hold; the corresponding window length chosen by the algorithm is large.
(3) Fixed window ZC-IF estimate obtained with a longer window length is smeared/oversmoothed than that obtained with a shorter window length.
(4) The average error, η, is the best with adaptive window length estimator for the case of the step IF and also "sum of sinusoids" IF. However, with triangular IF, we find that the AZC-IF estimate obtained with the adaptive window length has a few dB higher error compared to that obtained with a medium window length. Such a behaviour, which appears to be counterintuitive at first, is possible with any kind of IF, as simulations later will show. However, it must be noted that this is because of the choice of the set of window lengths. The set of window lengths chosen are dyadic in nature and hence the optimum MSE search is very coarse. It is possible that, in such a case, the adaptive algorithm determines a window length quite away from the optimum, which yields poorer performance than the medium window length. This may be overcome by finely searching the space of window lengths. This is discussed in the following section.

Coarse search versus fine search
We study the effect of discretizing the search space of windows on the performance of the algorithm. We consider the following window lengths: (1) medium window length: L = 51 samples, In the arithmetic and dyadic cases, L 0 is the initial window length encompassing (p + 1) farthest ZCs; the values of s max ands max are chosen such that L smax+1 and Ls max +1 , respectively, just exceed the given observation window length. For illustration, we chose the "sum of sinusoids" IF. The ZC-IF estimates corresponding to each of the above windows are obtained. The SNR was varied from 0 dB to 24 dB in steps of 2 dB. For each SNR, 100 Monte-Carlo realizations were run to obtain the statistics of the ZC-IF estimator. The squared bias, variance, and MSE corresponding to the sample instants n = 128 and n = 200 are plotted in Figure 8. From the figures, we observe that ZC-IF estimates obtained with medium window size (whose choice cannot be made a priori anyway) show "slightly" better performance than the adaptive window algorithms at low SNR. This is because the adaptation algorithm uses asymptotic expressions for bias and variance which are derived under a high SNR assumption and hence less appropriate at low SNR. At high SNR, however, the adaptive estimates are consistently better than fixed window estimates. Also of interest is the comparison between the arithmetic and dyadic window choices for the adaptive algorithm. Consistently, the AZC-IF estimates obtained with arithmetic window set (corresponding to fine search) outperform those obtained with the dyadic set (corresponding to coarse search) for low to moderate SNR. Sometimes, the improvement in performance can be as high as 12 dB (bottom row in Figure 8, corresponding to SNR = 8 dB). However, for high SNR, there seems to be no distinct advantage in doing a fine search. One can protract this observation to state that if the SNR estimate indicates a high value (roughly beyond 18 dB or so), a considerable computational saving can be achieved by using the dyadic window set.
These are time averages of instantaneous squared bias, variance, and MSE, respectively. Their importance is due to the fact that they give a single error measure taking into account pointwise errors. These can be useful indicators of the performance of IF estimators for a given data. These are shown in Figure 9. The significant improvement in performance offered by adaptive estimates compared to fixed window ZC-IF estimates stresses the significance of appropriate window length choice in ZC-IF estimation.

Comparison with other techniques
We compare the performance of the AZC-IF algorithm with adaptive spectrogram (ASPEC) and Wigner-Ville distribution (AWVD)-based IF estimates. For a discussion on adaptive spectrogram and adaptive WVD-based IF estimation, refer [10,11,24].
A dyadic window set was used for the three methods. For ASPEC and AWVD, the initial window size was chosen to be 2 samples, whereas for AZC, it is chosen as the window length encompassing (p + 1) farthest consecutive ZCs. This varies from realization to realization and is definitely greater than twice the sampling period and hence relatively, the search for AZC is more coarse than ASPEC and AWVD. The spectrogram and WVD are computed by zero-padding to compute a 4N-point discrete Fourier transform (DFT) (N is the length of the entire data available, N = 256 samples in the present examples). After locating a peak in the 4Npoint DFT, the actual position of the peak is further refined by fitting a quadratic polynomial to three samples about the DFT peak (two samples on either side of the peak and the DFT peak sample itself). This gives a peak location closer to the actual discrete-time Fourier transform (DTFT) peak than the DFT peak. This was done to minimize the contribution of the error in peak-picking to the error in the IF estimate. For the AZC algorithm, to locate the ZCs, the signal is oversampled four times (equivalently, four stages of the bandlimited interpolation plus bisection scheme) and linear interpolation between adjacent samples of opposite sign was performed to estimate the ZC. The cumulative MSE is shown in Figure 10. We infer that AZC technique for triangular IF estimation offers as much as 10 dB less error at low SNRs (and hence more robust) than ASPEC and AWVD-based estimates. For high SNRs, however, the performance of the AZC technique is about 5-6 dB poorer than ASPEC and AWVD. For the "sum of sinusoids" IF, the AZC technique performance is similar to the other two techniques at all SNRs. The experiment is repeated with the arithmetic window set. For ASPEC and AWVD, the window set consists of consecutive multiples of 2, whereas for AZC algorithm it consists of consecutive multiples of the initial window length (which is chosen as the window length encompassing (p +1) farthest ZCs). Therefore, relatively, the search is more coarse for AZC algorithm than ASPEC and AWVD algorithms. The results are shown in Figure 11. Here, AZC consistently outperforms ASPEC and AWVD algorithms. This is because, the squared bias of the ZC-IF estimator varies as L 2p which is quite faster than the corresponding variation for spectrogram and WVD [10,11,24]. This implies that a coarse search in the window space can severely affect the performance of AZC more than ASPEC and AWVD. Simulations (Figures 10 and 11) strongly support this argument and emphasize the need for appropriate discretization of the window search space for robust IF estimation using the AZC algorithm.

CONCLUSION
We have achieved robust estimation of arbitrary IF using real ZCs of the frequency modulated signal in an adaptive window framework. This approach combines in an interesting and useful manner nonlinear measurement (ZCs), nonuniform sampling, and adaptive window techniques, resulting in superior IF estimation. Comparative simulations show that the adaptive window ZC technique can provide as much as 5-10 dB performance advantage over adaptive spectrogram and Wigner-Ville distribution-based IF estimation techniques. Extension of the new technique for estimating arbitrary instantaneous frequencies of multicomponent signals (similar to [28]) is being explored.

APPENDIX
In this appendix, we give the detailed derivation for bias and variance of the ZC-IF estimator reported in Section 4. With a change of variable from t n to t − t n , each of the quantities in (9)  where ∆φ(t, t n ) = ∞ r=p+1 ((−t n ) r /r!)φ r (t), r! denoting the factorial of r. Retaining only the significant term in the infinite summation, we can write ∆φ(t, t n ) ≈ ((−t n ) p+1 /(p + 1)!)φ (p+1) (t). Since the power spectral density of noise, w(t), is known, it is easy to compute E {w(t − t n )w(t − t m )}, because rww(τ) = (σ 2 w /A 2 )(sin(Bπτ)/Bπτ), and E {w(t − t n )w(t − t m )} = rww(t n − t m ). However, this leads to very complicated expressions for the variance and covariance of the coefficient estimation errors. For the purposes of analysis and ease of computation, one can arrive at simplifed expressions by assuming that E {w(t − t n )w(t − t m )} ≈ (σ 2 w /A 2 )δ tn,tm . This is equivalent to assuming that the autocorrelation function of noise, w(t), is highly localized at zero lag and is negligible elsewhere (nearly white noise). Under this assumption, one can write the following approximate expressions (for 0 ≤ , k ≤ p):