Instantaneous frequency estimation of nonlinear ... - IEEE Xplore

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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 10, OCTOBER 2001

Instantaneous Frequency Estimation of Nonlinear Frequency-Modulated Signals in the Presence of Multiplicative and Additive Noise Braham Barkat

Abstract—This paper addresses the problem of estimating the instantaneous frequency (IF) of monocomponent nonlinear, not necessarily polynomial, frequency modulated (FM) signals affected by stationary multiplicative and additive noise. Both noise processes are assumed to be complex circular Gaussian and independent. The peak of the polynomial Wigner–Ville distribution (PWVD) is proposed here as an IF estimator. We derive analytical expressions for the bias and asymptotic variance of the estimator and propose an algorithm to select the optimal window length to resolve the bias-variance tradeoff in the IF estimation. Simulation results are presented to confirm the theoretical results. Index Terms—Instantaneous frequency, mean square error, multilplicative noise, polynomial Wigner–Ville distribution, time– frequency distributions.

I. INTRODUCTION

I

N many engineering applications such as radar, sonar, telecommunications, and engine diagnosis, the signals under consideration are known to be nonstationary, i.e., their spectral contents vary with time. Time–frequency analysis, among other methods, was proposed to deal with such signals [1]. Time–frequency distributions (TFDs) are natural extensions of the Fourier transform. They map a one-dimensional (1-D) signal, as a function of time only, to a two-dimensional (2-D) quantity, function of time, and frequency. A concept intimately related to TFDs is that of instantaneous frequency (IF) [2]. In many situations, the IF characterizes important physical parameters of the signal [3]. Therefore, accurate and effective estimation of this quantity is of great importance. The statistical performance of the IF estimator is also important since it provides the practitioner with a tool to judge the accuracy of the estimates. Diverse IF estimation methods have been developed for constant, or slowly varying, amplitudes frequency modulated (FM) signals embedded in additive noise. Some of these methods are parametric and some are nonparametric. In general, parametric methods use a signal model, and the goal, in this case, is to estimate some parameters in order to obtain the IF [4]. Nonparametric methods, on the other hand, do not require full knowledge of the signal. A well-known class of nonparametric methods, for IF estimation, is based on TFDs of the signal [5]–[7]. The use Manuscript received August 1, 2000; revised July 2, 2001. The associate editor coordinating the review of this paper and approving it for publication was Dr. Paulo S. R. Diniz. The author is with the School of Electrical and Electronics Engineering, Nanyang Technological University, Singapore (e-mail: [email protected]). Publisher Item Identifier S 1053-587X(01)08245-9.

of the time-frequency analysis stems from its ability to display the temporal localization of the signal’s spectral components. In many cases, it is assumed that the signal of interest has a constant amplitude. While this is a valid assumption in a wide range of scenarios, there are several important applications in which the constant amplitude assumption does not hold. Indeed, in many applications, the signal under consideration may be subjected to a random amplitude modulation that behaves as multiplicative noise. Examples include fading in wireless communications [8], fluctuating target in radar [9, ch. 11], and structural vibration of a spacecraft during launch and atmospheric turbulence [10]. In the literature, we may find several parametric methods to deal with such signals [11]–[13]. In this paper, we use a nonparametric method, namely, the polynomial Wigner–Ville distribution (PWVD) [14] to address the problem of multiplicative noise. In particular, we consider the IF estimation of monocomponent frequency modulated signals affected by stationary multiplicative and additive noise processes. Both noise processes are assumed complex circular Gaussian and independent. Here, we propose the peak of the PWVD as an IF estimator and study in detail its statistical performance. We show that the estimator bias and variance are highly signal dependent and tend to vary differently in function of the length of the window used in the implementation of the PWVD, that is, if the window length increases, the bias increases, and the variance decreases, whereas if the window length decreases, the bias decreases, and the variance increases. Based on this observation, we derive an expression for the optimal window length that minimizes the IF estimator mean squared error. The optimal window length is shown to be a function of the signal IF or its higher order derivatives. In practice, these quantities are, in general, unknown. This makes the use of the theoretical result very difficult. To avoid this problem, we propose an algorithm that can select the optimal window length to resolve the bias-variance tradeoff without having to know the IF law. Simulation results, using highly nonlinear FM signals, show that the proposed algorithm can estimate the signal IF accurately. The idea of the IF estimator proposed here can be cast in the framework of nonparametric regression and local polynomial approximation [15]–[17] and is an extension of the work in [6]. The paper is organized as follows. In the next section, a brief review of the PWVD is given. In Section III, we give details of the statistical evaluation. In Section IV, we outline the proposed algorithm along with some illustrative examples. Section V concludes the paper.

1053–587X/01$10.00 © 2001 IEEE

BARKAT: INSTANTANEOUS FREQUENCY ESTIMATION OF NONLINEAR FM SIGNALS

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Fig. 1. (Left) WVD and (right) sixth-order PWVD of a cubic FM signal.

Fig. 2. Sixth-order PWVD of a cubic FM signal corrupted by a complex Gaussian multiplicative noise whose variance is equal to 1 and whose mean is equal to (left) 2 and (right) 5, respectively.

II. POLYNOMIAL WIGNER–VILLE DISTRIBUTION In this section, we review the PWVD and give a brief outline of the discussed problem. The PWVD was designed to represent, in the time–frequency domain, polynomial FM signals of the form real numbers

(1)

as a continuum of delta functions around the signal IF. This time–frequency representation is defined as [14]

a member of the PWVDs class with parameters and . Full details of the design procedure can be reviewed in [14]. To illustrate the use of the PWVD, consider the analysis of a in (1)]. For this type constant amplitude cubic FM signal [ of signal, the WVD displays some artifacts (or interferences) due to the nonlinearity of the signal FM law (see the left plot of Fig. 1), and its peak does not occur at the signal IF but is always shifted by a certain amount, which is referred as the systematic bias [18]. To analyze this signal correctly, we suggest the use of the sixth-order PWVD (PWVD6), whose kernel is given by [14]

(4) (2) is an even integer, indicating the order of nonwhere is the time lag; and and linearity of the PWVD; are real coefficients. The coefficients and are calculated so that the PWVD is real and equal to

where

represents the IF of the signal

, i.e., (3)

. Note that the realness of the PWVD implies that In addition, note that the Wigner–Ville distribution (WVD) is

This distribution is not only interference-free (see the right plot of Fig. 1), but its peak occurs at the signal IF. Indeed, a general statistical performance evaluation showed that the PWVD peak-based IF estimator is unbiased when there is no mismatch between the constant amplitude polynomial FM signal and the PWVD order [7]. In practice, in addition to the additive noise, the signal under consideration may be subjected to a random amplitude modulation that is often referred to as multiplicative noise. Here, we show that the PWVD can handle such cases. For that, let us reconsider the previous cubic FM signal and assume that it has been corrupted (multiplied) by a complex Gaussian multiplicative noise with unit variance. The sixth-order PWVD of this noisy signal is displayed in Fig. 2 for two different mean values (two for the left

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plot and five for the right plot). We observe that the PWVD is indeed able to display the spectral content of the noisy signal. In what follows, we generalize our analysis to any nonlinear, not necessarily a polynomial, FM signal. In particular, we propose the peak of the PWVD as an IF estimator for FM signals affected by both multiplicative and additive noise and study in detail its statistical performance.

where given by

is the signal’s IF, and

is

(8)

III. BIAS AND VARIANCE OF THE IF ESTIMATOR In this section, we derive the analytical expressions of the bias and variance for the proposed estimator. Our primary interest is in estimating the IF of a nonlinear, not from necessarily a polynomial, FM signal the discrete-time observations

Now, using the PWVD design condition the distribution expression (7) becomes

(9)

(5) where integer; sampling period; discrete-time version of . and are assumed cirThe stationary processes cular complex Gaussian and independent with means and variand , respectively. ances given by as By writing the nonzero mean multiplicative noise , with being a zero-mean complex circular Gaussian noise process with variance , we can rewrite the expression in (5) as

(6) Imag Real and is a zero-mean circular complex . Gaussian noise with variance equal to Note that the instantaneous frequencies of the signals and are exactly the same. To deal with such signals, we need to introduce a real-valued , of time-varying length , in symmetric window the PWVD as follows: In (6),

(7) and where by Replacing in (7) the signal using Taylor’s expansion of the phase

for

. and around , we obtain

[14],

Note that for a given polynomial phase signal of order , the for odd values of PWVD is designed such that [14]. Therefore, if the signal under considerais of this type (i.e., a polynomial FM signal), and the tion is always appropriate PWVD is used, we see that (for ) or zero through either (for ). In this case, the expression of , i.e., the PWVD becomes the PWVD yields delta functions around the signal’s IF. However, if the signal is not a polynomial FM signal or if there is a mismatch between the polynomial FM signal and the PWVD order that we use, then an error is always present in the IF estimate, even when no noise is present. In the presence of noise, we should consider the statistical error too. It is for this general case that we will derive, in the following text, the expressions of the variance and bias of the estimator. at a fixed In general, the IF estimate of a noisy FM signal time instant is given by the frequency where the maximum of the PWVD occurs, that is, the IF estimate is given by solving for (10) Any error in the IF estimate may be due to one (or a combination) of the following: ; i) the computational estimation error ; ii) the error due to noise iii) the error due to the mismatch of the signal with the dis. tribution order Thus, by linearizing expression (10) with respect to these quantities, we obtain

(11) where where

means that the expressions are computed at the point , and .


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Evaluating each term in (11), using the same procedure as the one in [6] and some further computation, which is detailed in Appendix A, we find the bias and the variance to be Bias

(12)

Var

(13)

with

Fig. 3. Peak of the sixth-order PWVD is used as an IF estimator at the middle of the observation interval for a quadratic FM signal corrupted by complex Gaussian multiplicative and additive noise processes. The window length is equal to the signal length (h = N = 129). The continuous lines (superimposed) represent the theoretical variances, whereas “+” and “o” correspond to the estimated variances for A = 0:1 and A = 10, respectively.

where is the number of coefficients in the kernel of the PWVD, and is the multiplicity of each of these coefficients. A. Special Case: Polynomial FM Signals If the signal under consideration is a polynomial FM signal as the one given by (1) and if the appropriate PWVD is used, the bias in (12) is always zero through either (for ) or (for ). This result means that the PWVD-based IF estimator is always unbiased for a polynomial FM signal corrupted by multiplicative and additive noise. Furthermore, for a rectangular window, the bias and variance reduce to Bias

(14)

Var

(15)

To check the validity of the above theoretical results, let us consider the IF estimation of a quadratic FM signal at the middle of the signal interval. As there is no mismatch between the sixth-order PWVD whose kernel is defined in (4) and this signal, the peak of this distribution is used here as the IF estimator. The noiseless signal, which is sampled at with a number of samples , is modulated by a comand added to complex plex circular Gaussian noise , as suggested in (5). The additive and Gaussian noise multiplicative noise processes are assumed independent. In the simulations, the overall signal-to-noise ratio (SNR), which , is is defined as SNR varied in a 1-dB step from 0 to 15 dB. Monte Carlo simulations for 1000 realizations are run for each value of SNR . The and , are displayed in Fig. 3. results, for We observe that above a certain threshold, the estimated variances represented by “ ” and “ ” are in agreement with the derived theoretical ones given by (15) and represented by the continuous lines (superimposed). In this example, we fixed and chose the window length equal to the . We used other values for and signal length and considered and found similar results, i.e., the estimated results confirm the theoretical ones.

IV. OPTIMAL WINDOW-LENGTH SELECTION ALGORITHM In this section, we outline an algorithm to select an adaptive optimal window length in the implementation of the PWVD. The optimality is given in terms of minimum mean squared error of the IF estimate. From the general results of the previous section, namely (12) and (13), and assuming the window to be rectangular, we find the mean squared error of the IF estimator to be equal to m.s.e

(16) To obtain the above expression, we approximated the bias ), which is nonzero only in (12) by its first term (i.e., for the WVD. However, for a higher order PWVD, this term is zero (as well as all the terms up to the appropriate order of the PWVD used), and we should then approximate the bias by the first nonzero term in the sum (as explained in the previous sections). For example, the first nonzero term in the PWVD of and for the PWVD of order order six [14] corresponds to , and so on. This correction eight [14], it corresponds to in the bias approximation would change only the expression of the mean squared error in (16) (and the expression of the optimal length that follows) and will not affect the rest of the paper derivations. From (16), the optimal window length, in the sense of minimum mean squared error, is found equal to

Observe that the optimal window length is function of the signal’s IF (or its higher order derivatives). In most usual cases, the expression of the IF is unknown, and therefore, the above result becomes of little practical importance. In order to avoid

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TABLE I ALGORITHM FOR THE OPTIMAL WINDOW LENGTH SELECTION

Fig. 4.

Effect of varying the window on the estimator bias and variance.

this difficulty, we propose here an algorithm that can determine the optimal window length without having to know the IF law. The basic idea of the algorithm and its details can be found in [6]. The key idea of the algorithm is based on the fact that for a small size window length, the variance of the estimator is very large, and the bias is very small. However, for a large size window length, the variance is very small, and the bias is very large. Fig. 4 illustrates this point. In this figure, refer to the biases of the IF estimate, and Var Var Var refer to its variances for different window lengths. Therefore, if we compute the variance and the bias for a certain number of window lengths (taken from a continuous set of window lengths) beginning from the smallest possible value to the largest possible one, we arrive at a particular window length value that is referred as the optimal window length, where we achieve the best compromise in terms of the smallest mean squared error for the IF estimate. This smallest mean squared error is obtained when there is no more overlapping between two consecutive intervals of the variances [6]. Before we give a brief description of the algorithm, we should note the following details and definitions. For practical reasons, the set of window lengths used in the implementation of the PWVD is not continuous but discrete and finite, and we assume that the optimal window length (or at least a value very close to it) belongs to , which is defined as with

being a power of

In addition, it can be seen, at least asymptotically, that the IF is a random variable distributed around the true estimate with a bias Bias and a standard deviation , IF that is, we may write

where

is an IF estimate obtained for the window length , and is its standard deviation. The algorithm can now be described as follows. For a given time instant , we choose the smallest window length in to implement a slice of the PWVD. We estimate the IF by searching for the frequency that corresponds to the maximum of this slice. This value is used to compute the variance [from (13), where the amplitude as well as the noise variance are estimated from the signal data as in [6]] and the interval limits in (17). We save these results. Then, we choose the next window length value in the set and use it to compute a new slice of the PWVD. From this new slice, we obtain a new value for the IF estimate and a new confidence interval. Again, we save these results. We repeat the procedure for a certain number of window lengths, and and we stop the algorithm when we obtain . The first window length to guarantee and is declared as the optimal window length, and its corresponding IF estimate will be chosen as the signal IF estimate for that particular time instant. The whole procedure is repeated again for a new time instant . Table I outlines the algorithm. Note that the search for the optimal window length over the finite set is a simple optimization problem. However, the discretization of the window length inevitably leads to a suboptimal window length value. Fortunately, this loss of accuracy is not significant in many cases as the mean square error has a stationary point for the optimal window length and varies very slowly for window lengths close to it.

Bias where the inequality holds with a probability and where is a quantile of the standard Gaussian distribution. For instance, , the probability would be very close if we choose [6], [17]. to one Now, we define the confidence intervals of the random IF , where and are given by estimate as (17)

A. Examples 1) Example 1: In this example, we consider the IF estima. The signal is multion of a highly nonlinear FM signal tiplied by a complex circular Gaussian noise with mean and vari. Then, additive zero-mean Gaussian noise is ance added, as suggested in (5). Here, we choose and the overall signal-to-noise ratio SNR dB. The window set con. sidered is


Fig. 5.

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IF estimates of the noisy signal for a (left) small size (center), large size (proposed algorithm) and the adaptive window length, respectively.

Fig. 6. IF estimates of the same signal as in Fig. 5 using (left) the CFD estimator, (center) the LMS estimator, and (right) the RLS estimator.

Fig. 7.

Power of the received signal (left plot) and estimation of its IF using the time-frequency algorithm (right plot).

In Fig. 5, we plot the IF estimates for the noisy signal using, respectively, a small (left plot), large (center), and the optimal window length (right). For the small window length, the variance is seen to be large, which is in agreement with the theoretical derived variance in (13). For large variations in the phase, the bias is seen to be large, which is also in agreement with (12). Note the superiority of the proposed algorithm in reducing the variance and the bias. The dashed curve represents the true IF. For comparison purposes, the algorithm is compared with other existing IF estimation techniques such as the adaptive least mean squared LMS, the adaptive recursive mean square RMS, and the central finite difference CFD [19]. The estimation results for these algorithms, applied to the same noisy signal above, are displayed in Fig. 6. The superiority of the proposed algorithm over the others is obvious. 2) Example 2: In this example, we consider the IF estimation of a received signal in a flat fading digital communication transsystem. It can be shown that for a complex signal

mitted over a flat fading channel, the received complex signal can be described by (5), [20]. The dramatic drop in power of the signal, due to fading, makes it very difficult to be detected. For frequency shift keying (FSK) like signals, the signal information is contained in its IF. Thus, by using an appropriate tool such as the PWVD-based algorithm, to estimate the IF of the received signal, we may be able to retrieve the signal without having to use expensive and very complex receivers. To illustrate this point, consider the estimation of a binary FSK-like signal

, where . To account for the fading channel, is multiplied by a complex circular Gaussian noise with . Then, additive zero-mean Gaussian mean and variance is added. We choose the overall signal-to-noise ratio noise dB. SNR The drop in power of the received signal due to channel fading is displayed in the left plot of Fig. 7. Using the PWVDbased algorithm, we can retrieve the IF of the signal in an ac-

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curate way, as shown in the right plot of Fig. 7. In the figure, the IF of the transmitted signal is represented by a dotted line, whereas the IF of the received signal is represented by a continuous line.

The cross terms are obtained from the noisy signal by noting that its kernel

(18)

V. CONCLUSION In this paper, we have considered the IF estimation of monocomponent nonlinear, not necessarily polynomial, FM signals corrupted by multiplicative and additive noise. Both noises were assumed complex, circular, Gaussian, and independent of each other. We proposed the peak of the PWVD as an IF estimator. We derived analytical expressions for the bias and asymptotic variance for this estimator. These quantities are highly signal dependent and tend to vary inversely as a function of the analysis window length. An example, based on Monte Carlo simulation, was used to validate the theoretical derivations. In addition, we proposed an algorithm to select the optimal window length to resolve the bias-variance tradeoff without having to know the signal IF law. Examples of highly nonlinear FM signals show that the adaptive PWVD-based IF estimator is very accurate and outperforms IF estimators based on fixed window TFD.

can be written as

yielding

APPENDIX A In this Appendix, we develop the derivation that yields to the bias and variance expressions in (12) and (13). Evaluating each term in (11), we obtain

(19) where the final terms indicated by dots are terms that involve products of more than one noise term. We note that the noisy kernel is the sum of two expressions. The first expression corresponds to the noise-free kernel, or auto-terms, whereas the , is the cross terms. Assecond expression, namely, suming the power of the auto terms to be larger than the power of the discarded terms, we can write

(20) From (10), we have (21) with (22) (23)

where PWVD kernel.

(24)

represents the cross terms of the noisy and

, as defined in (20).


Solving (21) for the error

, we find

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To evaluate the bias, let us compute as follows: write

. Using (8), we can

(25) Thus

(26) Since the mean of the cross-terms is zero [7], we also have . Therefore (27)

Bias Var

Var

(28)

Now, we use the result [7] for for

where

to obtain

(30) Again, for lowing approximation:

and from (28), the variance of the IF estimate is found to be

and

, we have the fol-

Inserting the above result in (27), we find the expression of the bias given in (12).

Var ACKNOWLEDGMENT (29) For approximations:

and

, we have the following

The author would like to thank all the reviewers whose valuable comments and suggestions helped improve the quality of the paper. He also extends his thanks to Prof. B. Boashash for providing the TFSA package used to generate Figs. 1 and 2. REFERENCES

Inserting the above quantities in (29) and recalling that , we obtain the result given in (13), namely Var

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Braham Barkat received the degree of “Ingenieur d’Etat” in electronics in 1985 from the National Polytechnic Institute of Algiers (ENPA), Algiers, Algeria, the M.S. degree in control systems in 1988 from the University of Colorado, Boulder, and the Ph.D. degree from the Signal Processing Research Centre, Queensland University of Technology (QUT), Brisbane, Australia. In 1989, he joined University of Blida, Blida, Algeria, where he held a lecturer position in digital and advanced control systems. In 1996, he joined the Signal Processing Research Centre at Queensland University of Technology (QUT), Brisbane, Australia as a Senior Research Assistant and then as a Ph.D. candidate in signal processing. From September 1999 to November 2000, he had been a Postdoctoral Research Fellow, first at QUT and then at Curtin University of Technology, Perth, Australia. In November 2000, he joined the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, as an Assistant Professor. His research interests include time–frequency signal analysis, estimation and detection, statistical array processing, and signal processing in telecommunications.