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Int. J. Electron. Commun. (AEÜ) 68 (2014) 631–636

Contents lists available at ScienceDirect

International Journal of Electronics and Communications (AEÜ) journal homepage: www.elsevier.com/locate/aeue

Quasi maximum likelihood estimator of polynomial phase signals for compressed sensed data Igor Djurovic´ a , Vladimir V. Lukin b , Marko Simeunovic´ a,∗ , Braham Barkat c a b c

University of Montenegro, Electrical Engineering Department, Cetinjski put bb, 81000 Podgorica, Montenegro National Aerospace University, Department of Transmitters, Receivers and Signal Processing, Kharkov, Ukraine Petroleum Institute, Ruwais Building, Abu Dhabi, United Arab Emirates

a r t i c l e

i n f o

Article history: Received 11 September 2013 Accepted 21 January 2014 Keywords: Polynomial phase signals Parameter estimation Compressive sensing

a b s t r a c t Several papers in the literature cover parameter estimation of frequency modulated (FM) signals under reduced number of signal samples with respect to the Nyquist/Shannon criterion, i.e., within the compressive sensing (CS) framework. However, scope of these papers is mainly limited to sinusoids or sum of sinusoids. In this paper, the CS framework is extended to parameter estimation of higher order polynomial phase signals (PPSs) using the quasi-maximum likelihood (QML) estimator and robust short-time Fourier transform (STFT). The considered signal is assumed to be non-uniformly sampled PPS with smaller number of samples with respect to the Nyquist/Shannon criterion. However, the proposed technique can also be generalized to uniformly sampled signals with missing or unreliable samples. © 2014 Elsevier GmbH. All rights reserved.

1. Introduction Processing of undersampled data or data with missing samples has attracted signiﬁcant attention within the compressive sensing (CS) framework [1,2]. Namely, joint data acquisition and compression is possible if data records are sparse in the recording domain [3]. There exist several research papers dealing with the CS in the frequency domain related to parameter estimation of a sinusoidal signal [4–6]. In addition, reconstruction of samples of time-frequency (TF) representations based on compressed information is considered in [7]. Note that reconstructing a signal using data with missing samples is in fact equivalent to compressive sensing, which can be described as a technique for efﬁcient determination of entire signal from relatively small number of measurements. In [8,9], the robust Huber statistics [10] has been applied to the high frequency data and FM signals corrupted by impulsive noise. However, missing observations within CS framework can also be considered as unreliable signal samples due to high amount of noise [6]. Recently, it has been shown that parametric estimation of non-uniformly sampled frequency modulated (FM) signals can be more accurate than that of uniformly sampled data [11]. Previous research in the CS is extended in several directions in this paper. Firstly, polynomial phase signals (PPSs) are considered. Secondly, in addition to the estimation of parameters of uniformly

∗ Corresponding author. Tel.: +382 68656038. ´ [email protected] (V.V. Lukin), E-mail addresses: [email protected] (I. Djurovic), ´ [email protected] (B. Barkat). [email protected], [email protected] (M. Simeunovic), 1434-8411/$ – see front matter © 2014 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.aeue.2014.01.011

sampled data with missing samples, we consider undersampled data with non-uniform sampling. Thirdly, a recently proposed quasi maximum likelihood (QML) approach [12], used as a tool for PPS estimation from CS observations, has been modiﬁed to estimate signals with non-polynomial modulation. The paper is organized as follows. The background informations: the PPS deﬁnition, the sampling model, the robust discrete Fourier transform (DFT) and the robust short-time Fourier transform (STFT) frameworks are presented in Section 2. The QML algorithm for handling CS data (QML-CS) is proposed in Section 3, while its statistical study is presented in Section 4. Concluding remarks are given in Section 5.

2. Objective and theoretical background 2.1. Signal model and objective A polynomial phase signal (PPS) is deﬁned as

x(t) = A exp(j(t)) = A exp

ja0 + j

M

m

am t /m

,

(1)

m=1

where A is the amplitude, M is the polynomial phase order and a0 , a1 , . . ., aM are the phase parameters. Note that this is a compressed representation of the signal x(t) since by only knowing the

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PPS parameters the full description of x(t) can be known at any time instant t. The radian instantaneous frequency (IF) of x(t) is given by

ω(t) = (t) =

M

am t

m−1

.

(2)

m=1

If we deﬁne ωmax and ωmin to be the maximal and the minimal IFs in an arbitrary time interval [T1 , T2 ] with T = T2 − T1 , then we can write

ik ∈ {Im{x(tn ) exp(−jωtn )}|n ∈ [1, N]},

with ik ≤ ik+1 , rk ≤ rk+1 and x(n) = x(nt) = x(tn ) (t is the sampling period). In the L-DFT implementation, the parameter bk is usually chosen such that k bk = 1 and bk = bN+1−k . A widely used value for bk , when N is even, is given by [17]

⎧ ⎨ bk =

B = ωmax − ωmin , ωmax = max ω(t),

(3)

ωmin = min ω(t),

(4)

(8)

⎩

N − ˛(N − 2) , 2

1 , 2˛(N − 2) + 2

k∈

0,

elsewhere

N + 1 + ˛(N − 2) 2

(9)

where B is the signal bandwidth in the interval T. To satisfy the sampling theorem, within the considered time interval T, the total number of signal samples N has to be equal or larger than BT, i.e., N ≥ BT. However, this number N is usually significantly larger than (M + 2), the number of signal parameters (A, a0 , . . ., aM ) to estimate. In this paper, our objective is to estimate these parameters using only K samples where K is an integer verifying M + 2 ≤ K ≤ N.

where the operator · denotes the rounding to the closest integer. The value of the chosen ˛ (˛ ∈ [0, 0.5]) controls the amount of outlier rejection. For instance, small values of ˛ yield a large amount of outlier rejection but at the same time introduce some spectral signal distortion. From extensive computational analysis we observed, using trial and error, that the best value for ˛ is 0.35. Of course, alternative methods such as adaptive ones can be used to determine the optimal value for ˛ [16]. Note that the standard DFT is obtained for bk = 1/N, k = 1, . . ., N. The L-DFT is an example of the robust Huber’s statistics concept used to represent, ﬁlter and estimate the parameters of FM signals corrupted with impulsive noise [9,10,18].

2.2. Cramer-Rao lower bounds (CRLBs) for uniformly and non-uniformly sampled PPSs

2.4. L-short-time Fourier transform (L-STFT)

t∈[T1 ,T2 ]

t∈[T1 ,T2 ]

If the signal x(t) is uniformly sampled using N samples, the best achievable performance in the estimation of the phase parameters, a0 , . . ., aM , is given by the Cramer-Rao lover bound CRLBN (am ), m = 0, . . ., M [13,14]. However, if the signal x(t) is non-uniformly sampled using K ≤ N samples, whereby the samples are randomly drawn from x(t) using the uniform distribution p(ts ) =

1 1 = T T2 − T1

ts ∈ [T1 , T2 ],

with s = 1, 2, . . ., K

(5)

then, the best achievable performance in the estimation of the phase parameters is given by CRLBK (am ), m = 0, . . ., M [15] N CRLBK (am ) ≈ CRLBN (am ). K

(6)

In the sequel, for the sake of simplicity, we will assume that the non-uniform time instants ts , s = 1, . . ., K satisfy the relationship ts < ts+1 . 2.3. L-discrete Fourier transform (L-DFT) When the signal x(t) is uniformly sampled and some of its samples are missing the standard DFT becomes inadequate. As the missing samples can be treated as samples corrupted by impulsive noise [6], the best tool to use in this case is the robust DFT. Here, we will consider the L-DFT form of the robust DFT deﬁned as [16] XL (ω) =

N

bk [rk + jik ].

(7)

k=1

In this robust DFT, the quantity rk (k = 1, . . ., N) is deﬁned such that r1 corresponds to the ﬁrst element of the value-ordered set {Re{x(tn ) exp(− jωtn )}|n ∈ [1, N]}, r2 corresponds to the second element of the value-ordered set {Re{x(tn ) exp(− jωtn )}|n ∈ [1, N]} and so on. The quantity ik is deﬁned in a similar way using the valueordered set {Im{x(tn ) exp(− jωtn )}|n ∈ [1, N]}. Mathematically, we have rk ∈ {Re{x(tn ) exp(−jωtn )}|n ∈ [1, N]},

The L-DFT, applicable to stationary signals, has been generalized to non-stationary signals using the L-STFT deﬁned as [8,19]

STFT˛,h (ts , ω) =

h+1

bs,h [rs,h + jis,h ] k k k

(10)

k=1

where

rs,h ≤ rs,h , rs,h ∈ Re{x(tn ) exp(−jωtn )}|n ∈ s − k k+1 k

is,h ≤ is,h , is,h ∈ Im{x(tn ) exp(−jω(tn )}|n ∈ k k+1 k

h h ,s + 2 2

s−

h h ,s + 2 2

(11)

(12)

and s = h/2 +1, . . ., N − h/2. As we can see from the expressions, the L-STFT is implemented using windowing of the data to select a narrow time interval and subsequently applying it on the L-DFT. The window length (h + 1) used in the L-STFT is assumed to be odd, and are selected in the same way as those in the the coefﬁcients bs,h k L-DFT. Other TF representations exist in the literature and can be applied instead of the STFT [8,20]. 3. QML-CS algorithm 3.1. QML algorithm Many algorithms have been developed for the estimation of the parameters of a PPS [21–25]. Here, we consider the QML estimator [12,26] which consists of the following ﬁve basic steps: Step 1. Evaluate the STFTs corresponding to different selected window lengths. Step 2. For each evaluated STFT, estimate the corresponding signal IF. Step 3. From each estimated IF, obtain the PPS parameters estimates {ˆa1 , . . ., aˆ M } using polynomial regression described in [12]. Observe that these estimates are biased due to the biased nature of the STFT IF estimates. Also, note that for

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each window length we have a corresponding set of parameters {ˆa1 , . . ., aˆ M }. Step 4. Improve the parameter estimates {ˆa1 , . . ., aˆ M } to obtain f f f reﬁned estimates {ˆa0 , aˆ 1 , . . ., aˆ M } using the technique developed in [27]. f f f Step 5. For each reﬁned parameter set {ˆa0 , aˆ 1 , . . ., aˆ M }, evaluate the QML function deﬁned as

2 M f f m x(tn ) exp −jˆa0 − j aˆ m tn /m n

Existing PPS parameter estimation algorithms, including the QML algorithm presented above, have been developed for uniformly sampled signals that satisfy the sampling theorem. In the sequel, we will modify the QML algorithm in order to estimate the PPS parameters when (i) the sampling rate is random and nonuniform and (ii) the number of considered samples is smaller than the number of samples used in uniform sampling case. 3.2. Modiﬁed QML algorithm Here we consider a signal x(t) that has been non-uniformly sampled using K samples. The samples are randomly drawn from x(t) using the uniform distribution given by (5). 3.2.1. Step 1: L-STFT evaluation In the modiﬁed QML instead of using STFT we use the L-STFT. In this case, in the evaluation of the L-STFT we only consider the available time instants ts , s ∈ [1, K] as the center of our ﬁxed length window h. We note that the number of samples in the window around ts is not symmetrical (i.e., number of samples to the left of ts is not necessarily equal to the number of samples to the right of it) and this number can be different for each considered time instant ts . Thus, if we assume s,h to be the number of signal samples within the window interval h, then the L-STFT can be expressed as bs,h [rs,h + jis,h ] k k k

(13)

k=1

where rs,h ≤ rs,h , k k+1 s,h

s,h

ik ≤ ik+1 ,

rs,h ∈ k s,h

ik ∈

Re{x(tq ) exp(−jω(tq − ts )}|tq ∈

Im{x(tq ) exp(−jω(tq − ts )}|tq ∈

ts −

h h , ts + 2 2

ts −

h h , ts + 2 2

(14)

ω

.

(15)

(16)

Since a large number of signal samples are missing, it is expected that some of the IF estimates may be relatively far from the true IF values. To mitigate this, we propose to employ the median ﬁlter to remove these large errors in the IF estimates, i.e., ω ˆ ˛,h (ts ) = median{ω ˆ ˛,h (tq ) | q ∈ [s − m, s + m]},

where m is the median ﬁlter length.

−1

ˆ XTh ˛,h

(18)

Xh = [tsm−1 | m ∈ [1, M] × s ∈ [1, K]]. For the notation, now we have added the subscripts ˛ and h to the parameters aˆ i to become aˆ ˛,i,h to indicate that the parameter estimates depend on the selected values of ˛ and h. Again, we note that the estimates are biased because the L-STFT-based IF estimates are biased. 3.2.4. Step 4: reﬁnement procedure The reﬁnement procedure is a very important step in the QML algorithm. In order to improve the accuracy of the phase estimates obtained in Step 3, a reﬁnement algorithm described in [27] is used. However, this reﬁnement technique had to be adapted for non-uniformly and subsampled data. Thus, here we explain the main steps of the reﬁnement procedure that have been used in our proposed algorithm. First, the signal x(ts ) is dechirped using the parameters estimates {ˆa˛,1,h , . . ., aˆ ˛,M,h }

x˜ (ts ) = x(ts ) exp

(17)

−j

= A exp

M

aˆ ˛,m,h tsm /m

m=1

M

ja0 + j

am t m /m

−j

exp

m=1

x˜ (ts ) = A exp

ja0 − j

M

M

aˆ ˛,m,h tsm /m

(19)

m=1

ıa˛,m,h tsm /m

.

(20)

m=1

Due to errors in the estimation of the parameters, the difference between the true values and their estimates is non-zero, i.e., / 0, m = 0, . . ., M. The objective of the reﬁneıa˛,m,h = aˆ ˛,m,h − am = ment is to ﬁnd the values of ıa˛,m,h and to use them to correct the estimates aˆ ˛,m,h . This can be done by ﬁltering, downsampling, phase unwrapping of x˜ (ts ) and subsequent polynomial ﬁtting of the unwrapped phase, as presented below. The ﬁltering of x˜ (ts ), necessary to improve the signal to noise ratio, is performed using a moving average ﬁlter expressed by

3.2.2. Step 2: IF estimation The IF estimation can be performed in the same manner as in the case of the QML algorithm and other related TF-based techniques, i.e., using position of the TF representation maxima ω ˆ ˛,h (ts ) = argmax|STFT˛,h (ts , ω)|.

aˆ ˛,h = (XTh Xh )

where aˆ ˛,h is the vector of the estimated phase parameters T (t )|s ∈ [1, K]] and ˆ while ˆ ˛,h s ˛,h = [ω

and select the set that yields the maximum value of the evaluated function. This set is considered as the optimal set of PPS parameters.

s,h

3.2.3. Step 3: parameter estimation Here, the PPS parameter estimation from the IF estimates is performed in the same way as in the QML algorithm described earlier. Since the IF estimates are sparse and non-uniformly distributed (which is common issue in polynomial ﬁtting), these parameters are estimated using

aˆ ˛,h = [ˆa˛,1,h , . . ., aˆ ˛,M,h ]

m=1

STFT˛,h (ts , ω) =

633

xˆ (tq ) =

1 L

q+L/2−1

x˜ (tp ).

(21)

p=q−L/2

The downsampling of xˆ (tq ) is needed in order to stabilize the results of the estimation process. Namely, polynomial ﬁtting could become an unstable process for a large number of samples. Then, we use the polynomial ﬁtting with reduced number of samples x(tl ) = xˆ (tL(l+1/2) ) for l ∈ [0, K/L).

(22)

Remark. As an alternative, some more robust procedures can be applied [28]. Once the noise is suppressed, the phase of x(tl ) is unwrapped using v(tl ) = unwrap(angle(x(tl )) l ∈ [0, K/L) and polynomial ﬁtting is performed to estimate ıa˛,m,h , m = 0, . . ., M using ı a˜ = ( ) T

−1

T

v,

(23)

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I. Djurovi´c et al. / Int. J. Electron. Commun. (AEÜ) 68 (2014) 631–636

where v is a vector of components v(tl ), is a matrix given m a˜ is a vector deﬁned by = [tl |m ∈ [0, M] × l ∈ [0, K/L)], and ı by [ıˆa˛,m,h |m ∈ [0, M]] with ıˆa˛,m,h being the estimated values of ıa˛,m,h . Note that l is used for brevity as substitution for l = L(l + 1/2). Consequently, the improved phase parameter estimates are obtained as f

aˆ ˛,m,h = aˆ ˛,m,h + m

ıˆa˛,m,h , Lm

m = 1, . . ., M

f

aˆ ˛,0,h = ıˆa˛,0,h .

(24) (25)

Remark. It should be mentioned that the moving average ﬁlter length L is commonly chosen in the range L ∈ [5, 15]. Larger values of L provide better ﬁltering of data sequence; however, working with undersampled data means that we cannot use large L due to limited number of available samples. Therefore, in the case of the CS data, the accuracy improvement in the ﬁne estimation process is limited with respect to the standard sampling case.

3.2.5. Step 5: Selection of the best estimates f Note that in the proposed technique, the phase estimates aˆ ˛,m,h depend on the values of ˛, h and M (if we assume that M is unknown). Thus, in the proposed algorithm implementation we selected a set of values for ˛, a set of values for h, a set of values for M, and we re-run the algorithm for all possible combinations of (˛, f h, M). Thus, the resulting number of parameter sets {ˆa˛,m,h;M |m ∈ [0, M]} is equal to the number of possible combinations of (˛, h, M). The best set of parameters selected among all the obtained ones is the maximizer of the function:

2 M f f m x(tn ) exp −jˆa˛,0,h;M − j aˆ ˛,m,h;M tn /m . n

m=1

That is, the optimal set of parameter estimates is given by

2 M f f ˆ M) ˆ = arg max (˛, ˆ h, x(tn ) exp −jˆa˛,0,h;M − j aˆ ˛,m,h;M tnm /m . (˛,h,M) n

(26)

m=1

Now that all the steps have been described in details, we can summarize the proposed algorithm steps as follows:

for ˛ ∈ for h ∈ H Evaluation of STFT˛,h (t, ω) by using Eqs. (13) and (14). (t ) using Eqs. (16) and (17). ˆ ˛,h IF estimation ω s

f

Fine parameter estimation of aˆ ˛,m,h using Eqs. (19)–(25). Evaluation of the criterion function:

2 f f m x(tn ) exp −jˆa˛,0,h; − j aˆ ˛,m,h; tn /m . J(˛, h, ) = end end

(˛,h, )

aˆ m = aˆ

f ˆ M ˆ ˛,m, ˆ h;

ˆ for m ∈ [0, M].

(28)

At the ﬁrst glance, it looks like the proposed technique requires a 3D search for parameter estimation. However, if we ﬁx ˛, the search is performed over small number of elements of the set H and ﬁnally, if the order of the polynomial in the signal phase is known in advance, which is common in practice [13], the search over M is not needed. 4. Numerical examples 4.1. Example 1 The statistical study of the proposed method is performed for a sixth order PPS. This signal is of rather high order and conventional phase differentiation techniques, such as the high-order ambiguity function (HAF) and product HAF (PHAF), usually produce a mean squared error (MSE) that is 15–20 dB above the CRLB for a signal uniformly sampled according to the Shannon/Nyquist theorem. Suppose that the signal is uniformly sampled within the interval t ∈ [−1, 1). The signal amplitude is A = 1 while, in each trial, the phase parameters are randomly selected according to the uniform distribution within ranges: a6 ∈ [0, 2 ], a5 ∈ [0, 2 ], a4 ∈ [40 , 56 ], a3 ∈ [31.5 , 40.5 ], a2 ∈ [−56 , − 40 ], a1 ∈ [6 , 10 ] and a0 = 0. The trimming parameter ˛ is ﬁxed equal to 0.35 and the order of polynomial phase M is known in advance. The set of windows has the following widths H = 1/16 : 1/16 : 1/2 (8 windows in total). Length of the median ﬁlter is set to 2m + 1 =5. We have performed 300 trials for various numbers of non-uniformly sampled data K for three experiments: noise-free signal, SNR = 5 dB and SNR = 10 dB. Here, we can expect errors in the estimation process even for the noise-free signal due to spectral distortion effects associated with the robust STFT [17,29]. Fig. 1 depicts the MSE in the estimation of the two highest-order phase parameters. Note that similar results are obtained for all the other phase parameters. Also, we have conducted experiments for M < 6 and obtained similar outcomes. As it can be seen, the proposed technique for K close to N practically reaches the CRLB but as the number of considered samples reduces, the MSE of the proposed technique departs from the CRLB. For example, for K = 110 samples, difference between the MSE and the CRLB is less than 15 dB. This is smaller than the difference between the MSE in the case of the HAF based estimator and CRLB (see [23,25]). Even for K = 70, the proposed technique has MSE 18 dB higher than the CRLB but still there are no outliers in the parameters estimation. For K < 70 (more than 72% samples are missing) the proposed estimator starts to produce outliers.

In this example we consider a test signal with non-polynomial modulation:

Rough estimation of parameters aˆ ˛,h using Eq. (18).

end

ˆ M) ˆ = arg max J(˛, h, ) (˛, ˆ h,

4.2. Example 2

for M ∈

n

Estimation of parameters as

m=1

(27)

x(t) = exp(j24 sin(2 t)).

(29)

The IF of this signal is crucial since, in this case, we cannot use the parameters of the PPS model. The IF for the considered signal is ω(t) = 48 cos(2 t).

(30)

In this experiment, we have used the same ˛ value and the same set of window lengths as those in Example 1. However, for the polynomial order M, we have searched over the interval M ∈ [2, 12]. The

I. Djurovi´c et al. / Int. J. Electron. Commun. (AEÜ) 68 (2014) 631–636

100

635

N=256 M=6, estimation of a6

MSE[dB]

Noiseless SNR=5dB SNR=10dB CRLB for SNR=5dB CRLB for SNR=10dB

80 60 40 20 0 60

80

100

120

140

160

180

200

220

240 K

N=256 M=6, estimation of a

5

MSE[dB] 80

Noiseless SNR=5dB SNR=10dB CRLB for SNR=5dB CRLB for SNR=10dB

60

Fig. 3. (K, SNR) pairs for which there are no outliers in the IF estimation (white color) and with outliers (black color).

40

20

0 60

80

100

120

140

160

180

200

220

240 K

Fig. 1. MSE estimation for M = 6th order signal as function of K for three cases: noise-free signal; SNR = 5 dB; SNR = 10 dB and comparison with the corresponding CRLBs.

IF estimate is depicted in Fig. 2 for three characteristic SNR values and K. From Fig. 2 it can be seen that, for high SNR (SNR = 10 dB) and relatively small number of missing samples (K = 120), the obtained results are excellent, i.e., the estimated IF is very close to the exact IF. For lower SNR (SNR = 6 dB) and larger number of missing samples (K = 100), the results are worse with larger difference between the exact and estimated IF. However, we can say that reasonable accuracy has been archived since in all instants the obtained IF ω(t)

5. Conclusion The QML-based estimator of polynomial phase signals, sampled at a rate below the Nyquist/Shannon theorem limit, is proposed in this paper. The proposed technique is tested for two scenaria: a signal with known order of polynomial, and a signal that has no polynomial modulation. In both cases the proposed technique produces accurate results for a high percentage of missing samples.

IF estimation Exact IF K=80, SNR=4dB K=100, SNR=6dB K=120, SNR=8dB

150

estimate is close to the exact one. This cannot be claimed for K = 80 and SNR = 4 dB since signiﬁcant difference between the exact and estimated IF exists. The estimated IF for this case can be treated as an outlier. In order to ﬁnd (K, SNR) regions in which the modiﬁed QML algorithm produces estimates that are free of outliers, we have performed 300 trials of Monte Carlo simulations for various K and SNR values. We assume that an estimated IF is an outlier if the maximal absolute difference between the estimated and exact IF is larger than 20. The results of this statistical analysis are presented in Fig. 3. In this ﬁgure, the white regions correspond to (K, SNR) values where we have not obtained outliers in our trials, while the black color depicts the range where outliers are detected. It can be seen that for SNR = 10 dB, the proposed technique performs quite accurately for 50% of missing samples, while for SNR = 5 dB the proposed technique is able to operate accurately enough for 33% of missing samples.

100

Acknowledgements 50

Research of Igor Djurovic´ and Marko Simeunovic´ is supported in a part by the FP7 ForeMont project and the Ministry of Science of Montenegro.

0 −50

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−100 −150 −0.6

−0.4

−0.2

0

0.2

0.4

Fig. 2. IF estimation for various K and SNR.

0.6

t

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[5] Grifﬁn A, Hirvonen T, Mouchtaris A, Tsakalides P. Encoding the sinusoidal model of an audio signal using compressed sensing. In: IEEE International Conference on Multimedia and Expo. IEEE; 2009. p. 153–6. [6] Stankovic´ LJ, Stankovic´ S, Orovic´ I, Amin M. Robust time-frequency analysis based on the L-estimation and compressive sensing. IEEE Signal Processing Letters 2013;20(5):499–502. [7] Flandrin P, Borgnat P. Time-frequency energy distributions meet compressed sensing. IEEE Transactions on Signal Processing 2010;58(6): 2974–82. [8] Djurovic´ I, Stankovic´ LJ, Barkat B. Robust time-frequency distributions based on the robust short time Fourier transform. In: Annales des télécommunications, vol. 60. Berlin: Springer-Verlag; 2005. p. 681–97. [9] Katkovnik V. Robust m-periodogram. IEEE Transactions on Signal Processing 1998;46(11):3104–9. [10] Huber PJ. Robust statistics. Hoboken, NJ, USA: John Wiley & Sons, Inc.; 2004. Series in probability and statistics. [11] O’Shea P. Improving polynomial phase parameter estimation by using nonuniformly spaced signal sample methods. IEEE Transactions on Signal Processing 2012;60(7):3405–14. [12] Djurovic´ I, Stankovic´ LJ. STFT-based estimator of polynomial phase signals. Signal Processing 2012;92(11):2769–74. [13] Peleg S, Porat B. The Cramer-Rao lower bound for signals with constant amplitude and polynomial phase. IEEE Transactions on Signal Processing 1991;39(3):749–52. [14] Ristic´ B, Boashash B. Comments on “The Cramer-Rao lower bounds for signals with constant amplitude and polynomial phase”. IEEE Transactions on Signal Processing 1998;46(6):1708–9. [15] Legg JA, Gray DA. Performance bounds for polynomial phase parameter estimation with non-uniform and random sampling schemes. IEEE Transactions on Signal Processing 2000;48(2):331–7. [16] Roenko AA, Lukin VV, Djurovic´ I. An overview of the adaptive robust DFT. EURASIP Journal on Advances in Signal Processing 2010; 2010:3.

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International Journal of Electronics and Communications (AEÜ) journal homepage: www.elsevier.com/locate/aeue

Quasi maximum likelihood estimator of polynomial phase signals for compressed sensed data Igor Djurovic´ a , Vladimir V. Lukin b , Marko Simeunovic´ a,∗ , Braham Barkat c a b c

University of Montenegro, Electrical Engineering Department, Cetinjski put bb, 81000 Podgorica, Montenegro National Aerospace University, Department of Transmitters, Receivers and Signal Processing, Kharkov, Ukraine Petroleum Institute, Ruwais Building, Abu Dhabi, United Arab Emirates

a r t i c l e

i n f o

Article history: Received 11 September 2013 Accepted 21 January 2014 Keywords: Polynomial phase signals Parameter estimation Compressive sensing

a b s t r a c t Several papers in the literature cover parameter estimation of frequency modulated (FM) signals under reduced number of signal samples with respect to the Nyquist/Shannon criterion, i.e., within the compressive sensing (CS) framework. However, scope of these papers is mainly limited to sinusoids or sum of sinusoids. In this paper, the CS framework is extended to parameter estimation of higher order polynomial phase signals (PPSs) using the quasi-maximum likelihood (QML) estimator and robust short-time Fourier transform (STFT). The considered signal is assumed to be non-uniformly sampled PPS with smaller number of samples with respect to the Nyquist/Shannon criterion. However, the proposed technique can also be generalized to uniformly sampled signals with missing or unreliable samples. © 2014 Elsevier GmbH. All rights reserved.

1. Introduction Processing of undersampled data or data with missing samples has attracted signiﬁcant attention within the compressive sensing (CS) framework [1,2]. Namely, joint data acquisition and compression is possible if data records are sparse in the recording domain [3]. There exist several research papers dealing with the CS in the frequency domain related to parameter estimation of a sinusoidal signal [4–6]. In addition, reconstruction of samples of time-frequency (TF) representations based on compressed information is considered in [7]. Note that reconstructing a signal using data with missing samples is in fact equivalent to compressive sensing, which can be described as a technique for efﬁcient determination of entire signal from relatively small number of measurements. In [8,9], the robust Huber statistics [10] has been applied to the high frequency data and FM signals corrupted by impulsive noise. However, missing observations within CS framework can also be considered as unreliable signal samples due to high amount of noise [6]. Recently, it has been shown that parametric estimation of non-uniformly sampled frequency modulated (FM) signals can be more accurate than that of uniformly sampled data [11]. Previous research in the CS is extended in several directions in this paper. Firstly, polynomial phase signals (PPSs) are considered. Secondly, in addition to the estimation of parameters of uniformly

∗ Corresponding author. Tel.: +382 68656038. ´ [email protected] (V.V. Lukin), E-mail addresses: [email protected] (I. Djurovic), ´ [email protected] (B. Barkat). [email protected], [email protected] (M. Simeunovic), 1434-8411/$ – see front matter © 2014 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.aeue.2014.01.011

sampled data with missing samples, we consider undersampled data with non-uniform sampling. Thirdly, a recently proposed quasi maximum likelihood (QML) approach [12], used as a tool for PPS estimation from CS observations, has been modiﬁed to estimate signals with non-polynomial modulation. The paper is organized as follows. The background informations: the PPS deﬁnition, the sampling model, the robust discrete Fourier transform (DFT) and the robust short-time Fourier transform (STFT) frameworks are presented in Section 2. The QML algorithm for handling CS data (QML-CS) is proposed in Section 3, while its statistical study is presented in Section 4. Concluding remarks are given in Section 5.

2. Objective and theoretical background 2.1. Signal model and objective A polynomial phase signal (PPS) is deﬁned as

x(t) = A exp(j(t)) = A exp

ja0 + j

M

m

am t /m

,

(1)

m=1

where A is the amplitude, M is the polynomial phase order and a0 , a1 , . . ., aM are the phase parameters. Note that this is a compressed representation of the signal x(t) since by only knowing the

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PPS parameters the full description of x(t) can be known at any time instant t. The radian instantaneous frequency (IF) of x(t) is given by

ω(t) = (t) =

M

am t

m−1

.

(2)

m=1

If we deﬁne ωmax and ωmin to be the maximal and the minimal IFs in an arbitrary time interval [T1 , T2 ] with T = T2 − T1 , then we can write

ik ∈ {Im{x(tn ) exp(−jωtn )}|n ∈ [1, N]},

with ik ≤ ik+1 , rk ≤ rk+1 and x(n) = x(nt) = x(tn ) (t is the sampling period). In the L-DFT implementation, the parameter bk is usually chosen such that k bk = 1 and bk = bN+1−k . A widely used value for bk , when N is even, is given by [17]

⎧ ⎨ bk =

B = ωmax − ωmin , ωmax = max ω(t),

(3)

ωmin = min ω(t),

(4)

(8)

⎩

N − ˛(N − 2) , 2

1 , 2˛(N − 2) + 2

k∈

0,

elsewhere

N + 1 + ˛(N − 2) 2

(9)

where B is the signal bandwidth in the interval T. To satisfy the sampling theorem, within the considered time interval T, the total number of signal samples N has to be equal or larger than BT, i.e., N ≥ BT. However, this number N is usually significantly larger than (M + 2), the number of signal parameters (A, a0 , . . ., aM ) to estimate. In this paper, our objective is to estimate these parameters using only K samples where K is an integer verifying M + 2 ≤ K ≤ N.

where the operator · denotes the rounding to the closest integer. The value of the chosen ˛ (˛ ∈ [0, 0.5]) controls the amount of outlier rejection. For instance, small values of ˛ yield a large amount of outlier rejection but at the same time introduce some spectral signal distortion. From extensive computational analysis we observed, using trial and error, that the best value for ˛ is 0.35. Of course, alternative methods such as adaptive ones can be used to determine the optimal value for ˛ [16]. Note that the standard DFT is obtained for bk = 1/N, k = 1, . . ., N. The L-DFT is an example of the robust Huber’s statistics concept used to represent, ﬁlter and estimate the parameters of FM signals corrupted with impulsive noise [9,10,18].

2.2. Cramer-Rao lower bounds (CRLBs) for uniformly and non-uniformly sampled PPSs

2.4. L-short-time Fourier transform (L-STFT)

t∈[T1 ,T2 ]

t∈[T1 ,T2 ]

If the signal x(t) is uniformly sampled using N samples, the best achievable performance in the estimation of the phase parameters, a0 , . . ., aM , is given by the Cramer-Rao lover bound CRLBN (am ), m = 0, . . ., M [13,14]. However, if the signal x(t) is non-uniformly sampled using K ≤ N samples, whereby the samples are randomly drawn from x(t) using the uniform distribution p(ts ) =

1 1 = T T2 − T1

ts ∈ [T1 , T2 ],

with s = 1, 2, . . ., K

(5)

then, the best achievable performance in the estimation of the phase parameters is given by CRLBK (am ), m = 0, . . ., M [15] N CRLBK (am ) ≈ CRLBN (am ). K

(6)

In the sequel, for the sake of simplicity, we will assume that the non-uniform time instants ts , s = 1, . . ., K satisfy the relationship ts < ts+1 . 2.3. L-discrete Fourier transform (L-DFT) When the signal x(t) is uniformly sampled and some of its samples are missing the standard DFT becomes inadequate. As the missing samples can be treated as samples corrupted by impulsive noise [6], the best tool to use in this case is the robust DFT. Here, we will consider the L-DFT form of the robust DFT deﬁned as [16] XL (ω) =

N

bk [rk + jik ].

(7)

k=1

In this robust DFT, the quantity rk (k = 1, . . ., N) is deﬁned such that r1 corresponds to the ﬁrst element of the value-ordered set {Re{x(tn ) exp(− jωtn )}|n ∈ [1, N]}, r2 corresponds to the second element of the value-ordered set {Re{x(tn ) exp(− jωtn )}|n ∈ [1, N]} and so on. The quantity ik is deﬁned in a similar way using the valueordered set {Im{x(tn ) exp(− jωtn )}|n ∈ [1, N]}. Mathematically, we have rk ∈ {Re{x(tn ) exp(−jωtn )}|n ∈ [1, N]},

The L-DFT, applicable to stationary signals, has been generalized to non-stationary signals using the L-STFT deﬁned as [8,19]

STFT˛,h (ts , ω) =

h+1

bs,h [rs,h + jis,h ] k k k

(10)

k=1

where

rs,h ≤ rs,h , rs,h ∈ Re{x(tn ) exp(−jωtn )}|n ∈ s − k k+1 k

is,h ≤ is,h , is,h ∈ Im{x(tn ) exp(−jω(tn )}|n ∈ k k+1 k

h h ,s + 2 2

s−

h h ,s + 2 2

(11)

(12)

and s = h/2 +1, . . ., N − h/2. As we can see from the expressions, the L-STFT is implemented using windowing of the data to select a narrow time interval and subsequently applying it on the L-DFT. The window length (h + 1) used in the L-STFT is assumed to be odd, and are selected in the same way as those in the the coefﬁcients bs,h k L-DFT. Other TF representations exist in the literature and can be applied instead of the STFT [8,20]. 3. QML-CS algorithm 3.1. QML algorithm Many algorithms have been developed for the estimation of the parameters of a PPS [21–25]. Here, we consider the QML estimator [12,26] which consists of the following ﬁve basic steps: Step 1. Evaluate the STFTs corresponding to different selected window lengths. Step 2. For each evaluated STFT, estimate the corresponding signal IF. Step 3. From each estimated IF, obtain the PPS parameters estimates {ˆa1 , . . ., aˆ M } using polynomial regression described in [12]. Observe that these estimates are biased due to the biased nature of the STFT IF estimates. Also, note that for

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each window length we have a corresponding set of parameters {ˆa1 , . . ., aˆ M }. Step 4. Improve the parameter estimates {ˆa1 , . . ., aˆ M } to obtain f f f reﬁned estimates {ˆa0 , aˆ 1 , . . ., aˆ M } using the technique developed in [27]. f f f Step 5. For each reﬁned parameter set {ˆa0 , aˆ 1 , . . ., aˆ M }, evaluate the QML function deﬁned as

2 M f f m x(tn ) exp −jˆa0 − j aˆ m tn /m n

Existing PPS parameter estimation algorithms, including the QML algorithm presented above, have been developed for uniformly sampled signals that satisfy the sampling theorem. In the sequel, we will modify the QML algorithm in order to estimate the PPS parameters when (i) the sampling rate is random and nonuniform and (ii) the number of considered samples is smaller than the number of samples used in uniform sampling case. 3.2. Modiﬁed QML algorithm Here we consider a signal x(t) that has been non-uniformly sampled using K samples. The samples are randomly drawn from x(t) using the uniform distribution given by (5). 3.2.1. Step 1: L-STFT evaluation In the modiﬁed QML instead of using STFT we use the L-STFT. In this case, in the evaluation of the L-STFT we only consider the available time instants ts , s ∈ [1, K] as the center of our ﬁxed length window h. We note that the number of samples in the window around ts is not symmetrical (i.e., number of samples to the left of ts is not necessarily equal to the number of samples to the right of it) and this number can be different for each considered time instant ts . Thus, if we assume s,h to be the number of signal samples within the window interval h, then the L-STFT can be expressed as bs,h [rs,h + jis,h ] k k k

(13)

k=1

where rs,h ≤ rs,h , k k+1 s,h

s,h

ik ≤ ik+1 ,

rs,h ∈ k s,h

ik ∈

Re{x(tq ) exp(−jω(tq − ts )}|tq ∈

Im{x(tq ) exp(−jω(tq − ts )}|tq ∈

ts −

h h , ts + 2 2

ts −

h h , ts + 2 2

(14)

ω

.

(15)

(16)

Since a large number of signal samples are missing, it is expected that some of the IF estimates may be relatively far from the true IF values. To mitigate this, we propose to employ the median ﬁlter to remove these large errors in the IF estimates, i.e., ω ˆ ˛,h (ts ) = median{ω ˆ ˛,h (tq ) | q ∈ [s − m, s + m]},

where m is the median ﬁlter length.

−1

ˆ XTh ˛,h

(18)

Xh = [tsm−1 | m ∈ [1, M] × s ∈ [1, K]]. For the notation, now we have added the subscripts ˛ and h to the parameters aˆ i to become aˆ ˛,i,h to indicate that the parameter estimates depend on the selected values of ˛ and h. Again, we note that the estimates are biased because the L-STFT-based IF estimates are biased. 3.2.4. Step 4: reﬁnement procedure The reﬁnement procedure is a very important step in the QML algorithm. In order to improve the accuracy of the phase estimates obtained in Step 3, a reﬁnement algorithm described in [27] is used. However, this reﬁnement technique had to be adapted for non-uniformly and subsampled data. Thus, here we explain the main steps of the reﬁnement procedure that have been used in our proposed algorithm. First, the signal x(ts ) is dechirped using the parameters estimates {ˆa˛,1,h , . . ., aˆ ˛,M,h }

x˜ (ts ) = x(ts ) exp

(17)

−j

= A exp

M

aˆ ˛,m,h tsm /m

m=1

M

ja0 + j

am t m /m

−j

exp

m=1

x˜ (ts ) = A exp

ja0 − j

M

M

aˆ ˛,m,h tsm /m

(19)

m=1

ıa˛,m,h tsm /m

.

(20)

m=1

Due to errors in the estimation of the parameters, the difference between the true values and their estimates is non-zero, i.e., / 0, m = 0, . . ., M. The objective of the reﬁneıa˛,m,h = aˆ ˛,m,h − am = ment is to ﬁnd the values of ıa˛,m,h and to use them to correct the estimates aˆ ˛,m,h . This can be done by ﬁltering, downsampling, phase unwrapping of x˜ (ts ) and subsequent polynomial ﬁtting of the unwrapped phase, as presented below. The ﬁltering of x˜ (ts ), necessary to improve the signal to noise ratio, is performed using a moving average ﬁlter expressed by

3.2.2. Step 2: IF estimation The IF estimation can be performed in the same manner as in the case of the QML algorithm and other related TF-based techniques, i.e., using position of the TF representation maxima ω ˆ ˛,h (ts ) = argmax|STFT˛,h (ts , ω)|.

aˆ ˛,h = (XTh Xh )

where aˆ ˛,h is the vector of the estimated phase parameters T (t )|s ∈ [1, K]] and ˆ while ˆ ˛,h s ˛,h = [ω

and select the set that yields the maximum value of the evaluated function. This set is considered as the optimal set of PPS parameters.

s,h

3.2.3. Step 3: parameter estimation Here, the PPS parameter estimation from the IF estimates is performed in the same way as in the QML algorithm described earlier. Since the IF estimates are sparse and non-uniformly distributed (which is common issue in polynomial ﬁtting), these parameters are estimated using

aˆ ˛,h = [ˆa˛,1,h , . . ., aˆ ˛,M,h ]

m=1

STFT˛,h (ts , ω) =

633

xˆ (tq ) =

1 L

q+L/2−1

x˜ (tp ).

(21)

p=q−L/2

The downsampling of xˆ (tq ) is needed in order to stabilize the results of the estimation process. Namely, polynomial ﬁtting could become an unstable process for a large number of samples. Then, we use the polynomial ﬁtting with reduced number of samples x(tl ) = xˆ (tL(l+1/2) ) for l ∈ [0, K/L).

(22)

Remark. As an alternative, some more robust procedures can be applied [28]. Once the noise is suppressed, the phase of x(tl ) is unwrapped using v(tl ) = unwrap(angle(x(tl )) l ∈ [0, K/L) and polynomial ﬁtting is performed to estimate ıa˛,m,h , m = 0, . . ., M using ı a˜ = ( ) T

−1

T

v,

(23)

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where v is a vector of components v(tl ), is a matrix given m a˜ is a vector deﬁned by = [tl |m ∈ [0, M] × l ∈ [0, K/L)], and ı by [ıˆa˛,m,h |m ∈ [0, M]] with ıˆa˛,m,h being the estimated values of ıa˛,m,h . Note that l is used for brevity as substitution for l = L(l + 1/2). Consequently, the improved phase parameter estimates are obtained as f

aˆ ˛,m,h = aˆ ˛,m,h + m

ıˆa˛,m,h , Lm

m = 1, . . ., M

f

aˆ ˛,0,h = ıˆa˛,0,h .

(24) (25)

Remark. It should be mentioned that the moving average ﬁlter length L is commonly chosen in the range L ∈ [5, 15]. Larger values of L provide better ﬁltering of data sequence; however, working with undersampled data means that we cannot use large L due to limited number of available samples. Therefore, in the case of the CS data, the accuracy improvement in the ﬁne estimation process is limited with respect to the standard sampling case.

3.2.5. Step 5: Selection of the best estimates f Note that in the proposed technique, the phase estimates aˆ ˛,m,h depend on the values of ˛, h and M (if we assume that M is unknown). Thus, in the proposed algorithm implementation we selected a set of values for ˛, a set of values for h, a set of values for M, and we re-run the algorithm for all possible combinations of (˛, f h, M). Thus, the resulting number of parameter sets {ˆa˛,m,h;M |m ∈ [0, M]} is equal to the number of possible combinations of (˛, h, M). The best set of parameters selected among all the obtained ones is the maximizer of the function:

2 M f f m x(tn ) exp −jˆa˛,0,h;M − j aˆ ˛,m,h;M tn /m . n

m=1

That is, the optimal set of parameter estimates is given by

2 M f f ˆ M) ˆ = arg max (˛, ˆ h, x(tn ) exp −jˆa˛,0,h;M − j aˆ ˛,m,h;M tnm /m . (˛,h,M) n

(26)

m=1

Now that all the steps have been described in details, we can summarize the proposed algorithm steps as follows:

for ˛ ∈ for h ∈ H Evaluation of STFT˛,h (t, ω) by using Eqs. (13) and (14). (t ) using Eqs. (16) and (17). ˆ ˛,h IF estimation ω s

f

Fine parameter estimation of aˆ ˛,m,h using Eqs. (19)–(25). Evaluation of the criterion function:

2 f f m x(tn ) exp −jˆa˛,0,h; − j aˆ ˛,m,h; tn /m . J(˛, h, ) = end end

(˛,h, )

aˆ m = aˆ

f ˆ M ˆ ˛,m, ˆ h;

ˆ for m ∈ [0, M].

(28)

At the ﬁrst glance, it looks like the proposed technique requires a 3D search for parameter estimation. However, if we ﬁx ˛, the search is performed over small number of elements of the set H and ﬁnally, if the order of the polynomial in the signal phase is known in advance, which is common in practice [13], the search over M is not needed. 4. Numerical examples 4.1. Example 1 The statistical study of the proposed method is performed for a sixth order PPS. This signal is of rather high order and conventional phase differentiation techniques, such as the high-order ambiguity function (HAF) and product HAF (PHAF), usually produce a mean squared error (MSE) that is 15–20 dB above the CRLB for a signal uniformly sampled according to the Shannon/Nyquist theorem. Suppose that the signal is uniformly sampled within the interval t ∈ [−1, 1). The signal amplitude is A = 1 while, in each trial, the phase parameters are randomly selected according to the uniform distribution within ranges: a6 ∈ [0, 2 ], a5 ∈ [0, 2 ], a4 ∈ [40 , 56 ], a3 ∈ [31.5 , 40.5 ], a2 ∈ [−56 , − 40 ], a1 ∈ [6 , 10 ] and a0 = 0. The trimming parameter ˛ is ﬁxed equal to 0.35 and the order of polynomial phase M is known in advance. The set of windows has the following widths H = 1/16 : 1/16 : 1/2 (8 windows in total). Length of the median ﬁlter is set to 2m + 1 =5. We have performed 300 trials for various numbers of non-uniformly sampled data K for three experiments: noise-free signal, SNR = 5 dB and SNR = 10 dB. Here, we can expect errors in the estimation process even for the noise-free signal due to spectral distortion effects associated with the robust STFT [17,29]. Fig. 1 depicts the MSE in the estimation of the two highest-order phase parameters. Note that similar results are obtained for all the other phase parameters. Also, we have conducted experiments for M < 6 and obtained similar outcomes. As it can be seen, the proposed technique for K close to N practically reaches the CRLB but as the number of considered samples reduces, the MSE of the proposed technique departs from the CRLB. For example, for K = 110 samples, difference between the MSE and the CRLB is less than 15 dB. This is smaller than the difference between the MSE in the case of the HAF based estimator and CRLB (see [23,25]). Even for K = 70, the proposed technique has MSE 18 dB higher than the CRLB but still there are no outliers in the parameters estimation. For K < 70 (more than 72% samples are missing) the proposed estimator starts to produce outliers.

In this example we consider a test signal with non-polynomial modulation:

Rough estimation of parameters aˆ ˛,h using Eq. (18).

end

ˆ M) ˆ = arg max J(˛, h, ) (˛, ˆ h,

4.2. Example 2

for M ∈

n

Estimation of parameters as

m=1

(27)

x(t) = exp(j24 sin(2 t)).

(29)

The IF of this signal is crucial since, in this case, we cannot use the parameters of the PPS model. The IF for the considered signal is ω(t) = 48 cos(2 t).

(30)

In this experiment, we have used the same ˛ value and the same set of window lengths as those in Example 1. However, for the polynomial order M, we have searched over the interval M ∈ [2, 12]. The

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100

635

N=256 M=6, estimation of a6

MSE[dB]

Noiseless SNR=5dB SNR=10dB CRLB for SNR=5dB CRLB for SNR=10dB

80 60 40 20 0 60

80

100

120

140

160

180

200

220

240 K

N=256 M=6, estimation of a

5

MSE[dB] 80

Noiseless SNR=5dB SNR=10dB CRLB for SNR=5dB CRLB for SNR=10dB

60

Fig. 3. (K, SNR) pairs for which there are no outliers in the IF estimation (white color) and with outliers (black color).

40

20

0 60

80

100

120

140

160

180

200

220

240 K

Fig. 1. MSE estimation for M = 6th order signal as function of K for three cases: noise-free signal; SNR = 5 dB; SNR = 10 dB and comparison with the corresponding CRLBs.

IF estimate is depicted in Fig. 2 for three characteristic SNR values and K. From Fig. 2 it can be seen that, for high SNR (SNR = 10 dB) and relatively small number of missing samples (K = 120), the obtained results are excellent, i.e., the estimated IF is very close to the exact IF. For lower SNR (SNR = 6 dB) and larger number of missing samples (K = 100), the results are worse with larger difference between the exact and estimated IF. However, we can say that reasonable accuracy has been archived since in all instants the obtained IF ω(t)

5. Conclusion The QML-based estimator of polynomial phase signals, sampled at a rate below the Nyquist/Shannon theorem limit, is proposed in this paper. The proposed technique is tested for two scenaria: a signal with known order of polynomial, and a signal that has no polynomial modulation. In both cases the proposed technique produces accurate results for a high percentage of missing samples.

IF estimation Exact IF K=80, SNR=4dB K=100, SNR=6dB K=120, SNR=8dB

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estimate is close to the exact one. This cannot be claimed for K = 80 and SNR = 4 dB since signiﬁcant difference between the exact and estimated IF exists. The estimated IF for this case can be treated as an outlier. In order to ﬁnd (K, SNR) regions in which the modiﬁed QML algorithm produces estimates that are free of outliers, we have performed 300 trials of Monte Carlo simulations for various K and SNR values. We assume that an estimated IF is an outlier if the maximal absolute difference between the estimated and exact IF is larger than 20. The results of this statistical analysis are presented in Fig. 3. In this ﬁgure, the white regions correspond to (K, SNR) values where we have not obtained outliers in our trials, while the black color depicts the range where outliers are detected. It can be seen that for SNR = 10 dB, the proposed technique performs quite accurately for 50% of missing samples, while for SNR = 5 dB the proposed technique is able to operate accurately enough for 33% of missing samples.

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Acknowledgements 50

Research of Igor Djurovic´ and Marko Simeunovic´ is supported in a part by the FP7 ForeMont project and the Ministry of Science of Montenegro.

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References

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Fig. 2. IF estimation for various K and SNR.

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