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 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 6, NO. 1, MARCH 2012 1
 Time-Frequency Distributions based on Compact
 Support Kernels: Properties and Performance
 Evaluation
 Mansour Abed, Student Member, IEEE, Adel Belouchrani, Member, IEEE Mohamed Cheriet, Senior Member,
 IEEE and Boualem Boashash, Fellow, IEEE
 Abstract— This paper presents two new time-frequency distri-
 butions (TFDs) based on kernels with compact support (KCS)
 namely the separable (CB) (SCB) and the polynomial CB (PCB)
 TFDs. The implementation of this family of TFDs follows the
 method developed for the Cheriet-Belouchrani (CB) TFD. The
 mathematical properties of these three TFDs are analyzed and
 their performance is compared to the best classical quadratic
 TFDs using several tests on multi-component signals with linear
 and nonlinear frequency modulation (FM) components including
 the noise effects. Instead of relying solely on visual inspection
 of the time-frequency domain plots, comparisons include the
 time slices’ plots and the evaluation of the Boashash-Sucic’s
 normalized instantaneous resolution performance measure that
 permits to provide the optimized TFD using a specific methodol-
 ogy. In all presented examples, the KCS-TFDs show a significant
 interference rejection, with the component energy concentration
 around their respective instantaneous frequency laws yielding
 high resolution measure values.
 Index Terms— Time-frequency analysis, compact support ker-
 nel, separable compact support kernel, polynomial compact
 support kernel, performance evaluation, instantaneous frequency,
 quadratic TFDs.
 I. INTRODUCTION
 The majority of real-life signals are generally classified
 as nonstationary, i.e. as signals with time-varying spectra.
 In addition, signals in practice are often multi-component.
 Because of this, time-frequency distributions (TFDs) are the
 natural choice to analyze and process nonstationary signals
 accurately and efficiently by performing a mapping of one-
 dimensional signal x(t) into a two dimensional function of
 time and frequency TFDx(t, f).
 Herein, we are interested in the quadratic class of TFDs,
 also known in the literature as kernel-based transform [1]
 TFDx(t, f) = ∫ ∫ ∫+∞
 −∞
 ej2piη(s−t)φ(η, τ)x(s + τ/2)
 x∗(s − τ/2)e−j2pifτdηdsdτ (1)
 M. Abed is with the Electrical Engineering Department, Ecole Nationale
 Polytechnique, Algiers, and Laboratoire Signaux et Systèmes, University of
 Mostaganem, ALGERIA, e-mail: abed.mansour@univ-mosta.com
 A. Belouchrani is with the Electrical Engineering Department, Ecole
 Nationale Polytechnique, El Harrach, Algiers, ALGERIA, e-mail:
 adel.belouchrani@enp.edu.dz
 M. Cheriet is with Synchromedia, University of Quebec (ETS), 1100 Notre-
 Dam West, Montreal, Quebec, Canada, e-mail:cheriet@gpa.etsmtl.ca
 B. Boashash is with the University of Queensland, Centre for Clinical
 Research, Australia and Qatar University, Dept of Electrical Engineering,
 Qatar, e-mail:Boualem@qu.edu.qa
 Manuscript submitted on July 29, 2011.
 where φ(η, τ) is a two-dimensional kernel. This class of
 distributions could also be expressed as
 TFDx(t, f) = ∫+∞
 −∞
 ∫+∞
 −∞
 J(s − t, τ) x(s + τ/2)
 x∗(s − τ/2)e−j2pifτdsdτ (2)
 where
 J(s′, τ) =
 ∫ +∞
 −∞
 φ(η, τ)ej2piηs′dη (3)
 The advantage of expression 1 is to facilitate the compu-
 tation and the analysis of the considered TFD by reducing
 the number of integrals. On the other hand, the quantity J
 can be viewed as simply the inverse Fourier transform of
 φ(η, τ) with respect to η. Moreover, if we note by CJx(t, τ)
 the convolution of the instantaneous autocorrelation function
 y(t, τ) = x(t+ τ/2)x∗(t− τ/2) with G(t, τ) = J(−t, τ), i.e.
 CJx(t, τ) =
 ∫ +∞
 −∞
 y(s, τ)G(t − s)ds
 =
 ∫ +∞
 −∞
 x(s + τ/2)x∗(s − τ/2)J(s − t, τ)ds (4)
 then any quadratic TFD can be expressed as the Fourier
 transform of CJx(t, τ) with respect to τ . It is known in the
 art that the use of a quadratic class of distributions permits
 the definition of kernels whose main property is to reduce
 the interference patterns induced by the distribution itself. In
 [2], it was shown that kernels with compact support (KCS),
 derived from the Gaussian kernel, allow a tradeoff between
 a good autoterm resolution and a high cross term rejection.
 The Gaussian kernel suffers from information loss due to
 reduction in accuracy when the Gaussian is cut off to compute
 the time-frequency distribution, and the prohibitive processing
 time due to the mask’s width which is increased to minimize
 the accuracy loss [2]. On the contrary, kernels with compact
 support are found to recover this information loss and improve
 processing time and, at the same time, retains the most
 important properties of the Gaussian kernel [3]. These features
 are achieved thanks to the compact support analytical property
 of this type of kernels since they vanish themselves outside
 a given compact set. It turns out that through a control pa-
 rameter of the kernel width, the corresponding time-frequency
 distributions allow a better elimination of cross-terms while
 providing good resolution in both time and frequency.
 Motivated by these interesting properties, we propose in
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 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 6, NO. 1, MARCH 2012 2
 this contribution the use of two new kernels with compact
 support derived from the Gaussian kernel for time-frequency
 analysis namely the separable KCS (SKCS) [4] and the
 polynomial KCS (PKCS) [5]. Similarly to the CB TFD [6],
 the induced TFDs referred to as SCB TFD and PCB TFD,
 respectively are generated following a specific method that
 uses first the Hilbert transform for producing analytical signals
 from real samples of the original signal then computes the
 convolutions of the proposed compact support kernels and the
 instantaneous autocorrelation functions and finally applies a
 Fourier transform to determine information related to the en-
 ergy of the original signal with respect to time and frequency.
 In order to provide an objective assessment, the established
 comparisons between the KCS based TFDs and the most
 commonly used time-frequency representations are based on
 the Boashash-Sucic performance measure [7]. In this context,
 it is shown through several tests that the compact support
 kernels outperform the other ones even for the hard case of
 closely spaced noisy multi-component signals in the t-f plane.
 The paper is organized as follows. In the next section, we
 analyze the mathematical characteristics that the CB kernel
 satisfies in the time-frequency domain. In Sections III and
 IV, we detail the construction and the main properties of the
 two new proposed classes of quadratic distributions based on
 the SCB and PCB kernel respectively. Section V describes
 the performance evaluation of TFDs with special attention
 to the Boashash-Sucic objective performance measure used
 to select the optimum time-frequency representation in each
 studied case. Section VI is devoted to presenting comparative
 experimental results obtained by applications involving energy
 estimation of linear and nonlinear multi-component frequency
 modulated signals including the influence of noise. Finally,
 concluding remarks are given in Section VII.
 II. MATHEMATICAL PROPERTIES OF THE CB TFD
 The choice of the two-dimensional kernel is crucial in the
 definition of a quadratic TFD and it determines the proper-
 ties of the generated distribution e.g. real-valued, marginal
 conditions, instantaneous frequency (IF) as well as its overall
 performance in terms of energy concentration and resolution.
 In general the purpose of the kernel is to reduce the interfer-
 ence terms in the time-frequency distribution. However, Eq. (1)
 shows that the reduction of the interference patterns involves
 smoothing and thus results in a reduction of time-frequency
 resolution. Moreover, depending on the type of kernel, some
 of the desired properties of the time-frequency distribution
 are preserved while others are lost [8]. In what follows, we
 consider the main desirable properties verified by the CB
 kernel defined as [6]
 φCB(η, τ) =
 
 
 
 Ae
 C
 (η2 + τ2)/D2 − 1 if η
 2 + τ2
 D2
 < 1
 0 Otherwise
 (5)
 where D and A = eC are control parameters. The CB TFD
 is thus expressed as
 CBx(t, f) =
 ∫ +∞
 −∞
 ∫ +∞
 −∞
 JCB(s − t, τ)y(s, τ)e−j2pifτdsdτ
 (6)
 where
 JCB(s′, τ) = A
 ∫ √D2−τ2
 −
 √
 D2−τ2
 e
 
 
 C
 (η2 + τ2) /D2 − 1
 
 
 ej2piηs
 ′
 dη
 (7)
 As all quadratic time-frequency distributions, the CB TFD
 verifies translation covariance with respect to time and fre-
 quency. Furthermore, as shown in the Appendices A-E respec-
 tively, the CB TFD is always real-valued, conserves energy
 and does not satisfy the marginal properties, dilation covari-
 ance and perfect localization on linear chirp signals property.
 Moreover, from the definitions [9], [10], [11], unitarity, com-
 patibility with filterings and compatibility with modulations
 cannot be satisfied by any smoothed version of the WVD.
 III. MODIFICATION OF THE CB KERNEL: THE
 SEPARABLE CB (SCB)
 Recent results in the field of time-frequency signal analysis
 have shown that quadratic TFDs with separable kernels out-
 perform many other popular TFDs in resolving closely spaced
 components [12], [13], [14]. This type of kernels takes the
 following general form
 φ(η, τ) = φ1(η).φ2(τ) (8)
 In [4], a separable kernel family with compact support
 (SKCS) applied to image processing was introduced. The later
 is a separable version of the compact support kernel. Hence,
 the CB kernel also referred to as KCS can be modified to the
 separable form that we will call Separable Cheriet-Belouchrani
 (SCB) kernel yielding to a new time-frequency distribution of
 quadratic class referred to as SCB TFD. The derived SCB
 kernel is given by
 φSCB(η, τ) =
 
 
 
 φCB(η, 0)φCB(0, τ) if
 
 
 η2 < D2
 and
 τ2 < D2
 0 Otherwise
 (9)
 Thus
 φSCB(η, τ) =
 
 
 
 A2e
 CD2
 η2 − D2
 +
 CD2
 τ2 − D2 if
 
 
 η2 < D2
 and
 τ2 < D2
 0 Otherwise
 (10)
 The separable CB (SCB) TFD is thus expressed as
 SCBx(t, f) =
 ∫ D
 −D
 ∫ +∞
 −∞
 JSCB(s − t, τ)y(s, τ)e−j2pifτdsdτ
 (11)
 JSCB(s′, τ) = A2e
 CD2
 τ2 − D2
 ∫ D
 −D
 e
 CD2
 η2 − D2 ej2piηs
 ′
 dη
 Note that the SCB TFD satisfies all the mathematical
 properties verified by the CB TFD.
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 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 6, NO. 1, MARCH 2012 3
 IV. THE POLYNOMIAL KCS BASED TFD
 Because of its infinite support, the Gaussian kernel must be
 truncated to a finite window when implemented in a computer.
 Regardless of the size of the window, a discontinuity will
 be introduced at its borders that could lead to serious errors
 in the derivatives [15]. As a result, the use of the Gaussian
 kernel presents two practical limitations: information loss and
 derivative border effects owing to diminished accuracy, and the
 prohibitive processing time due to the mask size [5]. In order to
 avoid these drawbacks, two approaches exist: approximating
 the Gaussian kernel by a finite support kernel, or defining
 new kernels with properties close to the Gaussian. In [5], a
 new compact support kernel of polynomial form was proposed
 and applied to scale-space image processing. The new kernel,
 called PKCS, is not obtained by approximating the Gaussian,
 though it is derived from it. This compact support nature
 together with the possibility of controlling the kernel’s window
 width led us to propose a new kernel for time-frequency
 analysis called Polynomial Cheriet-Belouchrani (PCB) kernel
 yielding to a new quadratic time-frequency distrbution referred
 to as PCB TFD. The latter is implemented following the same
 procedure as for the CB TFD and the SCB TFD.
 The PCB kernel is defined as
 φPCB(η, τ) =
 
 
 γ+1
 piλ2γ+2
 (
 λ2 − (η2 + τ2))γ
 if
 (
 η2 + τ2
 )
 < λ2
 0 Otherwise
 (12)
 where λ is the radius of the kernel support and γ is considered
 to be a positive integer so that the resulting kernel has a
 polynomial form.
 The polynomial CB (PCB) TFD is thus formulated as
 PCBx(t, f) =
 ∫ +∞
 −∞
 ∫ +∞
 −∞
 JPCB(s−t, τ)y(s, τ)e−j2pifτdsdτ
 (13)
 JPCB(s′, τ) = γ + 1
 piλ2γ+2
 ∫ √λ2−τ2
 −
 √
 λ2−τ2
 (
 λ2 − (η2 + τ2))γ ej2piηs′dη
 The PCB TFD is real-valued and satisfies translation co-
 variance with respect to time and frequency. Table I gives
 comparisons between the most known kernel-based transforms
 and the KCS TFDs in terms of mathematical properties. It is
 important to note that there is a trade-off between the quantity
 of interferences and the number of good properties. In fact,
 many popular and valuable TFDs (e.g., the spectrogram) do
 not satisfy the marginal and the IF moment condition. What is
 more important in most practical applications is to maximize
 the energy concentration about the IF for mono-component
 signals and improve the resolution for multi-component signals
 [7]. The powerful point of KCS based TFDs is that they have
 by definition a limited width extend since they have a compact
 support. The kernel width is controlled through the parameter
 D for both the CB TFD and the SCB TFD and λ for the PCB
 TFD; and its peak is adjusted through the parameter A = eC
 for the CB and SCB TFDs and γ for the PCB TFD allowing
 a tradeoff between a good autoterm resolution and a sufficient
 cross-term suppression.
 V. PERFORMANCE EVALUATION OF
 TIME-FREQUENCY DISTRIBUTIONS
 Just like some spectral estimates are better than others, some
 time-frequency distributions outperform others when used to
 analyze certain classes of signals [16]-[19]. For example,
 the Wigner-Ville distribution (WVD) [1], [16] is known to
 be optimal for linear frequency modulated monocomponent
 signals since it achieves the best energy concentration around
 the signal IF law. The spectrogram [1], [16], on the other
 hand, results in an undesirable smoothing of the signal energy
 around its IF [16]. Consequently, the choice of the right
 TFD to analyze the given signal is not straightforward. An
 illustration example is shown in Fig. 1 where the bat echo
 location signal is represented in the t-f plane using the WVD,
 the spectrogram, the Born-Jordan distribution [1], the Choi-
 Williams distribution [20], the Zhao-Atlas-Marks distribution
 [21], the CB TFD, the SCB TFD and the PCB TFD. According
 to the common practice, determination of the best representing
 TFD is based on visual inspection of the eight plots so that the
 most appealing one is chosen. From Fig. 1, we can see that the
 KCS based TFDs and the spectrogram have cleaner plots (less
 interference and better componenent’s concentration) than the
 other distributions. Hence, efficient TFD concentration and
 resolution measurement can provide a quantitative criterion
 to evaluate performances of different distributions and can be
 used for adaptive and automatic parameters selection in t-f
 analysis [1]. Among the various objective measures that are
 discussed in literature, our attention is focused particularly on
 the Boashash-Sucic performance measure [7]
 Pi = 1 −
 1
 3
 As
 Am
 +
 Ax
 2Am
 + (1 − S) (14)
 TABLE I
 MATHEMATICAL PROPERTIES VERIFIED BY WDM, CMD, BJD, ZAMD, CB TFD, SCB TFD AND PCB TFD.
 WVD CMD BJD ZAMD CB TFD SCB TFD PCB TFD
 Real-valued x x x x x x x
 Marginal properties x x x
 Energy conservation x x x x x
 Translation covariance x x x x x x x
 Dilation covariance x x x
 Perfect localization on linear chirp signals x
 Unitarity x
 Compatibility with filterings x
 Compatibility with modulations x
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 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 6, NO. 1, MARCH 2012 4
 (a) (b)
 (c) (d)
 (e) (f)
 (g) (h)
 Fig. 1. TFDs of the bat echo location signal. (a) WVD, (b) Spectrogram (Hanning, L = 55), (c) BJD, (d) CWD (σ = 0.6), (e) ZAMD (α = 0.8), (f) CB
 TFD (D = 2.5, A = 1.4), (g) SCB TFD (D = 4, A = 1.4) and (h) PCB TFD (λ = 3, γ = 2).
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 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 6, NO. 1, MARCH 2012 5
 where Am, As, Ax are respectively the average amplitudes
 of the main-lobes, side-lobes and cross-terms of two consecu-
 tive signal components, with S = (B1+B2)/[2(f2−f1)] being
 a measure of the components’ separation in frequency (Bk and
 fk, k = 1, 2, are respectively the instantaneous bandwidth and
 the instantaneous frequency (IF) of the kth component).
 The components’ main lobes average instantaneous band-
 width is defined by the quantity Bi(t) = (B1(t) + B2(t))/2.
 Pi is close to 1 for well-performing TFDs and 0 for poorly-
 performing ones. An overall measure P is taken to be the
 median of the instantaneous measures Pi corresponding to
 different time slices in the relevant sections of the signals.
 The parameters in (14) can be computed automatically using
 the methodology described in [22].
 VI. EXPERIMENTAL RESULTS
 The performance of the KCS based TFDs is compared to the
 classical best known time-frequency distributions. Four exam-
 ples are considered and discussed in detail in order to evaluate
 each TFD and determine the best one in terms of concentration
 and resolution. The TFDs with smoothing parameters are first
 optimized and their relative overall performance measure P
 values are recorded in tables where
 P =
 N∑
 j=1
 Pi(t0 = j); (15)
 and N is the full range of time instants. Then, the maximum
 value among them is selected and it corresponds to the best
 performing TFD in representing the multi-component test
 signal.
 A. Example 1: Sum of 2 crossing linear FM signals
 Here, we deal with a multi-component signal s1(t) of
 duration T = 128 composed of two noiseless crossing chirps
 of frequency ranges f = [0.1−0.2] Hz and f = [0.2−0.1] Hz,
 respectively. The time-frequency representations of the signal
 s1(t) are given in Fig. 2 using several popular TFDs together
 with the CB TFD, the SCB TFD and the PCB TFD. It can
 be seen that the KCS based TFDs and the spectrogram have
 the greatest ability to remove the cross terms and present all
 clear curves in contrast to the other representations. Let us
 examine in depth the performance of each distribution. For
 this purpose, the considered TFDs are optimized with respect
 to the Boashash-Sucic’s criterion over the time interval [1, T ]
 except for the WVD and the BJD that have no smoothing
 parameters and then they cannot be optimized.
 The resulting P ’s values are recorded in Table II and they
 clearly reveal that the KCS based TFDs produce the best
 performance compared with the other time-frequency repre-
 sentations. Moreover, the CB TFD with control parameters
 D = 4 and A = 1.44 gives the largest value of P and hence
 is selected as the best performing TFD of the signal s1(t).
 B. Example 2: Sum of 2 parallel FM signals
 As a second illustration test, we consider a multicomponent
 signal s2(t) of length N = 128 that consists of two closely
 TABLE II
 OPTIMIZATION RESULTS FOR A SELECTION OF TFDS OF THE SIGNAL OF
 EXAMPLE 1 (TWO CROSSING CHIRPS TEST).
 TFD Optimal kernel Parameters P
 WVD N/A 0.6581
 Spectrogram Hanning, L = 85 0.8588
 BJD N/A 0.7072
 CWD σ = 0.45 0.7269
 ZAMD α = 0.8 0.6822
 CB TFD D = M/B = 4, A = eC = 1.44 0.8708
 SCB TFD D = M/B = 5, A = eC = 2.1 0.8678
 PCB TFD λ = 2, γ = 2 0.8626
 spaced parallel linear FMs with frequencies increasing from
 0.15 to 0.25 Hz and from 0.2 to 0.3 Hz, respectively. The
 signal s2(t) is analyzed in the t-f domain using the same selec-
 tion of TFDs as in example 1. The time-frequency plots of the
 optimized TFDs according to Boashash-Sucic’s performance
 measure are shown in Fig. 3, where we can see that the CB
 TFD, the SCB TFD and the PCB TFD have all clear plots
 since the two time-varying components of the signal s2(t)
 are well concentrated in their respective frequency ranges and
 the interferences between them are largely attenuated by the
 effects of the compact support nature of the three investigated
 kernels.
 In this example, we first compare the TFDs’ resolution
 performance at time instant t0 = 64; the middle of the
 signal duration, including the Modified B-distribution [23] as
 well. Table III reports the related performed measurements by
 refering to the Boashash-Sucic’s methodology that is used to
 compute the parameters of (14), whereas Fig. 4 shows the
 slices of a selection of TFDs at t0 = 64. It indicates that the
 SCB TFD with smoothing parameters D = 5 and A = 0.13 is
 the optimal TFD of the signal s2(t) at this time instant giving
 the largest value of Pi. Let us then search for the TFD that
 best resolves the two chirp components of the signal s2(t)
 over the entire time interval [1,128]. Table IV contains the
 optimization process and indicates that the KCS based TFDs
 outperform the other quadratic time-frequency distributions.
 Furthermore, it shows that the signal s2(t) is best presented
 in the t-f plane using the CB TFD with parameters D = 2
 and A = 0.11 since it has the largest value of P .
 C. Example 3: Effect of Additive noise
 In order to check the behavior of TFDs in the case of noisy
 multi-component signals, let us search for the optimal TFD
 of the two-component signal s2(t) considered in example 2,
 embedded in additive white Gaussian noise, with a signal-
 to-noise ratio of 10 dB. The test signal, denoted by s3(t),
 is analyzed in the t-f domain using a selection of quadratic
 TFDs. The time-frequency plots of the optimized TFDs under
 the constraints of Boashash-Sucic’s criterion are shown in Fig.
 5. Here again, from visual inspection, we can see that the KCS
 based TFDs perform much better that the other considered
 TFDs since they generate the most appealing plots. Table V
 records the numerical results of the optimization procedure
 over the entire time interval [1,128] and reveals that the opti-
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 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 6, NO. 1, MARCH 2012 6
 (a) (b)
 (c) (d)
 (e) (f)
 (g) (h)
 Fig. 2. Optimized TFDs over the entire time interval [1, 128] of the signal of example 1 composed of two crossing chirps with frequency ranges f = 0.1−0.2
 Hz and f = 0.2 − 0.1 Hz, respectively. (a) WVD, (b) Spectrogram (Hanning, L = 85), (c) BJD, (d) CWD (σ = 0.45), (e) ZAMD (α = 0.8), (f) CB TFD
 (D = M/B = 4, A = eC = 1.44), (g) SCB TFD (D = 5, A = 2.1) and (h) PCB TFD (λ = 2, γ = 2).
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 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 6, NO. 1, MARCH 2012 7
 (a) (b)
 (c) (d)
 (e) (f)
 (g) (h)
 Fig. 3. Optimized TFDs over the entire time interval [1, 128] of the signal of example 2 composed of two parallel LFMs with frequency ranges spreading
 from 0.15 to 0.25 Hz and 0.2 to 0.3 Hz, respectively. (a) WVD, (b) Spectrogram (Hanning, L = 85), (c) BJD,(d) CWD (σ = 0.45), (e) ZAMD (α = 0.8),
 (f) CB TFD (D = 2, A = 0.11), (g) SCB TFD (D = 5, A = 0.487) and (h) PCB TFD (λ = 1.5, γ = 1).
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 TABLE III
 PARAMETERS AND THE NORMALIZED INSTANTANEOUS RESOLUTION PERFORMANCE MEASURE Pi OF THE DIFFERENT TFDS FOR THE TIME INSTANT
 t0 = 64 RELATED TO EXAMPLE 2. THE FIRST SIX MEASUREMENTS ARE ADOPTED FROM [7].
 TFD (optimal parameter) AM(64) AS(64) AX(64) Bi(64) ∆fi(64) S(64) Pi(64)
 WVD 0.9153 0.3365 1 0.0130 0.0574 0.7735 0.6199
 Spectrogram (Hanning, L = 35) 0.9119 0.0087 0.5527 0.0266 0.0501 0.4691 0.7188
 BJD 0.9320 0.1222 0.3798 0.0219 0.0488 0.5512 0.7388
 CWD (σ = 2) 0.9355 0.0178 0.4415 0.0238 0.0493 0.5172 0.7541
 ZAMD (α = 2) 0.9146 0.4847 0.4796 0.0214 0.0420 0.4905 0.5661
 Modified B (β = 0.01) 0.9676 0.0099 0.0983 0.0185 0.0526 0.5957 0.8449
 CB TFD (D = M/B = 2, A = eC = 0.48) 0.9941 0.0314 0.0179 0.0159 0.0556 0.7143 0.8912
 SCB TFD (D = 5, A = 0.13) 0.9868 0.0183 0.0323 0.0159 0.0556 0.7143 0.8931
 (a) (b) (c) (d)
 (e) (f) (g)
 Fig. 4. Normalized slices of TFDs at t0=64 of the signal s2(t). (a) WVD, (b) Spectrogram (Hanning, L = 35), (c) BJD,(d) CWD (σ = 2), (e) ZAMD
 (α = 2), (f) CB TFD (D = 2, A = 0.48) and (g) SCB TFD (D = 5, A = 0.13). The first five plots are adopted from [7] and compare the TFDs (dashed)
 against the Modified B distribution (β = 0.01) (solid).
 TABLE IV
 OPTIMIZATION RESULTS FOR A SELECTION OF TFDS OF THE SIGNAL OF
 EXAMPLE 2.
 TFD Optimal kernel
 Parameters
 P
 WVD N/A 0.6449
 Spectrogram Hanning, L = 73 0.8232
 BJD N/A 0.6860
 CWD σ = 1.2 0.7228
 ZAMD α = 0.8 0.6856
 CB TFD D = 2, A = 0.11 0.8449
 SCB TFD D = 5, A = 0.487 0.8442
 PCB TFD λ = 1.5, γ = 1 0.8409
 TABLE V
 OPTIMIZATION RESULTS FOR A SELECTION OF TFDS OF THE SIGNAL OF
 EXAMPLE 3 (ROBUSTNESS TO NOISE TEST).
 TFD Optimal kernel
 Parameters
 P
 WVD N/A 0.6442
 Spectrogram Bartlett, L = 71 0.8222
 BJD N/A 0.6764
 CWD σ = 0.9 0.7173
 ZAMD α = 0.56 0.6422
 CB TFD D = 2.5, A = 0.11 0.8443
 SCB TFD D = 3, A = 0.28 0.8439
 PCB TFD λ = 2, γ = 1 0.8363
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 (a) (b)
 (c) (d)
 (e) (f)
 (g) (h)
 Fig. 5. Optimized TFDs over the full duration T = 128 of the signal of example 3 composed of two parallel LFMs with frequency ranges spreading from
 0.15 to 0.25 Hz and 0.2 to 0.3 Hz, respectively; embedded in 10 dB AWGN. (a) WVD, (b) Spectrogram (Bartlett, L = 71), (c) BJD,(d) CWD (σ = 0.9),
 (e) ZAMD (α = 0.56), (f) CB TFD (D = 2.5, A = 0.11), (g) SCB TFD (D = 3, A = 0.28) and (h) PCB TFD (λ = 2, γ = 1).
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 -mal TFD of the noisy signal s3(t) is the CB TFD with
 smoothing parameters D = 2.5 and A = 0.11 since it
 possesses the largest value of P .
 D. Example 4: Sum of 2 sinusoidal FM signals and 2 chirp
 signals.
 In this example, we consider a synthetic signal s4(t)
 consisting of two intersecting sinusoidal FMs and two non-
 parallel, non-intersecting chirps. The nonlinear components
 consist of an increasing and decreasing sinusoidal frequency
 modulated signals at t0 = 0, f(t0) = 0.35 Hz, having both
 a period T = 128 sec with smallest and highest frequencies
 equal to 0.25 Hz and 0.45 Hz respectively. The two chirps
 occupy the frequency ranges f = [0.16 − 0.19] Hz and
 f = [0.07 − 0.1] Hz, respectively. The smallest frequency
 separation between the linear and nonlinear components is
 within the range 0.18 − 0.25 Hz near 97 sec and it is low
 enough and is just avoiding intersection. The purpose here is
 to confirm again the effectiveness of the KCS based kernels in
 detecting closely spaced components in the case of mixtures
 of linear and nonlinear nonstationary signals. Fig. 6 shows the
 superiority of the KCS based kernels and the spectrogram over
 the other quadratic time-frequency distributions in resolving
 the four closely spaced components as well as in reducing the
 cross-terms. In this example, the Boashash-Sucic’s procedure
 is applied twice in order to measure the parameters for each
 of the pairs of consecutive components with equal amplitudes
 of each TFD time slice. The optimizing TFD’s parameters
 are chosen so that they produce the greatest value of the
 Boashash-Sucic’s overall performance measure for both the
 two linear chirps (P (1)) and the two sinusoidal FMs (P (2));
 the resulting P to maximize is equal to (P (1)+P (2))/2. Table
 VI presents the numerical results of the optimization procedure
 and indicates that the CB TFD with parameters A = 1.2;
 D = 3 is the optimal TFD for representing s4(t) since it
 produces the largest value of P .
 TABLE VI
 OPTIMIZATION RESULTS OF EXAMPLE 4.
 TFD Optimal kernel
 Parameters
 P (1) P (2) P
 WVD N/A 0.6428 0.6089 0.6258
 Spectrogram Hanning, L = 45 0.8741 0.8644 0.8692
 BJD N/A 0.6869 0.7623 0.7246
 CWD σ = 0.45 0.7602 0.7687 0.7644
 ZAMD α = 0.5 0.7352 0.7381 0.7366
 CB TFD D = 1.2, A = 3 0.8780 0.8786 0.8783
 SCB TFD D = 3, A = 4 0.8701 0.8802 0.8751
 PCB TFD λ = 0.2, γ = 4 0.8813 0.8701 0.8757
 VII. CONCLUSIONS
 Several time-frequency experimental tests were made to
 analyze linear and nonlinear FM laws with very closely
 spaced multi-components signals and noise effects. These tests
 showed that the KCS based TFDs outperform other well-
 known classical TFDs in terms of crossterms reduction while
 still achieving the best time-frequency resolution and then
 preserving high energy concentration around the components’
 instantaneous frequencies. The comparisons made are not
 based only on visual measure of goodness of TFD plots by
 looking for the most appealing one but are quantified using
 the Boashash-Sucic’s objective criterion that implies a deep
 inspection of each time slice. KCS based TFDs give in all stud-
 ied cases the largest performance measure value compared to
 the most known and powerful time-frequency representations.
 In addition, they reveal the most information about the time-
 varying test signals in the t-f plane in terms of detection of
 the components’ number, extraction of the IF laws from the
 TFD’s peaks, estimation of signal components bandwiths and
 evaluation of sidelobe and cross-term amplitudes. The later are
 the best eliminated using KCS kernels thanks to their compact
 support nature and the flexibility in tuning the kernel width
 and amplitude in order to reach their optimization. Note that
 controlling the kernel amplitude is more flexible using the CB
 and SCB kernels compared with the PCB kernel that uses, by
 definition, an integer tuning parameter.
 The combination of these results, together with the method
 and system implementation proposed in [6], opens the way
 for further promising development in high-performing DSP
 systems for practical measurement of nonstationary signals’
 energy. Future work will also attempt a more detailed and
 comprehensive comparison with other recently proposed high-
 resolution TFDs such as the MBD and BD [24] so as to
 guide the user in terms of how to select a specific TFD
 for a particular application; such considerations could not be
 included in this paper due to space limitations.
 VIII. APPENDICES
 A. Appendix A: Real-valued property
 A time-frequency distribution is real if
 TFDx(t, f) = TFD∗x(t, f) = <{TFDx(t, f)} ∀ t, f (16)
 By calculating the complex conjugate of (1) and making the
 change of integration variables τ ′ = −τ and η′ = −η we get
 TFD∗x(t, f) =
 ∫ ∫ ∫+∞
 −∞
 e−j2piη(s−t)φ∗(η, τ)y∗(s, τ)
 e+j2pifτdηdsdτ
 = −
 ∫
 −∞
 +∞
 ∫
 −
 ∫
 −∞
 +∞ e
 j2piη′(s−t)φ∗(−η′, −τ ′)y(s, τ ′)
 e−j2pifτ
 ′
 dη′dsdτ ′
 =
 ∫+∞
 −∞
 ∫+∞
 −∞
 ∫+∞
 −∞
 e−j2piη(s−t)φ∗(−η, −τ)y(s, τ)
 e−j2pifτdηdsdτ
 Thus, a real quadratic distribution is obtained if the corre-
 sponding kernel satisfies
 φ(η, τ) = φ∗(−η, −τ) ∀ η, τ ∈ < (17)
 Since the CB kernel is an even real function with respect to
 both η and τ , then
 φ∗CB(−η, −τ) = φCB(−η, −τ) = φCB(η, τ) (18)
 Hence, the CB TFD is always real-valued.
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 (a) (b)
 (c) (d)
 (e) (f)
 (g) (h)
 Fig. 6. Optimized TFDs over the time duration [1, 128] of the signal s4(t) composed of two non-parallel, non-intersecting chirps and two intersecting
 sinusoidal FMs. (a) WVD, (b) Spectrogram (Hanning, L = 45), (c) BJD,(d) CWD (σ = 0.45), (e) ZAMD (α = 0.5), (f) CB TFD (D = 1.2, A = 3), (g)
 SCB TFD (D = 3, A = 4) and (h) PCB TFD (λ = 0.2, γ = 4).
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 B. Appendix B: Marginal properties
 A time-frequency distribution TFDx(t, f) of x(t) obeys
 the marginal properties if it reduces to the spectrum and
 instantaneous power by integrating over t and f respectively,
 i.e.
 ∫ +∞
 −∞
 TFDx(t, f)dt = |X(f)|2 (19)∫ +∞
 −∞
 TFDx(t, f)df = |x(t)|2 (20)
 By integrating (1) over t we obtain
 I(f) =
 ∫ +∞
 −∞
 ∫ ∫ ∫ +∞
 −∞
 e−j2piη(s−t)φ(η, τ)y(s, τ)
 e−j2pifτdηdsdτdt(21)
 =
 ∫ ∫ ∫ +∞
 −∞
 [∫ +∞
 −∞
 e+j2piηtdt
 ]
 e−j2piηsφ(η, τ)
 y(s, τ)e−j2pifτdηdsdτ
 Since
 ∫+∞
 −∞
 e+j2piηtdt = δ(η) and δ(η)e−j2piηsφ(η, τ) =
 φ(0, τ)δ(η) it yields
 I(f) =
 ∫ ∫ +∞
 −∞
 [∫ +∞
 −∞
 δ(η)dη
 ]
 ︸ ︷︷ ︸
 1
 φ(0, τ)y(s, τ)
 e−j2pifτdsdτ
 =
 ∫ ∫ +∞
 −∞
 φ(0, τ)y(s, τ)e−j2pifτdsdτ (22)
 In the case of (the requirement for time marginal property)
 φ(0, τ) = 1, ∀τ (23)
 (22) becomes
 I(f) =
 ∫ ∫ +∞
 −∞
 x(s + τ/2)x∗(s − τ/2)e−j2pifτdsdτ (24)
 Recall that the Wigner-Ville distribution is defined as
 WVx(t, f) =
 ∫ +∞
 −∞
 x(t + τ/2)x∗(t − τ/2)e−j2pifτdτ (25)
 Hence, the integral (24) can be rewritten as follows
 I(f) =
 ∫ ∫ +∞
 −∞
 x(s + τ/2)x∗(s − τ/2)e−j2pifτdsdτ
 =
 ∫ +∞
 −∞
 WVx(s, f)ds
 I(f) = |X(f)|2 (26)
 since the WVD verifies the marginal properties. From the
 development above, we conclude that in order to a TFD of the
 quadratic class preserves the marginal property with respect to
 time, the kernel must satisfy condition (23), which is not the
 case of the CB kernel. By integrating (1) over f we obtain
 I(t) =
 ∫ +∞
 −∞
 ∫ ∫ ∫ +∞
 −∞
 e−j2piη(s−t) φ(η, τ) y(s, τ)
 e−j2pifτdηdsdτdf
 =
 ∫ ∫ ∫ +∞
 −∞
 ∫ +∞
 −∞
 e−j2pifτdfe−j2piη(s−t)φ(η, τ)
 y(s, τ)dηdsdτ
 Since
 ∫+∞
 −∞
 e−j2pifτdf = δ(τ) and δ(τ)φ(η, τ)y(s, τ) =
 δ(τ) φ(η, τ) y(s, 0) = φ(η, 0) x(s) x∗(s) δ(τ) =
 φ(η, 0) |x(s)|2 δ(τ) it yields
 I(t) =
 ∫ ∫ +∞
 −∞
 ∫ +∞
 −∞
 δ(τ)dτe−j2piη(s−t)φ(η, 0)
 |x(s)|2dsdη
 =
 ∫ ∫ +∞
 −∞
 e−j2piη(s−t)φ(η, 0)|x(s)|2dsdη (27)
 In the case of (the requirement for frequency marginal prop-
 erty)
 φ(η, 0) = 1, ∀η (28)
 it results
 I(t) =
 ∫ +∞
 −∞
 [∫ +∞
 −∞
 e−j2piη(s−t)dη
 ]
 |x(s)|2ds (29)
 From the delta function properties we have:∫+∞
 −∞
 e−j2piη(s−t)dη = δ(s − t), thus the integral (29)
 can be rewritten as follows
 I(t) =
 ∫ +∞
 −∞
 δ(s − t)|x(s)|2ds
 =
 ∫ +∞
 −∞
 δ(s − t)|x(t)|2ds
 = |x(t)|2
 [∫ +∞
 −∞
 δ(s′)ds′
 ]
 ︸ ︷︷ ︸
 1
 (s′ = s − t)
 I(t) = |x(t)|2 (30)
 Consequently, the CB kernel does not satisfy the marginal
 property with respect to frequency as well since condition (28)
 is not verified.
 C. Appendix C: Energy conservation
 A given TFD of x conserves energy if, by integrating it over
 time and frequency, we obtain the energy of x
 Ex =
 ∫ +∞
 −∞
 ∫ +∞
 −∞
 TFDx(t, f)dt df =
 ∫ +∞
 −∞
 I(f)df (31)
 Referring to (22) the right-side integral in (31) is given as
 follows∫ +∞
 −∞
 I(f)df =
 ∫ ∫ ∫ +∞
 −∞
 φ(0, τ)y(s, τ)e−j2pifτdsdτdf
 =
 ∫ +∞
 −∞
 ∫ +∞
 −∞
 [∫ +∞
 −∞
 e−j2pifτdf
 ]
 ︸ ︷︷ ︸
 δ(τ)
 φ(0, τ)y(s, τ)dsdτ
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 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 6, NO. 1, MARCH 2012 13
 Since δ(τ)φ(0, τ)y(s, τ) = δ(τ)φ(0, 0)y(s, 0) =
 φ(0, 0)x(s)x∗(s)δ(τ) = φ(0, 0)|x(s)|2δ(τ) it yields∫ +∞
 −∞
 I(f)df = φ(0, 0)
 ∫ +∞
 −∞
 δ(τ)dτ
 ︸ ︷︷ ︸
 1
 ∫ +∞
 −∞
 |x(s)|2ds
 = φ(0, 0)
 ∫ +∞
 −∞
 |x(s)|2ds (32)
 Hence, if one wants to preserve the energy conservation
 characteristic, the kernel must satisfy the condition
 φ(0, 0) = 1 (33)
 which is the case of the CB kernel.
 D. Appendix D: Dilation covariance
 A given TFD preserves dilations if
 z(t) = √kx(kt); k > 0 ⇒ TFDz(t, f) = TFDx(kt, fk )(34)
 Referring to (2), we have
 TFDx(kt, fk ) =
 ∫ ∫ +∞
 −∞
 J(s − kt, τ)y(s, τ)e−j2pi fk τdsdτ
 (35)
 and
 TFDz(t, f) =
 ∫ +∞
 −∞
 ∫ +∞
 −∞
 J(s − t, τ)z(s + τ/2)z∗(s − τ/2)
 e−j2pifτdsdτ
 = k
 ∫ +∞
 −∞
 ∫ +∞
 −∞
 J(s − t, τ)x(ks + kτ/2)x∗(ks − kτ/2)
 e−j2pifτdsdτ
 Let: s′ = ks and τ ′ = kτ . Then
 TFDz(t, f) = 1k
 ∫ ∫ +∞
 −∞
 J(s
 ′
 k −t,
 τ ′
 k )y(s
 ′
 , τ ′)e−j2pi fk τ ′ds′dτ ′
 (36)
 From (35) and (36), the dilation property is satisfied if
 J( sk − t,
 τ
 k ) = k J(s − kt, τ) (37)
 a condition that the CB TFD does not verify.
 E. Appendix E: Perfect localization on linear chirp signals
 This property is achieved if the following condition holds
 x(t) = ej2pi(f0+2βt)t ⇒ TFDx(t, f) = δ(f−(f0+βt)) (38)
 It is obvious that condition (38) only holds for the Wigner-
 Ville distribution since it is the only case where we get a sum
 of complex exponentiels (the kernel φWV (η, τ) = 1, ∀η, τ )
 WVx(t, f) =
 ∫ +∞
 −∞
 ej2pi[f0+2β(t+τ/2)](t+τ/2)e−j2pifτ
 e−j2pi[f0+2β(t−τ/2)] (t−τ/2)dτ
 =
 ∫ +∞
 −∞
 e−j2pi[f−(f0+βt)]τdτ
 = δ(f − (f0 + βt))
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 Mansour Abed received the Dipl.-Ing. Electronique
 degree in 2000 from the University of Sciences
 and Technology of Oran (USTO), Oran, Algeria,
 the M.sc. degree in communication engineering in
 2004 from the University of Technology, Baghdad,
 Iraq and the M.sc. degree in electrical engineering
 in 2006 from the University of Alexandria, Faculty
 of engineering, Alexandria, Egypt. He is a Maitre-
 Assistant at the department of electrical engineering,
 faculty of sciences and technology, University of
 Mostaganem since November 2006. He is currently
 pursuing the Ph.D. degree with the department of electronics, Ecole Nationale
 Polytechnique (ENP), El-Harrach, Algiers, Algeria.
 His research interests include time-frequency signal analysis, adaptive
 signal processing in wireless CDMA networks, spread spectrum systems,
 image processing and blind source separation.
 Adel Belouchrani (M’96) was born in Algiers,
 Algeria, on May 5, 1967. He received the State
 Engineering degree in 1991 from Ecole Nationale
 Polytechnique (ENP), Algiers, Algeria, the M.S.
 degree in signal processing from the Institut National
 Polytechnique de Grenoble (INPG), France, in 1992,
 and the Ph.D. degree in signal and image processing
 from Télécom Paris (ENST), France, in 1995. He
 was a Visiting Scholar at the Electrical Engineering
 and Computer Sciences Department, University of
 California, Berkeley, from 1995 to 1996. He was
 with the Department of Electrical and Computer Engineering, Villanova
 University, Villanova, PA, as a Research Associate from 1996 to 1997.
 From 1998 to 2005, he has been with the Electrical Engineering Department
 of ENP as Associate Professor. He is currently and since 2006 Full Professor
 at ENP.
 His research interests are in statistical signal processing and (blind) array
 signal processing with applications in biomedical and communications, time-
 frequency analysis, time-frequency array signal processing, wireless commu-
 nications, and FPGA implementation of signal processing algorithms.
 Mohamed Cheriet was born in Algiers (Algeria)
 in 1960. He received his B.Eng. from USTHB Uni-
 versity (Algiers) in 1984 and his M.Sc. and Ph.D.
 degrees in Computer Science from the University of
 Pierre et Marie Curie (Paris VI) in 1985 and 1988
 respectively. Since 1992, he has been a professor in
 the Automation Engineering department at the Ecole
 de Technologie Supérieure (University of Quebec),
 Montreal, and was appointed full professor there in
 1998. He co-founded the Laboratory for Imagery,
 Vision and Artificial Intelligence (LIVIA) at the
 University of Quebec, and was its director from 2000 to 2006. He also
 founded the SYNCHROMEDIA Consortium (Multimedia Communication in
 Telepresence) there, and has been its director since 1998. His interests include
 document image processing and analysis, OCR, mathematical models for
 image processing, pattern classification models and learning algorithms, as
 well as perception in computer vision.
 Dr. Cheriet has published more than 200 technical papers in the field,
 and has served as chair or co-chair of the following international confer-
 ences: VI’1998, VI’2000, IWFHR’2002, ICFHR’2008, and ISSPA’2012. He
 currently serves on the editorial board and is associate editor of several inter-
 national journals: IJPRAI, IJDAR, and Pattern Recognition. He co-authored
 a book entitled, "Character Recognition Systems: A guide for Students and
 Practitioners," John Wiley and Sons, Spring 2007.
 Dr. Cheriet is a senior member of the IEEE and the chapter founder and
 former chair of IEEE Montreal Computational Intelligent Systems (CIS).
 Boualem Boashash is a Fellow of the IEEE "for
 pioneering contributions to time-frequency signal
 analysis and signal processing education". He is
 also a Fellow of IE Australia and a Fellow of the
 IREE. After his Baccalaureat in Grenoble, France
 in 1973, Professor Boashash went on to get his
 Diplome d’ingenieur-Physique - Electronique from
 Lyon, France, in 1978, and then a DEA (Masters
 degree) from the University of Grenoble, France
 in 1979, followed by a Doctorate from the same
 university in May 1982. Between 1979 and 1982,
 Boualem Boashash was also with Elf-Aquitaine Geophysical Research Centre,
 Pau, France,as a research engineer. In 1982, he joined the Institut National des
 Sciences Appliquees de Lyon, France, where he was an Assistant Professor. In
 January 1984, he took a position at the University of Queensland, Australia,
 as a Lecturer, Senior Lecturer (1986) and Reader (1989). In 1990, he joined
 Bond University, Graduate School of Science and Technology, as Professor
 of Signal Processing. In 1991, he was invited to move to the Queensland
 University of Technology as the foundation Professor of Signal Processing,
 and then held several senior academic management positions. In 2006, he was
 invited by the University of Sharjah to be the Dean of Engineering and in
 2009, he joined Qatar University as Associate Dean for Academic affairs and
 Professor while still an Adjunct Professor at the University of Queensland,
 Brisbane, Australia.
 Professor Boashash has published over 500 technical publications, three
 research books and five text-books, over 30 book-chapters and supervised
 over 50 PhD students. His work has been cited over 7000 times (Google
 Scholar). He was the technical chairman of ICASSP 94 and played a
 leading role between 1985 and 1995 in the San Diego SPIE conference on
 Signal Processing, establishing the original special sessions on time-frequency
 analysis. Since 1985, He has been the Founder and General Chairman of the
 International Symposium on Signal Processing and its Applications (ISSPA)
 which is organized every two years.
 Professor Boashash was instrumental in developing the field of time-
 frequency signal analysis and processing via his research work and by
 organizing the first international conference on the topic at ISSPA 90 and
 other scientific meetings. He developed the first software package for time-
 frequency signal analysis first. Current version is being released as freeware.
 For more details, see his full CV available on request