5. The applications of wavelet transforms

The variable time-frequency resolution is an important property of the CWT which permits to use CWT for the analysis of the signals consisting of the slowly varying low-frequency components and the rapidly varying high-frequency components [7]. For this reason, the CWT is a

Suppose that the dilation parameter a and the translation parameter b are discrete and take a

The two-dimensional sequence d kð Þ ; l is defined as DWT of f tð Þ [7]. The values of DWT d kð Þ ; l

f tð Þ <sup>1</sup>

Comparison of CWT and DWT shows that the signal f tð Þ in the both cases is expressed in terms of dilations and translations of a single mother wavelet [6]. DWT is used in the multiresolution analysis (MRA) which is based on a hierarchy of approximations to functions in N various subspaces WN�<sup>1</sup>, WN�<sup>2</sup>, …, W<sup>1</sup> of a linear vector space VN ¼ WN�<sup>1</sup> ⊕ WN�<sup>2</sup> ⊕ :…, W<sup>1</sup> ⊕ V<sup>1</sup> [6]. In general case, the wavelet ψð Þt providing the DWT corresponding to the MRA must satisfy

Here δð Þ¼ n 1, n ¼ 0 and δð Þ¼ n 0, n 6¼ 0 and ϕð Þt is the scaling function ϕð Þt satisfying the

The scaling function ϕð Þt and the wavelet function ψð Þt are defined by the following equations,

n¼�∞

where c nð Þ,d nð Þ are sequences of scalars. It is seen that the scaling function ϕð Þt is determined by its own dyadic dilation and translation. For this reason, the equation for ϕð Þt is called a dilation Equation [6, 7, 17]. It can be shown that the DWT is equivalent to filtering a signal by a

<sup>2</sup><sup>k</sup>=<sup>2</sup> <sup>ψ</sup><sup>∗</sup> <sup>2</sup>�<sup>k</sup>

d kð Þ ; <sup>l</sup> <sup>2</sup>�k=<sup>2</sup>

l, where k and l are integers [6, 7, 17]. Then, Eq. (37) takes the form [7]:

ψ 2�<sup>k</sup>

, b <sup>¼</sup> <sup>2</sup><sup>k</sup>

dt <sup>¼</sup> <sup>1</sup>;hψð Þ<sup>t</sup> ;ψð Þ <sup>t</sup> � <sup>n</sup> i ¼ <sup>δ</sup>ð Þ <sup>n</sup> ; <sup>ψ</sup>ð Þ<sup>t</sup> ; <sup>ϕ</sup>ð Þ <sup>t</sup> � <sup>n</sup> � � <sup>¼</sup> 0 (40)

dt <sup>¼</sup> <sup>1</sup>; <sup>ϕ</sup>ð Þ<sup>t</sup> ; <sup>ϕ</sup>ð Þ <sup>t</sup> � <sup>n</sup> � � <sup>¼</sup> <sup>δ</sup>ð Þ <sup>n</sup> (41)

d nð Þϕð Þ 2t � n , n ¼ 0, � 1, � 2, … (42)

<sup>t</sup> � <sup>l</sup> � � (38)

<sup>t</sup> � <sup>l</sup> � �dt (39)

l [7]. Then DWT Wklð Þ a; b takes

unique tool for the study of the superoscillating signals described in Section 2.

f tðÞ¼ <sup>X</sup> k¼∞

Wklð Þ¼ a; b

are related to the values of CWT W að Þ ; <sup>b</sup> Eq. (18) at <sup>a</sup> <sup>¼</sup> <sup>2</sup><sup>k</sup>

k¼�∞

X l¼∞

l¼�∞

ð ∞

�∞

form <sup>a</sup> <sup>¼</sup> <sup>2</sup><sup>k</sup>

the form [6, 7]:

, b <sup>¼</sup> <sup>2</sup><sup>k</sup>

206 Wavelet Theory and Its Applications

the following conditions [7]:

ψð Þt dt ¼ 0;

ð ∞

�∞

n¼�∞

<sup>ϕ</sup>ðÞ¼ <sup>t</sup> <sup>X</sup><sup>∞</sup>

ð ∞

j j <sup>ψ</sup>ð Þ<sup>t</sup> <sup>2</sup>

�∞

ϕð Þt dt ¼ 1;

ð ∞

<sup>ϕ</sup>ð Þ<sup>t</sup> � � � � 2

�∞

c nð Þϕð Þ <sup>2</sup><sup>t</sup> � <sup>n</sup> ;ψðÞ¼ <sup>t</sup> <sup>X</sup><sup>∞</sup>

band of filters with nonoverlapping bandwidths differing by a factor of 2 [17].

ð ∞

�∞

following conditions [7]:

respectively [7]:

The different types of WT are widely used in different areas of mathematics and engineering [17]. The number of scientific books and articles concerning wavelet transforms (WT) applications is enormous and hardly observable. In this section, we briefly review some typical applications of wavelet transforms in optical communication systems and signal processing. Wavelet methods may complement the Fourier techniques due to their following specific features mentioned above [17]. Wavelets are functions of two parameters which represent the dilation and translation while the Fourier transform is characterized by the dilation only. In the case of wavelets, the width of the window through which the signal is observed is varying as a function of location. For a wavelet method, the window function in the time-frequency plane is nonuniform being a function of both time and frequency.

Wavelet transforms as a mathematical tool can be successfully used in the electromagnetic problems and signal processing applications [6, 7, 17–24]. Wavelet based signal processing represents a useful technique for the compression of certain classes of data demonstrating isolated band-limited properties [17]. Wavelets may be used as basis functions for the solution of Maxwell's equations in the integral or differential form [17]. Signal denoising process can be implemented by using wavelets with a smaller computational complexity as compared to the Fourier technique [17].

Wavelets can be successfully applied to signal and image processing including noise reduction, signal and image compression, signature identification, target detection, and interference suppression [6].

Wavelet packet transform (WPT) can be used in optical communications [20, 24]. WPTs are the generalization of wavelet transforms where the orthogonal basis functions are wavelet packets instead of ordinary wavelets [24]. Discrete WPT (DWPT) is used in the coherent optical orthogonal frequency division multiplexing (CO-OFDM) systems [24]. The detailed analysis of CO-OFDM communication systems can be found in [20, 24] and references therein. In a WPT-OFDM system, each channel occupies a wavelet packet, that is, a subcarrier in wavelet domain [24]. Inverse DWPT (IDWPT) is used at the transmitter which reconstructs the time domain signal from wavelet packets [24]. DWPT are used at the receiver in order to decompose the time domain signal into different wavelet packets by means of successive low-pass and high-pass filtering in the time domain [24].

We proposed a novel hierarchical architecture of the 1Tb=s transmission system based on DWPT-OFDM in order to reduce the computational complexity of the digital signal processing (DSP) algorithms [20]. We separated the low bit rate and high bit rate signal channels in such a way that the low bit rate signals are processed in the electrical domain, while the high bit rate signals are processed optically [20]. We have shown theoretically that the performance of the WPT based CO-OFDM can be significantly improved by increasing the spectral efficiency (SE) of the system and mitigating the channel chromatic dispersion [20].

Recently, some novel applications of different types of wavelet transforms have been reported. CWT can be applied for the improvement of the time-delay estimation (TDE) method in the different-wavelength based inteferometric vibration sensor in a fiber link [25].

The maximal overlap DWPT (MODWPT) has been used for the real-time estimation of root mean square (RMS) power value, active power, reactive power, apparent power, and power factor in power electronic systems [26].

and n ¼ 5. We used the Mexican hat mother wavelet Eq. (23) and the Morlet mother wavelet Eq. (24), since their oscillating temporal behavior is similar to the behavior of the

We investigated the scalogram of the energy contours for the spectral component of the signal Eq. (38) at the highest frequency fð Þ¼ ω0t cosð Þ ω0t and used the results for the analysis of the

The behavior of the component with the frequency ω0=2π in the time domain (lower box) and its scalogram (the upper box) are shown in Figure 3. The pseudo-frequency ωa=2π shown in

where ω<sup>c</sup> is the mother wavelet central frequency defined by the second expression (31) [6, 7]. It is seen from Figure 3 that the homogeneous scalogram in the time-pseudo-frequency plane is strictly periodic. The maximum energy at the scalogram corresponds to the pseudo-

The spectra, the temporal behavior, and the scalograms of the signal Eq. (38) for n ¼ 4 and n ¼ 5 are presented in Figures 4–7, respectively. The spectra shown in Figures 4 and 6 are obtained by using the Fourier transform. For this reason, the superoscillations are absent in the spectra shown in Figures 4 and 6, and the highest frequency corresponding to the maximum

The scalograms shown in the upper box of Figures 5 and 7 are obtained by evaluating the

The superoscillations with the time duration of about ΔT nð Þ¼ ¼ 4 8μs can be identified in the lower box of Figure 5. These superoscillations correspond to the frequency of about ω4=2π ¼ 1=ΔT nð Þ¼ ¼ 4 125KHz > ω0=2π ¼ 100KHz. The energy contours of these superoscillations in the time-pseudo-frequency plane shown in the upper box of Figure 5 correspond to the pseudo-frequency of about ωa4=2π ¼ ω4=2πa ≈ ð Þ 50 � 60 KHz > ωa=2π ≈ 30KHz. Their

Figure 3. The time dependence (lower box) and scalogram (upper box) of the monochromatic sinusoidal signal

CWT of the signal Eq. (38) with the Mexican hat mother wavelet Eq. (23).

ωa=2π ¼ ωc=2πa (44)

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209

superoscillating signal Eq. (38).

the scalogarm is defined as follows:

spectral amplitude is �100KHz in the both cases.

superoscillation features.

frequency ωa=2π ≈ 30KHz.

fð Þ¼ ω0t cosð Þ ω0t , ω0=2π ¼ 100KHz.

The time-reversal (TR) technique is used for the detection and localization of objects in microwave imaging [27]. TR technique is based on an assumption that in a lossless medium, for every wave component propagating away from a source point along a certain path there exists a corresponding time-reversed wave propagating along the same path back to the original point of the source [27]. This assumption is caused by the time invariance of the Maxwell's Equations [27]. TR can achieve super-resolution by using the multipath propagation [27]. However, TR in real media is deteriorated due to the dispersion and losses [27]. A compensation method based on CWT has been proposed which can overcome both the dispersion and attenuation of the electromagnetic wave propagating in a dispersive and lossy medium [27]. In this method, the adjustable-length windows are used in such a way that the long-time windows and short-time windows are applied at low and high frequencies, respectively [27]. Wavelets depend on both the time and frequency which results in the signal decomposition into different time and frequency components. The dispersion and attenuation of these components can be compensated by different filters. Unlike the short-time Fourier transform (STFT) method, the proposed CWT method can be applied in real-life scenarios, and its resolution is about three times higher than in other methods [27].

Online monitoring and control of power grid require the accurate and fast estimation of harmonics [28]. The WT has been widely used in the estimation of time-varying harmonics [28]. In particular, undecimated WPT (UWPT) is one of multiresolution techniques characterized by redundancy and time invariance which can be implemented by a set of filter banks [28]. Unlike DWPT, the UWPT does not perform downsampling on wavelet coefficients at each decomposition level preserving time-invariant property which permits the accurate estimation of the timevarying harmonics in one cycle of the fundamental frequency [28]. The comparison of the simulation results obtained by using the UWPT based method and the experimental results shows that the UWPT algorithm has better estimation accuracy for different types of signals [28].

We for the first time to our best knowledge applied CWT to the theoretical investigation of superoscillations which requires the dynamic time-frequency analysis of the strongly localized signals. CWT appeared to be a powerful mathematical tool for the identification of the superoscillation characteristic features.

#### 6. The simulation results and discussion

We theoretically investigated the superoscillations of the signal defined by the real part of expression (15):

$$\operatorname{Re} f\_n(\omega\_0 t, \omega/\omega\_0) = \operatorname{Re} \left\{ \cos \left( \frac{\omega \eta t}{n} \right) + i \frac{\omega}{\omega\_0} \sin \left( \frac{\omega \eta t}{n} \right) \right\}^n \tag{43}$$

The signal Eq. (38) is band limited by the maximum frequency ω0=2π as it was mentioned above. Without the loss of generality, we have chosen the frequency ω0=2π ¼ 100KHz, n ¼ 4 and n ¼ 5. We used the Mexican hat mother wavelet Eq. (23) and the Morlet mother wavelet Eq. (24), since their oscillating temporal behavior is similar to the behavior of the superoscillating signal Eq. (38).

The maximal overlap DWPT (MODWPT) has been used for the real-time estimation of root mean square (RMS) power value, active power, reactive power, apparent power, and power

The time-reversal (TR) technique is used for the detection and localization of objects in microwave imaging [27]. TR technique is based on an assumption that in a lossless medium, for every wave component propagating away from a source point along a certain path there exists a corresponding time-reversed wave propagating along the same path back to the original point of the source [27]. This assumption is caused by the time invariance of the Maxwell's Equations [27]. TR can achieve super-resolution by using the multipath propagation [27]. However, TR in real media is deteriorated due to the dispersion and losses [27]. A compensation method based on CWT has been proposed which can overcome both the dispersion and attenuation of the electromagnetic wave propagating in a dispersive and lossy medium [27]. In this method, the adjustable-length windows are used in such a way that the long-time windows and short-time windows are applied at low and high frequencies, respectively [27]. Wavelets depend on both the time and frequency which results in the signal decomposition into different time and frequency components. The dispersion and attenuation of these components can be compensated by different filters. Unlike the short-time Fourier transform (STFT) method, the proposed CWT method can be applied in real-life scenarios, and its

Online monitoring and control of power grid require the accurate and fast estimation of harmonics [28]. The WT has been widely used in the estimation of time-varying harmonics [28]. In particular, undecimated WPT (UWPT) is one of multiresolution techniques characterized by redundancy and time invariance which can be implemented by a set of filter banks [28]. Unlike DWPT, the UWPT does not perform downsampling on wavelet coefficients at each decomposition level preserving time-invariant property which permits the accurate estimation of the timevarying harmonics in one cycle of the fundamental frequency [28]. The comparison of the simulation results obtained by using the UWPT based method and the experimental results shows that the UWPT algorithm has better estimation accuracy for different types of signals [28]. We for the first time to our best knowledge applied CWT to the theoretical investigation of superoscillations which requires the dynamic time-frequency analysis of the strongly localized signals. CWT appeared to be a powerful mathematical tool for the identification of the

We theoretically investigated the superoscillations of the signal defined by the real part of

The signal Eq. (38) is band limited by the maximum frequency ω0=2π as it was mentioned above. Without the loss of generality, we have chosen the frequency ω0=2π ¼ 100KHz, n ¼ 4

n 

þ i ω ω0

<sup>n</sup>

sin <sup>ω</sup>0<sup>t</sup> n

(43)

Re<sup>f</sup> <sup>n</sup>ð Þ¼ <sup>ω</sup>0t; <sup>ω</sup>=ω<sup>0</sup> Re cos <sup>ω</sup>0<sup>t</sup>

factor in power electronic systems [26].

208 Wavelet Theory and Its Applications

superoscillation characteristic features.

expression (15):

6. The simulation results and discussion

resolution is about three times higher than in other methods [27].

We investigated the scalogram of the energy contours for the spectral component of the signal Eq. (38) at the highest frequency fð Þ¼ ω0t cosð Þ ω0t and used the results for the analysis of the superoscillation features.

The behavior of the component with the frequency ω0=2π in the time domain (lower box) and its scalogram (the upper box) are shown in Figure 3. The pseudo-frequency ωa=2π shown in the scalogarm is defined as follows:

$$
\omega\_a/2\pi = \omega\_c/2\pi a \tag{44}
$$

where ω<sup>c</sup> is the mother wavelet central frequency defined by the second expression (31) [6, 7]. It is seen from Figure 3 that the homogeneous scalogram in the time-pseudo-frequency plane is strictly periodic. The maximum energy at the scalogram corresponds to the pseudofrequency ωa=2π ≈ 30KHz.

The spectra, the temporal behavior, and the scalograms of the signal Eq. (38) for n ¼ 4 and n ¼ 5 are presented in Figures 4–7, respectively. The spectra shown in Figures 4 and 6 are obtained by using the Fourier transform. For this reason, the superoscillations are absent in the spectra shown in Figures 4 and 6, and the highest frequency corresponding to the maximum spectral amplitude is �100KHz in the both cases.

The scalograms shown in the upper box of Figures 5 and 7 are obtained by evaluating the CWT of the signal Eq. (38) with the Mexican hat mother wavelet Eq. (23).

The superoscillations with the time duration of about ΔT nð Þ¼ ¼ 4 8μs can be identified in the lower box of Figure 5. These superoscillations correspond to the frequency of about ω4=2π ¼ 1=ΔT nð Þ¼ ¼ 4 125KHz > ω0=2π ¼ 100KHz. The energy contours of these superoscillations in the time-pseudo-frequency plane shown in the upper box of Figure 5 correspond to the pseudo-frequency of about ωa4=2π ¼ ω4=2πa ≈ ð Þ 50 � 60 KHz > ωa=2π ≈ 30KHz. Their

Figure 3. The time dependence (lower box) and scalogram (upper box) of the monochromatic sinusoidal signal fð Þ¼ ω0t cosð Þ ω0t , ω0=2π ¼ 100KHz.

Figure 4. The spectrum of the signal Ref <sup>n</sup>ð Þ ω0t; ω=ω<sup>0</sup> for ω0=2π ¼ 100KHz, n ¼ 4.

It is seen from the lower box of Figure 7 that the superoscillations with the time duration of about ΔT nð Þ ¼ 5 ≈ 7:2μs < ΔT nð Þ ¼ 4 are identified for n ¼ 5. They have the frequency ω5=2π ¼ 1=ΔT

Figure 7. The time dependence (lower box) and scalogram (upper box) of the superoscillating signal Ref <sup>n</sup>ð Þ ω0t; ω=ω<sup>0</sup> ,

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211

The corresponding energy contours are identified in the scalogram (upper box of Figure 7) in the time intervals localized near t ¼ 0, � 25μs. The maxima of the corresponding energy contours are localized at the pseudo frequency of about ωa5=2π ¼ ω5=2πa ≈ ð Þ 100 � 130 KHz > ωa=2π ≈ 30KHz. Evidently, we can identify the higher frequency superoscillations by increas-

The scalograms for different mother wavelets are also different. In order to compare the CWT results consider the application of the Morlet mother wavelet Eq. (24) for the superoscillating

Comparison of Figures 7 and 8 shows that the spectral features of superoscillations are pronounced at the higher pseudo-frequencies of about 250 ð Þ � 300 GHz, because the central frequencies of the

Figure 8. The time dependence (lower box) and scalogram (upper box) of the superoscillating signal Ref <sup>n</sup>ð Þ ω0t; ω=ω<sup>0</sup> ,

ð Þ n ¼ 5 ≈ 139KHz > ω4=2π ¼ 125KHz.

ω0=2π ¼ 100KHz, n ¼ 5, the Mexican hat mother wavelet.

signal Ref <sup>n</sup>ð Þ ω0t; ω=ω<sup>0</sup> , ω0=2π ¼ 100KHz, n ¼ 5.

ω0=2π ¼ 100KHz, n ¼ 5, the Morlet mother wavelet.

ing n and using CWT.

Figure 5. The time dependence (lower box) and scalogram (upper box) of the superoscillating signal Ref <sup>n</sup>ð Þ ω0t; ω=ω<sup>0</sup> , ω0=2π ¼ 100KHz, n ¼ 4, the Mexican hat mother wavelet.

Figure 6. The spectrum of the signal Ref <sup>n</sup>ð Þ ω0t; ω=ω<sup>0</sup> for ω0=2π ¼ 100KHz, n ¼ 5.

maxima are strongly manifested at the time intervals with the center localized near t ¼ 0, � 20μs and the pseudo-frequencies higher than the ones corresponding to the highest pseudo-frequency shown in the upper box of Figure 3.

Figure 7. The time dependence (lower box) and scalogram (upper box) of the superoscillating signal Ref <sup>n</sup>ð Þ ω0t; ω=ω<sup>0</sup> , ω0=2π ¼ 100KHz, n ¼ 5, the Mexican hat mother wavelet.

It is seen from the lower box of Figure 7 that the superoscillations with the time duration of about ΔT nð Þ ¼ 5 ≈ 7:2μs < ΔT nð Þ ¼ 4 are identified for n ¼ 5. They have the frequency ω5=2π ¼ 1=ΔT ð Þ n ¼ 5 ≈ 139KHz > ω4=2π ¼ 125KHz.

The corresponding energy contours are identified in the scalogram (upper box of Figure 7) in the time intervals localized near t ¼ 0, � 25μs. The maxima of the corresponding energy contours are localized at the pseudo frequency of about ωa5=2π ¼ ω5=2πa ≈ ð Þ 100 � 130 KHz > ωa=2π ≈ 30KHz. Evidently, we can identify the higher frequency superoscillations by increasing n and using CWT.

The scalograms for different mother wavelets are also different. In order to compare the CWT results consider the application of the Morlet mother wavelet Eq. (24) for the superoscillating signal Ref <sup>n</sup>ð Þ ω0t; ω=ω<sup>0</sup> , ω0=2π ¼ 100KHz, n ¼ 5.

Comparison of Figures 7 and 8 shows that the spectral features of superoscillations are pronounced at the higher pseudo-frequencies of about 250 ð Þ � 300 GHz, because the central frequencies of the

Figure 8. The time dependence (lower box) and scalogram (upper box) of the superoscillating signal Ref <sup>n</sup>ð Þ ω0t; ω=ω<sup>0</sup> , ω0=2π ¼ 100KHz, n ¼ 5, the Morlet mother wavelet.

maxima are strongly manifested at the time intervals with the center localized near t ¼ 0, � 20μs and the pseudo-frequencies higher than the ones corresponding to the highest pseudo-frequency

Figure 5. The time dependence (lower box) and scalogram (upper box) of the superoscillating signal Ref <sup>n</sup>ð Þ ω0t; ω=ω<sup>0</sup> ,

shown in the upper box of Figure 3.

Figure 6. The spectrum of the signal Ref <sup>n</sup>ð Þ ω0t; ω=ω<sup>0</sup> for ω0=2π ¼ 100KHz, n ¼ 5.

ω0=2π ¼ 100KHz, n ¼ 4, the Mexican hat mother wavelet.

Figure 4. The spectrum of the signal Ref <sup>n</sup>ð Þ ω0t; ω=ω<sup>0</sup> for ω0=2π ¼ 100KHz, n ¼ 4.

210 Wavelet Theory and Its Applications

Mexican hat and Morlet mother wavelets are different. The real superoscillation frequencies can be obtained by pseudo-frequency multiplication by a scaling parameter a according to Eq. (39).

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The theoretical results of the wavelet analysis clearly show that the superoscillations with the local frequency larger that the band limit of the signal can be identified by using CWT.
