## Abstract

The phenomenon of superoscillation is the local oscillation of a band limited function at a frequency ω higher than the band limit. Superoscillations exist during the limited time intervals, and their amplitude is small compared to the signal components with the frequencies inside the bandwidth. For this reason, the wavelet transform is a useful mathematical tool for the quantitative description of the superoscillations. Continuous-time wavelet transform (CWT) of a transient signal ft is a function of two variables: one of them represents a time shift, and the other one is the scale or dilation variable. As a result, CWT permits the simultaneous analysis of the transient signals both in the time and frequency domain. We show that the superoscillations strongly localized in time and frequency domains can be identified by using CWT analysis. We use CWT with the Mexican hat and Morlet mother wavelets for the theoretical investigation of superoscillation spectral features and time dependence for the first time, to our best knowledge. The results clearly show that the high superoscillation frequencies, time duration, and energy contours can be found by using CWT of the superoscillating signals.

### Keywords

- wavelet transform
- superoscillations
- transient signals
- low-pass filter

## 1. Introduction

Superoscillating signals are band-limited signals that oscillate in some region faster than their largest Fourier component [1]. Superoscillatory functions may have interesting applications in quantum mechanics, signal processing, and optics (see, for instance, [1] and references therein). However, the superoscillation amplitude is usually so small compared to the typical values of the amplitude in non-superoscillating regions that the practical applications of the superoscillating functions depend on tailoring the functions in order to reduce such an effect [1]. It has been shown that the superoscillations amplitude decreases exponentially with the length of the superoscillating stretch [2]. Nevertheless, the existence of superoscillations and the possibility of encoding arbitrary amounts of information into an arbitrary short segment of a low-bandwidth signal do not contradict the information theory [2]. Taking into account the Shannon’s theorem concerning the information channel capacity, it appeared to be that the superoscillatory information can be compressed to an arbitrary extent under the condition that the signal power increases exponentially with the length of the superoscillatory part of the message [2]. Superoscillations can be designed by prescribing their amplitude and/or their derivative on a grid which is denser than the Nyquist density [3]. Four different ways to constrain the signal in order to render it superoscillatory have been described in Reference [3]: (1) amplitude constraints, without any restriction on the derivative; (2) derivative constraints, without restrictions on the amplitude; (3) the amplitude and the derivative constraints on staggered grids; and (4) the amplitude and the derivative constraints on aligned grids at one half density [3]. When a set of constraints is chosen to ensure a required superoscillation, the signal is optimized by minimizing its total energy within the subspace of all the superoscillatory functions obeying the same set of constraints [2]. Superoscillations can be constructed also by using the so-called direct approach. This approach is based on a signal that is a superposition of time shifted

It should be noted that it is impossible to infer the bandwidth of a finite energy signal

In this chapter, we constructed a family of complex valued superoscillating functions and investigated their behavior for different values of the maximum frequency

The chapter is organized as follows. The superoscillating function properties are discussed in Section 2. Some possible applications of superoscillations are reviewed in Section 3. CWT and discrete wavelet transform (DWT) definition and fundamental features are presented in Section 4. The applications of wavelet transform for the optical signal processing are briefly discussed in Section 5. The simulation results are presented and discussed in Section 6. Conclusions are presented in Section 7.

## 2. The properties of the superoscillating functions

The frequency limited functions and superoscillations occur in a number of scientific and technological applications such as foundations of quantum mechanics, information theory, optics, and signal processing which led also to work on the optimization and stability of superoscillations [8, 9, 10, 11, 12, 13, 14]. Some examples of superoscillatory functions have been proposed and investigated in the past [1, 2, 3, 4, 8, 10, 11]. In this section, we consider the generic example of the Aharonov, Popescu, Rohlich functions

Here, for general,

For small

Obviously, in the limiting case, the function determined by Eq. (3) is varying faster than the function determined by Eq. (2). Consider now the Fourier series for

where

Equations (4) and (5) contain only wavenumbers

Here, the local wavenumber

The relationship Eq. (6) can be proved immediately taking into account that:

and using the identities

The number of oscillations

Equation (6) shows in particular that in the superoscillatory region

Superoscillations is a week phenomenon such that there is no slightest indication of superoscillations in the power spectrum

where

The asymptotic spectrum Eq. (12) is a narrow Gaussian with the center at

The function

In the framework of the precise classical wave model, it has been shown how superoscillations can emerge and propagate into the far-field region [14]. The band-limited superoscillatory wave (the “red light”) is propagating along the

For the sake of definiteness, we consider the time-dependent superoscillating signal of the type Eq. (1) assuming that:

Substituting relationships Eq. (14) into Eq. (1) we obtain:

Expression (15) is the signal band limited by the frequency

For finite

Equation (17) shows that the band limited signal Eq. (15) oscillates with a frequency

## 3. The possible applications of superoscillations

Optical superoscillations can be used in the subwavelength imaging [15]. This super-resolution technology is based on a superoscillatory lens (SOL) which represents a nanostructured mask [15]. SOL illuminated with a coherent light source creates a focus at a distance which is larger than the near-field of the mask [15]. Indeed, the ability to focus beyond the diffraction limit is related to the superoscillation, since the band-limited functions in such a case oscillate faster than their highest Fourier components [11]. Superoscillatory binary masks do not use evanescent waves and focus at distances tens of wavelengths away from the mask [15]. The superoscillation-based imaging has the following advantages with respect to other technologies: (1) it is non-invasive which allows to place the object at a substantial distance from SOL; (2) it can operate at the wide range of wavelengths from X-rays to microwaves; and (3) the resolution of the SOL can be improved by refining the design, increasing the size of the superoscillatory mask and by increasing the dynamic range of the light detection [15]. SOL can be also used for the creation of sub-diffraction-limit optical needles [16]. An optical needle could be created by converting the central region of the SOL into an opaque area forming a shadow, and changing the diameter of the blocking region without varying the rest of SOL [16]. The possible applications of the sub-diffraction-limit optical needles are the far-field super-resolution microscopy and nanofabrication [16].

The possible applications of superoscillations for data compressions have been discussed [8]. However, the superoscillations are unstable in a way that tiny perturbations of a band-limited superoscillating function can induce very high-frequency components [8]. For this reason, the practical use of the superoscillations in imperfect communication channels is difficult [8].

## 4. The fundamental properties of CWT and DWT

There exist different types of a wavelet transform: CWT, discrete wavelet transform (DWT) [6, 7, 17], multi-wavelets [17, 18], and complex wavelets [19]. We applied these types of wavelets to the problems related to the signal processing in optical communication systems [20, 21, 22, 23]. We have found that CWT is the most appropriate for the analysis of superoscillations.

In this section, we consider some fundamental features of CWT. Unlike the Fourier transform and STFT, the CWT is characterized by the time and frequency selectivity [6, 7]. It can localize events both in time and in frequency in the entire time-frequency plane [6, 7]. That is why CWT is unique mathematical tool for the investigation of the superoscillations where the time-frequency analysis in different regions of the spectrum is necessary as it is mentioned earlier [2, 6]:

The CWT

Here

CWT

Defining

and substituting expression (19) into Eq. (18) we obtain [7]:

The energy conservation law for the mother wavelet has the form for all values of

Consider some typical mother wavelets [6, 7]. The Haar wavelet is a piecewise continuous function. It has the form:

The Mexican hat wavelet is obtained by taking the second derivative of the negative Gaussian function

The time dependence of the Mexican hat wavelet is shown in Figure 1.

The Morlet wavelet represents a sinusoidal function modulated by a Gaussian function given by [7]:

The time dependence of the Morlet wavelet is shown in Figure 2. It is a wavelet of an infinite duration, but most of the energy in this wavelet is confined to a finite interval [7].

CWT can be used in pattern detection and classification [6, 7]. Indeed, taking into account the definition of the inner product

one can say that CWT is a collection of the inner products of a signal

and CWT expression (20) we can write [7]:

The CWT is characterized by the time selectivity or the so-called windowing effect because the segment of

we can write for the CWT:

Consequently, for any given

Then, the corresponding

Expressions (31) provide the location of the center of

Expressions (31) and (32) can be used only for the mother wavelet

Combining expressions (33) we obtain:

It has been shown that the smallest time-bandwidth product is equal to 1/2., and condition Eq. (34) takes the form [6, 7]:

Equation (34) shows that the product of the wavelet duration and bandwidth is invariant to dilation. For small values of

The inverse CWT can be evaluated under the following sufficient condition for the mother wavelet Fourier transform

Then the inverse CWT has the form [7]:

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

Suppose that the dilation parameter

The two-dimensional sequence

Comparison of CWT and DWT shows that the signal

Here

The scaling function

where

## 5. The applications of wavelet transforms

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, 18, 19, 20, 21, 22, 23, 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

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].

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 time-varying 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):

The signal Eq. (38) is band limited by the maximum frequency

We investigated the scalogram of the energy contours for the spectral component of the signal Eq. (38) at the highest frequency

The behavior of the component with the frequency

where

The spectra, the temporal behavior, and the scalograms of the signal Eq. (38) for

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

It is seen from the lower box of Figure 7 that the superoscillations with the time duration of about

The corresponding energy contours are identified in the scalogram (upper box of Figure 7) in the time intervals localized near

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

Comparison of Figures 7 and 8 shows that the spectral features of superoscillations are pronounced at the higher pseudo-frequencies of about

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.

## 7. Conclusions

We for the first time to our best knowledge applied CWT for the theoretical analysis of superoscillations in the time and frequency domain. We discussed the basic properties of the superoscillating signals containing the components with the frequencies larger than the maximum frequency in the signal spectrum. We also considered some possible applications of superoscillations in optics and signal processing. The superoscillating components are extremely weak and short in the time domain. They cannot be identified by the Fourier transform since they require the time-frequency analysis. We discussed the fundamental properties of CWT and DWT and their typical applications. The CWT is a unique tool for the superoscillation studies because it provides the localization of the signal both in time and in the frequency domain. We used the Mexican hat and the Morlet mother wavelets for the CWT of the sinusoidal superoscillating signal because these mother wavelets are similar to the signal oscillations. The theoretical results clearly show that the superoscillation frequency, time duration, and energy contours can be identified by using the CWT of the corresponding signal. Generally, CWT with different mother wavelets can be used for the analysis of superoscillating signals with different structures.