4. The fundamental properties of CWT and DWT

to capture the superoscillations, the region near x ¼ 0 is selected with a Gaussian window of width L which is opened and closed with a Gaussian switching function over an interval near t ¼ 0 [14]. The window must faithfully transmit the red light including the superoscillaitons

For the sake of definiteness, we consider the time-dependent superoscillating signal of the type

<sup>n</sup> ; g <sup>¼</sup> <sup>ω</sup> ω0

ω0t n � �

Expression (15) is the signal band limited by the frequency ω<sup>0</sup> with the superoscillation manifested by a single peak of a width <sup>Δ</sup><sup>t</sup> <sup>¼</sup> <sup>ω</sup>�<sup>1</sup> while the ratio <sup>ω</sup>=ω<sup>0</sup> may be arbitrary larger

For finite n and under the first condition of Eq. (11) which now takes the form ω0t <

, expression (16) reduces to the following approximation:

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

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

þ i ω ω0

� � � � <sup>n</sup>

sin <sup>ω</sup>0<sup>t</sup> n

lim<sup>n</sup>!<sup>∞</sup> Re<sup>f</sup> <sup>n</sup>ð Þ¼ <sup>ω</sup>0t; <sup>ω</sup>=ω<sup>0</sup> cosð Þ <sup>ω</sup><sup>t</sup> (16)

f <sup>n</sup>ð Þ ω0t; ω=ω<sup>0</sup> ≈ cosð Þ ωt (17)

(14)

(15)

<sup>x</sup> <sup>¼</sup> <sup>ω</sup>0<sup>t</sup>

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

f <sup>n</sup>ð Þ¼ ω0t; ω=ω<sup>0</sup> cos

than the band limit ω<sup>0</sup> for the arbitrary long time depending on n.

3. The possible applications of superoscillations

[14].

Eq. (1) assuming that:

200 Wavelet Theory and Its Applications

than unity. It should be noted that:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n= ð Þ ω=ω<sup>0</sup>

r h i

<sup>2</sup> � <sup>1</sup>

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 [20–23]. 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 timefrequency analysis in different regions of the spectrum is necessary as it is mentioned earlier [2, 6]:

The CWT W að Þ ; b of any square integrable function f tð Þ with respect to a wavelet ψð Þt is defined as follows [7]:

$$\mathcal{W}(a,b) \equiv \bigcap\_{-\infty}^{\infty} f(t) \frac{1}{\sqrt{|a|}} \psi^\* \left( \frac{t-b}{a} \right) dt \tag{18}$$

Here a, b are real, the asterisk denotes complex conjugation, the energy signals f tð Þ,ψð Þ<sup>t</sup> <sup>⊂</sup> <sup>L</sup><sup>2</sup> ð Þ R , L2 ð Þ <sup>R</sup> is the set of square integrable functions such that <sup>Ð</sup> ∞ �∞ j j <sup>ψ</sup>ð Þ<sup>t</sup> <sup>2</sup> dt < ∞. The real- or complexvalue continuous-time function ψð Þt is called the mother wavelet. It satisfies the following condition that Ð ∞ �∞ ψð Þt dt ¼ 0 [7].

CWT W að Þ ; b is a function of two variables: (1) the scale or dilation variable a determines the amount of time scaling or dilation and (2) the translation or time shift variable b represents the shift of ψa, <sup>0</sup>ð Þt by an amount b along the time axis and indicates the location of the wavelet window along it [6, 7]. The inverse scaling parameter 1=a is a measure of frequency [6].

Defining

$$
\psi\_{a,b}(t) = \frac{1}{\sqrt{a}} \psi\left(\frac{t-b}{a}\right) \tag{19}
$$

ψðÞ¼ t exp �t

duration, but most of the energy in this wavelet is confined to a finite interval [7].

h i x tð Þ; y tð Þ ¼

definition of the inner product h i x tð Þ; y tð Þ of two finite energy signals as:

and dilated mother wavelets ψa, <sup>b</sup>ð Þt for all a, b: W að Þ¼ ; b f tð Þ;ψa, <sup>b</sup>ð Þt

factor a [7]. Comparing the definition of the cross-correlation function:

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

Figure 2. The Morlet wavelet.

Rx, <sup>y</sup>ð Þ� τ h i x tð Þ; y tð Þ � τ ¼

W að Þ¼ ; b f tð Þ; ψa,0ð Þ t � b

tion of the convolution of the input signal x tð Þ and the system impulse response h tð Þ:

<sup>2</sup> � �cos πt

The time dependence of the Morlet wavelet is shown in Figure 2. It is a wavelet of an infinite

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

ð ∞

�∞

one can say that CWT is a collection of the inner products of a signal f tð Þ and the translated

considered as the cross-correlation at lag b between f tð Þ and the mother wavelet dilated to scale

ð

D E <sup>¼</sup> Rf ,ψa, <sup>0</sup>

The CWT is characterized by the time selectivity or the so-called windowing effect because the segment of f tð Þ influencing the value of W að Þ ; b for any ð Þ a; b coincides with the interval over which ψa, <sup>b</sup>ð Þt has the bulk of its energy [7]. The CWT frequency selectivity can be described by its representation as a collection of linear, time-invariant filters with impulse responses which are dilations of the mother wavelet reflected about the time axis [7]. Indeed, using the defini-

ffiffiffiffiffiffiffi 2 ln2 ! <sup>r</sup>

Applications of Wavelet Transforms to the Analysis of Superoscillations

x tð Þy<sup>∗</sup>ð Þ<sup>t</sup> dt (25)

D E [7]. CWT can be also

http://dx.doi.org/10.5772/intechopen.76333

x tð Þy<sup>∗</sup>ð Þ <sup>t</sup> � <sup>τ</sup> dt (26)

ð Þb (27)

(24)

203

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

$$\mathcal{W}(a,b) \equiv \bigcap\_{-\infty}^{\infty} f(t)\psi\_{a,b}^\*(t)dt \tag{20}$$

The energy conservation law for the mother wavelet has the form for all values of a, b [7].

$$\int\_{-\infty}^{\infty} \left| \psi\_{a,b}(t) \right|^2 dt = \int\_{-\infty}^{\infty} |\psi(t)|^2 dt \tag{21}$$

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

$$\psi(t) = \begin{cases} 1, 0 \le t < 1/2 \\ -1, 1/2 \le t < 1 \\ 0, otherwise \end{cases} \tag{22}$$

The Mexican hat wavelet is obtained by taking the second derivative of the negative Gaussian function �exp �t <sup>2</sup> � �=2 [7]. It is given by [7]:

$$
\psi(t) = \begin{pmatrix} 1 - 2t^2 \end{pmatrix} \exp(-t^2) \tag{23}
$$

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]:

Figure 1. The Mexican hat wavelet.

Applications of Wavelet Transforms to the Analysis of Superoscillations http://dx.doi.org/10.5772/intechopen.76333 203

$$\psi(t) = \exp\left(-t^2\right)\cos\left(\pi t \sqrt{\frac{2}{\ln 2}}\right) \tag{24}$$

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 h i x tð Þ; y tð Þ of two finite energy signals as:

$$
\langle \mathbf{x}(t), \mathbf{y}(t) \rangle = \int\_{-\infty}^{\infty} \mathbf{x}(t) \mathbf{y}^\*(t) dt \tag{25}
$$

one can say that CWT is a collection of the inner products of a signal f tð Þ and the translated and dilated mother wavelets ψa, <sup>b</sup>ð Þt for all a, b: W að Þ¼ ; b f tð Þ;ψa, <sup>b</sup>ð Þt D E [7]. CWT can be also considered as the cross-correlation at lag b between f tð Þ and the mother wavelet dilated to scale factor a [7]. Comparing the definition of the cross-correlation function:

$$R\_{x,y}(\tau) \equiv \langle \mathbf{x}(t), y(t-\tau) \rangle = \int \mathbf{x}(t) y^\*(t-\tau) dt\tag{26}$$

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

Defining

202 Wavelet Theory and Its Applications

function. It has the form:

function �exp �t

Figure 1. The Mexican hat wavelet.

by [7]:

ψa, <sup>b</sup>ðÞ¼ t

W að Þ� ; b

ψa, <sup>b</sup>ð Þt � � �

ψðÞ¼ t

ð ∞

�∞

<sup>2</sup> � �=2 [7]. It is given by [7]:

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

1 ffiffi <sup>a</sup> <sup>p</sup> <sup>ψ</sup> <sup>t</sup> � <sup>b</sup> a � �

ð ∞

f tð Þψ<sup>∗</sup>

�∞

1, 0 ≤ t < 1=2 �1, 1=2 ≤ t < 1 0, otherwise

Consider some typical mother wavelets [6, 7]. The Haar wavelet is a piecewise continuous

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

The Morlet wavelet represents a sinusoidal function modulated by a Gaussian function given

<sup>2</sup> � �exp �<sup>t</sup>

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

�∞

The energy conservation law for the mother wavelet has the form for all values of a, b [7].

� � � 2 dt ¼ ð ∞

8 ><

>:

ψðÞ¼ t 1 � 2t

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

(19)

(22)

a, <sup>b</sup>ð Þt dt (20)

dt (21)

<sup>2</sup> � � (23)

$$\mathcal{W}(a,b) = \left\langle f(t), \psi\_{a,0}(t-b) \right\rangle = \mathcal{R}\_{f, \psi\_{a,0}}(b) \tag{27}$$

The CWT is characterized by the time selectivity or the so-called windowing effect because the segment of f tð Þ influencing the value of W að Þ ; b for any ð Þ a; b coincides with the interval over which ψa, <sup>b</sup>ð Þt has the bulk of its energy [7]. The CWT frequency selectivity can be described by its representation as a collection of linear, time-invariant filters with impulse responses which are dilations of the mother wavelet reflected about the time axis [7]. Indeed, using the definition of the convolution of the input signal x tð Þ and the system impulse response h tð Þ:

Figure 2. The Morlet wavelet.

$$h(t) \* \mathbf{x}(t) \equiv \int\_{-\infty}^{\infty} h(\tau)\mathbf{x}(t-\tau)d\tau\tag{28}$$

Δt ¼

Ð ∞ �∞

vuuuuut

Δωψð Þa of its dilation ψa,0ð Þt are given by [7]:

Combining expressions (33) we obtain:

Eq. (34) takes the form [6, 7]:

wavelet Fourier transform Ψð Þ ω [7]:

Then the inverse CWT has the form [7]:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

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

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

;Δω ¼

Expressions (31) and (32) can be used only for the mother wavelet ψð Þt and its Fourier transform Ψð Þ ω rapidly decaying in time and frequency, respectively, since the integrals in the numerators of these expressions should have finite values [7]. For the RMS duration Δt<sup>ψ</sup> and bandwidth Δωψ of the mother wavelet ψð Þt , the RMS duration Δtψð Þa and bandwidth

It has been shown that the smallest time-bandwidth product is equal to 1/2., and condition

Δtψð Þa Δωψð Þa ≥

Equation (34) shows that the product of the wavelet duration and bandwidth is invariant to dilation. For small values of a, the CWT is characterized by good time resolution and poor frequency resolution because the RMS duration of the dilated wavelet is small while the RMS bandwidth of the dilated wavelet is large [7]. For large values of a, the time resolution of the CWT is poor, and its frequency resolution is good. The CWT provides better frequency resolution for the low-frequency region of the spectrum and poorer frequency resolution for the high-frequency region of the spectrum [7]. It can be shown that the translation parameter b

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

does not influence the mother wavelet duration and bandwidth [7].

ð ∞

j j Ψð Þ ω 2

j j ω

a¼�∞

ð ∞

1

b¼�∞

�∞

f tðÞ¼ <sup>1</sup> C ð ∞

Ð ∞ �∞

vuuuuut

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Applications of Wavelet Transforms to the Analysis of Superoscillations

j j Ψð Þ ω 2 dω

j j Ψð Þ ω 2 dω

http://dx.doi.org/10.5772/intechopen.76333

(32)

205

ð Þ <sup>ω</sup> � <sup>ω</sup><sup>c</sup> <sup>2</sup>

Δtψð Þ� a j j a Δtψ;Δωψð Þ¼ a Δωψ=j j a (33)

Δtψð Þa Δωψð Þ¼ a ΔtψΔωψ ¼ c<sup>ψ</sup> ¼ const (34)

<sup>2</sup> (35)

dω � C; 0 < C < ∞ (36)

j j <sup>a</sup> <sup>2</sup> W að Þ ; <sup>b</sup> <sup>ψ</sup>a, <sup>b</sup>ð Þ<sup>t</sup> dadb (37)

1

Ð ∞ �∞

ð Þ <sup>t</sup> � tc <sup>2</sup>

Ð ∞ �∞

we can write for the CWT:

$$\mathcal{W}(a,b) = f(b) \* \psi\_{a,0}^\*(-b) \tag{29}$$

Consequently, for any given a, CWT W að Þ ; b is the output of a filter with the impulse response ψ∗ a, <sup>0</sup>ð Þ �b and input f bð Þ [7]. There exists a continuum of filters characterized by the scale factor a as a parameter [7]. We define the Fourier transform Ψð Þ ω of the mother wavelet ψð Þt :

$$\Psi(\omega) = F\{\psi(t)\} \equiv \int\_{-\infty}^{\infty} \psi(t) \exp(-i\omega t)dt\tag{30}$$

Then, the corresponding Q factor is determined as Q ¼ ωc=Δω, where ω<sup>c</sup> is center frequency of the Fourier transform Eq. (30) and Δω is the 3-dB bandwidth defined as the difference between the two frequencies on either side of the peak at which j j Ψð Þ ω <sup>2</sup> is exactly half its peak value j j Ψð Þ ω 2 max [7]. The Q factor is invariant with respect to the wavelet dilation, since Ff g ψð Þ t=a ¼ j j a Ψð Þ aω . The center frequency ω<sup>c</sup> of j j Ψð Þ ω <sup>2</sup> for any <sup>a</sup> is at 1=j j <sup>a</sup> times the center frequency of the mother wavelet ψð Þt , and its 3-dB bandwidth is 1=j j a times the 3-dB bandwidth of the mother wavelet ψð Þt which gives the same value of the Q factor as the one mentioned above [7]. Hence the continuum of filters mentioned above is a set of constant Q bandpass filters which results in the frequency selectivity of the CWT [7]. For large values of a, the corresponding filter has a frequency response with a low center frequency ω0, and the corresponding CWT W að Þ ; b captures the frequency content of the signal f tð Þ around this low frequency [7]. The bandpass filter shifts to higher frequencies region with the decrease of a in such a way that the CWT W að Þ ; b at small scales contains information about f tð Þ at the higher end of its frequency spectrum [7]. The time and frequency resolution of the CWT W að Þ ; b are based on the duration and bandwidth of the mother wavelet ψð Þt , respectively. The first moments tc and ω<sup>c</sup> of the mother wavelet ψð Þt and its Fourier transform Ψð Þ ω , respectively, are given by [7]:

$$t\_{\boldsymbol{\varepsilon}} \equiv \frac{\int\_{-\infty}^{\infty} t \left| \psi(t) \right|^{2} dt}{\int\_{-\infty}^{\infty} \left| \psi(t) \right|^{2} dt}; \boldsymbol{\omega}\_{\boldsymbol{\varepsilon}} = \frac{\int\_{-\infty}^{\infty} \left| \Psi(\omega) \right|^{2} d\omega}{\int\_{-\infty}^{\infty} \left| \Psi(\omega) \right|^{2} d\omega} \tag{31}$$

Expressions (31) provide the location of the center of ψð Þt and Ψð Þ ω along the time and frequency axes, respectively [6, 7]. A measure of the wavelet duration Δt, or the root mean square (RMS) duration, and the RMS bandwidth of the wavelet Δω are given by, [6, 7], respectively:

$$
\Delta t = \sqrt{\frac{\int\_{-\infty}^{\infty} \left(t - t\_c\right)^2 \left|\psi(t)\right|^2 dt}{\int\_{-\infty}^{\infty} \left|\psi(t)\right|^2 dt}};
\Delta \omega = \sqrt{\frac{\int\_{-\infty}^{\infty} \left(\omega - \omega\_c\right)^2 \left|\Psi(\omega)\right|^2 d\omega}{\int\_{-\infty}^{\infty} \left|\Psi(\omega)\right|^2 d\omega}}\tag{32}
$$

Expressions (31) and (32) can be used only for the mother wavelet ψð Þt and its Fourier transform Ψð Þ ω rapidly decaying in time and frequency, respectively, since the integrals in the numerators of these expressions should have finite values [7]. For the RMS duration Δt<sup>ψ</sup> and bandwidth Δωψ of the mother wavelet ψð Þt , the RMS duration Δtψð Þa and bandwidth Δωψð Þa of its dilation ψa,0ð Þt are given by [7]:

$$
\Delta t\_{\psi}(\mathfrak{a}) \equiv |a| \Delta t\_{\psi} ; \Delta \omega\_{\psi}(\mathfrak{a}) = \Delta a \omega\_{\psi} / |a| \tag{33}
$$

Combining expressions (33) we obtain:

h tð Þ∗x tðÞ�

we can write for the CWT:

204 Wavelet Theory and Its Applications

ψ∗

j j Ψð Þ ω 2

are given by [7]:

respectively:

ð ∞

hð Þτ x tð Þ � τ dτ (28)

a,0ð Þ �b (29)

ψð Þt expð Þ �iωt dt (30)

<sup>2</sup> for any <sup>a</sup> is at 1=j j <sup>a</sup> times the center frequency of

<sup>2</sup> is exactly half its peak value

(31)

�∞

W að Þ¼ ; <sup>b</sup> f bð Þ∗ψ<sup>∗</sup>

a as a parameter [7]. We define the Fourier transform Ψð Þ ω of the mother wavelet ψð Þt :

Ψð Þ¼ ω Ff g ψð Þt �

the two frequencies on either side of the peak at which j j Ψð Þ ω

tc �

Ð ∞ �∞

Ð ∞ �∞ <sup>t</sup>j j <sup>ψ</sup>ð Þ<sup>t</sup> <sup>2</sup> dt

j j <sup>ψ</sup>ð Þ<sup>t</sup> <sup>2</sup> dt ; ω<sup>c</sup> ¼

Expressions (31) provide the location of the center of ψð Þt and Ψð Þ ω along the time and frequency axes, respectively [6, 7]. A measure of the wavelet duration Δt, or the root mean square (RMS) duration, and the RMS bandwidth of the wavelet Δω are given by, [6, 7],

Ð ∞ �∞

Ð ∞ �∞

ωj j Ψð Þ ω 2 dω

j j Ψð Þ ω 2 dω

j j a Ψð Þ aω . The center frequency ω<sup>c</sup> of j j Ψð Þ ω

Consequently, for any given a, CWT W að Þ ; b is the output of a filter with the impulse response

a, <sup>0</sup>ð Þ �b and input f bð Þ [7]. There exists a continuum of filters characterized by the scale factor

ð ∞

�∞

max [7]. The Q factor is invariant with respect to the wavelet dilation, since Ff g ψð Þ t=a ¼

Then, the corresponding Q factor is determined as Q ¼ ωc=Δω, where ω<sup>c</sup> is center frequency of the Fourier transform Eq. (30) and Δω is the 3-dB bandwidth defined as the difference between

the mother wavelet ψð Þt , and its 3-dB bandwidth is 1=j j a times the 3-dB bandwidth of the mother wavelet ψð Þt which gives the same value of the Q factor as the one mentioned above [7]. Hence the continuum of filters mentioned above is a set of constant Q bandpass filters which results in the frequency selectivity of the CWT [7]. For large values of a, the corresponding filter has a frequency response with a low center frequency ω0, and the corresponding CWT W að Þ ; b captures the frequency content of the signal f tð Þ around this low frequency [7]. The bandpass filter shifts to higher frequencies region with the decrease of a in such a way that the CWT W að Þ ; b at small scales contains information about f tð Þ at the higher end of its frequency spectrum [7]. The time and frequency resolution of the CWT W að Þ ; b are based on the duration and bandwidth of the mother wavelet ψð Þt , respectively. The first moments tc and ω<sup>c</sup> of the mother wavelet ψð Þt and its Fourier transform Ψð Þ ω , respectively,

$$
\Delta t\_{\psi}(a)\Delta\omega\_{\psi}(a) = \Delta t\_{\psi}\Delta\omega\_{\psi} = c\_{\psi} = \text{const}\tag{34}
$$

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

$$
\Delta t\_{\psi}(a)\Delta a \omega\_{\psi}(a) \gtrsim \frac{1}{2} \tag{35}
$$

Equation (34) shows that the product of the wavelet duration and bandwidth is invariant to dilation. For small values of a, the CWT is characterized by good time resolution and poor frequency resolution because the RMS duration of the dilated wavelet is small while the RMS bandwidth of the dilated wavelet is large [7]. For large values of a, the time resolution of the CWT is poor, and its frequency resolution is good. The CWT provides better frequency resolution for the low-frequency region of the spectrum and poorer frequency resolution for the high-frequency region of the spectrum [7]. It can be shown that the translation parameter b does not influence the mother wavelet duration and bandwidth [7].

The inverse CWT can be evaluated under the following sufficient condition for the mother wavelet Fourier transform Ψð Þ ω [7]:

$$\int\_{-\infty}^{\infty} \frac{|\Psi(\omega)|^2}{|\omega|} d\omega \equiv \mathcal{C}; 0 < \mathcal{C} < \infty \tag{36}$$

Then the inverse CWT has the form [7]:

$$f(t) = \frac{1}{\mathbb{C}} \int\_{a = -\circ}^{\circ} \int\_{b = -\circ}^{\circ} \frac{1}{|a|^2} W(a, b) \psi\_{a, b}(t) da db \tag{37}$$

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 a and the translation parameter b are discrete and take a form <sup>a</sup> <sup>¼</sup> <sup>2</sup><sup>k</sup> , b <sup>¼</sup> <sup>2</sup><sup>k</sup> l, where k and l are integers [6, 7, 17]. Then, Eq. (37) takes the form [7]:

$$f(t) = \sum\_{k=-\nu}^{k=\nu} \sum\_{l=-\nu}^{l=\nu} d(k,l) 2^{-k/2} \psi \left( 2^{-k}t - l \right) \tag{38}$$

5. The applications of wavelet transforms

nonuniform being a function of both time and frequency.

Fourier technique [17].

high-pass filtering in the time domain [24].

of the system and mitigating the channel chromatic dispersion [20].

different-wavelength based inteferometric vibration sensor in a fiber link [25].

suppression [6].

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

Applications of Wavelet Transforms to the Analysis of Superoscillations

http://dx.doi.org/10.5772/intechopen.76333

207

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

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

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

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)

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

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 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> , b <sup>¼</sup> <sup>2</sup><sup>k</sup> l [7]. Then DWT Wklð Þ a; b takes the form [6, 7]:

$$\mathcal{W}\_{kl}(a,b) = \bigcap\_{-\infty}^{\infty} f(t) \frac{1}{2^{k/2}} \psi^\*(2^{-k}t - l)dt\tag{39}$$

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 the following conditions [7]:

$$\int\_{-\infty}^{\infty} \psi(t)dt = 0; \int\_{-\infty}^{\infty} \left| \psi(t) \right|^2 dt = 1; \langle \psi(t), \psi(t-n) \rangle = \delta(n); \left\langle \psi(t), \phi(t-n) \right\rangle = 0 \tag{40}$$

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 following conditions [7]:

$$\int\_{-\infty}^{\infty} \phi(t)dt = 1; \int\_{-\infty}^{\infty} \left| \phi(t) \right|^2 dt = 1; \left\langle \phi(t), \phi(t-n) \right\rangle = \delta(n) \tag{41}$$

The scaling function ϕð Þt and the wavelet function ψð Þt are defined by the following equations, respectively [7]:

$$\phi(t) = \sum\_{n = -\infty}^{\infty} c(n)\phi(2t - n); \psi(t) = \sum\_{n = -\infty}^{\infty} d(n)\phi(2t - n), n = 0, \pm 1, \pm 2, \dots \tag{42}$$

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