Analytical Expressions of Infinite Fourier Sine and Cosine Transform-Based Ramanujan Integrals *RS,C*(*m, n*) in Terms of Hypergeometric Series <sup>2</sup>*F*3(�)

*Showkat Ahmad Dar and M. Kamarujjama*

#### **Abstract**

In this chapter, we obtain analytical expressions of infinite Fourier sine and cosine transform-based Ramanujan integrals,

$$\mathbf{R}\_{\mathrm{S},\mathrm{C}}(m,n) = \int\_0^\infty \frac{\mathbf{x}^m}{-\mathbf{1} + \exp\left(2\pi\sqrt{\mathbf{x}}\right)} \frac{\sin}{\cos}(\pi m \boldsymbol{\infty}) d\boldsymbol{\infty},$$

in an infinite series of hypergeometric functions <sup>2</sup>*F*3ð Þ� , using the hypergeometric technique. Also, we have given some generalizations of the Ramanujan's integrals **<sup>R</sup>***<sup>S</sup>*,*<sup>C</sup>*ð Þ *<sup>m</sup>*, *<sup>n</sup>* in the form of integrals denoted by **<sup>I</sup>** <sup>∗</sup> *<sup>S</sup>*,*<sup>C</sup>*ð Þ *υ*, *b*, *c*, *λ*, *y* ,**J***<sup>S</sup>*,*<sup>C</sup>*ð Þ *υ*, *b*, *c*, *λ*, *y* , **K***<sup>S</sup>*,*<sup>C</sup>*ð Þ *υ*, *b*, *c*, *λ*, *y* and **I***<sup>S</sup>*,*<sup>C</sup>*ð Þ *υ*, *b*, *λ*, *y* . These generalized definite integrals are expressed in terms of ordinary hypergeometric functions <sup>2</sup>*F*3ð Þ� , with suitable convergence conditions. Moreover, as applications of Ramanujan's integrals **R***<sup>S</sup>*,*<sup>C</sup>*ð Þ *m*, *n* , some closed form of infinite summation formulas involving hypergeometric functions <sup>1</sup>*F*2, <sup>2</sup>*F*3ð Þ� , and <sup>0</sup>*F*<sup>1</sup> are derived.

**Keywords:** generalized hypergeometric function, infinite Fourier sine and cosine transforms, Ramanujan's integrals, Fox-Wright psi hypergeometric function, hypergeometric series

#### **1. Introduction**

Naturally, we call a function"special" when the function, just as the logarithm, the exponential and trigonometric functions (the elementary transcendental functions), belongs to the toolbox of the applied mathematician, the physicist, or the engineer. This branch of mathematics has a good history with great names such as Gauss, Euler, Fourier, Legendre, and Bessel. This chapter includes definitions, namely infinite Fourier sine and cosine transforms, Pochhammer's symbol and related results, generalized Gauss hypergeometric function and its special cases, Fox-Wright hypergeometric function and its convergence conditions, Hypergeometric form of elementary

functions, Gauss-Legendre multiplication formula and infinite series decomposition identity. In the literature [1–6], the analytical expressions of the Fourier sine and cosine transforms of *<sup>x</sup><sup>υ</sup>*�1nð Þ exp ð Þ� *bx* <sup>1</sup> are available in terms of Riemann's zeta function, the Psi function (Digamma function), hyperbolic function and Beta function. The analytical solution of the following infinite Fourier sine and cosine transforms based Ramanujan integrals ([7], p. 85, eq. (49) last line):

$$\mathbf{R}\_{\mathrm{S,C}}(m,n) = \int\_0^\infty \frac{\mathbf{x}^m}{\{-\mathbf{1} + \exp\left(2\pi\sqrt{\mathbf{x}}\right)\}} \frac{\sin}{\cos} (\pi m \mathbf{x}) d\mathbf{x},\tag{1}$$

are not given for all positive rational values of *n* and non-negative integral values of *m*.

#### **2. Definitions and preliminaries**

#### **2.1 Fourier sine and cosine transforms**

The infinite Fourier sine and cosine transforms of *g x*ð Þ over the interval 0, ½ Þ ∞ are defined by

$$F\_{\rm S,C}(\mathbf{g}(\infty);b) = \int\_0^\infty \mathbf{g}(\infty) \frac{\sin}{\cos} \left(b\mathbf{x}\right) d\mathbf{x} = \mathbf{G}\_{\rm S,C}(b), (b > \mathbf{0}).\tag{2}$$

For example, if *<sup>y</sup>* <sup>&</sup>gt;0, 0<sup>&</sup>lt; *Re*ð Þ*<sup>υ</sup>* <sup>&</sup>lt; 2 for Fourier sine transform of *<sup>x</sup>*�*<sup>υ</sup>* and *<sup>y</sup>*<sup>&</sup>gt; 0, <sup>0</sup><sup>&</sup>lt; *Re*ð Þ*<sup>υ</sup>* <sup>&</sup>lt;1 for Fourier cosine transform of *<sup>x</sup>*�*<sup>υ</sup>*, then the infinite Fourier sine and cosine transforms of *x*�*<sup>υ</sup>* ([3], p. 68) are given by

$$\int\_0^\infty x^{-\nu} \frac{\sin}{\cos}(\infty \eta) d\infty = \mathcal{y}^{\nu - 1} \Gamma(1 - \nu) \frac{\sin}{\cos} \left(\frac{\nu \pi}{2}\right) \tag{3}$$

Further, if *b*>0, �1< ℜð Þ*s* <1 for Fourier sine transform and *b*> 0, 0< ℜð Þ*s* < 1 for Fourier cosine transform, then the infinite Fourier sine and cosine transform of *xs*�<sup>1</sup> are given by [3, 5, 8].

$$\int\_0^\infty x^{s-1} \frac{\sin}{\cos}(b\infty) d\infty = \frac{\Gamma(s) \int\_0^\infty \left(\frac{\pi s}{2}\right)}{b^s}.\tag{4}$$

Moreover, if ℜð Þ *μ* > � 2 for Fourier sine transform and ℜð Þ *μ* > � 1 for Fourier cosine transform, then we can prove the following integral by using Maclaurin's expansion of exp �*ax<sup>ξ</sup>* � � and term by term integrating with the help of the result (4)

$$\int\_0^\infty x^\mu \exp\left(-a x^\xi\right) \frac{\sin}{\cos}(xy) dx = y^{-\mu-1} \sum\_{\ell=0}^\infty \left(-\frac{a}{y^\xi}\right)^\ell \frac{1}{\ell!} \Gamma(\mu+1+\xi\ell) \begin{array}{c} \sin\\ \cos \end{array} \left\{\frac{\pi}{2}(\mu+\xi\ell)\right\}.\tag{5}$$

where 0 <*ξ*<1, *a*>0 and *y* >0. The conditions ℜð Þ *μ* > � 2 and ℜð Þ *μ* > � 1 stated in the integrals (5) follows from the theory of analytic continuation [5, 8]. We have also verified the conditions ℜð Þ *μ* > � 2 and ℜð Þ *μ* > � 1, using Wolfram Mathematica software.

*Analytical Expressions of Infinite Fourier Sine and Cosine Transform-Based Ramanujan… DOI: http://dx.doi.org/10.5772/intechopen.108401*

#### **2.2 Generalized gauss hypergeometric function**

A natural generalization of the Gauss hypergeometric function <sup>2</sup>*F*1ð Þ*z* is the generalized hypergeometric function *pFq*ð Þ*z* with *p* numerator parameters *α*1, … ,*α<sup>p</sup>* and *q* denominator parameters *β*1, … ,*β<sup>q</sup>* defined by [9].

$${}\_{p}F\_{q}\left(\begin{matrix}a\_{1},\ldots,a\_{p};\\\beta\_{1},\ldots,\beta\_{q};\end{matrix}\bigg|z\right)=\sum\_{n=0}^{\infty}\frac{(a\_{1})\_{n}\ldots\left(a\_{p}\right)\_{n}}{(\beta\_{1})\_{n}\ldots\left(\beta\_{q}\right)\_{n}}\frac{z^{n}}{n!},\tag{6}$$

where *α<sup>j</sup>* ∈ ð Þ *j* ¼ 1, … , *p* , *β<sup>j</sup>* ∈ n� <sup>0</sup> ð Þ *j* ¼ 1, … , *q* and *p*, *q*∈ 0. Then the hypergeometric *pFq*ð Þ*z* function in (6) converges absolutely for ∣*z*∣< ∞ when *p* ≤*q* and for ∣*z*∣< 1 when *p* ¼ *q* þ 1. Furthermore, if we set,

$$\alpha := \left(\sum\_{j=1}^{q} \beta\_j - \sum\_{j=1}^{p} a\_j\right),\tag{7}$$

it is known that when *p* ¼ *q* þ 1 the function *pFq*ð Þ*z* is absolutely convergent for ∣*z*∣ ¼ 1 if ℜð Þ *ω* > 0, conditionally convergent for ∣*z*∣ ¼ 1 (*z* 6¼ 1) if �1< ℜð Þ *ω* <0 and divergent for ∣*z*∣ ¼ 1 if ℜð Þ *ω* ≤ � 1.

#### **2.3 Hypergeometric form of elementary functions**

The important special cases of *pFq*ð Þ*z* include (for example) the binomial series <sup>1</sup>*F*0ð Þ*z* given by [9].

$$\left(\left(1-z\right)^{-a}=\_{1}F\_{0}\left(\begin{array}{c}a;\\ -;\end{array}z\right)=\sum\_{n=0}^{\infty}\frac{(a)\_{n}}{n!}z^{n},\tag{8}$$

where ∣*z*∣<1, *a*∈ .

Elementary trigonometric functions ([10], p. 44, eq. (9) and eq. (10)) are given by

$$\cos z = {}\_0F\_1\left( \begin{array}{cc} \hline \text{---} \\ \hline \frac{1}{2}; & \frac{-z^2}{4} \end{array} \right),\tag{9}$$

$$z \sin z = z\_0 F\_1 \left( \begin{array}{c} \multicolumn{1}{c}{\smile}; \ -z^2 \\ \cline{2-5} \end{array} \right). \tag{10}$$

Lommel function ([10], p. 44, eq. (13)) is given by

$$s\_{\mu,\nu}(\mathbf{z}) = \frac{z^{\mu+1}}{(\mu-\nu+1)(\mu+\nu+1)} {}\_1F\_2\left( \begin{array}{c} \mu-\nu+3 \\ \hline 2 \end{array}, \frac{\mu+\nu+3}{2}; \frac{-z^2}{4} \right),\tag{11}$$

where *μ* � *υ*∈ n �f g 1, �3, �5, �7, … .

Struve function ([10], p. 44, eq. (16)) is given by

$$H\_{\nu}(\mathbf{z}) = \frac{2\binom{\mathbf{z}}{2}^{\nu+1}}{\sqrt{\pi}\,\Gamma\left(\nu+\frac{3}{2}\right)} {}\_{1}F\_{2}\left(\begin{array}{c} \mathbf{3} \\ \frac{3}{2}, \nu+\frac{3}{2}; \end{array}; \frac{-\mathbf{z}^{2}}{4}\right). \tag{12}$$

Modified Struve function ([10], p. 45, eq. (17)) is given by

$$L\_{\nu}(z) = \frac{2\left(\frac{\mu}{2}\right)^{\nu+1}}{\sqrt{\pi}\,\Gamma\left(\nu+\frac{3}{2}\right)} {}\_1F\_2\left(\begin{array}{c} 1; \\ \frac{3}{2}, \nu+\frac{3}{2}; \frac{2}{4} \end{array}; \frac{z^2}{4}\right). \tag{13}$$

#### **2.4 Pochhammer's symbol**

Here ð Þ*λ <sup>υ</sup>* ð Þ *λ*, *υ*∈ denotes the Pochhammer's symbol (or the shifted factorial, since 1ð Þ*<sup>n</sup>* ¼ *n*!Þ is defined, in general, by [10].

$$\Gamma(\lambda)\_{\nu} \coloneqq \frac{\Gamma(\lambda + \nu)}{\Gamma(\lambda)} = \begin{cases} \mathbf{1}, & (\nu = \mathbf{0}; \lambda \in \mathbb{C} \backslash \{\mathbf{0}\}) \\ \lambda(\lambda + \mathbf{1}) \dots(\lambda + n - \mathbf{1}), & (\nu = n \in \mathbb{N}; \lambda \in \mathbb{C}). \end{cases} \tag{14}$$

Algebraic property of Pochhammer symbol:

$$(\lambda)\_{m+n} = (\lambda)\_m (\lambda + m)\_n = (\lambda)\_n (\lambda + n)\_m. \tag{15}$$

#### **2.5 Gauss-Legendre multiplication formula**

For every positive integer *m* ([10], p. 22, eq. (26)), we have

$$(\lambda)\_{mn} = m^{mn} \prod\_{j=1}^{m} \left( \frac{\lambda + j - 1}{m} \right)\_n \qquad ; m \in \mathbb{N}, n \in \mathbb{N}\_0. \tag{16}$$

From the above result (16) with *λ* ¼ *mz*, it can be proved that

$$\Gamma(mz) = (2\pi)^{\frac{(1-m)}{2}} m^{mz-\frac{1}{2}} \prod\_{j=1}^{m} \Gamma\left(z + \frac{j-1}{m}\right),\tag{17}$$

where *<sup>z</sup>* 6¼ 0, � <sup>1</sup> *<sup>m</sup>* , � <sup>2</sup> *<sup>m</sup>* ,*:* … ; *m* ∈ .

The eq. (17) is known as Gauss-Legendre multiplication formula for Gamma function.

#### **2.6 Legendre's duplication formula**

When we put *m* ¼ 2 in the eq. (17), we get

$$
\sqrt{\pi}\Gamma(2z) = 2^{2x-1}\Gamma(z)\Gamma\left(z+\frac{1}{2}\right), \qquad 2z \in \mathbb{C}\backslash\mathbb{Z}\_0^-, \tag{18}
$$

which is known as Legendre's duplication formula.

*Analytical Expressions of Infinite Fourier Sine and Cosine Transform-Based Ramanujan… DOI: http://dx.doi.org/10.5772/intechopen.108401*

#### **2.7 Infinite series decomposition identity**

An infinite series decomposition identity ([11], p. 193, eq. (8)) is given by

$$\sum\_{\ell=0}^{\infty} \Omega(\ell) = \sum\_{j=0}^{N-1} \left\{ \sum\_{\ell=0}^{\infty} \Omega(N\ell + j) \right\},\tag{19}$$

where *N* is an arbitrary positive integer. Put *N* ¼ 4 in the above eq. (19), we get

$$\sum\_{\ell=0}^{\infty} \mathfrak{Q}(\ell) = \sum\_{j=0}^{3} \left\{ \sum\_{\ell=0}^{\infty} \mathfrak{Q}(4\ell + j) \right\},\tag{20}$$

$$\mathcal{I} = \sum\_{\ell=0}^{\infty} \Omega(4\ell) + \sum\_{\ell=0}^{\infty} \Omega(4\ell+1) + \sum\_{\ell=0}^{\infty} \Omega(4\ell+2) + \sum\_{\ell=0}^{\infty} \Omega(4\ell+3),\tag{21}$$

provided that all involved infinite series are absolutely convergent.

#### **2.8 Fox-Wright psi function of one variable**

A natural generalization of the hypergeometric function *pFq*ð Þ*z* is the Fox-Wright psi function of one variable with *p* pairs of numerator parameters ð Þ *α*1, *A*<sup>1</sup> , … , *αp*, *Ap* � � and *q* pairs of denominator parameters *β*<sup>1</sup> ð Þ , *B*<sup>1</sup> , … , *βq*, *Bq* � �, defined by [12, 13].

$$\begin{aligned} \ \_p\Psi\_q \left[ \begin{array}{ll} (a\_1, A\_1), \dots, (a\_p, A\_p); \\ (\beta\_1, B\_1), \dots, (\beta\_q, B\_q); \end{array} \right] &= \sum\_{k=0}^\infty \frac{\Gamma(a\_1 + kA\_1) \dots \Gamma(a\_p + kA\_p)}{\Gamma(\beta\_1 + kB\_1) \dots \Gamma(\beta\_q + kB\_q)} \frac{z^k}{k!}, & \text{(22)}\\ &= \frac{1}{2\pi\rho} \int\_L \frac{\Gamma(\zeta) \prod\_{i=1}^p \Gamma(a\_i - A\_i \zeta)}{\prod\_{j=1}^q \Gamma(\beta\_j - B\_j \zeta)} (-z)^{-\zeta} d\zeta, \end{aligned} \tag{23}$$

where *<sup>ρ</sup>* <sup>¼</sup> ffiffiffiffiffiffi �<sup>1</sup> <sup>p</sup> ,*z*<sup>∈</sup> ; parameters *<sup>α</sup>i*, *<sup>β</sup><sup>j</sup>* <sup>∈</sup> ; coefficients *Ai*, *Bj* <sup>∈</sup> ¼ �ð Þ <sup>∞</sup>, <sup>þ</sup><sup>∞</sup> in case of series (22) (or *Ai*, *Bj* ∈ <sup>þ</sup> ¼ ð Þ 0, þ∞ in case of contour integral (23)), *Ai* 6¼ 0 ð Þ *i* ¼ 1, 2, … , *p* ,*Bj* 6¼ 0 ð Þ *j* ¼ 1, 2, … , *q* . In eq. (22), the parameters *αi*, *β<sup>j</sup>* and coefficients *Ai*, *Bj* are adjusted in such a way that the product of Gamma functions in numerator and denominator should be well defined.

Suppose:

$$
\Delta^\* = \left(\sum\_{j=1}^q B\_j - \sum\_{i=1}^p A\_i\right),
\tag{24}
$$

$$\delta^\* = \left(\prod\_{i=1}^p |A\_i|^{-A\_i}\right) \left(\prod\_{j=1}^q |B\_j|^{B\_j}\right),\tag{25}$$

$$
\mu^\* = \sum\_{j=1}^q \beta\_j - \sum\_{i=1}^p a\_i + \left(\frac{p-q}{2}\right),
\tag{26}
$$

and

$$
\sigma^\* = \begin{pmatrix} \mathbf{1} + A\_1 + \dots + A\_p \end{pmatrix} - \begin{pmatrix} B\_1 + \dots + B\_q \end{pmatrix} = \mathbf{1} - \Delta^\*.\tag{27}
$$

Then we have the following convergence conditions of (22) or (23).

Case (1): When contour ð Þ *L* is a left loop beginning and ending at �∞, then *<sup>p</sup>*Ψ*q*½ �� given by (22) or (23) holds the following convergence conditions.


Case (2): When contour ð Þ *L* is a right loop beginning and ending at þ∞, then *<sup>p</sup>*Ψ*q*½ �� given by (22) or (23) holds the following convergence conditions.


Case (3): When contour ð Þ *L* is starting from *γ* � *i*∞ and ending at *γ* þ *i*∞ where *γ* ∈ ¼ �ð Þ ∞, þ∞ , then *<sup>p</sup>*Ψ*q*½ �� is also convergent under the following conditions.

 $\text{i. When } \sigma^\* > 0, \left| \arg(-z) \right| < \frac{\pi}{2}$  $\sigma^\*, \left| 0 < |z| < \infty$  $\varphi \neq 0.$ 

$$\text{iii. When } \sigma^\* = 0, \arg(-z) = 0, \text{ } 0 < |z| < \infty \text{ } z \neq 0 \text{ such that } -\gamma \Delta^\* + \Re(\mu^\*) > \frac{1}{2} + \gamma.$$

$$\text{iii. When } \gamma = 0, \sigma^\* = 0, \text{arg}(-z) = 0, \text{ } 0 < |z| < \infty \text{, such that } \Re(\mu^\*) > \frac{1}{2}.$$

In the available literature [7, 14–18] on Ramanujan's Mathematics, the analytical expression of Ramanujan's integrals **R***<sup>S</sup>*,*<sup>C</sup>*ð Þ *m*, *n* are not given. Therefore, the main object of this chapter is to evaluate the representation of **R***<sup>S</sup>*,*<sup>C</sup>*ð Þ *m*, *n* in an ordinary hypergeometric function <sup>2</sup>*F*3ð Þ� . Also, our contribution to Ramanujan's Mathematics is determined by the result in [19, 20]. Here in this chapter, we generalize Ramanujan's integrals **R***<sup>S</sup>*,*<sup>C</sup>*ð Þ *m*, *n* in the following forms:

**I** ∗ *<sup>S</sup>*,*<sup>C</sup>*ð Þ¼ *<sup>υ</sup>*, *<sup>b</sup>*, *<sup>c</sup>*, *<sup>λ</sup>*, *<sup>y</sup>* <sup>X</sup><sup>∞</sup> *k*¼0 Θð Þ*k k*! ð<sup>∞</sup> 0 *x<sup>υ</sup>*�<sup>1</sup> *e* �ð Þ *<sup>λ</sup>b*þ*ck* ffiffi *<sup>x</sup>* <sup>p</sup> sin cos ð Þ *xy dx*, **J***<sup>S</sup>*,*<sup>C</sup>*ð Þ¼ *υ*, *b*, *c*, *λ*, *y* ð<sup>∞</sup> 0 *x<sup>υ</sup>*�<sup>1</sup> *e* �*b<sup>λ</sup>* ffiffi *x* p *<sup>r</sup>*Ψ*<sup>s</sup>* ð Þ *α*1, *A*<sup>1</sup> , … , *α<sup>r</sup>* ð Þ , *Ar* ; *β*<sup>1</sup> ð Þ , *B*<sup>1</sup> , … , *β<sup>s</sup>* ð Þ , *Bs* ; *e* �*<sup>c</sup>* ffiffi *<sup>x</sup>* <sup>p</sup> � � sin cos ð Þ *xy dx*, **K***<sup>S</sup>*,*<sup>C</sup>*ð Þ¼ *υ*, *b*, *c*, *λ*, *y* ð<sup>∞</sup> 0 *x<sup>υ</sup>*�<sup>1</sup> *e* �*b<sup>λ</sup>* ffiffi *x* p *rFs α*1, … , *αr*; *β*1, … , *βs*; *e* �*<sup>c</sup>* ffiffi *<sup>x</sup>* <sup>p</sup> � � sin cos ð Þ *xy dx*, **I***<sup>S</sup>*,*<sup>C</sup>*ð Þ¼ *υ*, *b*, *λ*, *y* ð<sup>∞</sup> 0 *<sup>x</sup><sup>υ</sup>*�<sup>1</sup> �<sup>1</sup> <sup>þ</sup> exp *<sup>b</sup>* ffiffiffi *<sup>x</sup>* � � � � <sup>p</sup> �*<sup>λ</sup>* sin cos ð Þ *xy dx*,

*Analytical Expressions of Infinite Fourier Sine and Cosine Transform-Based Ramanujan… DOI: http://dx.doi.org/10.5772/intechopen.108401*

where f g <sup>Θ</sup>ð Þ*<sup>k</sup>* <sup>∞</sup> *<sup>k</sup>*¼<sup>0</sup> is a fixed sequence of the arbitrary real or complex numbers. Moreover, we also show how the main general theorem given below applies to obtaining new interesting results by suitable adjustments in parameters and variables.

#### **3. Ramanujan's integrals**

The analytical solution of the following integral of Ramanujan ([7], p. 85, eq. (49) last line):

$$\mathbf{R}\_{\mathcal{C}}(m,n) = \int\_0^{\infty} \mathfrak{x}^m \frac{\cos\left(\pi m \infty\right)}{\left\{-\mathbf{1} + \exp\left(2\pi\sqrt{\mathcal{X}}\right)\right\}} d\boldsymbol{\mathfrak{x}},\tag{28}$$

is not given for all positive rational values of *n* and non-negative integral values of *m*.

For particular values of *m* and *n* in Ramanujan's integral **R***C*ð Þ *m*, *n* , the following three integrals are given by ([7], p. 86, eq. (50)):

$$\mathbf{R}\_{\mathbb{C}}(\mathbf{1}, \mathbf{1}/2) = \int\_{0}^{\infty} \frac{\mathbf{x} \cos \left(\frac{\pi x}{2}\right)}{\left\{-\mathbf{1} + \exp \left(2\pi \sqrt{\mathbf{x}}\right)\right\}} d\mathbf{x} = \frac{\mathbf{1}\mathbf{3} - 4\pi}{8\pi^{2}},\tag{29}$$

$$\mathbf{R}\_{\mathbb{C}}(1,2) = \int\_{0}^{\infty} \frac{\mathbf{x} \cos(2\pi\mathbf{x})}{\{-\mathbf{1} + \exp(2\pi\sqrt{\mathbf{x}})\}} d\mathbf{x} = \frac{1}{64} \left(\frac{\mathbf{1}}{2} - \frac{\mathbf{3}}{\pi} + \frac{\mathbf{5}}{\pi^2}\right),\tag{30}$$

$$\mathbf{R}\_{C}(2,2) = \int\_{0}^{\infty} \frac{\mathbf{x}^{2} \cos(2\pi\mathbf{x})}{\{-1 + \exp(2\pi\sqrt{\mathbf{x}})\}} d\mathbf{x} = \frac{1}{256} \left(1 - \frac{5}{\pi} + \frac{5}{\pi^{2}}\right). \tag{31}$$

The following theorem is proved by Ramanujan ([7], p. 76, 77, eq. (10 and 10<sup>0</sup> )). Theorem 1.3.1. *Let n be real and positive. Then if*

$$R\_C(\mathbf{0}, n) = \Phi(n) = \int\_0^\infty \frac{\cos\left(\pi n \mathbf{x}\right)}{\left\{-\mathbf{1} + \exp\left(2\pi\sqrt{\mathbf{x}}\right)\right\}} d\mathbf{x},\tag{32}$$

and

$$\Upsilon(n) - \frac{1}{2\pi n} = \int\_0^\infty \frac{\sin\left(\pi n \pi\right)}{\left\{-1 + \exp\left(2\pi\sqrt{\pi}\right)\right\}} d\pi = R\_S(0, n), \tag{33}$$

then

$$\mathcal{R}\_C(0, n) = \Phi(n) = \frac{1}{n} \sqrt{\left(\frac{2}{n}\right)} \Upsilon\left(\frac{1}{n}\right) - \Upsilon(n), \tag{34}$$

and

$$\Upsilon(n) = \frac{1}{n} \sqrt{\left(\frac{2}{n}\right)} \Phi\left(\frac{1}{n}\right) + \Phi(n),\tag{35}$$

For particular values of *n*, some values of Ramanujan's integral ([7], p. 85 (eq. 48)) are given below

$$\mathbf{R}\_{C}(\mathbf{0},\mathbf{1}) = \Phi(\mathbf{1}) = \int\_{0}^{\infty} \frac{\cos\left(\pi\mathbf{x}\right)}{\left\{-\mathbf{1} + \exp\left(2\pi\sqrt{\mathbf{x}}\right)\right\}} d\mathbf{x} = \frac{2 - \sqrt{2}}{8},\tag{36}$$

$$\mathbf{R}\_{\rm C}(\mathbf{0},2) = \Phi(2) = \int\_{0}^{\infty} \frac{\cos\left(2\pi x\right)}{\{-1 + \exp\left(2\pi\sqrt{x}\right)\}} dx = \frac{1}{16},\tag{37}$$

$$\mathbf{R}\_{C}(\mathbf{0},\mathbf{4}) = \Phi(4) = \int\_{0}^{\infty} \frac{\cos\left(4\pi\mathbf{x}\right)}{\left\{-\mathbf{1} + \exp\left(2\pi\sqrt{\mathbf{x}}\right)\right\}} d\mathbf{x} = \frac{\mathbf{3} - \sqrt{\mathbf{2}}}{\mathbf{32}},\tag{38}$$

$$\mathbf{R}\_{\rm C}(\mathbf{0}, \boldsymbol{\Theta}) = \boldsymbol{\Phi}(\boldsymbol{\Phi}) = \int\_{0}^{\infty} \frac{\cos \left( 6 \pi \mathbf{x} \right)}{\left\{ -\mathbf{1} + \exp \left( 2 \pi \sqrt{\boldsymbol{\alpha}} \right) \right\}} d\boldsymbol{x} = \frac{\mathbf{1} \mathbf{3} - 4 \sqrt{\mathbf{3}}}{144},\tag{39}$$

$$\mathbf{R}\_{C}(\mathbf{0}, \mathbf{1}/2) = \Phi\left(\frac{1}{2}\right) = \int\_{0}^{\infty} \frac{\cos\left(\frac{\pi x}{2}\right)}{\left\{-\mathbf{1} + \exp\left(2\pi\sqrt{\mathbf{x}}\right)\right\}} d\mathbf{x} = \frac{\mathbf{1}}{4\pi},\tag{40}$$

$$\mathbf{R}\_{C}(\mathbf{0},2/\mathbf{5}) = \Phi\left(\frac{2}{\mathbf{5}}\right) = \int\_{0}^{\infty} \frac{\cos\left(\frac{2\pi x}{\mathbf{5}}\right)}{\left\{-\mathbf{1} + \exp\left(2\pi\sqrt{x}\right)\right\}} d\mathbf{x} = \frac{\mathbf{8} - \mathbf{3}\sqrt{\mathbf{5}}}{\mathbf{16}}.\tag{41}$$

By calculation of (7) and (9), we get the following infinite fourier sine transform of �<sup>1</sup> <sup>þ</sup> exp 2*<sup>π</sup>* ffiffiffi *<sup>x</sup>* <sup>p</sup> ð Þ

$$\int\_0^\infty \frac{\sin(\pi n \mathbf{x})}{\{-1 + \exp(2\pi \sqrt{\mathbf{x}})\}} d\mathbf{x} = \frac{1}{n} \sqrt{\left(\frac{2}{n}\right)} \Phi\left(\frac{1}{n}\right) + \Phi(n) - \frac{1}{2\pi n}.\tag{42}$$

For special values of *<sup>n</sup>* <sup>¼</sup> 1,2, <sup>1</sup> <sup>2</sup> in the above eq. (16) and using <sup>Φ</sup>ð Þ<sup>1</sup> ,Φð Þ<sup>2</sup> and <sup>Φ</sup> <sup>1</sup> 2 � �, we get after simplification the following three results:

$$\mathbf{R}\_S(\mathbf{0}, \mathbf{1}) = \int\_0^\infty \frac{\sin \left( \pi \alpha \right)}{\left\{ -\mathbf{1} + \exp \left( 2 \pi \sqrt{\alpha} \right) \right\}} d\mathbf{x} = \frac{\pi \sqrt{2} - 4}{8\pi},\tag{43}$$

$$\mathbf{R}\_{\rm S}(\mathbf{0}, \mathbf{2}) = \int\_{0}^{\infty} \frac{\sin \left(2\pi\mathbf{x}\right)}{\left\{-\mathbf{1} + \exp \left(2\pi\sqrt{\mathbf{x}}\right)\right\}} d\mathbf{x} = \frac{\pi - 2}{16\pi},\tag{44}$$

$$\mathbf{R}\_S(\mathbf{0}, \mathbf{1}/2) = \int\_0^\infty \frac{\sin\left(\frac{\pi x}{2}\right)}{\left\{-\mathbf{1} + \exp\left(2\pi\sqrt{x}\right)\right\}} d\mathbf{x} = \frac{\pi - 3}{4\pi}.\tag{45}$$

#### **4. Main general theorems on infinite Fourier sine and cosine transform**

In this section, we give some generalizations of the infinite Fourier sine and cosine transform-based Ramanujan integrals **R***<sup>S</sup>*,*<sup>C</sup>*ð Þ� in the form of infinite series of hypergeometric functions <sup>2</sup>*F*3ð Þ� . Moreover, we denote these generalizations by **<sup>I</sup>** <sup>∗</sup> *<sup>S</sup>*,*<sup>C</sup>*ð Þ� , **J***<sup>S</sup>*,*<sup>C</sup>*ð Þ� , **K***<sup>S</sup>*,*<sup>C</sup>*ð Þ� and **I***<sup>S</sup>*,*<sup>C</sup>*ð Þ� [21, 22].

Theorem 1.4.1. *Suppose* f g <sup>Θ</sup>ð Þ*<sup>k</sup>* <sup>∞</sup> *<sup>k</sup>*¼<sup>0</sup> *is a fixed sequence of arbitrary real or complex numbers and satisfy the conditions* ℜð Þ*υ* > � 1,*c*>0,*y*>0; *λ*>0,*b*>0ð Þ *or λ*< 0, *b*< 0 *.*

then we have

$$\mathbf{I}\_{\mathbf{S},\mathbf{C}}^{\*}(\nu,b,\varepsilon,\lambda,\chi) = \sum\_{k=0}^{\infty} \frac{\Theta(k)}{k!} \int\_{0}^{\infty} \varkappa^{\nu-1} e^{-(\lambda b+ck)\sqrt{\chi}} \frac{\sin}{\cos}(\chi\nu) d\mathbf{x},$$

(46)

*Analytical Expressions of Infinite Fourier Sine and Cosine Transform-Based Ramanujan… DOI: http://dx.doi.org/10.5772/intechopen.108401*

$$\mathbf{y} = \mathbf{y}^{-\nu} \sum\_{k=0}^{\infty} \frac{\Theta(k)}{k!} \sum\_{\ell=0}^{\infty} \frac{(-1)^{\ell} (\lambda b + ck)^{\ell} \Gamma \left(\nu + \frac{\ell}{2}\right)}{\mathbf{y}^{\frac{\ell}{2}} \ell!} \sin\left(\frac{\nu \pi}{2} + \frac{\ell \pi}{4}\right), \tag{47}$$

Now replacing ℓ by 4ℓ þ *j*, after simplification we get

**I** ∗ *<sup>S</sup>*,*C*ð Þ¼ *<sup>υ</sup>*, *<sup>b</sup>*, *<sup>c</sup>*, *<sup>λ</sup>*, *<sup>y</sup> <sup>y</sup>*�*<sup>υ</sup>*X<sup>∞</sup> *k*¼0 Θð Þ*k k*! X 3 *j*¼0 ð Þ �<sup>1</sup> *<sup>j</sup>* ð Þ *<sup>λ</sup><sup>b</sup>* <sup>þ</sup> *ck <sup>j</sup>* <sup>Γ</sup> *<sup>υ</sup>* <sup>þ</sup> *<sup>j</sup>* 2 � � *y j* <sup>2</sup> *j*! sin cos *υπ* <sup>2</sup> <sup>þ</sup> *<sup>j</sup><sup>π</sup>* 4 � � �2*F*<sup>3</sup> <sup>Δ</sup> 2; <sup>2</sup>*<sup>υ</sup>* <sup>þ</sup> *<sup>j</sup>* 2 � �; <sup>Δ</sup><sup>∗</sup> ð Þ 4; 1 <sup>þ</sup> *<sup>j</sup>* ; �1 64*y*<sup>2</sup> ð Þ *<sup>λ</sup><sup>b</sup> <sup>λ</sup>b*þ*<sup>c</sup> c* � � *k λb c* � � *k* ( )<sup>4</sup> 0 B@ 1 CA, (48) <sup>¼</sup> *<sup>y</sup>*�*<sup>υ</sup>*X<sup>∞</sup> *k*¼0 Θð Þ*k k*! X 3 *j*¼0 ð Þ �<sup>1</sup> *<sup>j</sup>* <sup>Γ</sup> *<sup>υ</sup>* <sup>þ</sup> *<sup>j</sup>* 2 � � *j*! sin cos *υπ* <sup>2</sup> <sup>þ</sup> *<sup>j</sup><sup>π</sup>* 4 � � *λb* ffiffi *y* p !*<sup>j</sup>* � *λb*þ*c c* � � *k λb c* � � *k* ( )*<sup>j</sup>* <sup>2</sup>*F*<sup>3</sup> <sup>Δ</sup> 2; <sup>2</sup>*<sup>υ</sup>* <sup>þ</sup> *<sup>j</sup>* 2 � �; <sup>Δ</sup><sup>∗</sup> ð Þ 4; 1 <sup>þ</sup> *<sup>j</sup>* ; �1 64*y*<sup>2</sup> ð Þ *<sup>λ</sup><sup>b</sup> <sup>λ</sup>b*þ*<sup>c</sup> c* � � *k λb c* � � *k* ( )<sup>4</sup> 0 B@ 1 CA, (49) ¼ <sup>Γ</sup>ð Þ*<sup>υ</sup>* sin cos *υπ* 2 � � *yυ* X∞ *k*¼0 Θð Þ*k <sup>k</sup>*! <sup>2</sup>*F*<sup>3</sup> *υ* 2 , *<sup>υ</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup> ; 1 4 , 1 2 , 3 4 ; �1 64*y*<sup>2</sup> ð Þ *<sup>λ</sup><sup>b</sup> <sup>λ</sup>b*þ*<sup>c</sup> c* � � *k λb c* � � *k* ( )<sup>4</sup> 0 BB@ 1 CCA � ð Þ *λb* Γ *υ* þ 1 2 � � sin cos *υπ* <sup>2</sup> <sup>þ</sup> *<sup>π</sup>* 4 � � *yυ*þ<sup>1</sup> 2 X∞ *k*¼0 Θð Þ*k k*! *λb*þ*c c* � � *k λb c* � � *k* ( ) �2*F*<sup>3</sup> 2*υ* þ 1 <sup>4</sup> , <sup>2</sup>*<sup>υ</sup>* <sup>þ</sup> <sup>3</sup> <sup>4</sup> ; 1 2 , 3 4 , 5 <sup>4</sup> ; �1 64*y*<sup>2</sup> ð Þ *<sup>λ</sup><sup>b</sup> <sup>λ</sup>b*þ*<sup>c</sup> c* � � *k λb c* � � *k* ( )<sup>4</sup> 0 BB@ 1 CCA þ ð Þ *<sup>λ</sup><sup>b</sup>* <sup>2</sup> Γð Þ *υ* þ 1 sin cos *υπ* 2 � � 2*yυ*þ<sup>1</sup> X∞ *k*¼0 Θð Þ*k k*! *λb*þ*c c* � � *k λb c* � � *k* ( )<sup>2</sup> �2*F*<sup>3</sup> *υ* þ 1 <sup>2</sup> , *<sup>υ</sup>* <sup>þ</sup> <sup>2</sup> <sup>2</sup> ; 3 4 , 5 4 , 3 2 ; �1 64*y*<sup>2</sup> ð Þ *<sup>λ</sup><sup>b</sup> <sup>λ</sup>b*þ*<sup>c</sup> c* � � *k λb c* � � *k* ( )<sup>4</sup> 0 BB@ 1 CCA � ð Þ *<sup>λ</sup><sup>b</sup>* <sup>3</sup> Γ *υ* þ 3 2 � � sin cos *υπ* <sup>2</sup> <sup>þ</sup> *<sup>π</sup>* 4 � � 6*yυ*þ<sup>3</sup> 2 X∞ *k*¼0 Θð Þ*k k*! *λb*þ*c c* � � *k λb c* � � *k* ( )<sup>3</sup> �2*F*<sup>3</sup> 2*υ* þ 3 <sup>4</sup> , <sup>2</sup>*<sup>υ</sup>* <sup>þ</sup> <sup>5</sup> <sup>4</sup> ; 5 4 , 3 2 , 7 <sup>4</sup> ; �1 64*y*<sup>2</sup> ð Þ *<sup>λ</sup><sup>b</sup> <sup>λ</sup>b*þ*<sup>c</sup> c* � � *k λb c* � � *k* ( )<sup>4</sup> 0 BB@ 1 CCA*:* (50)

Our result (30)or (31) or (32) is convergent in view of the convergence condition of *pFq*ð Þ� series, when *p*≤*q*, and ∀∣*z*∣< ∞.

Proof: The result (29) is obtained by the application of the integral (49) [with substitutions *<sup>μ</sup>* <sup>¼</sup> *<sup>υ</sup>* � 1, *<sup>a</sup>* <sup>¼</sup> *<sup>λ</sup><sup>b</sup>* <sup>þ</sup> *ck*, *<sup>ξ</sup>* <sup>¼</sup> <sup>1</sup> 2 ] in the R.H.S. of eq. (28). Also, we calculate the results (30) to (32) by using the infinite series decomposition identity (20) and (21) and algebraic properties of Pochhammer's symbols.

#### **4.1 Analytical expressions of infinite Fourier sine and cosine transforms**

Theorem 1.4.2. *Analytical expressions of the infinite Fourier sine and cosine transforms of x<sup>υ</sup>*�1*e*�*b<sup>λ</sup>* ffiffi *x* p *<sup>r</sup>*Ψ*s*½ �� *holds true for* ℜð Þ*υ* > � 1;*c*>0,*y*>0; *λ*>0,*b*>0 ð Þ *or λ*<0, *b*<0 *then we have*

$$J\_{S,C}(\nu, b, c, \lambda, \mathbf{y}) = \int\_0^\infty x^{\nu - 1} e^{-b\lambda\sqrt{x}} \, \_r\Psi\_i \left[ \begin{array}{cccc} (a\_1, \ A\_1), \ \dots, (a\_r, \ A\_r); & & & \mathbf{e}^{-\varepsilon\sqrt{x}} \\ (\beta\_1, \ B\_1), \dots, (\beta\_i, \ B\_i); & & & \mathbf{e}^{-\varepsilon\sqrt{x}} \end{array} \right] \, \cos(\mathbf{y}) d\mathbf{x}, \tag{51}$$

$$\begin{split} \mathbf{y} &= \mathbf{y}^{-\nu} \sum\_{k=0}^{\infty} \frac{\Gamma(a\_1 + kA\_1) \dots \Gamma(a\_r + kA\_r)}{\Gamma(\beta\_1 + kB\_1) \dots \Gamma(\beta\_r + kB\_r)k!} \sum\_{j=0}^{3} \frac{(-1)^j (\lambda b + ck)^j \Gamma\left(\nu + \frac{j}{2}\right)}{\mathbf{y}^{j\frac{1}{2}} j!} \times \\ & \quad \times \sum\_{\cos} \left(\frac{\nu \pi}{2} + \frac{j\pi}{4}\right) {}\_2F\_3 \left( \begin{matrix} \Delta \left(2; \frac{2\nu + j}{2}\right); \\\\ \frac{2}{\mathbf{6} \cdot \mathbf{y}^2} \left( \frac{\lambda b + ck}{c} \right)\_k \end{matrix} ; \end{split} \tag{52}$$

Here the parameters *αi*, *β<sup>j</sup>* ∈ and coefficients *Ai*, *Bj* ∈ ¼ �ð Þ ∞, þ∞ ; *Ai* ¼6 0 ð Þ *i* ¼ 1, 2, … , *r* , *Bj* ¼6 0 ð Þ *j* ¼ 1, 2, … , *s* and *<sup>r</sup>*Ψ*s*½ �� is the Fox-Wright psi function of one variable subject to suitable convergence conditions derived from conditions discussed in case (1) or case (2) or case (3) of the function *<sup>p</sup>*Ψ*q*½ �� given by (22) and (23). When *N* is positive integer then Δð Þ *N*; *λ* denotes the array of *N* parameters given by *λ <sup>N</sup>* , *<sup>λ</sup>*þ<sup>1</sup> *<sup>N</sup>* , … , *<sup>λ</sup>*þ*N*�<sup>1</sup> *<sup>N</sup>* . When *N* and *j* are independent variables then the notation Δð Þ *N*; *j* þ 1 denotes the set of *N* parameters given by *<sup>j</sup>*þ<sup>1</sup> *N* , *j*þ2 *<sup>N</sup>* , … , *j*þ*N <sup>N</sup>* . When *j* is dependent variable that is *<sup>j</sup>* <sup>¼</sup> 0,1,2,3, … ,*<sup>N</sup>* � 1, then the asterisk in <sup>Δ</sup><sup>∗</sup> ð Þ *<sup>N</sup>*; *<sup>j</sup>* <sup>þ</sup> <sup>1</sup> represents the fact that the (denominator) parameters *<sup>N</sup> <sup>N</sup>* is always omitted (due to the need of factorial in denominator in the power series form of hypergeometric function) so that the set <sup>Δ</sup><sup>∗</sup> ð Þ *<sup>N</sup>*; *<sup>j</sup>* <sup>þ</sup> <sup>1</sup> obviously contains only ð Þ *N* � 1 parameters ([10], Chap. 3, p. 214).

Proof: Let us consider <sup>Θ</sup>ð Þ¼ *<sup>k</sup>* <sup>Γ</sup>ð Þ *<sup>α</sup>*1þ*kA*<sup>1</sup> … <sup>Γ</sup>ð Þ *<sup>α</sup>r*þ*kAr* <sup>Γ</sup>ð Þ *<sup>β</sup>*1þ*kB*<sup>1</sup> … <sup>Γ</sup>ð Þ *<sup>β</sup>s*þ*kBs* ð Þ *<sup>k</sup>* <sup>¼</sup> 0,1,2,3, … in the eqs. (28) and (30), then after evaluation we get integral expressions (33) involving the Fox-Wright Psi function in the form of infinite series of an ordinary <sup>2</sup>*F*<sup>3</sup> hypergeometric function (34).

Theorem 1.4.3. *Analytical expressions of the infinite Fourier sine and cosine transforms of x<sup>υ</sup>*�<sup>1</sup>*e*�*b<sup>λ</sup>* ffiffi *x* p *rFs*ð Þ� *holds true for* ℜð Þ*υ* > � 1;*c* >0,*y* >0; *λ*>0,*b*> 0 *(or λ*< 0,*b*<0*) then we have*

$$\mathbf{K}\_{\mathbb{S},\mathbb{C}}(\nu, b, c, \lambda, y) = \int\_0^\infty x^{\nu - 1} e^{-b \lambda \sqrt{x}} {}\_r F\_t \left( \begin{array}{ccc} a\_1, \dots, a\_r; \\ \beta\_1, \dots, \beta\_s; \end{array} \right. \\ \left. e^{-c \sqrt{x}} \right) \sin \left( \begin{array}{ccc} \infty \text{y} \end{array} \right) d\mathbf{x}; \tag{53}$$

*Analytical Expressions of Infinite Fourier Sine and Cosine Transform-Based Ramanujan… DOI: http://dx.doi.org/10.5772/intechopen.108401*

$$\begin{split} \mathbf{y} &= \mathbf{y}^{-\nu} \sum\_{k=0}^{\infty} \frac{(a\_1)\_k \dots (a\_r)\_k}{(\beta\_1)\_k \dots (\beta\_i)\_k} \sum\_{j=0}^3 \frac{(-1)^j (\lambda b + ck)^j \,\Gamma\left(\nu + \frac{j}{2}\right)}{\mathbf{y}^j j!} \frac{\sin}{\cos} \left(\frac{\nu \pi}{2} + \frac{j\pi}{4}\right) \\ & \qquad \times {}\_2F\_3 \left(\Delta \left(2; \frac{2\nu + j}{2}\right); \qquad \frac{-1}{64\nu^2} \left\{\frac{(\lambda b)\left(\frac{\lambda b + c}{c}\right)\_k}{\left(\frac{\lambda b}{c}\right)\_k}\right\}^4\right), \end{split} \tag{54}$$

where the parameters *αi*, *β<sup>j</sup>* ∈ ð Þ *i* ¼ 1, 2, … , *r* ,ð Þ *j* ¼ 1, 2, … , *s* and *r*≤*s* þ 1. Proof: If *A*<sup>1</sup> ¼ … ¼ *Ar* ¼ *B*<sup>1</sup> ¼ … ¼ *Bs* ¼ 1 in (33) and (34), then we get the above integral expressions involving generalized hypergeometric function in the form of infinite series of an ordinary <sup>2</sup>*F*<sup>3</sup> hypergeometric function (36).

Corollary 1.4.4. *An infinite Fourier sine and cosine transforms of <sup>x</sup>υ*�<sup>1</sup> �<sup>1</sup> <sup>þ</sup> exp *<sup>b</sup>* ffiffiffi *<sup>x</sup>* <sup>p</sup> f g ð Þ �*<sup>λ</sup> holds true for* <sup>ℜ</sup>ð Þ*<sup>υ</sup>* <sup>&</sup>gt; � 1; *<sup>λ</sup>*>0,*b*><sup>0</sup> *and y*><sup>0</sup> *then we have*

$$I\_{\mathcal{S},\mathcal{C}}(\nu, b, \lambda, y) = \int\_0^\infty \frac{x^{\nu - 1}}{\left\{-1 + \exp\left(b\sqrt{\mathcal{X}}\right)\right\}^\lambda} \sin^\lambda(xy) d\mathbf{x},\tag{55}$$

$$\mathbf{y} = \mathbf{y}^{-\nu} \sum\_{k=0}^{\infty} \frac{(\lambda)\_k}{k!} \sum\_{\ell=0}^{\infty} \frac{(-\mathbf{1})^{\ell} (\lambda b + bk)^{\ell} \Gamma \left( \nu + \frac{\ell}{2} \right)}{\mathbf{y}^{\text{f}} \ell!} \frac{\sin}{\cos} \left( \frac{\nu \pi}{2} + \frac{\ell \pi}{4} \right), \tag{56}$$

$$\begin{split} \mathbf{y}^{-\nu} &= \mathbf{y}^{-\nu} \sum\_{k=0}^{\infty} \frac{(\lambda)\_{k}}{k!} \sum\_{j=0}^{3} \frac{(-1)^{j} (\lambda b + bk)^{j} \, \Gamma \left( \nu + \frac{j}{2} \right)}{j^{j} j!} \, \frac{\sin}{\cos} \left( \frac{\nu \pi}{2} + \frac{j \pi}{4} \right) \times \\ & \times {}\_{\lambda^{\*} 2} F\_{3} \left( \Delta \left( 2; \frac{2 \nu + j}{2} \right); \qquad \frac{-1}{64 y^{2}} \left\{ \frac{(\lambda b)(\lambda + 1)\_{k}}{(\lambda)\_{k}} \right\}^{4} \right), \end{split} \tag{57}$$

¼ <sup>Γ</sup>ð Þ*<sup>υ</sup>* sin cos *υπ* 2 � � *yυ* X∞ *k*¼0 ð Þ*λ <sup>k</sup> <sup>k</sup>*! <sup>2</sup>*F*<sup>3</sup> *υ* 2 , *<sup>υ</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup> ; 1 4 , 1 2 , 3 4 ; �1 64*y*<sup>2</sup> ð Þ *λb* ð Þ *λ* þ 1 *<sup>k</sup>* ð Þ*λ <sup>k</sup>* � �<sup>4</sup> 0 BB@ 1 CCA � ð Þ *λb* Γ *υ* þ 1 2 � � sin cos *υπ* <sup>2</sup> <sup>þ</sup> *<sup>π</sup>* 4 � � *y<sup>υ</sup>*þ<sup>1</sup> 2 X∞ *k*¼0 ð Þ *λ* þ 1 *<sup>k</sup> <sup>k</sup>*! <sup>2</sup>*F*<sup>3</sup> 2*υ* þ 1 <sup>4</sup> , <sup>2</sup>*<sup>υ</sup>* <sup>þ</sup> <sup>3</sup> <sup>4</sup> ; 1 2 , 3 4 , 5 4 ; �1 64*y*<sup>2</sup> ð Þ *λb* ð Þ *λ* þ 1 *<sup>k</sup>* ð Þ*λ <sup>k</sup>* � �<sup>4</sup> 0 BB@ 1 CCA þ ð Þ *<sup>λ</sup><sup>b</sup>* <sup>2</sup> Γð Þ *υ* þ 1 sin cos *υπ* 2 � � 2*y<sup>υ</sup>*þ<sup>1</sup> X∞ *k*¼0 ð Þ *λ* þ 1 *<sup>k</sup>* � �<sup>2</sup> ð Þ*<sup>λ</sup> <sup>k</sup> <sup>k</sup>*! <sup>2</sup>*F*<sup>3</sup> *υ* þ 1 <sup>2</sup> , *<sup>υ</sup>* <sup>þ</sup> <sup>2</sup> <sup>2</sup> ; 3 4 , 5 4 , 3 2 ; �1 64*y*<sup>2</sup> ð Þ *λb* ð Þ *λ* þ 1 *<sup>k</sup>* ð Þ*λ <sup>k</sup>* � �<sup>4</sup> 0 BB@ 1 CCA � ð Þ *<sup>λ</sup><sup>b</sup>* <sup>3</sup> Γ *υ* þ 3 2 � � sin cos *υπ* <sup>2</sup> <sup>þ</sup> *<sup>π</sup>* 4 � � 6*y<sup>υ</sup>*þ<sup>3</sup> 2 X∞ *k*¼0 ð Þ *λ* þ 1 *<sup>k</sup>* � �<sup>3</sup> *k*! ð Þ*λ <sup>k</sup>* � �<sup>2</sup> <sup>2</sup>*F*<sup>3</sup> 2*υ* þ 3 <sup>4</sup> , <sup>2</sup>*<sup>υ</sup>* <sup>þ</sup> <sup>5</sup> <sup>4</sup> ; 5 4 , 3 2 , 7 4 ; �1 64*y*<sup>2</sup> ð Þ *λb* ð Þ *λ* þ 1 *<sup>k</sup>* ð Þ*λ <sup>k</sup>* � �<sup>4</sup> 0 BB@ 1 CCA, (58)

Proof: If we consider Θð Þ¼ *k* ð Þ*λ <sup>k</sup>* and *c* ¼ *b* in (28), which yields

$$\mathbf{I}\_{\mathrm{S},\mathcal{C}}(\nu, b, \lambda, \mathcal{y}) = \int\_0^\infty x^{\nu - 1} e^{-(ib)\sqrt{x}} \left\{ \sum\_{k=0}^\infty \frac{(\lambda)\_k}{k!} e^{-(bk)\sqrt{x}} \right\} \frac{\sin}{\cos}(\mathcal{xy}) d\mathbf{x}.\tag{59}$$

Upon the use of binomial expansion (52) in the above eq. (41), then we get after evaluation (37). The results (38) to (40) are derived from (29) to (32) by putting Θð Þ¼ *k* ð Þ*λ <sup>k</sup>* and *c* ¼ *b*.

Corollary 1.4.5. *Analytical expressions of the Ramanujan's integrals R<sup>S</sup>*, *<sup>C</sup>*ð Þ *m*, *n holds true for non-negative integer m and positive rational number n [21, 22].*

*R<sup>S</sup>*,*<sup>C</sup>*ð Þ¼ *m*, *n* ð<sup>∞</sup> 0 *xm* �<sup>1</sup> <sup>þ</sup> exp 2*<sup>π</sup>* ffiffiffi *<sup>x</sup>* <sup>p</sup> f g ð Þ sin cos ð Þ *πnx dx*, <sup>¼</sup> ð Þ *<sup>n</sup><sup>π</sup>* �*m*�<sup>1</sup>X<sup>∞</sup> *k*¼0 X∞ ℓ¼0 1 ℓ! �ð Þ <sup>2</sup>*<sup>π</sup>* <sup>þ</sup> <sup>2</sup>*π<sup>k</sup>* ffiffiffiffiffi *nπ* p � �<sup>ℓ</sup> Γ *m* þ 1 þ ℓ 2 � � sin cos *mπ* 2 þ ℓ*π* 4 � �, (60) <sup>¼</sup> ð Þ *<sup>n</sup><sup>π</sup>* �*m*�<sup>1</sup>X<sup>∞</sup> *k*¼0 X 3 *j*¼0 1 *j*! �ð Þ <sup>2</sup>*<sup>π</sup>* <sup>þ</sup> <sup>2</sup>*π<sup>k</sup>* ffiffiffiffiffi *nπ* p � �*<sup>j</sup>* <sup>Γ</sup> *<sup>m</sup>* <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>j</sup>* 2 � � sin cos *mπ* <sup>2</sup> <sup>þ</sup> *<sup>j</sup><sup>π</sup>* 4 � �� �2*F*<sup>3</sup> Δ 2; 2*m* þ *j* þ 2 2 � �; <sup>Δ</sup><sup>∗</sup> ð Þ 4; 1 <sup>þ</sup> *<sup>j</sup>* ; �*π*<sup>2</sup> 4*n*<sup>2</sup> ð Þ2 *<sup>k</sup>* ð Þ1 *<sup>k</sup>* � �<sup>4</sup> 0 BB@ 1 CCA, (61) ¼ *m*! sin cos *mπ* 2 � � ð Þ *<sup>n</sup><sup>π</sup> <sup>m</sup>*þ<sup>1</sup> X∞ *k*¼0 <sup>2</sup>*F*<sup>3</sup> *m* þ 1 <sup>2</sup> , *<sup>m</sup>* <sup>þ</sup> <sup>2</sup> <sup>2</sup> ; 1 4 , 1 2 , 3 <sup>4</sup> ; � *<sup>π</sup>*<sup>2</sup> 4*n*<sup>2</sup> ð Þ2 *<sup>k</sup>* ð Þ1 *<sup>k</sup>* � �<sup>4</sup> 0 BBB@ 1 CCCA � 3 2 � � *m* sin cos *mπ* <sup>2</sup> <sup>þ</sup> *<sup>π</sup>* 4 � � ð Þ *<sup>π</sup> <sup>m</sup>*ð Þ *<sup>n</sup> <sup>m</sup>*þ<sup>3</sup> 2 X∞ *k*¼0 ð Þ2 *<sup>k</sup>* ð Þ1 *<sup>k</sup>* � �<sup>2</sup>*F*<sup>3</sup> 2*m* þ 3 <sup>4</sup> , <sup>2</sup>*<sup>m</sup>* <sup>þ</sup> <sup>5</sup> <sup>4</sup> ; 1 2 , 3 4 , 5 <sup>4</sup> ; �*π*<sup>2</sup> 4*n*<sup>2</sup> ð Þ2 *<sup>k</sup>* ð Þ1 *<sup>k</sup>* � �<sup>4</sup> 0 BBB@ 1 CCCA � ð Þ2 ð Þ *m* þ 1 ! sin cos *mπ* 2 � � ð Þ *<sup>π</sup> <sup>m</sup>*ð Þ *<sup>n</sup> <sup>m</sup>*þ<sup>2</sup> X∞ *k*¼0 ð Þ2 *<sup>k</sup>* ð Þ1 *<sup>k</sup>* � �<sup>2</sup> <sup>2</sup>*F*<sup>3</sup> *m* þ 2 <sup>2</sup> , *<sup>m</sup>* <sup>þ</sup> <sup>3</sup> <sup>2</sup> ; 3 4 , 5 4 , 3 <sup>2</sup> ; �*π*<sup>2</sup> 4*n*<sup>2</sup> ð Þ2 *<sup>k</sup>* ð Þ1 *<sup>k</sup>* � �<sup>4</sup> 0 BBB@ 1 CCCA þ 5 2 � � *m* sin cos *mπ* <sup>2</sup> <sup>þ</sup> *<sup>π</sup>* 4 � � ð Þ *<sup>π</sup> <sup>m</sup>*�<sup>1</sup> ð Þ *<sup>n</sup> <sup>m</sup>*þ<sup>5</sup> 2 X∞ *k*¼0 ð Þ2 *<sup>k</sup>* ð Þ1 *<sup>k</sup>* � �<sup>3</sup> <sup>2</sup>*F*<sup>3</sup> 2*m* þ 5 <sup>4</sup> , <sup>2</sup>*<sup>m</sup>* <sup>þ</sup> <sup>7</sup> <sup>4</sup> ; 5 4 , 3 2 , 7 <sup>4</sup> ; �*π*<sup>2</sup> 4*n*<sup>2</sup> ð Þ2 *<sup>k</sup>* ð Þ1 *<sup>k</sup>* � �<sup>4</sup> 0 BBB@ 1 CCCA, (62)

*Analytical Expressions of Infinite Fourier Sine and Cosine Transform-Based Ramanujan… DOI: http://dx.doi.org/10.5772/intechopen.108401*

Proof: The results (42) to (44) are obtained from (37), (38), (39) and (40) by putting *υ* ¼ *m* þ 1,*b* ¼ 2*π*, *λ* ¼ 1 and *y* ¼ *nπ*.

#### **5. Closed form infinite summation formulas**

We have derived some closed forms of infinite summation formulas involving hypergeometric functions <sup>0</sup>*F*1, <sup>1</sup>*F*2, and <sup>2</sup>*F*<sup>3</sup> [21, 22].

X∞ *k*¼0 <sup>1</sup>*F*<sup>2</sup> 1 ; 1 4 , 3 4 ; �*π*<sup>2</sup> 4 ð Þ2 *<sup>k</sup>* ð Þ1 *<sup>k</sup>* � �<sup>4</sup> 0 B@ 1 CA 2 6 4 3 7 5 *π* ffiffi 2 <sup>p</sup> <sup>X</sup><sup>∞</sup> *k*¼0 ð Þ2 *<sup>k</sup>* ð Þ1 *<sup>k</sup>* � �<sup>0</sup>*F*<sup>1</sup> ; 1 2 ; �*π*<sup>2</sup> 4 ð Þ2 *<sup>k</sup>* ð Þ1 *<sup>k</sup>* � �<sup>4</sup> 0 B@ 1 CA 2 6 4 3 7 5 þ *π*2 ffiffi 2 <sup>p</sup> <sup>X</sup><sup>∞</sup> *k*¼0 ð Þ2 *<sup>k</sup>* ð Þ1 *<sup>k</sup>* � �<sup>3</sup> <sup>0</sup>*F*<sup>1</sup> ; 3 2 ; �*π*<sup>2</sup> 4 ð Þ2 *<sup>k</sup>* ð Þ1 *<sup>k</sup>* � �<sup>4</sup> 0 B@ 1 CA 2 6 4 3 7 <sup>5</sup> <sup>¼</sup> *<sup>π</sup>* ffiffi 2 <sup>p</sup> � <sup>4</sup> <sup>8</sup> , (63)

$$\begin{aligned} &\sum\_{k=0}^{\infty} \left[ \,\_{1}F\_{2} \left( \begin{array}{c} 1 \\ 1 \\ 4 \end{array} , \frac{3}{4}; \end{aligned} \qquad \frac{-\pi^{2} \left\{ \begin{array}{c} (2)\_{k} \\ (1)\_{k} \end{array} \right\}^{4} \right) ] - \frac{\pi}{2} \sum\_{k=0}^{\infty} \left[ \left\{ \begin{array}{c} (2)\_{k} \\ (1)\_{k} \end{array} \right\}\_{0} \,\_{1}F\_{1} \left( \begin{array}{c} -; \\ \frac{1}{2} \end{array} , \begin{array}{c} -\frac{\pi^{2}}{16} \left\{ \begin{array}{c} (2)\_{k} \\ (1)\_{k} \end{array} \right\}^{4} \right) \right] \\ + \frac{\pi^{2}}{4} \sum\_{k=0}^{\infty} \left[ \left\{ \begin{array}{c} (2)\_{k} \\ (1)\_{k} \end{array} \right\}^{3} \,\_{0}F\_{1} \left( \begin{array}{c} - \\ \frac{1}{2} \end{array} , \begin{array}{c} -\frac{\pi^{2}}{16} \left\{ \begin{array}{c} (2)\_{k} \\ (1)\_{k} \end{array} \right\}^{4} \right) \right] \end{aligned} \tag{64}$$

$$\begin{split} &\sum\_{k=0}^{\infty} \left[ {}\_{1}F\_{2} \left( \begin{matrix} 1 & \vdots \\ 1 & \frac{3}{4} \end{matrix}; \begin{matrix} & -\pi^{2} \left\{ \frac{(2)\_{k}}{(1)\_{k}} \right\}^{4} \end{matrix} \right) \right] - \pi \sum\_{k=0}^{\infty} \left[ \left\{ \frac{(2)\_{k}}{(1)\_{k}} \right\} {}\_{0}F\_{1} \left( \begin{matrix} -; \\ \frac{1}{2}; \end{matrix}; \begin{matrix} & -\pi^{2} \left\{ \frac{(2)\_{k}}{(1)\_{k}} \right\}^{4} \end{matrix} \right) \right] \\ &+ 2\pi^{2} \sum\_{k=0}^{\infty} \left[ \left\{ \frac{(2)\_{k}}{(1)\_{k}} \right\}^{3} {}\_{0}F\_{1} \left( \begin{matrix} -; \\ \frac{3}{2}; \end{matrix}; \begin{matrix} & -\pi^{2} \left\{ \frac{(2)\_{k}}{(1)\_{k}} \right\}^{4} \end{matrix} \right) \right] = \frac{\pi-3}{8} . \end{split} \tag{17}$$

$$\begin{aligned} \text{(65)}\\ \sum\_{k=0}^{\infty} \left[ \begin{matrix} 1, \ \frac{3}{2} \\ \end{matrix} \begin{matrix} 2, \ \vdots \\ 4, \ \frac{1}{2}, \ \frac{3}{4}; \end{matrix} \right] \\ - \frac{3\pi}{2} \sum\_{k=0}^{\infty} \left[ \begin{pmatrix} \underbrace{(2)\_{k}}\_{1} \end{pmatrix} \begin{matrix} \underbrace{7}{4} \\ \end{matrix} \begin{matrix} \underbrace{7}{2} \\ \end{matrix} \begin{matrix} \underbrace{2}\_{1} \end{aligned} \right] \\ - \frac{3\pi}{2} \sum\_{k=0}^{\infty} \left[ \begin{pmatrix} \underbrace{(2)\_{k}}\_{1} \end{pmatrix} \begin{matrix} \underbrace{7}{4} \\ \end{matrix} \begin{matrix} \underbrace{7}{2} \\ \end{matrix} \begin{matrix} \underbrace{2}\_{1} \end{matrix} \begin{matrix} \underbrace{2}\_{1} \end{aligned} \right] \right] \\ + 5\pi^{2} \sum\_{k=0}^{\infty} \left[ \begin{pmatrix} \underbrace{(2)\_{k}}\_{1} \end{pmatrix}^{3} \begin{matrix} \underbrace{9}\_{1} \end{matrix} \begin{matrix} \underbrace{9}\_{1} \end{matrix} \begin{matrix} \underbrace{2}\_{1} \end{matrix} \right] \right] \\ + 5\pi^{2} \sum\_{k=0}^{\infty} \left[ \begin{pmatrix} \underbrace{(2)\_{k}}\_{1} \end{matrix} \begin{matrix} \underbrace{9}\_{1} \end{matrix} \begin{matrix} \underbrace{9}\_{1} \end{matrix} \begin{matrix} \underbrace{2}\_{1} \end{matrix} \begin{matrix} \underbrace{4}\_{1} \end{matrix} \right] \right] = \frac{1}{32} (4\pi - 13), \tag{63} \end{aligned} \tag{64}$$

$$\begin{split} &\sum\_{k=0}^{\infty} \left[ {}\_2F\_3 \left( \begin{matrix} 1, & \frac{3}{2} \\ & 1, & \frac{1}{2} \end{matrix}; \frac{-\pi^2}{16} \left\{ \frac{(2)\_k}{(1)\_k} \right\}^4 \right) \right] \\ &= \frac{3\pi}{4} \sum\_{k=0}^{\infty} \left[ \left\{ \frac{(2)\_k}{(1)\_k} \right\} {}\_1F\_2 \left( \begin{matrix} 7 & : & & -\pi^2 \left\{ (2)\_k \right\} \\ & 1 & : & & 16 \end{matrix} \left\{ \begin{matrix} (2)\_k \\ 4 \\ 2 \end{matrix} ; \frac{3}{4} \right\} \right) \right] \\ &+ \frac{5\pi^2}{8} \sum\_{k=0}^{\infty} \left[ \left\{ \frac{(2)\_k}{(1)\_k} \right\}^3 {}\_1F\_2 \left( \begin{matrix} 9 & : & & & -\pi^2 \left\{ (2)\_k \right\} \\ & 5 & : & & 16 \end{matrix} \left\{ \begin{matrix} 2 \\ (1)\_k \end{matrix} \right\} \right) \right] = \frac{\pi^2}{16} \left( \frac{3}{\pi} - \frac{1}{2} - \frac{5}{\pi^2} \right), \end{split} \tag{27}$$

X∞ *k*¼0 ð Þ2 *<sup>k</sup>* ð Þ1 *<sup>k</sup>* � �<sup>2</sup>*F*<sup>3</sup> 7 4 , 9 <sup>4</sup> ; 1 2 , 3 4 , 5 4 ; �*π*<sup>2</sup> 16 ð Þ2 *<sup>k</sup>* ð Þ1 *<sup>k</sup>* � �<sup>4</sup> 0 BB@ 1 CCA 2 6 6 4 3 7 7 5 � 16 5 X∞ *k*¼0 ð Þ2 *<sup>k</sup>* ð Þ1 *<sup>k</sup>* � �<sup>2</sup> <sup>2</sup>*F*<sup>3</sup> 2, <sup>5</sup> <sup>2</sup> ; 3 4 , 5 4 , 3 2 ; �*π*<sup>2</sup> 16 ð Þ2 *<sup>k</sup>* ð Þ1 *<sup>k</sup>* � �<sup>4</sup> 0 BB@ 1 CCA 2 6 6 4 3 7 7 5 þ 7*π* 6 X∞ *k*¼0 ð Þ2 *<sup>k</sup>* ð Þ1 *<sup>k</sup>* � �<sup>3</sup> <sup>2</sup>*F*<sup>3</sup> 9 <sup>4</sup> , <sup>11</sup> <sup>4</sup> ; 5 4 , 3 2 , 7 4 ; �*π*<sup>2</sup> 16 ð Þ2 *<sup>k</sup>* ð Þ1 *<sup>k</sup>* � �<sup>4</sup> 0 BB@ 1 CCA 2 6 6 4 3 7 7 <sup>5</sup> <sup>¼</sup> *<sup>π</sup>*<sup>2</sup> 60 5 *<sup>π</sup>* � <sup>5</sup> *<sup>π</sup>*<sup>2</sup> � <sup>1</sup> � �, (68)

$$\begin{split} &\sum\_{k=0}^{\infty} \left[ \left\{ \frac{(2)\_k}{(1)\_k} \right\}\_0 F\_1 \left( \begin{array}{c} \neg\\ \frac{1}{2}; \end{array} \right. - \frac{\pi^2}{4} \left\{ \frac{(2)\_k}{(1)\_k} \right\}^4 \right] \\ &- 2\sqrt{2} \sum\_{k=0}^{\infty} \left[ \left\{ \frac{(2)\_k}{(1)\_k} \right\}^2 \,\_1F\_2 \left( \begin{array}{c} 1; \\ 3 \end{array} \right. \frac{}{5} \sum\_{k=0}^{\infty} \frac{-\pi^2}{4} \left\{ \frac{(2)\_k}{(1)\_k} \right\}^4 \right] \\ &+ \pi \sum\_{k=0}^{\infty} \left[ \left\{ \frac{(2)\_k}{(1)\_k} \right\}^3 \,\_0F\_1 \left( \begin{array}{c} \neg\\ 3 \end{array} \right. \right. - \frac{\pi^2}{4} \left\{ \frac{(2)\_k}{(1)\_k} \right\}^4 \right] = \frac{\sqrt{2}-1}{4}, \end{split} \tag{69}$$

$$\begin{split} &\sum\_{k=0}^{\infty} \left[ \left\{ \frac{(2)\_{k}}{(1)\_{k}} \right\}\_{0} {}\_{0}F\_{1} \left( \begin{matrix} \multimap\\ 1 \end{matrix}; \qquad \frac{-\pi^{2}}{16} \left\{ \frac{(2)\_{k}}{(1)\_{k}} \right\}^{4} \right) \right] \\ &+ 2\sum\_{k=0}^{\infty} \left[ \left\{ \frac{(2)\_{k}}{(1)\_{k}} \right\}^{2} {}\_{1}F\_{2} \left( \begin{matrix} \multimap\\ 3 & 5 \end{matrix}; \qquad \frac{-\pi^{2}}{16} \left\{ \frac{(2)\_{k}}{(1)\_{k}} \right\}^{4} \right) \right] \\ &+ \frac{\pi}{2} \sum\_{k=0}^{\infty} \left[ \left\{ \frac{(2)\_{k}}{(1)\_{k}} \right\}^{3} {}\_{0}F\_{1} \left( \begin{matrix} \multimap\\ 3 & 1 \end{matrix}; \qquad \frac{-\pi^{2}}{16} \left\{ \frac{(2)\_{k}}{(1)\_{k}} \right\}^{4} \right) \right] = \frac{1}{4}, \end{split} \tag{70}$$

*Analytical Expressions of Infinite Fourier Sine and Cosine Transform-Based Ramanujan… DOI: http://dx.doi.org/10.5772/intechopen.108401*

X∞ *k*¼0 ð Þ2 *<sup>k</sup>* ð Þ1 *<sup>k</sup>* � �<sup>0</sup>*F*<sup>1</sup> ; 1 2 ; �*π*<sup>2</sup> 64 ð Þ2 *<sup>k</sup>* ð Þ1 *<sup>k</sup>* � �<sup>4</sup> 0 @ 1 A 2 4 3 5 � ffiffi 2 <sup>p</sup> <sup>X</sup><sup>∞</sup> *k*¼0 ð Þ2 *<sup>k</sup>* ð Þ1 *<sup>k</sup>* � �<sup>2</sup> <sup>1</sup>*F*<sup>2</sup> 1; 3 4 , 5 4 ; �*π*<sup>2</sup> 64 ð Þ2 *<sup>k</sup>* ð Þ1 *<sup>k</sup>* � �<sup>4</sup> 0 @ 1 A 2 4 3 5 þ *π* 4 X∞ *k*¼0 ð Þ2 *<sup>k</sup>* ð Þ1 *<sup>k</sup>* � �<sup>3</sup> <sup>0</sup>*F*<sup>1</sup> ; 3 2 ; �*π*<sup>2</sup> 64 ð Þ2 *<sup>k</sup>* ð Þ1 *<sup>k</sup>* � �<sup>4</sup> 0 @ 1 A 2 4 3 <sup>5</sup> <sup>¼</sup> <sup>3</sup> ffiffi 2 <sup>p</sup> � <sup>2</sup> <sup>4</sup> , (71)

$$\begin{split} &\sum\_{k=0}^{\infty} \left[ \left\{ \frac{(2)\_{k}}{(1)\_{k}} \right\} {}\_{0}F\_{1} \left( \begin{array}{ccc} -\raisebox{1.0pt}{ $\pi$ } \\ \frac{1}{2}; & \dfrac{-\pi^{2}}{144} \left\{ \frac{(2)\_{k}}{(1)\_{k}} \right\}^{4} \end{array} \right) \right] \\ & - \frac{2\sqrt{3}}{3} \sum\_{k=0}^{\infty} \left[ \left\{ \frac{(2)\_{k}}{(1)\_{k}} \right\}^{2} {}\_{1}F\_{2} \left( \begin{array}{ccc} 1; & & & -\pi^{2} \\ \frac{3}{4}, & \frac{5}{4}; & & \overline{144} \left\{ \frac{(2)\_{k}}{(1)\_{k}} \right\}^{4} \end{array} \right) \right] \\ & + \frac{\pi}{6} \sum\_{k=0}^{\infty} \left[ \left\{ \frac{(2)\_{k}}{(1)\_{k}} \right\}^{3} {}\_{0}F\_{1} \left( \begin{array}{ccc} -; & & & -\pi^{2} \\ \frac{3}{2}; & & \overline{144} \left\{ \frac{(2)\_{k}}{(1)\_{k}} \right\}^{4} \end{array} \right) \right] = \frac{13\sqrt{3} - 12}{12}, \end{split}$$

$$\begin{split} &\sum\_{k=0}^{\infty} \left[ \left\{ \frac{(2)\_{k}}{(1)\_{k}} \right\} {}\_{0}F\_{1} \left( \begin{array}{c} -; \\ \frac{1}{2}; \\ \end{array} \right. \left. -\pi^{2} \left\{ \frac{(2)\_{k}}{(1)\_{k}} \right\}^{2} {}\_{1}F\_{2} \left( \begin{array}{c} 1; \\ \frac{3}{4}, \\ \end{array} \right. \right. \left. \left. \begin{array}{c} 1; \\ \end{array} \right. \right. \\ &\left. + 2\pi \sum\_{k=0}^{\infty} \left[ \left\{ \frac{(2)\_{k}}{(1)\_{k}} \right\}^{3} {}\_{0}F\_{1} \left( \begin{array}{c} -; \\ \frac{3}{4}, \\ \end{array} \right. \right. \left. -\pi^{2} \left\{ \frac{(2)\_{k}}{(1)\_{k}} \right\}^{4} {}\_{1} \right] \right] = \frac{1}{8\pi}, \end{split}$$

(73)

(72)

$$\begin{split} &\sum\_{k=0}^{\infty} \left[ \left\{ \frac{(2)\_{k}}{(1)\_{k}} \right\} {}\_{0}F\_{1} \left( \begin{array}{ccc} - & & -25\pi^{2} \\ \hline \hline 2 \end{array} ; & \frac{-25\pi^{2}}{16} \left\{ \frac{(2)\_{k}}{(1)\_{k}} \right\}^{4} \right] \\ &-2\sqrt{5} \sum\_{k=0}^{\infty} \left[ \left\{ \frac{(2)\_{k}}{(1)\_{k}} \right\} {}\_{1}F\_{2} \left( \begin{array}{ccc} 1; & & & -25\pi^{2} \\ \hline \hline 4 & 5 & & \\ \end{array} ; & \frac{-25\pi^{2}}{16} \left\{ \begin{array}{ccc} - & & & -25\pi^{2} \\ \hline \end{array} \right\}^{4} \right] \\ &+\frac{5\pi}{2} \sum\_{k=0}^{\infty} \left[ \left\{ \frac{(2)\_{k}}{(1)\_{k}} \right\}^{3} {}\_{0}F\_{1} \left( \begin{array}{ccc} - & & & -25\pi^{2} \\ \hline \hline 2 & & & 16 \end{array} \right\}^{4} \right] = \frac{8\sqrt{5}-15}{100}. \end{split} \tag{74}$$

Proof: When *<sup>m</sup>* <sup>¼</sup> 0 with *<sup>n</sup>* <sup>¼</sup> 1,2, <sup>1</sup> <sup>2</sup> in the eqs. (42) to (44) and comparing with the eqs. (17), (18), and (19), we get the results (46), (47) and (48) respectively. In view of the hypergeometric functions (70), (71) and (72), we can express the above results (46) to (48) in terms of cosine, sine and Lommel functions. Our results (46) to (48) are convergent in view of the convergence condition of *pFq*ð Þ� series, when *p*≤*q*, and for all ∣*z*∣< ∞.

Similarly, we derive (49) to (51) by putting *<sup>m</sup>* <sup>¼</sup> 1, *<sup>n</sup>* <sup>¼</sup> <sup>1</sup> 2 ; *m* ¼ 1, *n* ¼ 2 and *m* ¼ 2, *n* ¼ 2 in the eqs. (42) and (44) and finally comparing with (3), (4) and (5). When *<sup>m</sup>* <sup>¼</sup> 0 with *<sup>n</sup>* <sup>¼</sup> 1,2,4,6, <sup>1</sup> 2 , 2 <sup>5</sup> in the eqs. (42) and (44) and comparing with (10) to (15), we get the rest of results (52) to (57) respectively. In view of the hypergeometric functions (53), (54) and (55), we can express the above results (52) to (57) in terms of cosine, sine and Lommel functions. Our results (49) to (57) are convergent in view of the convergence condition of *pFq*ð Þ� series, when *p*≤ *q*, and for all ∣*z*∣< ∞.

#### **6. Concluding remarks**

We have derived analytical expressions of the infinite Fourier sine and cosine transforms related to Ramanujan's integrals as an infinite sum of ordinary hypergeometric functions <sup>2</sup>*F*3, with suitable convergence conditions. Moreover, as applications of Ramanujan's integrals **R***S*ð Þ *m*, *n* , some closed form infinite summation formulas associated with hypergeometric functions <sup>1</sup>*F*2, <sup>2</sup>*F*<sup>3</sup> and <sup>0</sup>*F*<sup>1</sup> are evaluated. It is hoped that other such integrals can also be evaluated in a similar way. We conclude by remarking that various new results and applications can be obtained from our general theorem by appropriate choice of the parameters *υ*,*λ*,*b*,*c*,*y* and fixed sequence f g <sup>Θ</sup>ð Þ*<sup>k</sup>* <sup>∞</sup> *<sup>k</sup>*¼<sup>0</sup> in *<sup>I</sup>* <sup>∗</sup> *<sup>C</sup>* ð Þ *υ*, *b*, *c*, *λ*, *y* .

#### **Acknowledgements**

One of the authors (S.A. Dar) acknowledges financial support from the University Grants Commission of India for the award of a Dr. D.S. Kothari Post Doctoral Fellowship(DSKPDF) (Grant number F.4-2/2006 (BSR)/MA/20-21/0061).

#### **Author details**

Showkat Ahmad Dar\* and M. Kamarujjama Department of Applied Mathematics, Aligarh Muslim University, Aligarh, India

\*Address all correspondence to: showkatjmi34@gmail.com

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Analytical Expressions of Infinite Fourier Sine and Cosine Transform-Based Ramanujan… DOI: http://dx.doi.org/10.5772/intechopen.108401*

#### **References**

[1] Carslaw HS. Introduction to the Theory of Fourier's Series and Integrals. Martin's street, London: Macmillan and co., limited st; 1921

[2] Erdélyi A, Magnus W, Oberhettinger F, Tricomi FG. Higher Transcendental Functions. Vol. 1. New York, Toronto and London: McGraw-Hill; 1953

[3] Erdélyi A, Magnus W, Oberhettinger F, Tricomi FG. Tables of Integral Transforms. Vol. 1. New York, Toronto and London: McGraw-Hill; 1954

[4] Gradshteyn IS, Ryzhik IM. Table of Integrals, Series, and Products. 8th ed. USA: Academic Press is an imprint of Elsevier; 2015

[5] Oberhettinger F. Tables of Fourier Transforms and Fourier Transforms of Distributions. Berlin: Springer Verlag; 1990

[6] Sneddon NI. Fourier Transforms. Newyork: McGraw Hill Book Company, Inc; 1951

[7] Ramanujan S. Some definite integrals connected with Gauss's sums. Messenger of Mathematics;**XLIV**(2015): 75-86

[8] Oberhettinger F. Tables of Bessel Transforms. Berlin, Heidelberg, New York: Springer-Verlag; 1972

[9] Rainville ED. Special Functions. New York: Macmillan Company; 1960 Reprinted by Chelsea Publishing Company, Bronx, New York. 1971

[10] Srivastava HM, Manocha HL. A Treatise on Generating Functions. New York, Chichester, Brisbane and Toronto: Halsted Press (Ellis Horwood Limited, Chichester, U.K.), John Wiley and Sons; 1984

[11] Srivastava HM. A note on certain identities involving generalized hypergeometric series. Nederl. Akad. Wetensch. Proc. Ser. A 82=Indag. Math. 1979;**41**:191-201

[12] Kilbas AA, Saigo M. H-Transforms: Theory and Applications (Analytical Methods and Special Functions). Vol. 9. Boca Raton, London, New York, Washington, D.C.: CRC Press Company; 2004

[13] Kilbas AA, Saigo M, Trujillo JJ. On the generalized Wright function. Fractional Calculus and Applied Analysis. 2002;**5**(4):437-460

[14] Andrews GE, Berndt BC. Ramanujan's Lost Notebook. New York: Part IV. Springer-Verlag; 2013

[15] Berndt BC, Straub A. Certain integrals arising from Ramanujan's notebooks, symmetry, Integrability and geometry: Methods and applications. SIGMA. 2015;**11**(83):1-11

[16] Berndt BC. Integrals associated with Ramanujan and elliptic functions. The Ramanujan Journal. 2016;**41**:369-389

[17] Ramanujan S. Collected Papers. Cambridge: Cambridge University Press; 1927 reprinted by Chelsea, New York, 1962, reprinted by Amer. Math. Soc., Providence, RI, 2000

[18] Ramanujan S. The Lost Notebook and Other Unpublished Papers. New Delhi: Narosa; 1988

[19] Meyer JL. A generalization of an integral of Ramanujan. The Ramanujan Journal. 2007;**14**:79-88

[20] Qureshi MI, Khan IH. Ramanujan integrals and other definite integrals

associated with Gaussian hypergeometric functions. South East Asian Journal of Mathematics and Mathematical Sciences. 2005;**4**(1):39-52

[21] Qureshi MI, Dar SA. Generalizations of Ramanujan's integral associated with infinite Fourier cosine transforms in terms of hypergeometric functions and its applications. Kyungpook National University. 2020;**60**:781-795

[22] Qureshi MI, Dar SA. Generalizations of Ramanujan's integral associated with infinite Fourier sine transforms in terms of hypergeometric functions and its applications. Montes Taurus Journal of Pure and Applied Mathematics. 2021; **3**(3):216-226

#### **Chapter 5**

## Perspective Chapter: Cascaded-Resonator-Based Recursive Harmonic Analysis

*Miodrag D. Kušljević*

#### **Abstract**

It is well known that recursive algorithms for harmonic analysis have better characteristics in terms of monitoring the change of the spectrum in comparison to methods based on the processing of blocks of consecutive samples, such as, for example, discrete Fourier transform (DFT). This property is particularly important when applying spectral estimation in real-time systems. One of the recursive algorithms is the resonator-based one. The approach of the parallel cascades of multiple resonators (MR) with the common feedback has been generalized as the cascaded-resonator (CR)-based structure for recursive harmonic analysis. The resulting filters of the CR structure can be finite impulse response (FIR) type or the infinite impulse response (IIR) ones as a computationally more efficient solution, optimizing the frequency responses of all harmonics simultaneously. In the case of the IIR filter, the unit characteristic polynomial present in the FIR filter is replaced with an optimized characteristic polynomial of the transfer function. Such a change does not lead to an increase in computing requirements and changes only the resonator gain values. By using a conveniently linearized iterative algorithm for stability control purpose, based on the Rouche's theorem, the iterative linear-programming-based or the constrained linear least-squares (CLLS) optimization techniques can be used.

**Keywords:** cascaded-resonator (CR)-based filter, constrained linear least squares (CLLS), discrete Fourier transformation (DFT), Taylor-Fourier transformation (TFT), harmonic analysis, IIR filter, linear programming (LP), multiple-resonator (MR)-based filter

#### **1. Introduction**

In recent years, a lot of various algorithms for harmonic analysis have been proposed in the literature. Good surveys of some techniques are presented in Refs. [1, 2]. The discrete Fourier transform (DFT)-based method, as a mainstream approach, is widely used for harmonic analysis, thanks to its low computational burden, especially with the fast Fourier transform (FFT). However, errors arise when the power system is operating at off-nominal frequency, especially under dynamic conditions. Harmonic estimates under oscillating conditions were recently proposed in several studies. A huge volume of papers has been written on harmonics tracking in power

systems. The focus of recent literature has been on preprocessing and postprocessing methods for fixed-sample-rate algorithms surrounding a core DFT (or similar) analysis with a fixed number of samples [3, 4].

Idea of considering a dynamic model to better estimate the fundamental and harmonic phasors has been emerging in Refs. [4–9], and its importance has been pointed out in Ref. [10]. In Ref. [9], the discrete Taylor-Fourier transform (TFT) was proposed as an extension of the full DFT. The TFT by using a dynamic model of the signal extends and improves estimations obtained by DFT [9, 11]. This transformation corresponds to an FIR filter bank with a maximally flat frequency response. Each filter in the bank has maximum flat gain around the harmonic frequency and near-ideal attenuation around the other harmonics. This results in less distortion of the signal and less influence of disturbances present in the signal. In this way, the periodicity restriction assumed by the Fourier analysis is mitigated. As result, so obtained reconstruction is more accurate than the reconstruction obtained through DFT. When harmonics are narrow-band pass signals with spectral density confined into the flat-gain harmonic intervals, the coefficients of the TFT provide good estimates of the first derivatives of their complex envelopes. The digital TFT formulation in a matrix form that facilitates its implementation with the FFT to reduce the computational complexity of its straightforward implementation has been given in [11].

The multiple-resonator (MR)-based recursive estimators have been introduced in Ref. [12]. In Ref. [13], the MR-based observer structure is proposed for the implementation of TFT. Their good properties are provided by their parallel form, a recursive implementation, and good sensitivity properties assured by the infinite loop gain at the resonator frequencies [14]. Multiple zeros also provide reinforcing of the required attenuations and zero-gain flatness at the harmonic components with a high overall attenuation in the stopbands. For the known frequency of the periodic signal, the estimator based on resonators with common feedback enables the estimation of Fourier components even in cases when the sampling rate is not synchronized with the signal frequency. Also, this harmonic analyzer shows robustness in real conditions where there is noise and nonlinearity of the analog part of the equipment. MR-based harmonic analysis provides better performances of the spectral estimation than the single-resonator-based observer that corresponds to the classical DFT estimator.

This approach has been generalized as the cascaded-resonator (CR)-based structure for harmonic analysis. In Ref. [15], the cascaded-dispersed-resonator-based (CDR-based) structure for harmonic analysis is proposed. Although the design objectives in Refs. [15, 16] are different, the design technique is the same in both cases. In Ref. [16], the task is to replace multiple resonators with a cascade of single resonators. In this way, for the design purpose, it is possible to use the classic Lagrange interpolation technique instead of the more complex Hermitian interpolation. The condition that the poles are distributed in a narrow band around the resonant frequencies, as close as possible to each other, which is however limited by numerical accuracy. In Ref. [15], the task is to arrange the poles in the cascade in such a way as to enable optimal attenuation in the entire range around the harmonic frequencies. Practically, the only difference is in the arrangement of the resonator poles around the harmonic frequencies. The frequency deviation issue can be resolved by adaptive estimators based on the actual frequency feedback. This approach has drawbacks as a stability issue, due to an internal delay. Instead of that, usage of the external module for the fundamental frequency estimation is proposed in Ref. [17].

#### **2. Cascaded-resonator-based structure harmonic analysis**

**Figure 1** shows the block diagram of the *K*-type CR-based harmonic analyzer. The structure includes ð Þ *K* þ 1 ð Þ 2*M* þ 2 resonators with poles f g *zm*,*k*, *m* ¼ �*M*, … , 0, … , *M* þ 1, *k* ¼ 0, 2, … *K* , placed in the 2*M* þ 2 cascades each of them having *K* þ 1 cascaded complex poles on the unit circle around related harmonic frequency [13, 15, 16]. Each resonator has its belonging complex gains *gm*,*k*. A complete set of resonator cascades is connected in parallel in a common feedback loop. The gains of the transfer functions at the resonator frequencies are equal to unity due to the infinite loop gain at these frequencies. The number of cascades (for the coherent sampling) is 2*M* þ 2 (*M* is a number of harmonics) and depends on *ω*1, because the condition *Mω*<sup>1</sup> <*π* has to be satisfied. The *ω*<sup>1</sup> ¼ 2*π f* <sup>1</sup>*= f <sup>S</sup>*, *f* <sup>1</sup> and *f <sup>S</sup>* are the nominal fundamental frequency and sampling rate. The overall system order is ð Þ *K* þ 1 ð Þ 2*M* þ 2 .

We have in every *m*th channel of the structure, as an internal transfer function

$$H\_m(\mathbf{z}) = \frac{V\_m^F(\mathbf{z})}{E(\mathbf{z})} = \mathbf{z}^{-1} \sum\_{k=0}^K \frac{\mathbf{g}\_{m,k}^{\prime}}{\prod\_{i=0}^{k-1} (\mathbf{1} - \mathbf{z}\_{m,i} \mathbf{z}^{-1})}, \quad \mathbf{g}\_{m,k}^{\prime} = \prod\_{i=k}^K \mathbf{g}\_{m,i} \tag{1}$$

where *<sup>m</sup>* ¼ �*M*, … , 0, … , *<sup>M</sup>* <sup>þ</sup> 1, *<sup>k</sup>* <sup>¼</sup> 0, 1, … ,*K*, and <sup>Q</sup>*<sup>k</sup>*�<sup>1</sup> *<sup>i</sup>*¼<sup>0</sup> <sup>1</sup> � *zm*,*iz*�<sup>1</sup> ð Þ¼ 1 for *<sup>k</sup>* <sup>¼</sup> 0. *<sup>V</sup><sup>F</sup> <sup>m</sup>*ð Þ*z* is the total feedback signal corresponding to *m*th channel, composed as the linear combination of the output of each resonator, i.e., each channel contributes to the filter output with *K* þ 1 complex weights *gm*,*<sup>k</sup>*, *k* ¼ 0, 1, … ,*K* n o.

**Figure 1.** *Block diagram of the K-type CR-based harmonic analyzer.*

The closed transfer function for every channel *m* has the form of [15, 16].

$$T\_{m,0}(\mathbf{z}) = \frac{V\_{m,0}(\mathbf{z})}{V\_1(\mathbf{z})} = \mathbf{g}'\_{m,0} \frac{\mathbf{z}^{-1} P\_m(\mathbf{z})}{A(\mathbf{z})}, \quad P\_m(\mathbf{z}) = \prod\_{n = -M \atop n \neq m}^{M+1} \prod\_{i=0}^{K} (1 - \mathbf{z}\_{n,i} \mathbf{z}^{-1}), \tag{2}$$

$$T\_{m,k}(z) = \frac{V\_{m,k}(z)}{V\_1(z)} = \mathbf{g}'\_{m,k} \frac{z^{-1} P\_m(z) \prod\_{i=0}^{k-1} (1 - z\_{m,i} z^{-1})}{A(z)} \tag{3}$$

$$A(\mathbf{z}) = \prod\_{n=-M}^{M+1} \prod\_{i=0}^{K} \left( \mathbf{1} - \mathbf{z}\_{n,i} \mathbf{z}^{-1} \right) + \mathbf{z}^{-1} \sum\_{n=-M}^{M+1} \left[ P\_n(\mathbf{z}) \sum\_{k=0}^{K} \left( \mathbf{g}\_{n,k}' \prod\_{i=0}^{k-1} (\mathbf{1} - \mathbf{z}\_{n,i} \mathbf{z}^{-1}) \right) \right] \tag{4}$$

It can be seen that all poles of the resonators are mapped to the zeros of the transfer function *Tm*,0ð Þ*z* due to the common feedback, with the exception of the poles belonging to the cascade of the harmonic *m*, which are automatically canceled by the poles that generated them. In differentiators transfer functions f g *Tm*,*<sup>k</sup>*ð Þ*z* , *k* ¼ 1, … , *K* , ð Þ *k* 6¼ 0 , zeros *zm*,*<sup>i</sup>* f g , *i* ¼ 0, … , *k* � 1 , originated from poles in *m*th channel, exist providing zero gain.

From Eq. (2), it is obvious that the filter corresponding to *m*th-channel provides the maximally flatness property in the stop band around the remaining harmonic frequencies. For small pole displacements, Δ*f* frequency response reshaping is negligible in comparison to the multiple-resonator case [16]. It is important to mention that the lower border of Δ*f* is limited by the computational accuracy. On the other hand, avoiding of the multiple poles allows design by the direct usage of the classical Lagrange interpolation formula rather than the Hermite one.

Although the characteristic polynomial of the transfer functions can be chosen in different ways, under some conditions it is possible to choose one so that the error is driven to zero in exactly ð Þ *K* þ 1 ð Þ 2*M* þ 2 samples. This is provided by what is called a dead-beat observer, for which the coefficients are calculated from the condition that the observer has deadbeat settling, i.e., it finds the unknown state within at most ð Þ *K* þ 1 ð Þ 2*M* þ 2 steps. That leads to FIR filters (with *A z*ð Þ¼ 1) in each channel. This way, although the structure is realized by resonators, which are IIR filters, the resulting filters in each channel are FIR type. **Figure 2** shows frequency responses of *T*1,0ð Þ¼ *z g*<sup>0</sup> 1,0*z*�<sup>1</sup>*P*1ð Þ*<sup>z</sup>* (corresponding to the fundamental component) of the deadbeat observer for the first up to the sixth order of resonator multiplicity (*K* ¼ 0, 1, … , 5). It is observed that (quasi) MR structures with a higher order multiplicity of poles provide smaller sidelobes and thus ensure a lower sensitivity to noise and to harmonic and interharmonic disturbances. The negative effect is the increase in the order of the filter, which increases the group delay and response time of the filter, as well as the numerical complexity. Due to this feature, large values of the resonator multiplicity could be inconvenient in the control application. The case *K* ¼ 0 corresponds to a classic DFT estimator, while the cases *K* >0 correspond to the TFT.

Frequency responses of zeroth-, first-, and second-order differentiator discrete FIR filters corresponding to the transfer functions *Tm*,*<sup>k</sup>*ð Þ*z* related to the fundamental component (*m* ¼ 1, *k* ¼ 0, 1, 2), for *K* ¼ 2 and *K* ¼ 3 (the third- and forth-order resonator structure) are given in **Figure 3**.

In order to obtain wider flatness intervals in the pass band, the feedback signals *VF <sup>m</sup>*ð Þ*z* could be used for harmonic estimation instead of *Vm*,0ð Þ*z* [13]. The global

*Perspective Chapter: Cascaded-Resonator-Based Recursive Harmonic Analysis DOI: http://dx.doi.org/10.5772/intechopen.108402*

#### **Figure 2.**

*Frequency responses for T*1,0ð Þ*z (the zeroth differentiator of the first harmonic) for K* ¼ 0, 1, … , 5 *(the first- to sixth-order resonator structure).*

#### **Figure 3.**

*Frequency response of the zeroth-, first-, and second-order discrete FIR filters corresponding to Tm*,*<sup>k</sup>*ð Þ*z , (m* ¼ 1 *and k* ¼ 0, 1, 2*), related to the fundamental component, for fs* ¼ 800*Hz and f* <sup>1</sup> ¼ 50*Hz, for (a) K* ¼ 2 *estimator (the third-order resonator structure) and (b) K* ¼ 3 *estimator (the fourth-order resonator structure).*

transfer function of the feedback loop is a sum of the transfer functions of all *K* þ 1 differentiators *T<sup>F</sup> <sup>m</sup>*ð Þ¼ *<sup>z</sup> <sup>V</sup><sup>F</sup> <sup>m</sup>*ð Þ*<sup>z</sup> <sup>=</sup>V z*ð Þ¼ <sup>P</sup>*<sup>K</sup> <sup>k</sup>*¼<sup>0</sup>*Tm*,*<sup>k</sup>*ð Þ*<sup>z</sup>* . The frequency responses of the estimation of the fundamental component obtained by *Vm*,0ð Þ*z* and estimation obtained by *V<sup>F</sup> <sup>m</sup>*ð Þ*z* are given together in **Figure 4a**. The good properties of the filters corresponding to the transfer functions *T<sup>F</sup> <sup>m</sup>*ð Þ*z* are related to the phase responses. Frequency responses have a zero phase response in the frequency bands around the harmonic frequencies, which means that in those frequencies the group delay is equal to zero. Bad properties are high resonant gains at the edges of bandwidths and high sidelobes. The zero flat gains in the stop band are preserved, although their intervals are narrowed. It should be mentioned that the peaks of the interharmonic gains and the side lobes increase by the multiplicity of the resonators (**Figure 4b**).

**Figure 4.**

*Frequency responses of (a) T*1,0ð Þ*<sup>z</sup> and T<sup>F</sup>* <sup>1</sup> ð Þ*<sup>z</sup> for K* <sup>¼</sup> <sup>2</sup> *estimator (the third-order cascade) and (b) T<sup>F</sup>* <sup>1</sup> ð Þ*z for K* ¼ 1*, K* ¼ 2 *and K* ¼ 3*, related to the fundamental component, for fs* ¼ 800*Hz and f* <sup>1</sup> ¼ 50*Hz.*

#### **2.1 Optimization problem statement**

In order to adapt the achieved digital differentiators to their ideal frequency responses around the harmonic frequencies, it is possible to modify the filters transfer functions. An optimization technique is utilized to reshape frequency responses of the filters transfer functions, avoiding resonant frequency peaks and reducing a group delay simultaneously, that can be rather important in control applications. The optimization task can also be different, e.g., maximization of the selectivity.

The transfer function of the extended structure, including the compensation FIR filter *B z*ð Þ, for the *m*th component channel is as follows:

$$T\_{m,0}^{AB}(\mathbf{z}) = \frac{V\_{m,0}(\mathbf{z})}{V\_1(\mathbf{z})} = B(\mathbf{z})T\_{m,0}(\mathbf{z}) = \left[\mathbf{g}\_{m,0}'\mathbf{z}^{-1}\mathbf{P}\_m(\mathbf{z})\right] \frac{B(\mathbf{z})}{A(\mathbf{z})} = \left[\mathbf{g}\_{m,0}'\mathbf{z}^{-1}\mathbf{P}\_m(\mathbf{z})\right] \frac{\mathbf{q}\_B\mathbf{x}\_B}{\mathbf{1} + \mathbf{q}\_A\mathbf{x}\_A} \tag{5}$$

where

$$\begin{aligned} B(\mathbf{z}) &= b\_0 + b\_1 \mathbf{z}^{-1} + \dots + b\_{N\_B - 1} \mathbf{z}^{-(N\_B - 1)} + b\_{N\_B} \mathbf{z}^{-N\_B}, \\ A(\mathbf{z}) &= \mathbf{1} + a\_1 \mathbf{z}^{-1} + \dots + a\_{N\_A - 1} \mathbf{z}^{-(N\_A - 1)} + a\_{N\_A} \mathbf{z}^{-N\_A}, \end{aligned}$$

$$\mathbf{x}\_B = \begin{bmatrix} b\_0 \ b\_1 \cdots \quad b\_{N\_B - 1} & b\_{N\_B} \end{bmatrix}^T, \\ \mathbf{x}\_A &= \begin{bmatrix} a\_1 & a\_2 & \cdots \ a\_{N\_A - 1} & a\_{N\_A} \end{bmatrix}^T, \ \mathbf{x} = \begin{bmatrix} \mathbf{x}\_B^T & \mathbf{x}\_A^T \end{bmatrix}^T,$$

$$\mathbf{q}\_B = \begin{bmatrix} \mathbf{1} & \mathbf{z}^{-1} & \mathbf{z}^{-2} & \mathbf{z}^{-(N\_B - 1)} & \mathbf{z}^{-N\_B} \end{bmatrix}, \ \mathbf{q}\_A = \begin{bmatrix} \mathbf{z}^{-1} & \mathbf{z}^{-2} & \mathbf{z}^{-(N\_A - 1)} & \mathbf{z}^{-N\_A} \end{bmatrix}.$$

The polynomial *A z*ð Þ does not cause any additional computation and only the polynomial *B z*ð Þ represents an additional numerical burden. Even more, in some cases, it is possible to choose *B z*ð Þ¼ *b*<sup>0</sup> which causes only one additional multiplication. Nevertheless, for the purposes of design, we will consider the IIR filter *B z*ð Þ*=A z*ð Þ as a common compensation for the total set of FIR filters *Pm*ð Þ*z* , *m* ¼ �*M*, … , 0, … , *M* þ 1.

With a given weighting function *W*ð Þ *ω* , the weighted Chebyshev error between the desired and actual frequency responses is defined as follows:

*Perspective Chapter: Cascaded-Resonator-Based Recursive Harmonic Analysis DOI: http://dx.doi.org/10.5772/intechopen.108402*

$$J^1(\mathbf{x}) = \max\_{\boldsymbol{\alpha} \in \Omega} \quad \mathcal{W}(\boldsymbol{\alpha}) \left| T^{AB} \left( \boldsymbol{\varepsilon}^{j\boldsymbol{\alpha}} \right) - H^d \left( \boldsymbol{\varepsilon}^{j\boldsymbol{\alpha}} \right) \right| \tag{6}$$

where *H<sup>d</sup> ej<sup>ω</sup>* � � is the desired frequency response in angular frequency *ω* specified in the frequency region *Ω* (or a union of several compact frequency bands) of the interests and 0 ≤ ω ≤ π.

The sum of squares of absolute values of errors in *NF* angular frequencies as follows:

$$J^2(\mathbf{x}) = \sum\_{i=1}^{N\_F} \mathcal{W}(\mathbf{z}\_i) \left[ T^{AB}(\mathbf{z}\_i) - H^d(\mathbf{z}\_i) \right] \tag{7}$$

where *zi* <sup>¼</sup> *<sup>e</sup> <sup>j</sup>ω<sup>i</sup>* .

In order to minimize the error *J* 1 ð Þ *x* defined in Eq. (6), a new variable *δ* can be introduced and the problem reformulated as follows:

#### minimize *δ* subject to *E*ð Þ *ω<sup>i</sup>* j j≤*δ*, *ω<sup>i</sup>* ∈ *Ω*, *i* ¼ 1, 2, ⋯, *NF* (8)

where *E z*ð Þ¼*<sup>i</sup> W z*ð Þ*<sup>i</sup> <sup>T</sup>AB*ð Þ� *zi <sup>H</sup><sup>d</sup>*ð Þ *zi* � �, *zi* <sup>¼</sup> exp ð Þ *<sup>j</sup>ω<sup>i</sup>* , for the total number *NF* of points defined in *Ω*.

Further, it is:

$$\frac{1}{|A(z\_i)|} \left| T(z\_i)B(z\_i) - A(z\_i)H^d(z\_i) \right| \le \delta \tag{9}$$

In Ref. [18], a suitable method has been described to linearize the error function *J* 1 ð Þ *x* such that the design problem can be solved by the linear programming (LP) method. However, this method neglects the denominator part *A z*ð Þ*<sup>i</sup>* j j. In Ref. [19], the performance of the LP method was improved by eliminating the above drawback by using the following iterative constraints scheme:

$$\frac{W(z\_i)}{|A^{(k-1)}(z\_i)|} \Big| T(z\_i)B(z\_i) - A^{(k)}(z\_i)H^d(z\_i) \Big| \le \delta \tag{10}$$

For the sake of notational simplicity, we denote

$$\frac{W(z\_i)}{|A^{(k-1)}(z\_i)|} \left| \left[ T(z\_i) \mathbf{q}\_{\mathbf{B}} \big|\_{x=z\_i}, \quad -H^d(z\_i) \mathbf{q}\_{\mathbf{A}} \big|\_{x=z\_i} \right] \mathbf{x} - H^d(z\_i) \right| \le \delta \tag{11}$$

The vector of unknown coefficients *x* is expanded with an additional variable *δ*, so that the expanded vector of unknowns is obtained:

$$\mathbf{x}^{\delta} = \begin{bmatrix} \mathbf{x} & \delta \end{bmatrix}^{T}. \tag{12}$$

The constraints defined by inequality (11) refer to the frequency ranges in which the error optimization is performed. In addition, sometimes it is necessary to keep the error within predefined limits, such as for example the gains in the stopbands and/or the transition bands:

$$\frac{1}{|A^{(k-1)}(z\_i)|} \left| \left[ T(z\_i) \mathbf{q}\_{\mathbf{B}} \big|\_{x=x\_i}, \quad -H^d(z\_i) \mathbf{q}\_{\mathbf{A}} \big|\_{x=z\_i} \right] \mathbf{x} - H^d(z\_i) \right| \le l\_i \tag{13}$$

*zi* ¼ exp ð Þ *jω<sup>i</sup>* , *i* ¼ 1, 2, ⋯, *NG*

*li* , *i* ¼ 1, 2, ⋯, *NG*, are fixed borders of the absolute values of the set of complex error *E z*ð Þ along assemblies *zi* ¼ exp ð Þ *jω<sup>i</sup>* , *i* ¼ 1, 2, ⋯, *NG*.

If one wants to ensure unity gain in harmonic frequencies, the following condition must be met:

$$\left. \frac{B(z)}{A(z)} \right|\_{z=z\_m} = \mathbf{1}, \text{ i.e.} \quad B(z\_m) = A(z\_m). \tag{14}$$

where *m* ¼ �*M*, … , 0, … , *M* þ 1*:*

In a matrix form, it can be written as follows:

$$\boldsymbol{q}\_m \mathbf{x} = \mathbf{1}.\tag{15}$$

where *q* ¼ *qB* �*qA* � �, *qm* <sup>¼</sup> *<sup>q</sup>* � � *<sup>z</sup>*¼*zm* . Complex equality constraints (15) can be written as follows

$$\begin{bmatrix} \operatorname{Re}\left\{\boldsymbol{q}\_{m}\right\} \\ \operatorname{Im}\left\{\boldsymbol{q}\_{m}\right\} \end{bmatrix} \mathbf{x} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \ m = -M, \dots, 0, \dots, M+1. \tag{16}$$

#### **2.2 Linearization of constraints**

The inequalities (11) and (13) are nonlinear. The convex semi-infinite programming can be applied [20], thanks to the quadratic property of the functions. Furthermore, a convenient approximation of these inequalities by the system of the linear ones [21–25] allows us to solve this constrained optimization problem through the LP or the constrained linear least-squares (CLLS) optimization technique.

It is valid:

$$|E(\mathbf{z}\_i)| = |E(\mathbf{z}\_i)| \left(\cos^2 a\_i + \sin^2 a\_i\right) \\
= \operatorname{Re}\{E(\mathbf{z}\_i)\} \cos a\_i + \operatorname{Im}\{E(\mathbf{z}\_i)\} \sin a\_i \\
\tag{17}$$

where *α<sup>i</sup>* ¼ argf g *E z*ð Þ*<sup>i</sup>* .

Since *α<sup>i</sup>* is not known a priory, the nonlinear constraints in Eq. (8) can be approximated by the system of linear constraints:

$$\operatorname{Re}\{E(z\_i)\}\cos a\_{i,j} + \operatorname{Im}\{E(z\_i)\}\sin a\_{i,j} \le \delta \tag{18}$$

where *i* ¼ 1, 2, ⋯, *NF*. If we choose *L* equidistantly distributed angles then it is *α<sup>i</sup>*,*<sup>j</sup>* ¼ *α<sup>i</sup>*,0 þ ð Þ *j* � 1 2*π=L*, where *j* ¼ 1, 2, ⋯, *L*. **Figure 5** shows that approximations by square and octagon (*L* ¼ 4 and *L* ¼ 8, respectively) allow only rough approximations. A higher accuracy is obtained by increasing *L*. Herein, *L* ¼ 32 is used.

Let us define as following for frequency point *i*:

*Perspective Chapter: Cascaded-Resonator-Based Recursive Harmonic Analysis DOI: http://dx.doi.org/10.5772/intechopen.108402*

#### **Figure 5.**

*Approximation of a cycle with a square and an octagon.*

$$\boldsymbol{W}^{\boldsymbol{A}}(\boldsymbol{z}\_{i}) = \frac{\boldsymbol{W}(\boldsymbol{z}\_{i})}{\left|\boldsymbol{A}^{(k-1)}(\boldsymbol{z}\_{i})\right|}, \quad \mathbf{g}\_{i} = \left[\boldsymbol{T}(\boldsymbol{z}\_{i})\boldsymbol{q}\_{\mathcal{B}}\big|\_{\boldsymbol{x}=\boldsymbol{x}\_{i}}, \quad -\boldsymbol{H}^{d}(\boldsymbol{z}\_{i})\boldsymbol{q}\_{\mathcal{A}}\big|\_{\boldsymbol{x}=\boldsymbol{x}\_{i}}\right].\tag{19}$$

Hence, Eq. (13) can be linearized and written in a matrix form

$$\mathbf{W}^{A}(z\_{i})\mathbf{A}\_{i}^{\prime}\mathbf{x} \le \mathbf{W}^{A}(z\_{i})\mathbf{b}\_{i}^{\prime} + l\_{i}\mathbf{1}\_{L\times 1}, \ \ i = \mathbf{1}, 2, \cdots, N\_{G} \tag{20}$$

*:*

where matrix *A*<sup>0</sup> *<sup>i</sup>* and vector *b*<sup>0</sup> *<sup>i</sup>* are given by

$$\mathbf{A}'\_{i} = \begin{bmatrix} \operatorname{Re}\left\{\mathbf{g}\_{i}\right\}\cos a\_{i1} + \operatorname{Im}\left\{\mathbf{g}\_{i}\right\}\sin a\_{i1} \\ \operatorname{Re}\left\{\mathbf{g}\_{i}\right\}\cos a\_{i2} + \operatorname{Im}\left\{\mathbf{g}\_{i}\right\}\sin a\_{i2} \\ \vdots \\ \operatorname{Re}\left\{\mathbf{g}\_{i}\right\}\cos a\_{iL} + \operatorname{Im}\left\{\mathbf{g}\_{i}\right\}\sin a\_{iL} \end{bmatrix}, \mathbf{b}'\_{i} = \begin{bmatrix} \operatorname{Re}\left\{H^{d}(\mathbf{z}\_{i})\right\}\cos a\_{i1} + \operatorname{Im}\left\{H^{d}(\mathbf{z}\_{i})\right\}\sin a\_{i2} \\ \operatorname{Re}\left\{H^{d}(\mathbf{z}\_{i})\right\}\cos a\_{i2} + \operatorname{Im}\left\{H^{d}(\mathbf{z}\_{i})\right\}\sin a\_{i2} \\ \vdots \\ \operatorname{Re}\left\{H^{d}(\mathbf{z}\_{i})\right\}\cos a\_{iL} + \operatorname{Im}\left\{H^{d}(\mathbf{z}\_{i})\right\}\sin a\_{iL} \end{bmatrix}$$

and *li* is a constraint limit of the error in the point *zi*.

Using matrix notation, and collecting inequality linearization systems in all settled frequency points, (20) becomes the following linear form:

$$\mathbf{A}'\mathbf{x} \le \mathbf{b}' \quad \text{or} \quad \begin{bmatrix} \mathbf{A}' & \mathbf{0}\_{(N\_G L)\times 1} \end{bmatrix} \mathbf{x}^\delta \le \mathbf{b}' \tag{21}$$

where matrix *A*<sup>0</sup> and vector *b*<sup>0</sup> are given by

$$\mathbf{A}' = \begin{bmatrix} W^{A}(z\_{1})\mathbf{A}'\_{1} \\ W^{A}(z\_{2})\mathbf{A}'\_{2} \\ \vdots \\ W^{A}(z\_{N\_{G}})\mathbf{A}'\_{N\_{G}} \end{bmatrix}, \ \mathbf{b}' = \begin{bmatrix} W^{A}(z\_{1})\mathbf{b}'\_{1} + l\_{1}\mathbf{1}\_{L \times 1} \\ W^{A}(z\_{2})\mathbf{b}'\_{2} + l\_{2}\mathbf{1}\_{L \times 1} \\ \vdots \\ W^{A}(z\_{N\_{G}})\mathbf{b}'\_{N\_{G}} + l\_{N\_{G}}\mathbf{1}\_{L \times 1} \end{bmatrix}$$

In case of the (11), we have:

$$\left[\left.\boldsymbol{W}^{A}(\boldsymbol{z}\_{i})\boldsymbol{\mathsf{A}}\_{i}^{\prime}\right.\quad -\mathbf{1}\_{L\times 1}\right]\boldsymbol{\mathsf{x}}^{\delta}\leq\boldsymbol{W}^{A}(\boldsymbol{z}\_{i})\boldsymbol{b}\_{i}^{\prime},\quad i=1,2,\cdots,N\_{F}\tag{22}$$

or in a matrix notation

*A*<sup>0</sup> �**1**ð Þ� *NFL* <sup>1</sup> � �*x<sup>δ</sup>* ≤ *b*<sup>0</sup> (23)

*:*

where

$$\mathbf{A}' = \begin{bmatrix} W^A(z\_1)\mathbf{A}'\_1\\ W^A(z\_2)\mathbf{A}'\_2\\ \vdots\\ W^A(z\_{N\_F})\mathbf{A}'\_{N\_F} \end{bmatrix}, \quad \mathbf{b}' = \begin{bmatrix} W^A(z\_1)\mathbf{b}'\_1\\ W^A(z\_2)\mathbf{b}'\_2\\ \vdots\\ W^A(z\_{N\_F})\mathbf{b}'\_{N\_F} \end{bmatrix}.$$

#### **2.3 Design (optimization) approach 1: CLLS minimization**

An objective is to find a minimum of the sum of squares of absolute values of *hx* � *d* in the assembly of the *NF* selected frequencies subject to the vector *x*

$$\min\_{\mathbf{x}} \sum\_{i=1}^{N\_F} |h\_i \mathbf{x} - d\_i|^2 \tag{24}$$

where *hi* ¼ *W z*ð Þ*<sup>i</sup> gi <sup>=</sup> <sup>A</sup>*ð Þ *<sup>k</sup>*�<sup>1</sup> ð Þ *zi* � � � � and *di* <sup>¼</sup> *W z*ð Þ*<sup>i</sup> <sup>H</sup><sup>d</sup>*ð Þ *zi <sup>=</sup> <sup>A</sup>*ð Þ *<sup>k</sup>*�<sup>1</sup> ð Þ *zi* � � � �. If we apply the following equality

$$\left| \left| h\_i \mathbf{x} - d\_i \right| \right|^2 = \text{Re}^2 \{ h\_i \mathbf{x} - d\_i \} + \text{Im}^2 \{ h\_i \mathbf{x} - d\_i \} = \left\| \mathbf{C}\_i \mathbf{x} - d\_i \right\|\_2^2 \tag{25}$$

where *Ci* <sup>¼</sup> Re f g *hi* Imf g *hi* � �, *di* <sup>¼</sup> Re f g *di* Imf g *di* � �, (24) can be written in a matrix form:

$$\min\_{\mathbf{x}} \left\| \mathbf{C} \mathbf{x} - \mathbf{d} \right\|\_{2}^{2}. \tag{26}$$

where *C* and *d* include *Ci* and *di*, respectively, *i* ¼ 1, 2, ⋯, *NF*.

The constrained linear least squares (CLLS) is an optimization problem that deals with the maximization or minimization of a linear function called the objective function subject to linear constraints. Summarizing (16), (21), and (26), the CLLS problem is formalized as follows:

$$\min\_{\mathbf{x}} \frac{1}{2} \left\lVert \mathbf{Cx} - \mathbf{d} \right\rVert\_{2}^{2} \quad \text{subject to} \quad \mathbf{Ax} \le \mathbf{b} \quad \left( \text{and} \begin{bmatrix} \text{Re } \{q\_{m} \} \\ \text{Im} \{q\_{m} \} \end{bmatrix} \mathbf{x} = \begin{bmatrix} \mathbf{1} \\ \mathbf{0} \end{bmatrix} \right), \tag{27}$$
  $\mathbf{(m = -M, \dots, 0, \dots, M+1)}$ 

where *A* and *b* include *A*<sup>0</sup> and *b*<sup>0</sup> , respectively, defined in (21), for all frequency points in which the constraints are defined.

#### **2.4 Design (optimization) approach 2: minimax optimization**

The LP optimization problem can be formalized in the following way:

$$\begin{array}{ll}\text{minimize} & \mathbf{c} \ \mathbf{x}^{\delta} \quad \text{subject to} \ \mathbf{A} \mathbf{x}^{\delta} \le \mathbf{b} \quad \left(\text{and} \begin{bmatrix} \text{Re}\left\{\boldsymbol{q}\_{m}\right\} & \mathbf{0} \\ \text{Im}\left\{\boldsymbol{q}\_{m}\right\} & \mathbf{0} \end{bmatrix} \mathbf{x}^{\delta} = \begin{bmatrix} 1 \\ \mathbf{0} \end{bmatrix}\right), & \text{(28)} \\ \mathbf{m} = -\mathbf{M}, \ldots, \mathbf{0}, \ldots \mathbf{M} + \mathbf{1}) \end{array} \tag{28}$$

*Perspective Chapter: Cascaded-Resonator-Based Recursive Harmonic Analysis DOI: http://dx.doi.org/10.5772/intechopen.108402*

where **<sup>c</sup>** <sup>¼</sup> **01**�ð Þ *NA*þ*NB*þ**<sup>1</sup>** <sup>1</sup> � �, and *<sup>A</sup>* and *<sup>b</sup>* include *<sup>A</sup>*<sup>0</sup> and *<sup>b</sup>*<sup>0</sup> , respectively, for all frequency points in which the constraints or objective functions are defined in (21) and (23), respectively.

#### **3. IIR cascaded-resonator-based harmonic analysis**

In accordance with the prevailing trends in works dealing with this issue, in the initial works [13, 16, 22, 23, 25, 26] the resulting filters of CR structures were of the FIR type. Later, in [27], IIR filters were used, which represent a computationally more efficient solution [28, 29]. The unit characteristic polynomial of the transfer function is replaced by the optimized one. Such a change does not lead to an increase in the volume of numerical calculations and only requires a change in the gain values associated with the resonators. Since the optimization of frequency characteristics for all harmonics is carried out at the same time, it is possible to obtain frequency responses of the same shape. By using a linearized iterative scheme [30] based on Rouche's theorem with the aim of stability control, it is possible to use iterative optimization techniques based on LP or CLLS.

#### **3.1 Problem statement**

The task of optimization is to design a filter *B z*ð Þ*=A z*ð Þ where the order of the characteristic polynomial *A z*ð Þ is *NA* ¼ ð Þ *K* þ 1 ð Þ 2*M* þ 2 and the polynomial *B z*ð Þ is of order *NB*. We seek to find a causal stable rational function *TAB <sup>m</sup>*,0ð Þ¼ *z g*<sup>0</sup> *<sup>m</sup>*,0*z*�<sup>1</sup>*Pm*ð Þ*<sup>z</sup>* h i*B z*ð Þ*=A z*ð Þ for *<sup>m</sup>* ¼ �*M*, … , 0, … , *<sup>M</sup>* <sup>þ</sup> 1 that best approximates *<sup>H</sup><sup>d</sup> <sup>m</sup> <sup>e</sup><sup>j</sup><sup>ω</sup>* � �.

In order to make the notation as simple and short as possible, let us form a virtual transfer function so that in each bandwidth centered in *m f* <sup>1</sup> with width of *f* <sup>1</sup>, i.e., for *<sup>f</sup>* <sup>∈</sup> *m f* <sup>1</sup> � *<sup>f</sup>* <sup>1</sup>*=*<sup>2</sup> � , *m f* <sup>1</sup> <sup>þ</sup> *<sup>f</sup>* <sup>1</sup>*=*<sup>2</sup> � , it corresponds to the transfer function belonging to the harmonic *m*. It follows:

$$T^{A}(z) = \frac{T(z)}{A(z)}, \quad T(z) = \mathbf{g}'\_{m,0} z^{-1} P\_m(z) \tag{29}$$

for *<sup>f</sup>* <sup>∈</sup> *m f* <sup>1</sup> � *<sup>f</sup>* <sup>1</sup>*=*<sup>2</sup> � , *m f* <sup>1</sup> <sup>þ</sup> *<sup>f</sup>* <sup>1</sup>*=*<sup>2</sup> � , *m* ¼ �*M*, … , 0, … , *M* þ 1. In addition, we define a unique transfer function

$$T^{AB}(z) = B(z)T^A(z) = B(z)\frac{T(z)}{A(z)}\tag{30}$$

Similarly, a virtual unique desired transfer function in an angular frequency *ω* has the following form [27]:

$$H^d(e^{jw}) = \begin{cases} \begin{array}{c} e^{-j2\pi\left(\int f - mf\_1\right)/f\_{\text{S}}}, \text{ for} \newline f \in \left[mf\_1 - f\_{\text{PB}}, mf\_1 + f\_{\text{PB}}\right] \\ 0, \text{ for} \; f \in \left[mf\_1 - f\_1/2, mf\_1 - f\_{\text{SB}}\right] \cup \left[mf\_1 + f\_{\text{SB}}, mf\_1 + f\_1/2\right] \end{array} \end{cases} \tag{31}$$

where pass and stop bands are defined by *f PB* and *f SB*, respectively. A desired group delay in the passband is denoted as *τ*.

#### **3.2 Stability constraint**

To obtain a stable IIR filter *T z*ð Þ, stability constraint must be imposed on the coefficient vector *xA*. In [30], the more convenient stability condition which is based on Rouche's theorem was proposed.

*Rouche's Theorem. If f z*ð Þ *and g z*ð Þ *are analytic inside and on a closed contour C, and* j j *g z*ð Þ <j j *f z*ð Þ *on C, then f z*ð Þ *and g z*ð Þþ *f z*ð Þ *have the same number of zeros inside C.* Let

$$f(\mathbf{z}) = \mathbf{z}^{N\_A} \mathbf{A}(\mathbf{z}) = \mathbf{z}^{N\_A} + a\_1 \mathbf{z}^{N\_A - 1} + \dots + a\_{N\_A - 1} \mathbf{z} + a\_{N\_A},\tag{32}$$

$$\mathbf{g}(\mathbf{z}) = \mathbf{z}^{N\_A} \Delta(\mathbf{z}) = \delta\_0 \mathbf{z}^{N\_A} + \delta\_1 \mathbf{z}^{N\_A - 1} + \dots + \delta\_{N\_A - 1} \mathbf{z} + \delta\_{N\_A},\tag{33}$$

where Δð Þ*z* is the update of the characteristic polynomial *A z*ð Þ of the transfer function at each iteration step. Since the functions *f z*ð Þ and *g z*ð Þ are analytic, except at z = ∞, and have the same zeros as *A z*ð Þ and Δð Þ*z* , according to Rouche's theorem, if the polynomial *<sup>A</sup>*ð Þ *<sup>k</sup>*�<sup>1</sup> ð Þ*<sup>z</sup>* in the iteration step, *<sup>k</sup>* � 1 has all its zeros inside a circle of radius *ρ* ð Þ 0<*ρ* <1 with the center at the origin of the complex plane, then also the polynomial in the iteration step *k* given by [30]

$$A^{(k)}(\mathbf{z}) = A^{(k-1)}(\mathbf{z}) + a\Delta^{(k)}(\mathbf{z}), \ \mathbf{0} < a < \mathbf{1} \tag{34}$$

will retain the zeros within this circle provided that in step *k* the following condition satisfied

$$\left|\Delta^{(k)}(\mathbf{z})\right| \le \left|A^{(k-1)}(\mathbf{z})\right|, \ |\mathbf{z}| = \rho. \tag{35}$$

If (34) is included in (35), we get

$$\left| \bar{A}^{(k)}(\mathbf{z}) - \bar{A}^{(k-1)}(\mathbf{z}) \right| \le a \left| A^{(k-1)}(\mathbf{z}) \right| \tag{36}$$

where *<sup>A</sup>*ð Þ*<sup>k</sup>* ð Þ¼ *<sup>z</sup> <sup>A</sup>*ð Þ*<sup>k</sup>* ð Þ� *<sup>z</sup>* 1, *<sup>A</sup>*ð Þ *<sup>k</sup>*�<sup>1</sup> ð Þ¼ *<sup>z</sup> <sup>A</sup>*ð Þ *<sup>k</sup>*�<sup>1</sup> ð Þ� *<sup>z</sup>* 1, or in a matrix notation:

$$\left| \left[ \mathbf{0}\_{1 \times (N\_{\mathbf{B}} + \mathbf{1})}, \quad \mathbf{q}\_{\mathbf{A}} \big|\_{x = x\_i} \right] \mathbf{x} - \bar{\mathbf{A}}^{(k-1)}(x\_i) \right| \le a \big| A^{(k-1)}(x\_i) \big| \tag{37}$$

As for the initial value of the vector *<sup>x</sup>*, it is simplest to take *<sup>x</sup>*ð Þ <sup>0</sup> <sup>¼</sup> **<sup>0</sup>**, when all the roots of the polynomial *<sup>A</sup>*ð Þ <sup>0</sup> ð Þ*<sup>z</sup>* lie within the circle of radius *<sup>ρ</sup>* (0<sup>&</sup>lt; *<sup>ρ</sup>* <sup>&</sup>lt;1).

If constraint (37) is applied to a sufficiently dense set of points lying on a circle of radius *ρ* ð Þ j j *z* ¼ *ρ* , of total length *NS*, we get

$$A^S x \le b^S \tag{38}$$

where matrix *A*<sup>S</sup> *<sup>i</sup>* and vector *<sup>b</sup>*<sup>S</sup> *<sup>i</sup>* are given by

$$\mathbf{A}\_{i}^{S} = \begin{bmatrix} \mathbf{0}\_{1 \times (N\_{B}+1)}, \operatorname{Re}\left\{ \boldsymbol{q}\_{A} \big|\_{x = x\_{i}} \right\} \cos a\_{i1} + \operatorname{Im}\left\{ \boldsymbol{q}\_{A} \big|\_{x = x\_{i}} \right\} \sin a\_{i1} \\ \mathbf{0}\_{1 \times (N\_{B}+1)}, \operatorname{Re}\left\{ \boldsymbol{q}\_{A} \big|\_{x = x\_{i}} \right\} \cos a\_{i2} + \operatorname{Im}\left\{ \boldsymbol{q}\_{A} \big|\_{x = x\_{i}} \right\} \sin a\_{i2} \\ \vdots \\ \mathbf{0}\_{1 \times (N\_{B}+1)}, \operatorname{Re}\left\{ \boldsymbol{q}\_{A} \big|\_{x = x\_{i}} \right\} \cos a\_{iL} + \operatorname{Im}\left\{ \boldsymbol{q}\_{A} \big|\_{x = x\_{i}} \right\} \sin a\_{iL} \end{bmatrix},$$

*Perspective Chapter: Cascaded-Resonator-Based Recursive Harmonic Analysis DOI: http://dx.doi.org/10.5772/intechopen.108402*

$$\begin{aligned} \mathbf{b}\_{i}^{\mathcal{S}} &= \begin{bmatrix} \operatorname{Re}\left\{\bar{A}^{(k-1)}(\mathbf{z}\_{i})\right\}\cos a\_{i1} + \operatorname{Im}\left\{\bar{A}^{(k-1)}(\mathbf{z}\_{i})\right\}\sin a\_{i1} + a \left| A^{(k-1)}(\mathbf{z}\_{i})\right| \\ \operatorname{Re}\left\{\bar{A}^{(k-1)}(\mathbf{z}\_{i})\right\}\cos a\_{i2} + \operatorname{Im}\left\{\bar{A}^{(k-1)}(\mathbf{z}\_{i})\right\}\sin a\_{i2} + a \left| A^{(k-1)}(\mathbf{z}\_{i})\right| \\ &\vdots\\ \operatorname{Re}\left\{\bar{A}^{(k-1)}(\mathbf{z}\_{i})\right\}\cos a\_{iL} + \operatorname{Im}\left\{\bar{A}^{(k-1)}(\mathbf{z}\_{i})\right\}\sin a\_{iL} + a \left| A^{(k-1)}(\mathbf{z}\_{i})\right| \end{bmatrix}. \end{aligned}$$

Thus, the set of constraints (38) is added to the set of the above constraint conditions. In this way, the iterative methods mentioned above solve the LP or CLLS problem by taking into account constraints (38) in each iteration step. A fixed step size *α* can be used, while a gradual decrease (e.g., exponential) can help the convergence of the solution.

#### **3.3 Resonators' gains calculation**

After the polynomial *A z*ð Þ having been determined, the direct usage of the Lagrange interpolation formula provides the closed-form formulas [26]. It should be taken into account that these formulas are valid only in the case of single resonators. If a quasi-MR-based analyzer is designed, the resonator poles connected to the same harmonic should be arranged close enough to each other with a minimum distance that is limited by numerical precision. Its lower border depends on the resonator multiplicity and the sampling rate. A chosen displacement of 0*:*1 Hz allows a fair approximation for sampling frequencies up to 6.4 kHz (*M* ¼ 63 for *f* <sup>1</sup> ¼ 50 Hz) and *K* ¼ 5 [16].

A generalized closed-form formula for gains calculation for any *K* for previously chosen polynomial *A z*ð Þ is given as follows:

$$\mathbf{g}'\_{m,k} = \frac{A(\mathbf{z}) - \mathbf{z}^{-1} \mathbf{P}\_m(\mathbf{z}) \sum\_{j=0}^{k-1} \left[ \mathbf{g}'\_{m,j} \prod\_{i=0}^{j-1} (\mathbf{1} - \mathbf{z}\_{m,i} \mathbf{z}^{-1}) \right]}{\mathbf{z}^{-1} \mathbf{P}\_m(\mathbf{z}) \prod\_{i=0}^{k-1} (\mathbf{1} - \mathbf{z}\_{m,i} \mathbf{z}^{-1})} \tag{39}$$

As a final result, the designed resonator gains are as follows:

$$\mathbf{g}\_{m,k} = \mathbf{g}'\_{m,k} / \mathbf{g}'\_{m,k+1}, \left( \mathbf{g}'\_{m,K+1} = \mathbf{1} \right), \ m = -M, \ \dots, \ \mathbf{0}, \ \dots, M+1, k = \mathbf{0}, \mathbf{1}, \ \dots, K.$$

It should be mentioned that polynomial *B z*ð Þ can be conveniently implemented by adding its roots as poles in additional parallel channels to the basic structure (see **Figure 1**). In this case, the existing formulas for gains calculation are valid only for the identical structure of the extension cascades (they have to consist *K* þ 1 resonators). Otherwise, the formulas are not valid and need a completely new derivation.

#### **3.4 Design example**

In the next section, three demonstration examples, with frequency responses and pole-zeros maps, of the designed *K* ¼ 2 type CR-based harmonic analyzer are shown, for *f <sup>S</sup>* ¼ 800 Hz and *f* <sup>1</sup> ¼ 50 Hz. For a clear readability, lower values of *f <sup>S</sup>* ¼ 800 Hz and *M* ¼ 7 are selected. The following parameters are prescribed in all three examples:

#### **Figure 6.**

*(a) Frequency responses of g*<sup>0</sup> 3,0*z*�<sup>1</sup>*P*3ð Þ*<sup>z</sup> ,B z*ð Þ*=A z*ð Þ *and TAB* 3,0ð Þ*z and (b) Pole-zero map of T*3,0ð Þ*z and B z*ð Þ *(for the third harmonic), for NA* ¼ ð Þ *K* þ 1 ð Þ 2*M* þ 2 *, NB* ¼ ð Þ *K* þ 1 ð Þ 2*M* þ 2 *, and ρ* ¼ 0*:*95, *f PB* ¼ 1*:*7 Hz *and τ=0.9 K*ð Þ þ 1 ð Þ 2*M* þ 2 *:*

*f SB* ¼ 17 Hz, *lTB* ¼ 1*:*005, *W z*ð Þ¼ 1, *α*<sup>0</sup> ¼ 0*:*5, *ρ* ¼ 0*:*95, *NA* ¼ ð Þ *K* þ 1 ð Þ 2*M* þ 2 . *<sup>x</sup>*ð Þ <sup>0</sup> <sup>¼</sup> **<sup>0</sup>***:* Other parameters were variated, depending on the chosen optimization criteria. It should mention that a large variety of optimization scenarios is possible, allowing design spectrum analyzers for a wide scope of different applications.

#### *3.4.1 Example 1: flat-top passbands*

The filters with a wider flatness in the pass band allow better signal tracking in the dynamic conditions. Since it is difficult task to provide the tracking of the parameters changes together with good attenuation in the stopband, a relatively high order of *NB* ¼ ð Þ *K* þ 1 ð Þ 2*M* þ 2 is settled. The desired group delay (in samples) in the passband is *τ*=0.9ð Þ *K* þ 1 ð Þ 2*M* þ 2 *: f PB* ¼ 1*:*7 Hz*: l PB* <sup>¼</sup> <sup>0</sup>*:*01. Obtained frequency responses (**Figure 6**) show that the passband flatness is not derogated, while the selectivity and attenuation in the stopbands are increased thanks to the zeros of the polynomial *B z*ð Þ which are located between the existing multiple zeros of the resonator structure that had been obtained through the common feedback. A cost is an increased total group delay which causes higher latency.

#### *3.4.2 Example 2: narrow selective passbands*

In this example, the requests for passband and transition bands are omitted, which decrease a numerical burden. To obtain high selectivity, *NB* ¼ ð Þ *K* þ 1 ð Þ 2*M* þ 2 is kept. The obtained frequency responses (see **Figure 7**) show that selectivity and attenuation in the stopbands are increased. This is achieved thanks to the poles of the transfer function located on a circle of radius 0*:*95 (j j *z* ¼ 0*:*95) very close to the zeros located in the harmonic frequencies, as well as the additional zeros of the polynomial *B z*ð Þ. Such high selectivity caused a large increase in a group delay.

*Perspective Chapter: Cascaded-Resonator-Based Recursive Harmonic Analysis DOI: http://dx.doi.org/10.5772/intechopen.108402*

*(a) Frequency responses of g*<sup>0</sup> 3,0*z*�<sup>1</sup>*P*3ð Þ*<sup>z</sup> ,B z*ð Þ*=A z*ð Þ *and TAB* 3,0ð Þ*z and (b) Pole-zero map of T*3,0ð Þ*z and B z*ð Þ *(for the third harmonic), for NA* ¼ ð Þ *K* þ 1 ð Þ 2*M* þ 2 *, NB* ¼ ð Þ *K* þ 1 ð Þ 2*M* þ 2 *, and ρ* ¼ 0*:*95, *f PB* ¼ 0 Hz*.*

#### **Figure 8.**

*(a) Frequency responses of g*<sup>0</sup> 3,0*z*�<sup>1</sup>*P*3ð Þ*<sup>z</sup> ,B z*ð Þ*=A z*ð Þ *and TAB* 3,0ð Þ*z and (b) Pole-zero map of T*3,0ð Þ*z and B z*ð Þ *(for the third harmonic), for NA* ¼ ð Þ *K* þ 1 ð Þ 2*M* þ 2 *, NB* ¼ 0*, and ρ* ¼ 0*:*95, *f PB* ¼ 0 Hz*.*

#### *3.4.3 Example 3: numerically cost-effective solution*

This example is very similar to the previous one with different that now is *NB* ¼ 0, which means that there is no extension to the existing resonator structure with common feedback. Obtained frequency responses (see **Figure 8**) show that selectivity and attenuation in the stopbands are smaller than in the previous case, however, with a smaller group delay too.

**Figure 9.** *Block diagram of the K-type CR-based harmonic analyzer with postprocessing.*

#### **4. FIR cascaded-resonator-based harmonic phasor estimation**

Instead of the common simultaneous compensation of the frequency responses for all harmonics through the compensating filter *B z*ð Þ placed in the front of the parallel resonator structure, a more flexible solution is shown in **Figure 9** with postprocessing by the set of compensators *Qm*ð Þ*z* designed particularly for each harmonic *m*. In this case, *A z*ð Þ¼ 1 is chosen to allow the use of linear optimization techniques such as LP and CLLS.

In order to obtain an algorithm that can be utilized in a wide range of signal dynamics in a unified way and improve the frequency response, a linear combination of the differentiators' outputs in the cascade can be used [22, 25, 26]. The goal of this compromised solution was to propose a tracking-mode harmonic estimation technique. In Ref. [31], it is shown that this estimation technique exhibiting maximally flat frequency responses can be efficiently used for implementation of P-Class Compliant PMU in accordance with IEC/IEEE Standard 60255-118-1:2018 for harmonic phasors estimation. In this approach, the order of the resulted compensation filter was low and equals to the pole multiplicity. In Refs. [21, 23, 24], the proposed approach was generalized to any necessary order through the postprocessing compensation FIR filters applied to the output signals obtained by the CR structure. The drawback of this approach is that we have to use as many postprocessing FIR filters as there are harmonic phasors that we need to estimate (one estimator per one harmonic phasor). On the other hand, the advantage is that it is possible to obtain a filter bank, surrounding the core CR structure, with a set of different compensation filters corresponding to different signal dynamics.

The transfer function for every *m*th channel has the form of

$$T\_{m,0}^{Q}(\mathbf{z}) = \frac{V\_{m,0}^{Q}(\mathbf{z})}{V(\mathbf{z})} = T\_{m,0}(\mathbf{z}) \mathbf{Q}\_{m}(\mathbf{z}) = \left[ \mathbf{g}\_{m,0}' \mathbf{z}^{-1} \mathbf{P}\_{m}(\mathbf{z}) \right] \mathbf{q}\_{Q} \mathbf{x}\_{\mathbf{Q}}.\tag{40}$$

*Perspective Chapter: Cascaded-Resonator-Based Recursive Harmonic Analysis DOI: http://dx.doi.org/10.5772/intechopen.108402*

where *Qm*ð Þ¼ *<sup>z</sup> qm*,0 <sup>þ</sup> *qm*,1*z*�<sup>1</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> *qm*,*NQ*�1*z*�ð Þ *NQ*�<sup>1</sup> <sup>þ</sup> *bm*,*NQ <sup>z</sup>*�*NQ* . *qQ* ¼ 1 *z*�<sup>1</sup> *z*�<sup>2</sup> ⋯ *z*�ð Þ *NQ*�<sup>1</sup> *z*�*NQ* h i, *xQ* <sup>¼</sup> *<sup>q</sup>*<sup>0</sup> *<sup>q</sup>*<sup>1</sup> <sup>⋯</sup> *qNQ*�<sup>1</sup> *qNQ* � �<sup>T</sup> . Eq (40) has the same form as Eq. (5) with the following constraints: *A z*ð Þ¼ 1, *NA* ¼ 0, *xA* ¼ ½�, *qA* ¼ ½�, *x* ¼ *xQ* , *q* ¼ *qQ* . *qB* is replaced by *qQ* , and *xB* by *xQ :* Since *NA* ¼ 0, the stability constraints are not present. The desired frequency response is related only to the actual harmonic *m* and does not consider the frequency responses of the other ones.

#### **4.1 Total vector gradient (TVG) calculation**

The response time and delay of the estimator are directly correlated with the group delay (GD) of the filter. Due to the more complex calculation of GD, it is possible to use the gradient of the transfer function *dT<sup>Q</sup> <sup>m</sup>*,0ð Þ*z =dz*, which is called the total vector gradient (TVG) here. In a flat range with small amplitude changes, TVG and GD are proportional, and optimization of one leads to optimization of the other. The first derivative of the transfer function *T<sup>Q</sup> <sup>m</sup>*,0ð Þ*z* is as follows:

$$\begin{split}d\boldsymbol{T}\_{m,0}^{\mathbb{Q}}(\boldsymbol{z})/d\boldsymbol{z} &= \mathbf{g}\_{m,0}^{\prime} \left[ -\boldsymbol{z}^{-2} \mathbf{Q}\_{m}(\boldsymbol{z}) \mathbf{P}\_{m}(\boldsymbol{z}) + \boldsymbol{z}^{-1} \mathbf{P}\_{m}(\boldsymbol{z}) d\mathbf{Q}\_{m}(\boldsymbol{z})/d\boldsymbol{z} + \boldsymbol{z}^{-1} \mathbf{Q}\_{m}(\boldsymbol{z}) d\mathbf{P}\_{m}(\boldsymbol{z})/d\boldsymbol{z} \right] \\ &= \mathbf{g}\_{m,0}^{\prime} \boldsymbol{z}^{-1} [\boldsymbol{P}\_{m}(\boldsymbol{z}) d\mathbf{Q}\_{m}(\boldsymbol{z})/d\boldsymbol{z} + \boldsymbol{\Psi}\_{m}(\boldsymbol{z}) \mathbf{Q}\_{m}(\boldsymbol{z})] \end{split} \tag{41}$$

where

$$\frac{dQ\_m(z)}{dz} = -q\_{m,1}z^{-2} - 2q\_{m,2}z^{-3} - \dots - N\_Q q\_{m,N\_Q} z^{-\binom{N\_Q+1}{2}}.$$

$$\Psi\_m(z) = -z^{-1}P\_m(z) + dP\_m(z)/dz$$

$$\frac{dP\_m(z)}{dz} = P\_m(z) \sum\_{\substack{i=-M\\i \neq m}}^{M+1} \sum\_{k=0}^K \frac{z\_{i,k}z^{-2}}{1 - z\_{i,k}z^{-1}}.$$

Eq. (41) can be written in a matrix form as follows:

$$d T\_{m,0}^{Q}(z)/dz = \boldsymbol{\Psi}\_{m} \boldsymbol{\mathfrak{x}}\_{Q,m}.\tag{42}$$

where

$$\begin{aligned} \boldsymbol{\Psi}\_{\mathfrak{m}} &= \begin{bmatrix} \boldsymbol{\Psi}\_{\boldsymbol{m},0}(\boldsymbol{z}) & \boldsymbol{\Psi}\_{\boldsymbol{m},1}(\boldsymbol{z}) & \cdots & \boldsymbol{\Psi}\_{\boldsymbol{m},N\_{Q}-1}(\boldsymbol{z}) & \boldsymbol{\Psi}\_{\boldsymbol{m},N\_{Q}}(\boldsymbol{z}) \end{bmatrix}, \\ \boldsymbol{\Psi}\_{\boldsymbol{m},n}(\boldsymbol{z}) &= \boldsymbol{\mathcal{g}}\_{\boldsymbol{m},0}^{\prime} \boldsymbol{z}^{-(n+1)} \left[ \boldsymbol{\Psi}\_{\boldsymbol{m}}(\boldsymbol{z}^{-1}) - \boldsymbol{n} \boldsymbol{z}^{-1} \boldsymbol{P}\_{\boldsymbol{m}}(\boldsymbol{z}^{-1}) \right], & n = \mathbf{0}, \mathbf{1}, \ldots, N\_{Q}. \end{aligned}$$

#### **4.2 Optimization criteria**

Selections of the object and functions and constraints can be very different depending on the optimization criteria scenario. Herein will be considered three criteria summarized in **Table 1** [21, 24]. In the first Criterion 1, the cost function in which absolute values are minimized is the transfer function *T<sup>Q</sup> <sup>m</sup>*,0ð Þ*z* in the stop bands.


#### **Figure 10.**

*Frequency responses for the basic (T*3,0ð Þ*<sup>z</sup> ) and reshaped (T<sup>Q</sup>* 3,0ð Þ*z ) transfer function for K* ¼ 2*, for fs* ¼ 1*:*6 kHz*, NQ* <sup>¼</sup> <sup>16</sup> *and (a) lSB* <sup>3</sup> ∈f g 0*:*1, 0*:*01, 0*:*001 *and (b) TVG* j j*max* ∈f g 24, 32, 48 *.*

Weighting function *Wm*ð Þ *ω* is settled to 1. The function which absolute values are kept under settled limits is an error in the passband *Em*ð Þ¼ *zi <sup>T</sup><sup>Q</sup> <sup>m</sup>*,0ð Þ*z z*¼*zi* � *<sup>T</sup><sup>d</sup> <sup>m</sup>*ð Þ *zi* where

*Td <sup>m</sup>*ð Þ¼ *zi* exp ð Þ �*jτ ω*ð Þ *<sup>i</sup>* � *ω<sup>m</sup> :* In addition to this, a control of overshoots in the transition bands is performed.

Criterion 2 is similar with Criterion 1 with the difference that the absolute value of the TVG in the harmonic frequency is limited, that is *dT<sup>Q</sup> <sup>m</sup>*,0ð Þ*z =dz z*¼*zm* ≤j j *TVG max*,

where j j *TVG max* is a maximally allowed absolute total vector gradient. Like in Criterion 1, the limitation of overshoots in the transition bands is necessary.

Criterion 3 minimizes the absolute value of the TVG in the harmonic frequency subject to the limitation of the gain in the stopband. Similarly with the previous cases, the limitation of overshoots in the transition bands is necessary.

*Perspective Chapter: Cascaded-Resonator-Based Recursive Harmonic Analysis DOI: http://dx.doi.org/10.5772/intechopen.108402*

#### **4.3 Design example**

In order to illustrate the described algorithms, examples overtaken from [21] are shown for Criteria 2 and 3 defined in **Table 1**. **Figures 10** and **11** show the frequency responses of the transfer function of the third harmonic *T<sup>Q</sup>* 3,0ð Þ*z* in the case of *K* ¼ 2. For Criterion 3, the maximum allowed gains in the stopband was selected as *l SB <sup>m</sup>* ∈ f g 0*:*1, 0*:*01, 0*:*001 , which corresponds to attenuations of 20, 40, 60 f g dB. For Criterion 2, the maximum value of j j *TVG* in the harmonic frequencies *zm*, j j *TVG max* ∈ f g 24, 32, 48 was selected. Zoomed amplitude and j j *TVG* characteristics around the harmonic frequency are shown in the inset figures at the bottom of the figures. It can be seen that the higher value of *NQ* gives a smaller value of j j *TVG* and wider bandwidth. It is also visible that for smaller values j j *TVG* sidelobes are larger,

#### **Figure 11.**

*Frequency responses for the basic (T*3,0ð Þ*<sup>z</sup> ) and reshaped (T<sup>Q</sup>* 3,0ð Þ*z ) transfer function for K* ¼ 2*, for fs* ¼ 1, 6 kHz*, NQ* <sup>¼</sup> <sup>32</sup> *and (a) lSB* <sup>3</sup> ∈f g 0*:*1, 0*:*01, 0*:*001 *and (b) TVG* j j*max* ∈f g 24, 32, 48 *.*

#### **Figure 12**.

*Frequency responses for the basic (T*3,0ð Þ*<sup>z</sup> ) and reshaped (T<sup>Q</sup>* 3,0ð Þ*z ) transfer function for K* ¼ 1*, for fs* ¼ 1, 6 kHz*, NQ* <sup>¼</sup> <sup>16</sup> *and (a) lSB* <sup>3</sup> ∈f g 0*:*1, 0*:*01, 0*:*001 *and (b) TVG* j j*max* ∈f g 16, 24, 32 *.*

which reduces robustness to interharmonics and noise. In addition, the bandwidth increases for smaller values of j j *TVG* .

**Figure 12** shows the frequency responses of the transmission functions *T*3,0ð Þ*z* and *TQ* 3,0ð Þ*z* for different values of given parameters (a) *l SB <sup>m</sup>* and (b) j j *TVG max*, in the case of *K* ¼ 1. In this case, the total j j *TVG* is smaller than in the case of *K* ¼ 2. The order of the compensation filter *NQ* ¼ 16 is smaller, so the bandwidth is narrower. In the case of *l SB* <sup>3</sup> ¼ 0*:*001, the optimization problem has no solution.

#### **5. Conclusions**

CR-based algorithms for harmonic analysis and estimation of harmonic phasors are described in this chapter. The resulting filters for extracting harmonic signals can be of the FIR or IIR type. Algorithms for the optimization of frequency responses are presented and corresponding examples of synthesis are given. Linearized mathematical models were used, which enabled the use of linear optimization methods such as LP and CLLS. When designing the IIR analyzers, a linearized iteration scheme based on the Rouche's theorem was used to control the stability of the system. As for the optimization algorithms, they can potentially be improved by various modifications such as, for example, by nesting optimization loops related to different constraint conditions and/or objectives, and adaptation of iteration steps. It is notable that approximating result could be obtained heuristically thanks to the characteristic position of the pole and zeros. In addition, it seems that closed-form calculation expressions derivation could be possible. On the other hand, the FIR-type algorithm particularly optimizes frequency responses through the postprocessing compensation FIR filters applied to the output signals obtained by the CR structure. This approach allows the usage of a set of compensation filters corresponding to different signal dynamics.

### **Conflict of interest**

The author declares no conflict of interest.

### **Author details**

Miodrag D. Kušljević Termoelektro Enel ad, Belgrade, Serbia

\*Address all correspondence to: miodrag.kusljevic@te-enel.rs

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Perspective Chapter: Cascaded-Resonator-Based Recursive Harmonic Analysis DOI: http://dx.doi.org/10.5772/intechopen.108402*

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### *Edited by Mohammad Younus Bhat*

In the world of physical science, important physical quantities like sound, pressure, electrical current, voltage, and electromagnetic fields vary with time. Such quantities are labeled as signals/waveforms and include oral signals, optical signals, acoustic signals, biomedical signals, radar, and sonar. Time-frequency analysis is a vital aid in signal analysis, which is concerned with how the frequency of a function (or signal) behaves in time, and it has evolved into a widely recognized applied discipline of signal processing. This book discusses the Fourier transform (FT), which is one of the most valuable and widely used integral transforms that converts a signal from time versus amplitude to frequency versus amplitude. It is one of the oldest tools in the time-frequency analysis of signals. The book includes five chapters that discuss general Fourier transforms as well as new and novel transforms such as hybrid transforms, quadratic-phase Fourier transforms, fractional Fourier transforms, linear canonical transforms, and more.

Published in London, UK © 2023 IntechOpen © Daniel Olah / iStock

Time Frequency Analysis of Some Generalized Fourier Transforms

Time Frequency Analysis

of Some Generalized

Fourier Transforms

*Edited by Mohammad Younus Bhat*