**2. Theory of relationships**

In this approach, relations between various parameters of the radio sources would be derived based on established laws and theories under certain assumptions. The theories of relationships are outlined.

#### **2.1 Theory of temporal evolution in radio sources**

The standard relationship for an ideal temporal evolution model of extragalactic radio sources can be studied using the relationship between the two key parameters known as the observed linear size (D) and the spectral luminosity (Pv) in a general power-law function as [22]:

$$D \approx D\_0 P\_v^{\pm q} \tag{1}$$

where D0 (a constant) represents D at z � 0 while the slope (q) is the temporal evolution parameter.

Meanwhile, from the Friedmann-Robertson-Walker universe, the radio angular size–redshift ð Þ θ–z relation is described [23, 24] as

$$\theta = \frac{D(\mathbf{1} + \mathbf{z})^2}{d\_L},\tag{2}$$

where D can be represented using the unit of kpc and dL is the luminosity distance expressed [24, 25] as

$$d\_L = \frac{2c}{H\_0 \Omega\_0^2} \left\{ \Omega\_0 x + (\Omega\_0 - 2) \left[ (\Omega\_0 x + 1)^{\frac{1}{2}} - 1 \right] \right\},\tag{3}$$

and c is the speed of light. Moreover, the radio luminosity (P) of a radio source at redshift (z) can be defined as a function of the spectral flux density (*Sv*) at observing frequency (*ν*) according to the [26] as:

$$P = P\_0 (\mathbf{1} + z)^{\beta} \tag{4}$$

where α stands for spectral index. Thus, from [27] work, the relationship between P and z can be approximated to a simple power-law function of the form:

$$P = P\_0(\mathbf{1} + z)^{\beta} \tag{5}$$

where β is the slope of the P–z which is supposed to be constant over all values of z for a given sample of sources. In this scenario, a significant correlation for all values of z in the line with β >0 is expected in a P–z data of different samples of extragalactic radio sources, due to luminosity selection effects in flux density limited samples [27].

Besides, researchers [23, 24, 28] suggest that linear sizes of extragalactic radio sources evolve with cosmological epoch in the form:

$$D \approx D\_0 (\mathbf{1} + \mathbf{z})^{-k} \tag{6}$$

where D0 is the normalized linear size which depends on the assumed cosmology and k represents the evolution parameter which could be a function of both cosmological evolution and luminosity selection effects.

Alternatively, with different values of Ω0, the observed θ–z data of extragalactic radio sources deviate significantly from the standard Friedmann models [29], as a result of an entanglement of two effects, namely linear size evolution [29] and luminosity selection effect [30]. Hence, the linear size evolution of extragalactic radio sources can therefore be expressed as a function of both redshift and luminosity in the form of [24, 30, 31]

$$D\_{(\mathcal{P}, \mathfrak{z})} \approx P^{\pm q} (\mathfrak{1} + \mathfrak{z})^{-m} \tag{7}$$

where m is the residual cosmological evolution parameter defined [24, 27] as:

$$\mathbf{m} = \mathbf{k} \pm \mathbf{q} \clubsuit \tag{8}$$

When the effect of luminosity is controlled and this depends on the product of temporal evolution (q) and luminosity selection effect (β).

It is certain from above relation that if luminosity selection effect is above cosmological evolution, then, when the effect of luminosity is corrected, m < 0, so that Eq. (8) becomes

$$
\mathbf{k} - \mathbf{q}\boldsymbol{\beta} < \mathbf{0} \tag{9}
$$

and *q*> *k β*. For such sources, a significant positive D–P correlation is envisaged, suggesting a linear relationship between the two parameters. On the other hand, if there is a residual linear size evolution at certain values of k, q < 0 and m > 0, then *q*< *k β*, indicating a significant D–P anti-correlation and also implies that the luminosity decreases with the expansion of the sources. Hence, the values of k are expected to vary in a given sample. In this scenario, the temporal evolution model for an assumed cosmology can be constrained using the D–z plane.

It is also believed that D should not increase with z for all z. Although, considering the fact that radio sources are not just rigid rods and the θ–z plane depends on the assumed cosmology, which is also, characterized by the value of Ω0, in principle, this

may not always be the case. Hence, the P–D track of all inclusive radio source samples could be defined [24, 29] as:

$$P = P\_{\text{max}} \exp\left[\pm \left(\frac{D}{D\_c} - 1\right)^2\right] \tag{10}$$

where P max is the maximum luminosity and Dc is the critical linear size at which the P max is emitted by a radio source. Rearranging Eq. (10) gives

$$D = D\_c \left[ \mathbf{1} \mp \left\{ \ln \left( \frac{P}{Pc} \right) \right\}^{1/2} \right] \tag{11}$$

Eq. (11) comprises of two separable components corresponding to k <0 ð Þ � and k> 0 ð Þ þ *:* The D–P relation (c.f. Eq. (11)) is shown in **Figure 7**.

However, using Eq. (3) in (2), the linear size can be defined as:

$$D = \frac{D\_0 \left\{ \Omega\_0 \mathbf{z} + (\mathbf{\Omega}\_0 - \mathbf{\mathcal{I}}) \left[ (\mathbf{\Omega}\_0 \mathbf{z} + \mathbf{1})^{1/2} - \mathbf{1} \right] \right\}}{\left(\mathbf{1} + \mathbf{z}\right)^2},\tag{12}$$

where *D*0 ¼ 2*cθ H*0Ω2 0 is the intrinsic linear size and a constant. Hence, for inflationary universe, Ω0 ¼ 1, [32], the linear size of a radio source depends on z in the form [27]:

$$D \approx \frac{D\_0 \left\{ \left( 1 + z \right) - \sqrt{1 + z} \right\}}{\left( 1 + z \right)^2} \tag{13}$$

Eq. (13) indicates two components of z: z < 1 and z > 1, where D increases with increasing z in the first so that k > 0, while the revised is the case in the last component for all D and k. On the other hand, assuming Ω0 ¼ 0, on supposition of a low-density universe, for which dL is given as [33]:

$$d\_L = \frac{2c}{H\_0} \left[ (\mathbf{1} + \mathbf{z})^2 - \mathbf{1} \right],\tag{14}$$

Eq. (2) yields,

$$\theta = \frac{DH\_0(\mathbf{1} + z)^2}{2c\left[\left(\mathbf{1} + z\right)^2 - \mathbf{1}\right]}.\tag{15}$$

Hence, D can be expressed in relation with z as

$$D = D\_0 \left[ 1 - \frac{1}{\left(1 + z\right)^2} \right] \tag{16}$$

where the intrinsic radio size, *D*0 ¼ *θ*2*c H*0 . The variation of D with z for both cosmological models is shown in **Figure 3**.

It is obvious from **Figure 3** that there is increase in D as z increases up to a certain maximum point known as a critical value of DC ≈0*:*146 D0 (kpc) at a redshift

*Evolution of Radio Source Components and the Quasar/Galaxy Unification Scheme DOI: http://dx.doi.org/10.5772/intechopen.106244*

**Figure 3.** *Variation of D with z for* Ω0 ¼ 1 *(a) and* Ω0 ¼ 0 *(b) cosmologies.*

maximum called critical zc ≈1, and decreases after for Ω0 ¼ 1. This implies that for parameters, z < 1, k > 0 and q > 0, there is a positive temporal evolution, while for z >1, k < 0 and q < 0 a negative temporal evolution is envisaged which is in good agreement with the predictions of Eq. (13).

On the other hand, there is increase in D up to a critical point Dc≈0*:*8D0 after which it remains constant for Ω0 ¼ 0. It is now clear that in the two cosmological models, there is indication of Dc at which luminosity is maximum, suggesting that for any assumed cosmological model, the Dc value obtained from D–P turnover can still be found using the D–z plane of the same data. Hence, in any assumed cosmological model, there will be an expected range of the Dc value of 0*:*15D0 to 0*:*8D0 bounded by the two cosmologies. In this scenario, temporal evolution in the current sample of EGRSs would be modeled in terms of the current inflationary one with Ω0 ¼ 1.

#### *2.1.1 Relativistic beaming based on orientation and radio source asymmetries*

The standard relation for explaining an ideal relativistic beaming and orientation effects for extragalactic sources is often carried using a key parameter known as coredominance, (R) defined [34] as:

$$R = \frac{P\_C}{P\_E} = \frac{P\_{\text{SGHz}}^C}{P\_{1.4GHz}^E} \left( \imath \nearrow \right)^{-a\_E} (1+z)^{-a\_E} \tag{17}$$

where PC represents core-luminosity at 5 GHz, PE is the extended/lobe luminosity at 1.4 GHz and *αE* is the lobe spectral index. However, if relativistic beaming effect at small orientation angle, then R can be expressed in terms of the jet speed (β) and inclination angle (*ϕ*) in the form [35]:

$$R = \frac{P\_C}{P\_E} = \frac{R\_T}{2} \left[ (\mathbf{1} - \beta \cos \phi)^{-n+a} + (\mathbf{1} + \beta \cos \phi)^{-n+a} \right] \tag{18}$$

where *RT* is the value of R at *ϕ* ¼ 90° and n is a parameter that depends on the assumed flow model of the radio jet. For radiating plasma with continuous jet n = 2, otherwise n = 3 if the jet consists of blobs.

An obvious outcome of relativistic beaming and orientation effects in AGNs is the wide range of asymmetry observed in their radio structures. The radio source

asymmetry can be explained using the arm-length ratio (Q), defined as the ratio of the distance, from the central engine, of a plasma element emitting radio waves on the approaching jet side (dapp) to that on the receding jet side (drec), [36, 37] given as:

$$Q = \frac{d\_{app}}{d\_{rec}} = \frac{1 + \beta \cos \phi}{1 - \beta \cos \phi} \tag{19}$$

On the other hand, [38, 39] suggested that,

$$x = \frac{Q - 1}{Q + 1} \tag{20}$$

where, *x* represents the index of the asymmetry.

This *x* parameter which believed to have better relationship with orientation when compared to Q can further be defined in terms of the viewing angle as [40, 41]:

$$
\infty = \beta \cos \phi \tag{21}
$$

The relativistic beaming in AGN at small angles to the line-of-sight is fundamentally characterized by beaming enhancement factor (δ) expressed [40, 42] as:

$$\delta = \chi^{-1} (1 - \beta \cos \phi)^{-1} = \tag{22}$$

where, *γ* is the bulk Lorentz factor of the jet [8, 42, 43] defined as:

$$\chi = \frac{1}{\left(1 - \rho^2\right)^{\frac{1}{2}}} \tag{23}$$

Also, assuming ϕ ¼ 00 in Eq. (18) and analyzing further, [44, 45] gives

$$R\_{\text{max}} \approx R\_T \gamma^2 \left(2\gamma^2 - 1\right) \approx 2R\_T \gamma^4 \tag{24}$$

While at α ¼ 0 and β � 1, for high luminosity radio-loud AGN sources emitting at small angle to the line-of-sight of the observer, (to a first approximation) Eq. (18) reduces to

$$\cos \phi\_m = \mathbf{1} - \left(\frac{2Rm}{R\_T}\right)^{-\frac{1}{u}} \tag{25}$$

where Rm is the mean value of the R-distribution and ϕm is the mean observation angle.

Thus, it can be shown from Eq. (23) that the asymmetry parameter *x* can be expressed in terms of the beaming enhancement factor as:

$$\boldsymbol{\omega} = \frac{\delta \boldsymbol{\gamma} - \mathbf{1}}{\delta \boldsymbol{\gamma}} \tag{26}$$

Eq. (26) implies that there is an association of relativistic beaming and radio source asymmetry. In asymmetric sources with Q > 1, *x*–D anti-correlation is expected if

geometric projection at small viewing angles is responsible for the observed asymmetry. Following [40, 41], we assume a linear *x*–D relation of the form:

$$
\infty = \varkappa\_{\text{max}} - \lambda D,\tag{27}
$$

where *x max* represents the maximum *x* for a sample at D � 0ð Þ ϕc 6¼ 0 to the lineof-sight [44] and λ is the slope. Thus, if β � 1 for the relativistic jets, analysis of Eq. (20) down to Eq. (28) for optimum beaming gives [40, 45]:

$$
\Phi\_c \approx \sin^{-1}(\mathbb{Y}\_r) \approx \cos^{-1} \mathbb{x}\_m,\tag{28}
$$

Hence, if relativistic beaming at small viewing angles is responsible for the observed structural asymmetry, the critical viewing angle ϕ*c* as well as the Lorentz factor γ can be obtained using the *x*–D data.

#### **2.2 Statistical analyses and results**

#### *2.2.1 The source samples*

The data used in the present analysis were drawn from a well-defined source sample of [46] compilation which contains required information on the two objects of interest–ESS quasars and ESS radio galaxies, [34] compilation of 542 extragalactic radio sources and the deep VLA sample of FSRQs compiled by [47]. In these samples, there is wide dispersion in the distributions of the observed parameters.

#### *2.2.2 Distributions of observed radio parameters*

The distributions of the linear size, D, of the extended steep-spectrum sources (ESSs) in logarithm scales are represented in **Figure 4a**. The graph shows D-values of 1896.3, 17.1 and 1879 kpc for the maximum, minimum and range respectively for ESS quasars, while D-values of 5853.30, 29.80 and 5823.5 kpc were obtained for maximum, minimum and range respectively for ESS galaxies. The entire sample yields D-values of 5853.3, 17.1 and 5836.2 kpc for maximum, minimum and range respectively. Further analyses yield median D-values of 148.90, 323.59 and 201.90 kpc for ESS quasars, ESS galaxies and entire sample respectively. The mean D-values of 144.54 � 7.52 kpc and 288.40 � 33.66 kpc were obtained respectively for ESS quasars and radio galaxies.

The distributions of the redshift, z of the ESSs in logarithm scales are represented in **Figure 4b**. The graph shows z-values of 2.88, 0.05 and 2.83 for the maximum, minimum and range respectively for ESS quasars, while z-values of 3.22, 0.006 and 3.21 were obtained for maximum, minimum and range respectively for ESS galaxies. The entire sample yields z-values of 3.22, 0.006 and 3.21 for maximum, minimum and range respectively. Further analyses yield median z-values of 1.89, 1.17 and 1.62 for ESS quasars, ESS galaxies and entire sample respectively. The mean z-values of 1.95 � 0.01 and 1.30 � 0.02 were obtained respectively for ESS quasars and radio galaxies.

The distributions of the radio luminosity, P of the ESSs in logarithm scales are represented in **Figure 4c**. The graph shows P-values of 27.90, 25.71 and 2.18 WHz�1 for the maximum, minimum and range respectively for ESS quasars, while P-values of 28.01, 24.97 and 3.04 WHz�1 were obtained for maximum, minimum and range respectively for ESS galaxies. The entire sample yields P-values of 28.01, 24.97 and

**Figure 4.** *Distribution of (a) D, (b) z and (c) P respectively for ESS quasars (lines) and ESS galaxies (plane).*

3.04 WHz�1 for maximum, minimum and range respectively. Further analyses yield median P-values of 27.04, 26.39 and 26.84 WHz�1 respectively for ESS quasars, ESS galaxies and the entire data. The mean P-values of 27.01 � 0.01 WHz�1 and 26.39 � 0.02 WHz�1 were obtained respectively for ESS quasars and radio galaxies **Figure 4**.

### *2.2.3 D: P/z correlation*

**Figure 5** represents the scatter plot of linear size, D against the redshift, z. The median value data in seven redshift bins is superimposed on the plot. Critical investigation of the plot shows that on average, the linear size increases with increasing redshift up to a value logDc = 2.5 kpc (Dc = 316.23 kpc) at zc ¼ 1, after which it

*Evolution of Radio Source Components and the Quasar/Galaxy Unification Scheme DOI: http://dx.doi.org/10.5772/intechopen.106244*

**Figure 5.** *Scatterplot of logD (kpc) against z for entire sources (circle) with median values (square) superimposed [24].*

#### **Table 1.**

*D–P/z regression analyses results for both z < 1 and* z≥1*=*P<Pc *and* P ≥Pc *[24].*

decreases with increasing redshift [24]. This is in agreement with the prediction made in **Figure 3(a)**. Hence, the present data obviously proved consistency with the inflationary model of the universe ð Þ Ω0 ¼ 1 . The median values give significant trends with correlation coefficients of +0.95 and � 0.90 for zc ¼ 1 and zc ≥1 respectively. Results of the regression analyses of the D–P/z data for zc ¼ 1 and zc ≥1 are summarized in **Table 1**.

In modelling the temporal evolution of the sample, **Figure 6** represents the scatter plot of projected linear size (D) and the radio luminosity (P). Similarly, the median value data in eight uniform luminosity bins are superimposed on the plot. There is an obvious trend indicating that the linear size first increases

with increasing luminosity up to a certain value and thereafter decreases with increasing luminosity. The median value data showed very significant trends. This suggests that the turnover occurs at a critical point of luminosity, logPc ¼ 26*:*33 WHz�1 and logDc = 2.51 kpc (316.23 kpc) [24]. A summary of the results of these regression analyses of the D–P and D–z data for sources with P ≤ Pc and P > Pc is presented in **Table 1**.

The results in **Table 1** did not show any obvious trend in D–P relation for sources with P< Pc. However, for sources with P≥Pc, there is a fairly strong correlation. Therefore, the weak trend found in the region below z = 1 and P< Pc is believed to be due to the effects of luminosity selection. In this scenario, the low redshift samples, z < 1 have more impacts on average luminosity-redshift plane than the high redshift, z≥ 1 counterparts in any flux density limited samples.

#### **Figure 6.**

*Scatterplot of logD(kpc) against logP (*WHz‐1*) for both sources with* P≤Pc *(circle),* P≥Pc *(square) and median (triangle) values superimposed [24].*

#### *2.2.4 Luminosity selection effect on temporal evolution model*

**Figure 7** represents the P–z scatter plot of the sample. There is a turnover at critical point logPc ¼ 26*:*33 W/Hz and z � 0.05 in the P–z plane. This point of P is presumably the value of luminosity in **Figure 6** that will correspond to Dc at zc � 1. However, the critical luminosity, Pc found in the P–z plane is inconsistent with sources at z = 1. Hence, the luminosity-redshift relation of the current data did not assume Ω0 ¼ 1 model but Ω0 ¼ 0 cosmological model (low-density universe) [24].

In **Figure 6**, the effect of Eqs. (10) and (11) in the light of the temporal evolution model was considered. Hence, the data are grouped into two; P < Pc and P≥Pc. The results of the regression analyses are shown in **Table 1**. These opine that the linear size, D of radio sources increases up to critical radio luminosity, logPc ¼ 26*:*33WHz�1 and decreases thereafter. This is in agreement with Eq. (10), hence suggesting that eqn. (10) is approximately correct to zero order. This is applicable when critical linear size Dc is used. Dc ¼ 316kpc is obtained from the D - z plane at the turning point. This Dc corresponds to theoretical Dc ¼ 0*:*14D0 at zc ¼ 1 for Ω0 ¼ 1 in **Figure 3a** and logPc = 26.33 WHz�1 in **Figure 7**. The indication of this is that D0 (� 2100 kpc in the

**Figure 7.** *Scatterplot of logP (W/Hz) against log (1 + z) for all sample.*

### *Evolution of Radio Source Components and the Quasar/Galaxy Unification Scheme DOI: http://dx.doi.org/10.5772/intechopen.106244*

observed data) is approximately the linear size at the earliest epoch (at z � 0.02 in the observed data of the present sample). The correlation coefficient, r � +0.4, �0.5 and � 0.9 respectively for sources with P < Pc, P ≥ Pc and the median values respectively were obtained. These suggest that there are positive and negative correlations in the D–P track at P < Pc and P≥Pc respectively, indicating that the Dc ¼ 316kpc of the observed samples is consistent with Dc = 0.14D0 just in accordance with the theory for the inflationary model of the universe, Ω0 ¼ 1 only, at zc ¼ 1 and Pc ¼ 26*:*33WHz�1 , proving Eq. (11) to be perfectly correct. Hence, temporal evolution in extragalactic radio sources can be explained.

### *2.2.5 The* x *– D relationship*

There is a fairly significant trend in the *x*–D plot, which is obvious at the upper envelope function. This yields: x = 0.35–0.0006 D with a correlation coefficient r � � 0.5, chance probability ρ � 10�10 and critical viewing angle, *ϕ*c ≈70o, which corresponds to γ ¼ 1*:*1. The upper envelope function gave correlation coefficient, r � 0.9, *ϕ*c ≈ 48o and γ ¼ 1*:*3. The analyses for separate objects, radio galaxy and quasar sub-samples, of upper envelope functions yield *ϕ*c ≈59o; γ = 1.2 and *ϕ*c ≈33o; γ ¼ 1*:*8, for radio galaxies and quasars, respectively. These results correspond to angular separation *ϕ*sep of � 26o , based on the upper envelope functions [40]. The results are shown in **Table 2** and **Figure 8**.

The results opine that the relativistic beaming and source orientation effects are the major cause of large-scale structural asymmetries observed among powerful

#### **Table 2.**

x*–D regression analyses results [40].*

#### **Figure 8.**

*Scatterplot of the fractional separation difference (x) as against projected linear size (D) for quasars (•) and radio galaxies (Δ) [40].*

extragalactic radio sources, which are more obvious in core-dominated sources with large core-to-lobe luminosity ratio.

#### **2.3 General discussions and conclusions**

#### *2.3.1 Discussions*

The study of the redshift effect dependence of radio size in extragalactic radio sources has been of great interest since inception of the universe. This is obviously important in cosmological studies more especially in testing of the evolution in the extragalactic radio sources. In this scenario, a theoretical model that can best interpret the D–P track is developed. According to [48] the analyzes obtained, the high redshift radio sources show high bending angles, distorted and smaller structures than the low redshift radio sources. According to the researchers, the effects on the radio source evolution depend mostly on intracluster/ or interstellar medium. A plausible model was first developed by [49] for strong linear size evolution for radio galaxies. This model explained that the typical radio sources, both giant galaxies and sub-galactic quasars, evolve at high redshifts, z.

The temporal evolution of radio linear size could be interpreted in the light of observed D–z plane for radio source samples. Hence, the D–z correlation helps in constraining cosmological/temporal models. According to [50], the amount of linear size evolution required to interpret the observed θ–z data can be in the range of k = 2.0 and 1.5 to k = 1.2 and 0.75 for Ω ¼ 1 and 0 respectively. A clear investigation from the median values in **Figure 5** shows that the D–z curve of the observed data is in good agreement with that of the theoretical D–z plane of **Figure 3**. In other word, for Ω ¼ 1, there is an increase in the linear size of radio sources up to a certain point known as critical linear size, Dc which corresponds to critical luminosity, Pc at critical redshift, zc � 1 and thereafter decreases. This steep change in P–z relation of extragalactic radio sources may occur around z = 0.3 [27, 33] or z = 1 (see also [46], Fig. 2) [51]. This is due to the luminosity selection effect occurred as a result of Malmquist bias in flux density limited samples [27]. In this effect, the present work adopted samples based on zc ¼ 1 as predicted earlier with the theory. This raise and fall of the radio sources in the D–z plane are consistent with the observational results showing that radio galaxies and quasars exhibit different D–P correlations.

In this scenario, the median values of D–P of the entire data superimposed in **Figure 6** clearly show that the observed D–P data of the sample are consistent with the theoretical curves (c.f. **Figure 3a**). Hence, this suggests that the P–D data of the present sample could be used to constrain the temporal evolution model.

A cursory look in **Table 1** shows that there is steep change in q as predicted above in the subsample around z ≥ 1 with fairly trends with a significant correlation in the median values obtained from the eight appropriate luminosity bins of the entire sample. The results show that the apparent lack of D–P correlation in the samples below z = 1 was due to the luminosity selection effects in the observed data. In other words, in flux density limited samples, the low redshift (z < 1) sources exhibit more dependence on luminosity as a function of redshift than the high redshift (z > 1) source counterparts. Below z = 1, the P–z weak correlation is expected, while Beyond z = 1, the luminosities have a much milder dependence on z. The regression analyses yield q = +0.002 for all samples around z ≤ 1 and q = �1.59 at z = ≥ 1. Hence, at z < 1, there is a positive temporal evolution model while at z ≥ 1, negative temporal evolution is obtained for zc � 1 and Ω0 ¼ 1 as predicted. This shows that the temporal

*Evolution of Radio Source Components and the Quasar/Galaxy Unification Scheme DOI: http://dx.doi.org/10.5772/intechopen.106244*

evolution model can be well understood using Ω0 ¼ 1 than Ω0 ¼ 0. In this scenario, the temporal evolution is positive when z < 1, k > 0 and q > 0, and otherwise for z > 1, k < 0 and q < 0. Thus, the linear size increases as a function of radio luminosity up to a maximum value called critical D-value, Dc ¼ 316kpc in consistence with the maximum theoretical linear size, D max ¼ 0*:*14D0 at zc � 1 and Pc ¼ 26*:*33WHz�1 and thereafter decreases with increasing luminosity, as predicted earlier in this work.

Several authors have argued that the unified scheme will vanish out if the derived positive D–P dependence for radio galaxies (D � Pq for q � 0.3) contrasts with the negative D–P dependence in quasars [52]. Ubachukwu and Onuora [50] suggested an inverse correlation of D–P of the form D � P�0*:*52. They found out that radio galaxies locate at low redshifts with q � 0.3 [31], while radio quasars located at high redshifts with q � �0.5 [48]. The D–P turnover in extragalactic radio sources occurs around Dc = 100kpc [29]. It was argued [24] that at D � 1Mpc and critical luminosity, Pc � 26WHz�1 , extragalactic radio sources evolve and thereafter decrease.

In *x*–D regression analyses, there is an apparent lack of *x*–D anti-correlation in the entire samples and radio galaxies subsample. There was a fairly *x*–D anti-correlation in quasars subsample which was obvious in the upper envelope function with r � �0.5, 0.30 and � 0.3 for the entire sample, radio galaxies and quasars respectively. Hence, the results suggest that orientation effects may be more necessary in explaining the properties of the radio quasars population than that of the radio galaxies populations in which the expected *x*–D anti-correlation was not observed and also implies that the two objects are the same, the difference is just the angle at which the observer took in viewing them.


**3. Results**

#### *3.1. Radiographical Evaluation of the Freeze Foaming Process*

By acquiring two 2D pictures per second (500 ms exposure time) a real-time observation and analysis of the foaming process is possible. Due to the superposition of structural phenomena, the thickness of the sample was reduced to 5 mm for radiographically evaluations. To qualify the changes between the steps, manual tracking is conducted by overlaying each picture with a grid, and following the movement of distinct points in the sample. The resulting coordinates can be converted to changes in actual values for the size or height of the foam, and can be correlated with the pressure at the moment of picture acquisition. This method of evaluation was applied for three different sample temperatures and a pressure reduction rate of 50 and 10 mbar/s, respectively. As an example, the plotted results for 10 mbar/s are shown in Figure 3, as experiments at different pressure reduction rates behave similarly.

**Figure 3.** Percentage growth as a function of pressure for 5, 23, and 40 ◦C, at 10 mbar/s.

Independent of the initial suspension temperature, foaming starts between 450 and 550 mbar. However, the growth, as well as the pressure at which the foaming stops, are highly influenceable by the initial temperature. A lower temperature leads to an inhibited foaming, which results in a lower overall growth. Even though the foaming process itself continues up to lower pressures, the lower foaming rate cannot be compensated. For example, a suspension with a pressure reduction rate of 10 mbar/s and an initial temperature of 5 ◦C stops foaming at 15 mbar, while a 40 ◦C suspension already stops at 35 mbar. Suspensions undergoing 50 mbar/s exhibit a very similar behavior. Looking at foaming rates over pressure, the highest suspension temperature also results in the highest growth rates values at higher pressures (Figure 4).

**Figure 4.** Foaming rate as a function of pressure for different temperatures at 10 mbar/s.

Samples with an initial temperature of 23 ◦C exhibit a lower maximum at lower pressures while 5 ◦C samples showed an almost constant foaming rate and, therefore, no identifiable maximum.

To evaluate and verify the findings from the radiographical evaluation, the results were compared to data obtained from the freeze dryer at IKTS (Table 1) [24]. Apart from 5 ◦C suspensions, their findings support the trend identified at the ILK. For suspensions with an initial temperature of 5 ◦C, the IKTS identified a foaming rate maximum at pressures between 40 and 60 mbar. However, the freeze dryer is equipped with an additional condenser, which is not available in the in situ device. It is not possible, so far, to achieve the foaming rate maximum and finish the pressure-induced foaming process.

**Table 1.** Comparison of the pressures of beginning, end, and maximum of foaming at temperatures of 5, 23, and 40 ◦C by in situ μCTs (ILK) and by freeze dryer (IKTS) [24].

Through analysis of radiographic images, edge effects on the foaming process can be verified. Figure 5 illustrates their impact on a 10 mbar/s sample at 23 ◦C. Both edges, as well as the center, were manually tracked using the method described earlier, and the local growth rates were determined. As expected, the edges exhibit a much slower growth when compared to the center. Possible reasons include wall friction, as well as a drying of the suspension. This behavior is especially observable in suspensions with an initial temperature of 40 ◦C, which also develops a compact layer on top of the suspension.

**Figure 5.** Edge effects on foaming at 23 ◦C (10 mbar/s).

#### *3.2. CT Evaluation of the Freeze Foaming Process*

To examine the developmental stages of foaming, μCT and the improved testing device is used to create a virtual and reconstructed volume (VGStudio 2.0, Volume Graphics GmbH, Heidelberg, Germany) of the foam's structure. During the CT measurements, the sample is rotated 360◦, and after an angle of 0.25◦, one image is taken. Those pictures can be reconstructed with the program "Phoenix datos", which is generating a 3D model. This model can be imported into "VGStudioMax 3.0" (Volume Graphics GmbH). In this program, it is possible to perform various analyses, such as defect analysis or foam structure analysis. A region of interest (ROI) is selected, and a surface determination is performed automatically. A threshold value is determined to be able to separate material from background. The porosity can be determined from this data.

In order to acquire sufficient CT data, given that Freeze Foaming is a fast process, the exposure time is reduced from 500 to 250 ms. In addition, the number of images for the holding steps is reduced from 1440 to 720. The following parameters were selected for the evaluation of the foam structure: Threshold—80%, Accuracy—Fast; Direction of analysis—Right; Analysis mode—Background; Features—Advanced cell properties.

The first step was a complete foaming at two pressure reduction rates (10 mbar/s and 50 mbar/s). The results and their porosities are shown in Figure 6.

**Figure 6.** Cross-sections and porosities of completely foamed suspensions at two pressure reduction rates and three temperatures.

Increasing the pressure reduction rate from 10 to 50 mbar/s results in only a slight increase in porosity (Figure 6, right). However, the orientation of the pores seems to be significantly influenced by the pressure reduction rate. Suspensions foamed at 50 mbar/s exhibit vertically elongated pores, while those foamed at 10 mbar/s are oriented more horizontally. Due to the higher reduction rate inflicted on the process-induced air, the velocity of the inflating bubbles increases, thus forming vertical pores. In both cases, this is especially visible for 23 ◦C samples. Samples with an initial temperature of 40 ◦C show a decrease in porosity during foaming, due to their low viscosity, and the foam collapses before freezing by reaching the triple point.

#### *3.3. Stages of the Foaming Process*

Given the results of radiographic imaging, five holding stages were identified for analyzing the stages of the foaming progress (at 30, 40, 50, 70, and 100 mbar). The pressure reduction rate was adjusted to 10 mbar/s. Cross-sections of those scans are shown in Figure 7 (5 ◦C) and Figure 8 (23 ◦C). For each evaluation, three CT scans were executed to observe pores, and their development—the suspension (1000 mbar), the holding stage at its target pressure, and the final foam structure (5 mbar). Air bubbles that have been introduced into the mold during suspension filling have a large influence on the foam structure. They grow even larger during foaming, and develop significantly larger pores. Due to their high viscosity, this growth is inhibited in 5 ◦C tempered suspensions. Furthermore, the maximum foaming rate takes place at lower pressures (40–60 mbar) [24]. As a large amount of water evaporates, the pressure reduction rate drops, and the time to reach the target pressure of <5 mbar is too long. This process-induced growth inhibition results in a significantly lower porosity. In general, suspensions with an initial temperature of 5 ◦C exhibit a lower porosity after foaming, due to a higher viscosity and a lower amount of escaping water vapor. On the other hand, suspensions with an initial temperature of 40 ◦C exhibit a viscosity too low to be stable during the CT scan and, therefore, were not monitored.

**Figure 7.** Holding stages (I: 100 mbar, II: 70 mbar, III: 50 mbar, IV: 40 mbar, V: 30 mbar) of a 23 ◦C tempered suspension with a pressure reduction rate of 10 mbar/s ((**a**) suspension; (**b**) holding stage; (**c**) foam).

**Figure 8.** Holding stages (I: 100 mbar, II: 70 mbar, III: 50 mbar, IV: 40 mbar, V: 30 mbar) of a 5 ◦C tempered suspension with a pressure reduction rate of 10 mbar/s ((**a**) suspension; (**b**) holding stage; (**c**) foam).

Using the software VGStudioMax, which allows access to volume-based data, the pore size distributions were determined for each holding stage. An exemplary distribution for a 23 ◦C sample is depictured in Figure 9a. The growth starts slowly, accelerates to a maximum, and then slows down again. The lowest variance in pore size distribution can be found at 50 mbar on a 23 ◦C sample, with a relative curve width (b = d90/d10) of 36.2 (b100mbar = 53.1, b70mbar = 36.2, b40mbar = 153.5, b30mbar = 160.9). When the pressure drops below 50 mbar, the pore size distribution becomes flatter and wider, indicating a ripening process. Figure 9b shows the increasing porosity as a function of the decreasing pressure for the three investigated temperatures. Due to their high viscosity, 40 ◦C foamed samples could not be investigated with regard to holding stages and foam formation, because the foam structures collapsed during the investigation.

**Figure 9.** Pore size distribution of a 23 ◦C sample (**a**) and porosity (**b**) at 5, 23, and 40 ◦C samples of different holding stages; pressure reduction rate: 10 mbar/s.

#### *3.4. Mechanical Properties*

Due to the dependence of porosity and pore size distribution of Freeze Foamed samples on process parameters, mechanical properties should vary as well. To evaluate their behavior under load, cylindrical sintered samples were manufactured and subjected to standardized compression tests. Recorded tension–compression curves are depicted in Figure 10.

**Figure 10.** Compression tests on sintered samples (40 ◦C suspension; 5 ◦C suspension; devolatilized 23 ◦C suspension; 23 ◦C suspension).

The fracture behavior clearly exhibits a dependency on suspension temperature (5, 23, and 40 ◦C) and pretreatment (devolatilized and not devolatilized suspensions during the manufacturing process). Samples with a narrow pore size distribution [24] (5 ◦C and 23 ◦C devolatilized [23]) possess a pronounced maximum of force. On the other hand, specimens with a less uniform distribution of pores [24] (40 ◦C and 23 ◦C not devolatilized) show a more constant force level, and only reach about half the maximum force when compared to more homogeneous samples. Mechanical properties of the samples manufactured using Freeze Foaming are strongly influenced by microporosities inside the struts [23]. However, as the resolution of the CT scans were insufficient to examine their structure, they could not be taken into account here.

Furthermore, in situ compression tests, at selected load levels, were conducted to examine failure phenomenology (Figures 11–13).

**Figure 11.** Recorded force–travel curves of compression tests conducted on an in situ μCT; 5 ◦C.

**Figure 12.** Recorded force-travel curves of compression tests conducted on an in situ μCT; 23 ◦C not devolatilized.

**Figure 13.** Recorded force-travel curves of compression tests conducted on an in situ μCT; 40 ◦C.

For more inhomogeneous samples, material failure starts to occur between 25 N (40 ◦C) and 50 N (23 ◦C) (Figures 12 and 13). The density of fractures constantly rises, with increasing deformation, until a partial structure failure develops at relatively low load levels of 50 N (40 ◦C) and 110 N (23 ◦C not devolatilized). On the other hand, more homogeneous foam structures show a maximum load up to 200 N, even after the first signs of material failure (Figure 11). Detailed examinations show that especially cracks on the surface lead to material failure. This is a sign of an uneven sample surface, and results in a non-uniform load.


2. A literature review of the definition of a business cycle and methods for extracting cyclic components

indicators. In most cases, the authors of scientific publications use the real GDP (gross domestic product), as an indicator for investigation of macroeconomic business cycle. The development of this direction is the study of the interplay of the "total" business cycles identified in the GDP dynamics by countries. The second direction is the definition and quantitative description of the cyclic components in the dynamics of indicators characterizing the processes in individual sectors or spheres of the economy. The "specific cycles" thus estimated are the basis for the identification and quantification of the so-called common cycle using cluster analysis methods. In this case, the "common" (or as it is also called the "reference") cycle is represented as a multivariate value of the actual reaction of individual industries or economic subsystems, e.g., financial, investment, labor market, etc., observed with the help of statistical indicators. This chapter proposes a method for identifying and quantifying the common business cycle as a directly unmeasurable phenomenon, which manifests itself in fluctuations in the dynamics of the specific indicators of industries and economic systems, but has an objective and independent economic nature. In the author's opinion, specific cycles, even if they have a leading character with respect to the general cycle, are its economic consequence. The chapter suggests methods and their applications for identifying and quantifying the macroeconomic business cycle as a latent system-wide phenomenon, as well as methods for estimating intercountry synchronization and harmonization of common cycles. At the same time, the author gives her own definition of these forms of the interaction of business cycles. This approach is different to the view expressed in many publications about the identity of the concepts of synchronization, harmonization,

The chapter in addition to this introduction includes two main sections and a conclusion. The first section is devoted to the review of scientific publications that disclose the concept of the "business cycle," and also this section presents the author's systematization of the methods outlined in a number of publications for identifying cyclic components in the dynamics of macroeconomic indicators. In the second section of the chapter, an algorithm for quantifying the overall business cycle based on the principal component method is proposed, and methods for estimating synchronization and harmonizing business cycles are substantiated. The conclusion of the chapter contains a concentrated expression of scientific novelty of the author's methodological proposals of the business cycle quantification and the features of the algorithms for evaluating their synchronization and harmonization at

The proof of the concept proposed in this chapter and the approbation of the corresponding algorithm were realized on the basis of statistics of Eurostat and Rosstat. Most of the examples in this chapter are compiled from the results of calculations for Germany and Russia, which allows a comparative analysis provided that there is a significant difference in the duration and stages of the history of the market economy that have a significant impact on the forma-

Approbation of the proposed algorithm can be based on R-packages or the "STATISTICA"

concordance, and correlation of business cycles.

28 Statistics - Growing Data Sets and Growing Demand for Statistics

tion of sustainable multi-year business cycles.

the macroeconomic level.

program.

The number of scientific papers devoted to the theory and methods of quantifying business cycles is measured in hundreds. Nevertheless, there is a basis for their distribution into two large parts.

This basis is the definition of the concept of fluctuations in economic activity. This determines the choice of mathematical and statistical methods used to determine business cycles (Kijek [20]).

The first part includes the papers which are based on Burns and Mitchell's definition of a cycle [3]. According to them, a business cycle represents the four distinct phases of "aggregate economic activity" development that evolve from one into another: expansion, recession, depression, and revival.

The second part is the works based on a different view of business cycles, which was presented by Lucas [23], who does not interpret cycles as inevitable transitions between different phases of the cycle. He sees the business cycle as a process of oscillation of GNP around a long-term trend.

Some authors define the business cycle in accordance with Burns and Mitchell and then continue to measure the fluctuations of the macroeconomic aggregate values (GNP, GDP, industrial production, investment, and so on) relatively to the long-term trend.

This approach cannot be adopted without a preliminary study confirming that the observed cyclicality of the national economy is a violation of macroeconomic equilibrium rather than synchronous periodic fluctuations of various economic activities without changing of their balance.

The two basic approaches to the definition of the concept of a business cycle, mentioned above, were the basis for developing analytical tools for recognizing cycles.

Harding and Pagan [17] define three directions ("traditions") in the deviation of approaches presented in the literature on the development of cycle indicators from information available in a time-continuous random variable (yt).

The main idea of the first direction is changing of the initial time series (yt) by a series of binary random variable St, which includes turning points. In this case, peaks (troughs) are considered as local maxima (minima) in the series yt, and they are taking the value unity and zero otherwise:

$$\begin{aligned} \mathsf{V}\mathsf{t} &= \mathsf{1}\left(y\_{t} < y\_{t\pm 1}\leq \mathsf{j}\leq \mathsf{k}\right); \\ \mathsf{h}\mathsf{t} &= \mathsf{1}\left(y\_{t} < y\_{t\pm 1}\leq \mathsf{j}\leq \mathsf{k}\right). \end{aligned} \tag{1}$$

where ∨t (∧t) are binary variables and k is the length of period for local maxima (minima) estimation.

Harding and Pagan [18] show that for Burns and Mitchell's specific cycle dating procedures it is necessary to set k = 5 for monthly data or k = 2 for quarterly data, but it is not the single variant. Mathematics and IT providing of extracting and estimation of turning points in structure of economic time series have been actively developing.

Other two directions, or "traditions," by Harding and Pagan [17] are combined because they both are based on the prior transformation of yt so as to remove a permanent component, leaving only a transitory one (zt). The first direction of them provides for the isolation of the cycle on the basis of an analysis of the presence or absence of peaks in the spectral density. The second direction combines the methods of extracting a cycle by the results of the analysis of serial correlations in zt series.

The abovementioned approach to the systematization of methods for extracting and quantifying cycles in long-term economic dynamics, set forth in papers of Harding and Pagan, is accepted by many scientists, but is not the only one. An example of a somewhat different classification of methods for isolating stable fluctuations reflecting business cycles is presented in the paper of Kijek [20].

According to this author's view, three dominant approaches are used to distinguish cycles. Two of them coincide with the groups of methods singled out by Harding and Pagan (the last two "traditions"). They are (1) the presentation of the time series as a difference-stationary process and the application of autoregressive integrated moving average (ARIMA) models and (2) the presentation of economic dynamics as a trendstationary process and treat it as a sum (but also it can be multiple interconnection) of polynomial as deterministic trend and stochastic deviation around it. The last one is considered as a residual cyclical component, which includes business cycle's pattern and random deviation.

The third concept of time series decomposition according to Kijek [20] is the use of frequency filters, methodology, and software which are widely represented in the statistical literature.

The above methods are detailed and combined in different documents, but we must take into account that "The crucial question is not which method is more appropriate but whether different concepts of cycle are likely to produce alternative information which can be used to get a better perspective into economic phenomena and to validate theories" (Canova [4]).

Figure 1. System of methods for extraction of cyclic components from the seasonal adjusted time series of macroeconomic

Statistical Methodology for Evaluating Business Cycles with the Conditions of Their Synchronization…

http://dx.doi.org/10.5772/intechopen.75580

31

indicators.

The purpose of this paper is to create and demonstrate the results of the application of the algorithm of business cycle synchronization and harmonization proceed from economic content definition of business cycle by Burns and Mitchell and methods of estimation of economic fluctuations presented in publications. The main idea of it is based on the two key features in Burns and Mitchell's definition of business cycles. They are determined in many articles in the same variant. For example, in the paper of Diebold and Rudebusch ([8], p. 1), we can read "The first (key feature ) is the comovement among individual economic variables…In their analysis, Burns and Mitchell considered the historical concordance of hundreds of series, including those measuring commodity output, income, prices, interest rates, banking transactions, and transportation services…The second prominent element of Burns and Mitchell's definition of

business cycles is their division of business cycles into separate phases or regimes."

We believe that the first point should be based on another focus of "aggregate economic activity" in determining the business cycles of Burns and Mitchell: not only as a combination of the abovementioned indicators characterizing aggregate economic activity but above all the multiple effect of the entry of various economic activities into one and the same phase of the business cycle (the second key features in Burns and Mitchell's definition of business cycles).

Some authors include in the classification of methods for extracting from the time series of macroeconomic indicators of cyclic components adequate to real business cycles and groups of methods based on the evaluation of regression model parameters (such as the Fourier series models and multifactor models of the dependence of the cyclicality of GDP on the dynamics of factor variables).

We consider that it is not the methods of extraction of business cycles, but the methods for the next step of business cycle analysis—the quantification of cycle's characteristics (such as its phases, amplitude, duration of the period, and others).

The results of grouping of methods for extraction of cyclic components from the time series of macroeconomic indicators in conditions of its primly seasonal adjusting are presented in Figure 1. They include the group of "turning point" methods: methods for spectral density analysis and their realization with band-pass filtering (the statistical tools that pass frequencies within a certain range and reject frequencies outside that range) and methods for determining and subsequently evaluating cyclic components. We will not dwell on detrending methods in detail, since they are detailed in the articles, including in connection with the algorithms for isolating and quantifying the cycles (e.g., Canova [4]).

Statistical Methodology for Evaluating Business Cycles with the Conditions of Their Synchronization… http://dx.doi.org/10.5772/intechopen.75580 31

Other two directions, or "traditions," by Harding and Pagan [17] are combined because they both are based on the prior transformation of yt so as to remove a permanent component, leaving only a transitory one (zt). The first direction of them provides for the isolation of the cycle on the basis of an analysis of the presence or absence of peaks in the spectral density. The second direction combines the methods of extracting a cycle by the results of the analysis of

The abovementioned approach to the systematization of methods for extracting and quantifying cycles in long-term economic dynamics, set forth in papers of Harding and Pagan, is accepted by many scientists, but is not the only one. An example of a somewhat different classification of methods for isolating stable fluctuations reflecting business cycles is presented

According to this author's view, three dominant approaches are used to distinguish cycles. Two of them coincide with the groups of methods singled out by Harding and Pagan (the last two "traditions"). They are (1) the presentation of the time series as a difference-stationary process and the application of autoregressive integrated moving average (ARIMA) models and (2) the presentation of economic dynamics as a trendstationary process and treat it as a sum (but also it can be multiple interconnection) of polynomial as deterministic trend and stochastic deviation around it. The last one is considered as a residual cyclical component, which includes business cycle's pattern and

The third concept of time series decomposition according to Kijek [20] is the use of frequency filters, methodology, and software which are widely represented in the statistical

Some authors include in the classification of methods for extracting from the time series of macroeconomic indicators of cyclic components adequate to real business cycles and groups of methods based on the evaluation of regression model parameters (such as the Fourier series models and multifactor models of the dependence of the cyclicality of GDP on the dynamics of

We consider that it is not the methods of extraction of business cycles, but the methods for the next step of business cycle analysis—the quantification of cycle's characteristics (such as its

The results of grouping of methods for extraction of cyclic components from the time series of macroeconomic indicators in conditions of its primly seasonal adjusting are presented in Figure 1. They include the group of "turning point" methods: methods for spectral density analysis and their realization with band-pass filtering (the statistical tools that pass frequencies within a certain range and reject frequencies outside that range) and methods for determining and subsequently evaluating cyclic components. We will not dwell on detrending methods in detail, since they are detailed in the articles, including in connection with the algorithms for

phases, amplitude, duration of the period, and others).

isolating and quantifying the cycles (e.g., Canova [4]).

serial correlations in zt series.

30 Statistics - Growing Data Sets and Growing Demand for Statistics

in the paper of Kijek [20].

random deviation.

literature.

factor variables).

Figure 1. System of methods for extraction of cyclic components from the seasonal adjusted time series of macroeconomic indicators.

The above methods are detailed and combined in different documents, but we must take into account that "The crucial question is not which method is more appropriate but whether different concepts of cycle are likely to produce alternative information which can be used to get a better perspective into economic phenomena and to validate theories" (Canova [4]).

The purpose of this paper is to create and demonstrate the results of the application of the algorithm of business cycle synchronization and harmonization proceed from economic content definition of business cycle by Burns and Mitchell and methods of estimation of economic fluctuations presented in publications. The main idea of it is based on the two key features in Burns and Mitchell's definition of business cycles. They are determined in many articles in the same variant. For example, in the paper of Diebold and Rudebusch ([8], p. 1), we can read "The first (key feature ) is the comovement among individual economic variables…In their analysis, Burns and Mitchell considered the historical concordance of hundreds of series, including those measuring commodity output, income, prices, interest rates, banking transactions, and transportation services…The second prominent element of Burns and Mitchell's definition of business cycles is their division of business cycles into separate phases or regimes."

We believe that the first point should be based on another focus of "aggregate economic activity" in determining the business cycles of Burns and Mitchell: not only as a combination of the abovementioned indicators characterizing aggregate economic activity but above all the multiple effect of the entry of various economic activities into one and the same phase of the business cycle (the second key features in Burns and Mitchell's definition of business cycles). Сonsequently, at the macroeconomic level, the economic cycle is an integral result of business cycles of different economic activities that are at different or equal (to a greater or lesser extent) phases of the "common"1 cycle.

Harding and Pagan [16] graphically showed and explained the ratio of the peak (A) and the trough (C) as an example of a stylized recession (Figure 2). The height of the triangle in this graph is the amplitude, and the base is the duration. In this article, the authors also rightly notice that the knowledge of these two elements for any cycle makes it possible to calculate the area of the triangle and thereby estimate (say) the total loss in the output from the peak to the

Statistical Methodology for Evaluating Business Cycles with the Conditions of Their Synchronization…

For the purposes of quantification of the synchronization and harmonization of business cycles on the macroeconomic level and further cross-country comparison, we suggest the following

At the first stage of our algorithm, we extract trend, cyclical, and irregular components of the initial seasonal and calendar adjusted time series of quarterly indicators of gross value added by the types of activity (sectors) by country. Examples of implementation of the proposed algorithm of band-pass filtering and the following analysis are based on the seasonal and calendar adjusted time series of quarterly indicators of gross value

tors used are presented in fixed prices and converted to the natural logarithm, so they reflect the relative growth of the value added, forming the output of gross domestic

According to the approach of Baxter and King [1], the ideal band-pass filter should satisfy the

1. The filter must extract the specified range of periodicity, which means that it passes through time series components with periodic oscillations between low and high frequencies, defining a specific cycle. Baxter and King recommend a filter that approximates periodic oscillations between 6 and 32 quarters (according to the definition of the Mitchell

2. An ideal band-pass filter should not introduce a phase shift, i.e., do not change the time

3. The expression of the discrepancy between the exact and approximate filters should be

4. The application of the band-pass filter must extract of a deterministic trend from a time series and result in a stationary time series, even when applied to trending data.

5. The filter should allocate the same components of the business cycle, regardless of the length of the observation period. Baxter and King notes that "Technically, this means that

stamps of the turning points at any frequency.

expressed mathematically.

Eurostat: http://ec.europa.eu/eurostat

the moving averages we construct." 6. Method of filtering must be operational.

Russian Federal State Statistics Service (Rosstat): http://www.gks.ru

. The indica-

33

http://dx.doi.org/10.5772/intechopen.75580

added by the types of activity (sector) represented by Eurostat2 and Rosstat3

trough.

product (GDP).

six requirements:

cycle).

2

3

algorithm, presented in Figure 3.

According to this idea, we included following two points in the practical part of this article:


*4.3.2 PD*0 v/s *N*

1.'Th' case with Rayleigh distributed fading vectors (*ηP*, *μP*, *σP* = 1, 0 dB,

*Comparative ROC for the second group of fusion rules for different measured large scale parameters (varying ηP, μP and σP) with S* ¼ 8*, N* ¼ 8 *and Rayleigh distributed fading vector. Results for no shadowing condition,*

2.SC scenario with Rician distributed fading vectors (*ηP*, *μP*, *σP* = 2.72,

*Comparative ROC for all fusion rules for the* SC *environment with S* ¼ 8*, N* ¼ 8 *in Rician fading condition.*

*Results for Rayleigh fading-only condition ('Th') are plotted for comparison.*

*s* with ½*Ks*, min ,*Ks*, max � ¼ ½ � 0*:*5, 4 ),

0 dB; **h***n*,*s* � N ð Þ 0, 1 ),

*Wireless Sensor Networks - Design, Deployment and Applications*

*denoted by 'Th' are also plotted for comparison.*

**Figure 11.**

**Figure 12.**

**204**

1.22 dB, 2.4 dB; **h**Rice

In **Figures 15** and **16**, we show system probabilities of detection, *PD*0 with two groups of fusion rules as an interpolated function of the number of receive antennas *N* under *PF*0 ≤0*:*01.

• *Impact of measurement environment*: If both large and small scale channel parameters are varied, the probability of detection *PD*0 with MRC and CV-MMSE rules saturates with the increase in *N* over the SC scenario, but increases

#### **Figure 13.**

*Comparative ROC for all fusion rules for the* DC *environment with S* ¼ 8*, N* ¼ 8 *in TWDP fading condition. Results for Rayleigh fading-only condition ('Th') are plotted for comparison.*

**Figure 14.**

*Comparative ROC for all fusion rules for the* SI *environment with S* ¼ 8*, N* ¼ 8 *in double-Rayleigh (DR) fading condition. Results for Rayleigh fading-only condition ('Th') are plotted for comparison.*

**5. Conclusions**

**Figure 16.**

50% cases.

**207**

specific rule and channel SNR.

dynamic (people moving around) environments.

*impact of both large scale and small scale channel parameters.*

*Cross-Layer Inference in WSN: From Methods to Experimental Validation*

*DOI: http://dx.doi.org/10.5772/intechopen.93848*

This chapter summarizes design of sub-optimal fusion rules propounded for decision fusion at a DFC equipped with multiple antennas. Such rues are more efficient than exact LLR based optimal fusion rule for practical implementation. The sub-optimal fusion rules offer a plethora of choices for fusing sensor decisions at the DFC energy efficiently with lower requirement of system knowledge and computational complexity, thereby eliminating all problems with fixed point implementation. All these rules still significantly benefit from the addition of multiple antennas at the DFC, with a saturation on performance depending on the

*PD*0 *v/s N for the second group of fusion rules with S* ¼ 8 *for different measurement environments reflecting the*

We also investigate and study the practical implications of employing distributed MIMO based WSN, especially in the light of the recently proposed decision fusion algorithms for DFC equipped with multiple integrated antennas. A detailed measurement campaign is conducted for an indoor-to-outdoor distributed MIMO scenario with transmit antennas, representing sensors, deployed in a wide variety of indoor environments and receive antennas mounted on top of an outdoor tower, thereby replicating a DFC. Measurements are accumulated both in static and

For each measurement scenario, large and small scale statistics are derived from the accumulated data, and average values of pathloss and shadowing variations are

calculated. Fading distributions derived from the recorded channel impulse responses (CIRs) are found to closely match the double Rayleigh distribution in 21.4% cases, the TWDP distribution in 28.6% cases and the Ricean distribution in

Large and small scale channel parameters calculated from the accumulated measurements are used to model the MAC scenario over which performance of the formulated fusion rules is analyzed for virtual MIMO-based WSN. All the suboptimal fusion rules, on an average, exploit diversity offered by multiple antennas

**Figure 15.**

*PD*0 *v/s N for the first group of fusion rules with S* ¼ 8 *for different measurement environments reflecting the impact of both large scale and small scale channel parameters.*

with *N* for the DC and SI scenarios at a rate slower with higher *N*, as is evident in **Figures 15** and **16**. *PD*0 with CV-ML and Max-Log rules increases proportionately with *N* for all scenarios. It is worth-mentioning that this set of performances is limited to the chosen channel of 20 dB, and cannot be generalize to any value of channel SNR.

*Cross-Layer Inference in WSN: From Methods to Experimental Validation DOI: http://dx.doi.org/10.5772/intechopen.93848*

**Figure 16.**

*PD*0 *v/s N for the second group of fusion rules with S* ¼ 8 *for different measurement environments reflecting the impact of both large scale and small scale channel parameters.*

## **5. Conclusions**

This chapter summarizes design of sub-optimal fusion rules propounded for decision fusion at a DFC equipped with multiple antennas. Such rues are more efficient than exact LLR based optimal fusion rule for practical implementation. The sub-optimal fusion rules offer a plethora of choices for fusing sensor decisions at the DFC energy efficiently with lower requirement of system knowledge and computational complexity, thereby eliminating all problems with fixed point implementation. All these rules still significantly benefit from the addition of multiple antennas at the DFC, with a saturation on performance depending on the specific rule and channel SNR.

We also investigate and study the practical implications of employing distributed MIMO based WSN, especially in the light of the recently proposed decision fusion algorithms for DFC equipped with multiple integrated antennas. A detailed measurement campaign is conducted for an indoor-to-outdoor distributed MIMO scenario with transmit antennas, representing sensors, deployed in a wide variety of indoor environments and receive antennas mounted on top of an outdoor tower, thereby replicating a DFC. Measurements are accumulated both in static and dynamic (people moving around) environments.

For each measurement scenario, large and small scale statistics are derived from the accumulated data, and average values of pathloss and shadowing variations are calculated. Fading distributions derived from the recorded channel impulse responses (CIRs) are found to closely match the double Rayleigh distribution in 21.4% cases, the TWDP distribution in 28.6% cases and the Ricean distribution in 50% cases.

Large and small scale channel parameters calculated from the accumulated measurements are used to model the MAC scenario over which performance of the formulated fusion rules is analyzed for virtual MIMO-based WSN. All the suboptimal fusion rules, on an average, exploit diversity offered by multiple antennas

with *N* for the DC and SI scenarios at a rate slower with higher *N*, as is evident

proportionately with *N* for all scenarios. It is worth-mentioning that this set of

in **Figures 15** and **16**. *PD*0 with CV-ML and Max-Log rules increases

generalize to any value of channel SNR.

*impact of both large scale and small scale channel parameters.*

**Figure 14.**

**Figure 15.**

**206**

performances is limited to the chosen channel of 20 dB, and cannot be

*PD*0 *v/s N for the first group of fusion rules with S* ¼ 8 *for different measurement environments reflecting the*

*Comparative ROC for all fusion rules for the* SI *environment with S* ¼ 8*, N* ¼ 8 *in double-Rayleigh (DR)*

*fading condition. Results for Rayleigh fading-only condition ('Th') are plotted for comparison.*

*Wireless Sensor Networks - Design, Deployment and Applications*

at the DFC to achieve considerable gain in performance. Among all the rules, CV-ML performs worst and CV-MMSE performs best in all scenarios. MRC, EGC and Max-Log perform in between the two extremes of CV-ML and CV-MMSE. In this case, EGC performs better than MRC and MRC performs better than Max-Log.

**References**

[1] R. Jiang, S. Misra, B. Chen, and A. Swami. Robust suboptimal decision fusion in wireless sensor networks. Proceedings in IEEE Military Communications Conference, 2005. MILCOM 2005. Oct. 2005, pp. 2107–2113.

*DOI: http://dx.doi.org/10.5772/intechopen.93848*

*Cross-Layer Inference in WSN: From Methods to Experimental Validation*

[8] F. Li, J. S. Evans, and S. Dey. Decision fusion over non-coherent fading multi-access channels. IEEE Transactions on Signal Processing. 59 (9), pp. 4367–4380, Sep. 2011.

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Applications of Nanoimprint lithography

**3.** Contamination of substrate

200 Advances in Micro/Nano Electromechanical Systems and Fabrication Technologies

*3.5.2. Nano-imprint lithography*

Nanoimprint lithography (NIL) is an emerging process that can produce sub-10nm features. It is a simple process that uses a mould to emboss the resist with the required pattern. After embossing the resist, compressed resist material is removed using anisotropic etching and the substrate exposed. It can produce features at extremely small resolutions that cover a large area with a high throughput and relatively low cost, which is main advantage of this technique. It can be adapted to transfer all components needed to create a thin film transistor on a plastic substrate. It involves pressing and heating a thin film between a patterned template and a substrate. Upon heating, the patterned film adheres only to the substrate [ref Fig. 10]. This has high throughput and is relatively inexpensive compared to developing extreme deep UV lithography for commercial viability. It is also flexible enough to be used at chip level with several layers or at the wafer level when single layer is required. It can give resolutions lower than 10nm with high throughput at low cost. One of the current barriers to production at these resolutions is the development of mould. It can be used for fabricating nanoscale photo-

detectors, silicon quantum-dot, quantum wire and ring transistors (Chou, S.Y. 1996)

**Figure 10.** Schematic diagram of the steps involved in the nanoimprint lithographic process [Courtesy: Source: AZo‐

**4.** Stamp deformation

Nano]


**Author details**

Rahmouni Abdelkader\* and Belbachir Mohammed

\*Address all correspondence to<EMAIL_ADDRESS>
Laboratory of Polymer Chemistry, Department of Chemistry, Faculty of Sciences, Oran1, University Ahmed Benbella, Oran, Algeria


*2.5. Acellular Bioactivity*

The acellular bioactivity of PLLA-SBA2 fibres, related to the surface deposition of HCA layers, was evaluated by immersing the samples inserted in scaffold holders (CellCrownTM 24, Scaffdex, Sigma Aldrich, Munich, Germany) in simulated body fluid (SBF) solution [43] for 1, 7, 14 and 21 days at 37 ◦C, on an oscillating tray in an incubator. A falcon tube with only SBF was used as control. After each time point, samples were washed with distilled water, dried at room temperature before SEM-EDS characterization and Fourier Transform Infrared Spectroscopy (FTIR) analysis in attenuated total reflectance (ATR) mode. PLLA neat fibres were considered as control. FTIR spectra were recorded by a spectrometer (IRAffinity-1S, Shimadzu, Kyoto, Japan), repeating 32 scans over the wavenumber range 4000–500 cm<sup>−</sup>1, with a resolution of 4 cm−1.


*2.1. Analysis of N. edaphicum CCNP1411 Genome*

Total DNA has been isolated from *N*. *edaphicum* CCNP1411, and the whole genome sequence has been determined. Identified replicons of *N*. *edaphicum* CCNP1411 genome consist of the circular chromosome of 7,733,505 bps, and five circular plasmids (Table 1). Within the total size of 8,316,316 bps genome (chromosome and plasmids, Figure 2), we have distinguished, according to annotation, the total number of 6957 genes from which 6458 potentially code for proteins (CDSs), 415 are classified

as pseudo-genes and 84 are coding for non-translatable RNA molecules. Pseudo-genes can be divided into subcategories due to the shift in the coding frame (180), internal stop codons (77), incomplete sequence (228), or occurrence of multiple problems (63). Genes coding for functional RNAs consist of those encoding ribosomal (rRNA) (9), transporting (tRNA) (71) and regulatory noncoding (ncRNA) (4), all embedded on the chromosome. Out of total coding and pseudo-genes sequences (6873), the vast majority (5846) initiates with the ATG start codon, while GTG and TTG occur less frequently (561 and 217 times, respectively). The frequencies of stop codons were set out as follows: TAA (3455), TAG (1750), TGA (1526). Coding and pseudo-genes sequences are distributed almost equally on the leading and complementary strand, including 3408 and 3465 sequences, respectively.

**Table 1.** Composition and coverage of *N*. *edaphicum* CCNP1411 genome.

**Figure 2.** Map of the *N*. *edaphicum* CCNP1411 genome where chromosome (chr) and five plasmids (pNe\_1–5) are presented. The ORFs are indicated with grey arrows split into two rings outermost showing ORFs on direct strand and inner showing complementary strand ORFs. The middle circle shows GC content (black) and the innermost circle shows GC skew (green and purple). Genes for putative NRPS/PKS are marked on the chromosome in their proper position (red within a black circle), with closeup on NRPS in position 7,609,981–7,643,289 putatively coding for Ncp biosynthetic gene cluster.