**3. Theoretical background of array antenna beam steering**

As noted in the Introduction, phased array antennas are now widely used in radar equipment due to the possibility of fast electron beam scanning and increased failure-resistant feature compared to continuous aperture antennas and mechanical scanning. The application of PAAs in radar allows achieving high speed of viewing the service area and tracking high-speed maneuvering objects [1]. Besides, PAAs ensure the operability of the radar system in a complicated interference situation due to the adaptive formation of a complex-shape radiation pattern [4]. In many cases, the use of array antenna let reduce the weight of the radar system and lower its total cost. In addition to radar, mmWave array antennas capable of operating in ultra-wide frequency range are considered as one of a key enabling technology for designing RS of 5G network, as noted in the previous section. There, a formation of a narrow steered beam by means of the antenna array makes it possible to increase the directive gain to compensate for the attenuation in the mmWave-band. Besides, the use of a narrow beam would reduce the interference effects from other closely spaced transmitters, and also provide the possibility of spatial multiplexing to increase throughput while simultaneously exchanging information with several STs.

As described above, electronic scanning in the PAA is provided by a beamforming network, which includes phase shifters, or delay lines. The BFN supports a continuous or discrete beam movement in space due to phase control or signal time delay between the array elements. Below, a short theoretical study using ideal models will be presented pursuing the goal to define the complementary input data for the posterior design of the specific photonics-based BFNs for the ultra-wide mmWave-band PAA exploiting widespread microwave-electronic computer-aided design (CAD) environment NI AWRDE.

In general, the array antenna is a collection of antenna elements connected to the transmitter/receiver through RF feeds, which includes a BFN. A typical arrangement of antenna elements is shown in **Figure 4**.

The radiation pattern of an array of identical antenna elements (Eq. (1)) is the product of the diagram of an individual element *<sup>f</sup>*(θ,*<sup>φ</sup>*) and the array factor *<sup>F</sup>*(θ,*<sup>φ</sup>*), depending on the mutual arrangement of the elements

$$D(\Theta, \varphi) \, = \, f(\Theta, \varphi) \, \ast F(\Theta, \varphi), \tag{1}$$

where θ is an elevation angle, φ is an azimuth. The array factor has the form:

$$F(\theta,\varphi) \quad = \,\,\Sigma\sigma\_i \exp\left(-j k r\_i \cdot r\right),\tag{2}$$

where *ai* is the transmission gain (weight) in the channel of the feed network connected to the *i*-th element, *ri* is the radius vector of the *i*-th element, *r* is the radius-vector specifying the direction (θ,*<sup>φ</sup>*), *k* is the wave number, referred to the operation wavelength λ as *k*=2*π*/λ.

When analyzing various configurations of antenna arrays and feed networks, only specific variant of the array factor is often considered, assuming the antenna elements to be isotropic, with the radiation pattern *<sup>f</sup>*(θ,*<sup>φ</sup>*)=const. In the process of the examination, the following basic parameters are analyzed: antenna directivity, mainlobe's half power or null-to-null beamwidth, beam-direction angular error, and maximum sidelobe level [1]. Radiation patterns are usually represented for the azimuth *φ* and elevation θ sections in a Cartesian or polar coordinate system, or in 3D form.

As mentioned above, for scanning PAAs, there are two main ways of designing the feed networks: based on phase shifters and delay lines. Mathematically, the difference between these techniques can be represented as follows. First, expanding *<sup>k</sup>* and multiplying radius-vectors *ri* <sup>∙</sup> *<sup>r</sup>*, Eq. (2) can be rewritten as

$$F(\theta,\varphi) \quad = \,\,\Sigma a\_i \,\, \exp\left(-j\frac{a\_0}{\mathcal{C}}r\_i \cdot r\right) \quad = \,\,\Sigma a\_i \,\, \exp\left(-j\alpha \,\, \tau\_i\right),\tag{3}$$

where *ω* is the angular operating frequency and *<sup>τ</sup><sup>i</sup>* is the time delay that must be introduced into the channel of the feed network connected to the *i*-th element to obtain the required array factor. For the narrow-band case with a central operating frequency *ω*0, Eq. (3) takes the forms

$$F(\Theta, \varphi) \quad = \ \Sigma a\_i \exp\left(-j \, \alpha\_0 \, \tau\_i\right) \quad = \ \Sigma a\_i \exp\left(-j \, \Phi\_i\right), \tag{4}$$

**53**

exceed λ*min*/2.

*Design and Optimization of Photonics-Based Beamforming Networks for Ultra-Wide…*

monly not exceeding 10%, depending on the criterion used [23].

convenient for calculating the beamwidth in degrees:

determined according to:

with the direction of the beam *<sup>φ</sup>* 0 is:

limited by the expression:

length *L* and in the antenna, broadside is determined in radians according to:

taking into account the broadening effect at large deviation angles.

space. In the absence of such a plane, the gain is reduced accordingly.

feed network connected to the *i*-th element, to obtain the required array factor. Obviously, if the frequency deviates from *<sup>ω</sup>* 0, the phase shifts in the channels of the feed network will not provide the required *<sup>F</sup>*(*θ*,*<sup>φ</sup>*). This phenomenon called "beam squint" leads to an error in the direction of the maximum of the PAA pattern, and also to a certain increase in the level of the sidelobes. Nevertheless, despite the beam squint, the BFN based on phase shifter due to the ease of implementation has become widespread in narrowband PAAs, i.e., with a fractional bandwidth, com-

is the phase shift, which must be introduced into the channel of the

In addition, the important parameters of the PAA are the directive gain and the beamwidth closely related to it. In the simplest case of a linear equidistant array, the full width at half-maximum (FWHM) of the beam, <sup>∆</sup> 0,5, is inversely proportional to the aperture

∆0,5 = 0.886 λ/*L* (5)

∆0,5 = 101.5/*N* (6)

It should be noted that the beamwidth increases with the deviation from the PAA broadside direction. For this reason, the variants of the antenna array construction are often compared along the beamwidth in the direction of the normal,

The directive gain, which is also considered in the direction of the broadside, and for the array of N elements, the distance, between which is equal to λ/2, is

*D* = 2 ∗ *N*, (7)

where the factor of 2 is caused by the presence of the conducting plane in the majority of PAAs, which guides (reflects) the entire radiation flux into one half-

In the mmWave-band, it is sometimes not possible to arrange array elements at the intervals of <sup>λ</sup> *min*/2, where <sup>λ</sup> *min* is the minimum wavelength corresponding to the upper operating frequency. When the interval between elements in the radiation pattern increases, the grating lobes limiting the range of scanning angles might arise. Mathematically, this effect is due to the periodicity of the exponent in Eq. (2). For a linear array, the condition for the occurrence of a grating lobe in the azimuth *<sup>φ</sup>*

*kd*(sin*φ* − sin*φ*0) = *m* ∗ 2*π*, (8)

where *d* is the distance between antenna elements, and *m* is an integer. It can be shown from Eq. (2) that the range of scanning angles ±<sup>φ</sup>*max* for the distance *d* is

*d*/λ*min* ≤ (1 + sin*φmax*)−1 (9)

From Eq. (9), it follows that to ensure the absence of grating lobes when scanning in the range of *<sup>φ</sup>max* <sup>=</sup> 90°, the distance between antenna elements *d* should not

used in practice and will be considered below, Eq. (5) can be reduced to a form

For an array of *N* elements with the distance between them of λ/2, which is often

*DOI: http://dx.doi.org/10.5772/intechopen.80899*

where <sup>Φ</sup> *<sup>i</sup>*

**Figure 4.** *Array antenna of arbitrary geometry.*

*Design and Optimization of Photonics-Based Beamforming Networks for Ultra-Wide… DOI: http://dx.doi.org/10.5772/intechopen.80899*

where <sup>Φ</sup> *<sup>i</sup>* is the phase shift, which must be introduced into the channel of the feed network connected to the *i*-th element, to obtain the required array factor. Obviously, if the frequency deviates from *<sup>ω</sup>* 0, the phase shifts in the channels of the feed network will not provide the required *<sup>F</sup>*(*θ*,*<sup>φ</sup>*). This phenomenon called "beam squint" leads to an error in the direction of the maximum of the PAA pattern, and also to a certain increase in the level of the sidelobes. Nevertheless, despite the beam squint, the BFN based on phase shifter due to the ease of implementation has become widespread in narrowband PAAs, i.e., with a fractional bandwidth, commonly not exceeding 10%, depending on the criterion used [23].

In addition, the important parameters of the PAA are the directive gain and the beamwidth closely related to it. In the simplest case of a linear equidistant array, the full width at half-maximum (FWHM) of the beam, <sup>∆</sup> 0,5, is inversely proportional to the aperture length *L* and in the antenna, broadside is determined in radians according to:

$$
\Delta\_{0,5} = \text{ } \text{ } \text{88\\$} \,\lambda/L \tag{5}
$$

For an array of *N* elements with the distance between them of λ/2, which is often used in practice and will be considered below, Eq. (5) can be reduced to a form convenient for calculating the beamwidth in degrees:

$$
\Delta\_{0,5} = \text{ 101.5/N} \tag{6}
$$

It should be noted that the beamwidth increases with the deviation from the PAA broadside direction. For this reason, the variants of the antenna array construction are often compared along the beamwidth in the direction of the normal, taking into account the broadening effect at large deviation angles.

The directive gain, which is also considered in the direction of the broadside, and for the array of N elements, the distance, between which is equal to λ/2, is determined according to:

$$D \quad = \text{ } \mathbf{2} \ast \mathbf{N},\tag{7}$$

where the factor of 2 is caused by the presence of the conducting plane in the majority of PAAs, which guides (reflects) the entire radiation flux into one halfspace. In the absence of such a plane, the gain is reduced accordingly.

In the mmWave-band, it is sometimes not possible to arrange array elements at the intervals of <sup>λ</sup> *min*/2, where <sup>λ</sup> *min* is the minimum wavelength corresponding to the upper operating frequency. When the interval between elements in the radiation pattern increases, the grating lobes limiting the range of scanning angles might arise. Mathematically, this effect is due to the periodicity of the exponent in Eq. (2). For a linear array, the condition for the occurrence of a grating lobe in the azimuth *<sup>φ</sup>* with the direction of the beam *<sup>φ</sup>* 0 is:

$$kd(\sin\varphi - \sin\varphi\_0) = m \ast 2\pi,\tag{8}$$

where *d* is the distance between antenna elements, and *m* is an integer. It can be shown from Eq. (2) that the range of scanning angles ±<sup>φ</sup>*max* for the distance *d* is limited by the expression:

$$d/\lambda\_{\rm min} \le \left\{ \mathbf{1} + \sin \rho\_{\rm max} \right\}^{-1} \tag{9}$$

From Eq. (9), it follows that to ensure the absence of grating lobes when scanning in the range of *<sup>φ</sup>max* <sup>=</sup> 90°, the distance between antenna elements *d* should not exceed λ*min*/2.

*Array Pattern Optimization*

ment of antenna elements is shown in **Figure 4**.

depending on the mutual arrangement of the elements

where θ is an elevation angle, φ is an azimuth.

The array factor has the form:

connected to the *i*-th element, *ri*

operation wavelength λ as *k*=2*π*/λ.

where *ai*

3D form.

In general, the array antenna is a collection of antenna elements connected to the

The radiation pattern of an array of identical antenna elements (Eq. (1)) is the product of the diagram of an individual element *<sup>f</sup>*(θ,*<sup>φ</sup>*) and the array factor *<sup>F</sup>*(θ,*<sup>φ</sup>*),

*D*(θ,*φ*) = *f*(θ,*φ*) ∗ *F*(θ,*φ*), (1)

*F*(θ,*φ*) = ∑*ai exp*(−*jk ri* ∙ *r*), (2)

radius-vector specifying the direction (θ,*<sup>φ</sup>*), *k* is the wave number, referred to the

When analyzing various configurations of antenna arrays and feed networks, only specific variant of the array factor is often considered, assuming the antenna elements to be isotropic, with the radiation pattern *<sup>f</sup>*(θ,*<sup>φ</sup>*)=const. In the process of the examination, the following basic parameters are analyzed: antenna directivity, mainlobe's half power or null-to-null beamwidth, beam-direction angular error, and maximum sidelobe level [1]. Radiation patterns are usually represented for the azimuth *φ* and elevation θ sections in a Cartesian or polar coordinate system, or in

As mentioned above, for scanning PAAs, there are two main ways of designing the feed networks: based on phase shifters and delay lines. Mathematically, the difference between these techniques can be represented as follows. First, expanding

ω

introduced into the channel of the feed network connected to the *i*-th element to obtain the required array factor. For the narrow-band case with a central operating

*F*(θ,*φ*) = ∑*ai* exp(−*j*ω<sup>0</sup> τ*i*) = ∑*ai* exp(−*jΦi*), (4)

*<sup>k</sup>* and multiplying radius-vectors *ri* <sup>∙</sup> *<sup>r</sup>*, Eq. (2) can be rewritten as

where *ω* is the angular operating frequency and *<sup>τ</sup><sup>i</sup>*

*F*(θ,*φ*) = ∑*ai* exp(−*j* \_\_

frequency *ω*0, Eq. (3) takes the forms

is the transmission gain (weight) in the channel of the feed network

is the radius vector of the *i*-th element, *r* is the

<sup>с</sup> *ri* ∙ *r*) = ∑*ai* exp(−*j*τ*i*), (3)

is the time delay that must be

transmitter/receiver through RF feeds, which includes a BFN. A typical arrange-

**52**

**Figure 4.**

*Array antenna of arbitrary geometry.*

#### *Array Pattern Optimization*

The phase shifters or delay lines used in BFNs can be steered continuously or in discrete steps. Because of the operation convenience, the later type is most widely exploited. Such BFNs are usually controlled by binary codes and are steered in the range from 0 to (2 *<sup>R</sup>* <sup>−</sup> 1) <sup>∗</sup> <sup>δ</sup>, where δ is the sampling period (SP) that determines the error of the BFN operation due to discreet time sampling; *R* is the number of bits of the binary phase shifter or delay line. For phase shifters, the parameter *R* is selected in such a way as to cover the phase shift range of 360° with a step δ. For delay lines, the number of bits must provide the necessary time-delay setting <sup>τ</sup>*max* for deviating the PAA diagram by the maximum operating angle. For a given number of bits *R*, the SP is determined by:

$$
\delta \ge \quad \pi\_{\text{max}} / \left( \mathbf{2}^{\mathbb{R}} - \mathbf{1} \right) \tag{10}
$$

The sampling error within ±δ/2 affects primarily the direction and level of the sidelobes. For a PAA with a small number of elements, there is a primary error in the position of the beam. On the contrary, for large PAAs, the sampling error does not affect the position of the beam, since it has an average value of 0. Reducing the directivity for large PAAs due to the error of phase quantization and amplitude errors in the channels of the feed network might be described statistically in the form:

$$D\_- = D\_0/\left(\mathbb{1} + \sigma\_\Phi^2 + \sigma\_A^2\right),\tag{11}$$

where *D*0 is the directivity without taking into account the quantization error, *σΦ 2 ,σ<sup>A</sup> 2* are the variances of the phase quantization error or the amplitude error, respectively. Thereafter, the average level of the sidelobes might increase in more than (*σΦ <sup>2</sup> <sup>+</sup> <sup>σ</sup><sup>A</sup>* 2 ) times. It should be noted that the sidelobe closest to the main beam has the maximum level. Its level is determined by the distribution of the transmission gain *ai* in the channels of the feed network, and for uniform distribution is −13 dB relative to the main beam level.

The influence of the PAA's parameters considered above was examined for the example of a linear equidistant array of 16 isotropic elements (**Figure 5**) designed for operation at the frequency range of 57–76 GHz. To ensure scanning without grating lobes, the distance between the elements of 2 mm was selected in accordance with Eq. (9), starting from the minimum wavelength of 4 mm. This distance leads to beamwidth of 6.3° at highest frequency and antenna gain of 15 dB in accordance with Eqs. (6) and (7), respectively.

The beam FWHM of the PAA under test varies in accordance with Eq. (5) from 8.4° at 57 GHz to 6.3° at 76 GHz. The necessity of transmitting an ultra-wideband signal with a fractional bandwidth of up to 30% by means of this PAA will inevitably lead to significant distortion of the normalized radiation pattern (NRP) shape when using a phase-shifter based BFN. We will illustrate this with the example of a signal with a 19 GHz bandwidth and a central frequency of 66.5 GHz, for which phase shifts for a beam direction of 30° are calculated according to Eq. (4). The NRPs formed by the BFN with ideal phase shifters at 57, 66.5, and 76 GHz are shown in **Figure 6**. It can be seen from the figure that the deviation of the maximum of the NRP from the desired direction is 4–6°when the frequency is varied within the operating range.

**55**

**Figure 7.**

*transmitting a signal with a 30% fractional bandwidth.*

described by Eq. (11).

**Figure 6.**

*Design and Optimization of Photonics-Based Beamforming Networks for Ultra-Wide…*

Moreover, at the edges of the frequency band, the power radiated by the PAA in the given direction falls by 4 dB. Because of the squint effect, the different frequency components of the signal will be radiated in different directions. As the result, a receiver would experience a decrease in amplitude at the edges of the RF signal frequency spectrum up to 4 dB, which causes a distortion of the waveform and the occurrence of reception errors. Signal components emitted in undesired directions also might cause interference in the receivers, for which this signal was not intended. For comparison, **Figure 7** shows the NRPs formed by the BFN with delay lines at the same frequencies. It is seen from the figure that the squint phenomenon is completely absent in the entire frequency range. This well-known benefit ensures the superiority of the use of TTD-steered BFNs in PAAs with a relative bandwidth of more than 10% [23], in spite of the fact that TTD solution in principle is more expensive, bulky, and technically complex than the BFN on phase shifters. Thus, for the BFN under consideration, TTD-steered technique is the only suitable solution that ensures operability in the frequency range from 57 to 76 GHz. Essential schematic simplification can be achieved if binary switchable delay lines (BSDL) are used instead of continuously tuned ones. To determine the parameters of such a BSDL providing scanning angle of ±45°, the maximum time delay calculated in accordance with Eq. (3) is 70.8 ps. Following Eq. (10), the use of 4, 5 and 6-bit BFNs makes it possible to obtain a sampling period of 4.7, 2.3, and 1.1 ps, respectively. The growth of SP leads to a corresponding increase of the variance for the phase quantization error, which contributes to distortions and directivity decay

*Normalized radiation patterns formed by the BFN with ideal delay lines at 57, 66.5, and 76 GHz in the case of* 

*Normalized radiation patterns formed by the BFN with ideal phase shifters at 57, 66.5, and 76 GHz in the case* 

*DOI: http://dx.doi.org/10.5772/intechopen.80899*

*of transmitting a signal with a 30% fractional bandwidth.*

**54 Figure 5.** *Configuration of the PAA under test.*

*Design and Optimization of Photonics-Based Beamforming Networks for Ultra-Wide… DOI: http://dx.doi.org/10.5772/intechopen.80899*

**Figure 6.**

*Array Pattern Optimization*

the range from 0 to (2

number of bits *R*, the SP is determined by:

*D* = *D*0/(1 + σΦ

accordance with Eqs. (6) and (7), respectively.

The phase shifters or delay lines used in BFNs can be steered continuously or in discrete steps. Because of the operation convenience, the later type is most widely exploited. Such BFNs are usually controlled by binary codes and are steered in

the error of the BFN operation due to discreet time sampling; *R* is the number of bits of the binary phase shifter or delay line. For phase shifters, the parameter *R* is selected in such a way as to cover the phase shift range of 360° with a step δ. For delay lines, the number of bits must provide the necessary time-delay setting <sup>τ</sup>*max* for deviating the PAA diagram by the maximum operating angle. For a given

*δ* ≥ τ*max*/(2*<sup>R</sup>* − 1) (10)

The sampling error within ±δ/2 affects primarily the direction and level of the sidelobes. For a PAA with a small number of elements, there is a primary error in the position of the beam. On the contrary, for large PAAs, the sampling error does not affect the position of the beam, since it has an average value of 0. Reducing the directivity for large PAAs due to the error of phase quantization and amplitude errors in the channels of the feed network might be described statistically in the form:

where *D*0 is the directivity without taking into account the quantization error, *σΦ*

are the variances of the phase quantization error or the amplitude error, respectively.

feed network, and for uniform distribution is −13 dB relative to the main beam level. The influence of the PAA's parameters considered above was examined for the example of a linear equidistant array of 16 isotropic elements (**Figure 5**) designed for operation at the frequency range of 57–76 GHz. To ensure scanning without grating lobes, the distance between the elements of 2 mm was selected in accordance with Eq. (9), starting from the minimum wavelength of 4 mm. This distance leads to beamwidth of 6.3° at highest frequency and antenna gain of 15 dB in

It should be noted that the sidelobe closest to the main beam has the maximum level. Its

The beam FWHM of the PAA under test varies in accordance with Eq. (5) from 8.4° at 57 GHz to 6.3° at 76 GHz. The necessity of transmitting an ultra-wideband signal with a fractional bandwidth of up to 30% by means of this PAA will inevitably lead to significant distortion of the normalized radiation pattern (NRP) shape when using a phase-shifter based BFN. We will illustrate this with the example of a signal with a 19 GHz bandwidth and a central frequency of 66.5 GHz, for which phase shifts for a beam direction of 30° are calculated according to Eq. (4). The NRPs formed by the BFN with ideal phase shifters at 57, 66.5, and 76 GHz are shown in **Figure 6**. It can be seen from the figure that the deviation of the maximum of the NRP from the desired direction is 4–6°when the frequency is varied within the operating range.

Thereafter, the average level of the sidelobes might increase in more than (*σΦ*

level is determined by the distribution of the transmission gain *ai*

<sup>2</sup> + σ*<sup>A</sup>*

<sup>2</sup> ), (11)

*2 ,σ<sup>A</sup> 2*

*<sup>2</sup> <sup>+</sup> <sup>σ</sup><sup>A</sup>* 2 ) times.

in the channels of the

*<sup>R</sup>* <sup>−</sup> 1) <sup>∗</sup> <sup>δ</sup>, where δ is the sampling period (SP) that determines

**54**

**Figure 5.**

*Configuration of the PAA under test.*

*Normalized radiation patterns formed by the BFN with ideal phase shifters at 57, 66.5, and 76 GHz in the case of transmitting a signal with a 30% fractional bandwidth.*

Moreover, at the edges of the frequency band, the power radiated by the PAA in the given direction falls by 4 dB. Because of the squint effect, the different frequency components of the signal will be radiated in different directions. As the result, a receiver would experience a decrease in amplitude at the edges of the RF signal frequency spectrum up to 4 dB, which causes a distortion of the waveform and the occurrence of reception errors. Signal components emitted in undesired directions also might cause interference in the receivers, for which this signal was not intended. For comparison, **Figure 7** shows the NRPs formed by the BFN with delay lines at the same frequencies. It is seen from the figure that the squint phenomenon is completely absent in the entire frequency range. This well-known benefit ensures the superiority of the use of TTD-steered BFNs in PAAs with a relative bandwidth of more than 10% [23], in spite of the fact that TTD solution in principle is more expensive, bulky, and technically complex than the BFN on phase shifters. Thus, for the BFN under consideration, TTD-steered technique is the only suitable solution that ensures operability in the frequency range from 57 to 76 GHz.

Essential schematic simplification can be achieved if binary switchable delay lines (BSDL) are used instead of continuously tuned ones. To determine the parameters of such a BSDL providing scanning angle of ±45°, the maximum time delay calculated in accordance with Eq. (3) is 70.8 ps. Following Eq. (10), the use of 4, 5 and 6-bit BFNs makes it possible to obtain a sampling period of 4.7, 2.3, and 1.1 ps, respectively. The growth of SP leads to a corresponding increase of the variance for the phase quantization error, which contributes to distortions and directivity decay described by Eq. (11).

#### **Figure 7.**

*Normalized radiation patterns formed by the BFN with ideal delay lines at 57, 66.5, and 76 GHz in the case of transmitting a signal with a 30% fractional bandwidth.*

### *Array Pattern Optimization*


**Table 2.**

*FoMs for BSDL with various number of bits.*

**Figure 8.** *Examples of radiation pattern distortion due to quantization errors.*

One possible way to characterize the amount of distortion is to simulate all possible scan angles and calculate maximum sidelobe level (SLL) using the above calculated SP values [4]. Peak and average values obtained through the entire scan range can be considered as figures of merit (FoMs) that determine the performance of BSDL. **Table 2** lists the results of calculations carried out by the above technique.

**Figure 8** exemplifies the distortions for BSDLs with different number of bits, which affect NRPs in different directions of scanning range.

When using a 4-bit BSDL, a sharp increase in the level of the sidelobes, a decay in directivity, and a deviation of the beam position from the desired one are observed. However, it is acceptable to use a 5-bit BSDL with a sampling period of 2.3 ps, for this case.

To summarize, the following outcomes could be concluded:

