**4. Performance of the array**

In this section, the performance of steered beam adaptive array with uniform inter-element spacing of (λ/2) will be studied with respect to various operating conditions,<sup>2</sup> including pointing error, feedback loop gain, input *SNR*/element, input *INR*/element, and DOAs and bandwidths of the desired and interference signals.

Consider first the case where there is no interference. **Figure 4** shows the output SINR of a three-element array as a function of pointing error (*θperr*), which is the difference between the DOA of the desired signal (*θd*) and the direction of the main beam of the array (*θmax*). Several curves are shown for different input *SNR*'s/ element. The curves are computed for feedback loop gain *K* = 0.1, and zero desired signal bandwidth (*Bd* = 0).

From **Figure 5**, it can be seen that the sensitivity of the array to pointing errors increases with increasing *SNR*. For example, if *SNR* = 5 dB, the array provides output *SINR* <sup>&</sup>gt; 5B if the pointing error margin lies within �15*<sup>o</sup>* , but for *SNR* = 40 dB,

**Figure 4.** *Output SINR vs. <sup>θ</sup>perr. Three-element array, <sup>θ</sup><sup>d</sup>* <sup>¼</sup> <sup>0</sup>*o, K = 0.1, and Bd = 0. No interference.*

<sup>2</sup> The nonuniformly spaced adaptive arrays will be studied in the next chapters.

the array output exceeds 5 dB only if the pointing error lies within �0*:*8*<sup>o</sup>* . The explanation of this behavior is illustrated in **Figure 5**, where typical radiation patterns are plotted using Eq. (24), under the same conditions as in **Figure 4**.

In **Figure** <sup>5</sup>**(a)**, the pointing error *<sup>θ</sup>perr* <sup>¼</sup> *<sup>θ</sup><sup>d</sup>* � *<sup>θ</sup>max*Þ ¼ <sup>0</sup>*<sup>o</sup>* , and in **Figure** 5**(b)**, *<sup>θ</sup>perr* <sup>¼</sup> <sup>5</sup>*<sup>o</sup>* . For *<sup>θ</sup>perr* <sup>¼</sup> <sup>0</sup>*<sup>o</sup>* , it can be seen that as the *SNR* increases, the overall radiation pattern magnitude is reduced. As the pattern amplitude drops, both the desired signal power and the thermal noise power drop in proportion. Hence, the output *SINR* remains relatively unaffected by this change in pattern amplitude. For *<sup>θ</sup>perr* <sup>¼</sup> <sup>5</sup>*<sup>o</sup>* , the pattern behavior is quite different. Here, it can be seen that as the *SNR* increases, the array increasingly suppresses the desired signal. Since it can do this without lowering the overall pattern amplitude, the result is to reduce the desired signal power without reducing the thermal noise power. This accounts for the behavior seen in **Figure 4**. However, in most system designs, achieving maximum gain is not the most important objective. What matters most is achieving *SINR* at the output of the array that exceeds a certain threshold.

The effect of increasing the number of array elements is depicted in **Figure 6**. When comparing the curves in this figure with those in **Figure 4**, it can be seen that the five-element array is more sensitive to pointing errors than the three-element

#### **Figure 5.**

*Radiation patterns of three-element array. K = 0.1, with different SNR's. No interference. (a) Pointing error (θperr* <sup>¼</sup> *<sup>θ</sup><sup>d</sup>* � *<sup>θ</sup>max*Þ ¼ <sup>0</sup>*o. (b) Pointing error (error (θperr* <sup>¼</sup> *<sup>θ</sup><sup>d</sup>* � *<sup>θ</sup>max*Þ ¼ <sup>5</sup>*o.*

**Figure 6.** *Output* SINR *vs. pointing error. Five-element array, <sup>θ</sup><sup>d</sup>* <sup>¼</sup> <sup>0</sup>*o, K = 0.1. No interference.*

array. For example, when *SNR* = 5 dB, the array provides output *SINR* > 5 dB, as long as �9*<sup>o</sup>* <sup>≤</sup>*θperr* <sup>≤</sup>9*<sup>o</sup>* .

The explanation of this behavior lies in the fact that the directivity of the array increases by increasing the number of its elements, as illustrated in **Figure 7**. Here, the 3-dB beamwidth of the main beam for the five-element array is less than that of the three-element array.

A different perspective on these results may be gained by plotting the output *SINR* as a function of input *SNR*. **Figure 8** shows several curves for the threeelement array with different values of pointing error (*θperr*). This figure shows that the beam-pointing error that can be tolerated is essentially a matter of dynamic range. For example, if *<sup>θ</sup>perr* <sup>¼</sup> <sup>5</sup>*<sup>o</sup>* , the output *SINR* is greater than 5 dB only for

#### **Figure 7.**

*Normalized radiation patterns of three- and five-element arrays. <sup>θ</sup><sup>d</sup>* <sup>¼</sup> <sup>0</sup>*o, pointing error (θperr* <sup>¼</sup> *<sup>θ</sup><sup>d</sup>* � *<sup>θ</sup>max*Þ ¼ <sup>0</sup>*<sup>o</sup> (K = 0.1). No interference.*

#### **Figure 8.**

*Output SINR vs. input SNR/element. Three-element array. <sup>θ</sup><sup>d</sup>* <sup>¼</sup> <sup>0</sup>*o, K = 0.1, with different values of pointing error (θperr* ¼ *θ<sup>d</sup>* � *θmax). No interference.*

1 dB <sup>≤</sup>*SNR*≤19 dB, whereas if *<sup>θ</sup>perr* <sup>¼</sup> <sup>0</sup>*:*1*<sup>o</sup>* , the output *SINR* > 5 dB for 0 dB≤ *SNR*≤56 dB. Therefore, the greater the desired signal dynamic range we wish to accommodate, the less pointing error *θperr* we should have.

Until now, we assumed that the feedback loop gains of the array *K* = 0.1. However, the effect of *K* on the performance of the array is illustrated in **Figure 9**.

#### **Figure 9.**

*Output SINR vs. feedback loop gain (K). Three-element array, <sup>θ</sup><sup>d</sup>* <sup>¼</sup> <sup>0</sup>*o, and SNR = 30 dB, with different values of pointing error (θperr* <sup>¼</sup> *<sup>θ</sup><sup>d</sup>* � *<sup>θ</sup>max). (a) No interference. (b) One 40-dB INR interference signal @*30*<sup>o</sup> .*

In **Figure** 9**(a)** where interference does not exist, the output *SINR* deteriorates rapidly by increasing *K* because the array tends to suppress the desired signal more, which results in more sensitivity to pointing errors.

On the other hand, when interference exists, **Figure** 9**(b)** shows that low values of *K*, as well as high values of K, have a negative effect on the performance*.* This is due to the capability of the array to effectively decrease null interference signals by decreasing *K*. From **Figure 9**, it can be concluded that the most appropriate value for the feedback loop gain is 00*:*1≤ *K* ≤0*:*1. In this chapter, we used *K* = 0.1 as a representative value.

In **Figure 4**, we presented the performance of the array in the absence of interference signals. Now consider the case when interference is present. The equivalent results of **Figure 4** are presented in **Figure 10**, where one interference signal is incident at *<sup>θ</sup><sup>I</sup>* <sup>¼</sup> <sup>10</sup>*<sup>o</sup>* with *INR* = 30 dB and zero bandwidth *BI* <sup>¼</sup> 0.

These curves differ from those in **Figure 4** in several respects. First, for *SNR* < 10 dB, *θperr* has less effect on the *SINR* (except for *θperr* ¼ *θI*) than it did with no interference. Second, for higher *SNR*s, the *SINR* again becomes sensitive to *θperr*, but much less than that without interference. For example, in **Figure 10** with *SNR* = 30 dB, *SIN*<sup>R</sup> <sup>&</sup>gt; 5 dB for �8*:*25*<sup>o</sup>* <sup>≤</sup>*θperr* <sup>≤</sup> <sup>3</sup>*:*75*<sup>o</sup>* , whereas in **Figure 4**, *SNR* = 30 dB yields *SINR* <sup>&</sup>gt; 5 dB only for �2*:*55*<sup>o</sup>* <sup>≤</sup>*θperr* <sup>≤</sup>2*:*55*<sup>o</sup>* . The reason for this difference is that, in general, with the presence of interference, the array uses its degrees of freedom to form nulls toward the interference. Therefore, the array cannot null the desired signal, as it could without interference.

For further illustration of the performance of the array, **Figure 11** shows the output *SINR* for different DOAs of the desired signal. In this figure, it can be seen that the *SINR* decreases as the DOA of the desired signal gets closer to the DOA of the interference signal. This is because, as expected, the desired signal is increasingly suppressed by the null formed toward the interference signal.

The corresponding results for different DOAs of the interference signal can be seen in **Figure 12**. Again, the *SINR* drops as the interference DOA gets closer to the DOA of the desired signal. **Figure 13** provides extra explanation for these results: the depth of the null toward the interference signal decreases as the DOA of the interference signal gets closer to the DOA of the desired signal. Therefore, more interference power appears at the output of the array.

#### **Figure 10.**

*Output SINR vs. pointing error. Three-element array, <sup>θ</sup><sup>d</sup>* <sup>¼</sup> <sup>0</sup>*o, K = 0.1. One 30-dB INR interference signal @ <sup>θ</sup><sup>I</sup>* <sup>¼</sup> <sup>10</sup>*o, BI* <sup>¼</sup> <sup>0</sup>*:*

*Steered Beam Adaptive Antenna Arrays DOI: http://dx.doi.org/10.5772/intechopen.100168*

**Figure 11.**

*Output SINR vs. direction of the main beam* ð Þ *θmax . Three-element array, K = 0.1. One 30-dB INR interference signal at <sup>θ</sup><sup>I</sup>* <sup>¼</sup> <sup>10</sup>*o. SNR = 30 dB with DOAs of <sup>θ</sup><sup>d</sup>* ¼ �10*<sup>o</sup>*, � <sup>5</sup>*<sup>o</sup>*, � <sup>0</sup>*<sup>o</sup>*, 5*<sup>o</sup>*, 10*<sup>o</sup>*, 15*o. Bd* <sup>¼</sup> 0, *BI* <sup>¼</sup> <sup>0</sup>*.*

*Output SINR vs. pointing error* ð Þ *<sup>θ</sup>max . Three-element array, <sup>θ</sup><sup>d</sup>* <sup>¼</sup> <sup>0</sup>*o,* <sup>K</sup> *= 0.1, Bd* <sup>¼</sup> 0, *BI* <sup>¼</sup> <sup>0</sup>*. One 30-dB* INR *interference signal with DOA @θ<sup>I</sup>* <sup>¼</sup> <sup>5</sup>*<sup>o</sup>*, 10*<sup>o</sup>*, 15*<sup>o</sup> .*

**Figure 14** depicts the output *SINR* vs. the DOA of the interference signal for different values of *INR*, and **Figure 15** depicts the output *SINR* vs. the *INR* of the interference signal with different DOAs. Here, the pointing error *<sup>θ</sup>perr* <sup>¼</sup> <sup>0</sup>*<sup>o</sup>* .

From these two figures, it can be seen that for the case of *<sup>θ</sup>perr* <sup>¼</sup> <sup>0</sup>*<sup>o</sup>* , the output *SINR* decreases as the interference signal gets closer to the desired signal (**Figure 14**) and as its *INR* increases (**Figure 15**). It can be noticed that the interference signal is more effectively nulled when its DOA is far away from the DOA of the desired signal. When *<sup>θ</sup>perr* 6¼ <sup>0</sup>*<sup>o</sup>* , we get the similar curves as in **Figures 14** and **15** (not shown here), but with lower output *SINR*. The reason is that the complex weights for forming a radiation pattern are selected for the center frequency.

#### **Figure 13.**

*Normalized radiation patterns for three-element array. <sup>θ</sup><sup>d</sup>* <sup>¼</sup> <sup>0</sup>*o, pointing error (θperr* <sup>¼</sup> *<sup>θ</sup><sup>d</sup>* � *<sup>θ</sup>max*Þ ¼ <sup>0</sup>*o, Bd* <sup>¼</sup> 0, *BI* <sup>¼</sup> <sup>0</sup>*. One 30-dB* INR *interference signal with DOA at <sup>θ</sup><sup>I</sup>* <sup>¼</sup> <sup>5</sup>*<sup>o</sup>*, 15*<sup>o</sup>:*

**Figure 14.** *Output* SINR *vs. DOA of the interference signal. Three-element array, <sup>θ</sup><sup>d</sup>* <sup>¼</sup> *<sup>θ</sup>max* <sup>¼</sup> <sup>0</sup>*o, SNR = 30 dB, K = 0.1, and Bd* ¼ 0, *BI* ¼ 0*.*

**Figure 16** shows the output *SINR* versus the input *SNR* for several values of pointing error *θperr*, taking into consideration the effect of the interference signal bandwidth (*BI*). Here, it can be seen that the output *SINR* decreases by increasing the interference bandwidth (*BI*), due to the decreased efficiency in the null formation toward the interference.

By comparing **Figure 16** with **Figure 8** (where the interference is absent), it can be seen that the performance of the array has been enhanced in the interference case in terms of the dynamic range of the desired signal that can be accommodated

#### **Figure 15.**

*Output* SINR *vs. interference-to-noise ratio (*INR*). Three-element array, <sup>θ</sup><sup>d</sup>* <sup>¼</sup> *<sup>θ</sup>max* <sup>¼</sup> <sup>0</sup>*o. SNR = 30 dB, K = 0.1, and Bd* ¼ 0, *BI* ¼ 0*, with different DOAs of the interference signal (θI*Þ*.*

#### **Figure 16.**

*Output SINR vs. input SNR/element. Three-element array, K = 0.1, and Bd* ¼ 0*, with different values of pointing error (θperr* <sup>¼</sup> *<sup>θ</sup><sup>d</sup>* � *<sup>θ</sup>max). One 30-dB INR interference signal with DOA@* <sup>30</sup>*<sup>o</sup> with different bandwidths (BI).*

for a given *<sup>θ</sup>perr*. For example, for *<sup>θ</sup>perr* <sup>¼</sup> <sup>1</sup>*<sup>o</sup>* and *BI* <sup>¼</sup> 0, the interference-free case provides output SINR greater than 5 dB only for 1 dB≤ *SNR*≤34*:*2 dB, whereas in the presence of interference, the output *SINR* > 5 dB for 1 dB≤*SNR*≤ 43*:*5 dB.

The interference bandwidth *BI* decreases with a decreasing *INR*. This can be concluded from the comparison between **Figures 16** and **17**, where the effect of *BI* is insignificant in the latter case.

The effect of the interference signal bandwidth and its *INR* is further explained in **Figure 18**. Here, it can be seen that the effect of the bandwidth of the interference signal is more significant when its *INR* is higher.

#### **Figure 17.**

*Output SINR vs. input SNR/element. Three-element array, K = 0.1, and Bd* ¼ 0*, with different values of pointing error (θperr* <sup>¼</sup> *<sup>θ</sup><sup>d</sup>* � *<sup>θ</sup>max). One 5-dB* INR *interference signal with DOA@* <sup>30</sup>*<sup>o</sup> and bandwidths (*0% ≤*BI* ≤20%*).*

#### **Figure 18.**

*Output SINR vs. interference signal bandwidth (BI). Three-element array, K = 0.1, and Bd* ¼ 0*, pointing error (θperr* <sup>¼</sup> *<sup>θ</sup><sup>d</sup>* � *<sup>θ</sup>max). One interference signal with DOA@* <sup>30</sup>*<sup>o</sup> and different INRs.*

In contrast, for low *INR*s (<5 dB), the effect of the interference bandwidth is unnoticeable. Additionally, the interference bandwidth has more impact on the output *SINR* as its DOA gets closer to the DOA of the desired signal. This is illustrated in **Figure 19** for the case of *<sup>θ</sup>perr* <sup>¼</sup> <sup>0</sup>*<sup>o</sup>* . The effect of the interference bandwidth *BI* on the depth of the nulls formed toward the interference signals is explained in **Figures 20** and **21**.

**Figure 20** shows that the depth of the null formed at 30*<sup>o</sup>* has a depth of –61 dB when the bandwidth of the interference signals is 0%, while it gets shallower (–48 dB) when the bandwidth of the interference signals increases (20% of the carrier frequency). This also applies to the null formed at �50*<sup>o</sup>* , but with less change in the depth (as explained in **Figure 19**).

*Output SINR vs. interference signal bandwidth (BI). Three-element array, K = 0.1, and Bd* ¼ 0*, pointing error (θperr* ¼ *θ<sup>d</sup>* � *θmax). One interference signal with INR = 30 dB and different DOAs.*

#### **Figure 20.**

*Radiation patterns of a three-element array with different interference bandwidths B*ð Þ*<sup>I</sup> . Input SNR = 10 dB, Bd* ¼ 0*, K = 0.1, pointing error (θperr* ¼ *θ<sup>d</sup>* � *θmax). Two interference signals with identical bandwidths, INR's = 30 dB at* <sup>30</sup>*<sup>o</sup>* and � <sup>50</sup>*o.*

For more explanation, **Figure 21** illustrates the depths of the two nulls (formed at 30*<sup>o</sup>* and � <sup>50</sup>*<sup>o</sup>* ), as a function of the interference signals bandwidth. For simplicity, we assumed here that the two bandwidths are identical. It can be seen that the nulls get shallower as the bandwidth of the interference signals increases and the close the DOA of the interference signal to the DOA of the desired signal, the more change in the null depth occurs.

As previously discussed, the *N*-element steered beam adaptive arrays have *N* � 1 degrees of freedom. Hence, they have the capability of efficiently nulling up to *N* � 1 interference signals. However, if more than *N* � 1 interference signals are incident on the array, the array cannot form nulls toward the interference signals. Instead, it

#### **Figure 21.**

*Null depths formed by three-element array @* <sup>30</sup>*<sup>o</sup>* and � <sup>50</sup>*<sup>o</sup> as a function of the interference bandwidth. Input SNR = 10 dB, Bd* ¼ 0*, K = 0.1, pointing error (θperr* ¼ *θ<sup>d</sup>* � *θmax). Two interference signals with identical bandwidths, INR's = 30 dB at* <sup>30</sup>*<sup>o</sup>* and � <sup>50</sup>*o.*

tries to form a radiation pattern such that the output *SINR* is maximized. This is shown in **Figure 22** where 6 and 12 30-dB interference signals are incident on a seven-element uniformly spaced array with uniform spacings of (λ/2).

As observed in **Figure 22**, in the case of the six interference signals with DOAs of �60*<sup>o</sup>* , � 50, � <sup>40</sup>*<sup>o</sup>* , � <sup>45</sup>*<sup>o</sup>* , � <sup>30</sup>*<sup>o</sup>* , 20*<sup>o</sup>* , 30*<sup>o</sup>* , and 60*<sup>o</sup>* , we can see that the array has effectively nulled these signals, while in the case of the twelve interference signals

#### **Figure 22.**

*Radiation patterns for the seven-element uniformly spaced adaptive array with a different number of interference signals with INR's = 30 dB. SNR = 20 dB with pointing error (θperr*Þ ¼ <sup>0</sup>*o.*

## **Figure 23.**

*Performance of the seven-element uniformly spaced adaptive array with a different number of interference signals and pointing errors. INR's = 30 dB. SNR = 20 dB.*

with DOAs of <sup>20</sup>*<sup>o</sup>* , <sup>30</sup>*<sup>o</sup>* , <sup>60</sup>*<sup>o</sup>* , <sup>45</sup>*<sup>o</sup>* , <sup>10</sup>*<sup>o</sup>* , 10*<sup>o</sup>* , 20*<sup>o</sup>* , 30*<sup>o</sup>* , 45*<sup>o</sup>* , 50*<sup>o</sup>* , 60*<sup>o</sup>* , and 75*<sup>o</sup>* , the array could not form nulls toward these signals, but it arranged the radiation pattern in such a way that the least amount of interference power is allowed (e.g., by forming a null between two close interference signals).

However, the presence of a large number of interference signals adds more interference power to the array output, which lowers the output *SINR*. This is illustrated in **Figure 23**, where the output *SINR* for the six effectively nulled interference signals is very close to the situation where interference signals do not exist. Clearly, both of these cases show much better performance of the adaptive array when 12 interference signals are present. Significantly, the array has more sensitivity to pointing errors in the absence of interference signals. This explains the enhancement of the output *SINR* for *θperr* as the input *SNR* increases.
