Author: Peter Delos, Bob Broughton, Jon Kraft, ADI

**brief introduction**

In the first part, we introduce the concept of phased array, beam steering and array gain. In the second part, we discuss the concept of grating lobe and beam squint. In the third part, we first discuss the sidelobe and the influence of taper on the whole array. Taper is to manipulate the amplitude of a single element on the overall antenna response.

In the first part, taper is not applied, and it can be seen from the figure that the first side lobe is – 13dbc. Taper provides a method to reduce antenna sidelobe, but it will reduce antenna gain and main lobe beam width. After a brief introduction to taper, we will elaborate on several key points related to antenna gain.

**Fourier transform: rectangular function A kind of sinc function **

In electrical engineering, there are various ways to transform a rectangular function in one domain into a sinc function in another domain. The most common form is that the rectangular pulse in time domain is converted into the spectrum component of sinc function. The conversion process is reversible. In broadband applications, the wideband waveform can also be converted into narrow pulses in time domain. The phased array antenna also has the similar characteristics: the rectangular weighting along the plane axis of the array radiates the pattern according to the sine function.

Applying this property, the first sidelobe represented by sinc function is only – 13dbc, which is problematic. Figure 1 shows this principle.

Figure 1. The rectangular pulse in the time domain generates a sinusoidal function in the frequency domain, and the first sidelobe is only – 13dbc.

**Taper (or weighting)**

In order to solve the sidelobe problem, we can use weighted processing in the whole rectangular pulse. This is very common in FFT, and the taper option in phased array directly simulates the weighting in FFT. Unfortunately, there are some disadvantages in weighting. Although it can reduce the side lobe, it needs to widen the main lobe. Figure 2 shows some examples of weighting functions.

Figure 2. An example of a weighting function.

**Waveform and antenna analogy**

The conversion from time to frequency is common, and most electrical engineers will understand. However, it is not clear at the beginning how to use antenna pattern analogy for engineers who have just contacted phased array. For this reason, we use field excitation instead of time domain signal, and use space domain instead of frequency domain output.

**Time domain → field domain**

·V (T) – voltage is a function of time

·E (x) – the field strength is a function of the position in the aperture

Frequency domain → space domain

·Y (f) – power spectral density is a function of frequency

·G (q) – antenna gain as a function of angle

Figure 3 shows these principles. Here, we compare two different weighted radiation energies applied to the array. Fig. 3a and Fig. 3C show the field domain. Each point represents the amplitude of one element in this n = 16 array. Outside the antenna, there is no radiation energy, and the radiation starts from the edge of the antenna. In Fig. 3a, the field strength changes abruptly, while in Fig. 3C, the field strength increases with the distance from the antenna edge. The effects on radiation energy are shown in Fig. 3b and Fig. 3D, respectively.

Fig. 3. A chart showing the conversion of narrower elements into radiation energy weighting;

(A) All the components are weighted uniformly; (b) the sinusoidal function radiates in space; (c) all components are weighted by Hamming window; and (d) the radiation side lobe is reduced to 40 DBC at the cost of broadening the main beam.

In the next section, we will introduce two additional error terms that affect antenna pattern performance. The first is mutual coupling. In this paper, we only point out the existence of this problem and give the number of EM models used to quantify the impact. The second is the quantization sidelobe due to the limited precision in phase-shift control. We deal with the quantization error more deeply and quantize the side lobe.

**Mutual coupling error**

All equations and array factor graphs discussed here assume that the elements are identical and each element has the same radiation pattern. But this is not the case. One of the reasons is mutual coupling, that is, coupling between adjacent components. The radiation performance of the components dispersed in the array will be greatly changed compared with the components arranged closely with each other. The elements at the edge of the array and those at the center of the array are in different environments. In addition, the mutual coupling between the components will also change when the beam is turned. All these effects will produce an additional error term, which needs to be considered by antenna designers. In practical design, a lot of energy is needed to use electromagnetic simulator to characterize the radiation effects under these conditions.

**Beam angle resolution and quantized sidelobe**

There is another drawback of phased array antenna. The resolution of time delay unit or phase shifter used for beam steering is limited. This usually uses discrete-time (or phase) steps to achieve digital control. However, how to determine the resolution or bit number of delay unit or shifter to achieve the desired beam quality?

Contrary to common understanding, the beam angle resolution is not equal to the phase shifter resolution. From equation 1 (equation 2 in the second part), we can see the relationship as follows:

We can use the phase shift in the whole array to express this relationship. We need to replace the array width d with the element spacing D. Then, if we replace the phase shifter Φ LSB with? 6？ 2 Φ, we can roughly estimate the beam angle resolution. For a linear array of n elements arranged at half wavelength intervals, the beam angle resolution is shown in equation 2.

This is the beam angle resolution away from the line of sight, which describes the beam angle when half of the array phase shift is zero and the other half phase shift is the LSB of the phase shifter. If programming to less than half the angle of the LSB is not possible. Figure 4 shows the beam angle of a 30 element array using a 2-bit phase shifter (the phase LSB increases gradually). Note that the beam angle increases until half of the elements phase shift the LSB and then return to zero when all elements phase shift the LSB. This makes sense when the beam angle changes through the phase difference in the array. Note that, as previously calculated, the peak value of this characteristic is θ res.

Figure 4.30 relationship between beam angle and number of elements for linear array of elements at LSB.

Figure 5. Relationship between beam angle resolution and array size when phase shifter resolution is 2-bit to 8-bit.

Figure 5 shows the relationship between θ res and array diameter (element spacing is λ / 2) at different phase shifter resolutions. This shows that even a very coarse 2-bit phase shifter with a 90 ° LSB can achieve 1 ° resolution in an array of 30 elements in diameter. In the first part, we use equation 10 to solve the 30 element, λ / 2 spacing condition, and the main lobe beam width is about 3.3 °, which means that even with this very coarse phase shifter, we have enough resolution. So what are the results of using a higher resolution phase shifter? From the analogy between time sampling system (data converter) and spatial sampling system (phased array antenna), it can be seen that the higher resolution data converter produces lower quantization background noise. Higher resolution phase / time shifters result in lower quantization sidelobe levels (qsll).

Figure 6 shows the phase shifter settings and phase errors for a 2-bit 30 element linear array programmed with a θ res beam resolution angle. Half of the array is set to zero phase shift and the other half is set to 90 ° LSB. Note that the error (the difference between the ideal quantization phase shift and the actual quantized phase shift) curve is zigzag.

Figure 6. Element phase shift and phase error in the array.

Fig. 7 shows the antenna pattern of the same antenna when turning 0 ° and steering beam resolution angle. Note that severe pattern degradation occurs due to the quantization error of the phase shifter.

Fig. 7. Antenna pattern with quantized sidelobes at minimum beam angle.

When the maximum quantization error occurs in the aperture, all other elements are zero error, and the adjacent components are spaced LSB / 2, the worst quantization sidelobe occurs. This represents the maximum period of the maximum possible quantization error and aperture error. Figure 8 shows this when using a 2-bit 30 element.

Figure 8. Worst case antenna quantization sidelobe case – 2 bits.

This occurs at a predictable beam angle as shown in equation 3.

Where n “2bits”, and N is an odd number. For 2-bit systems, this occurs 4 times between ± 14.5 ° and ± 48.6 ° range. Figure 9 shows the antenna pattern of the system at n = 1, q = + 14.5 °. Note that there is a significant – 7.5db quantized sidelobe at – 50 °.

Figure 9. Worst case antenna quantization sidelobe: 2 bits, n = 1, 30 elements.

In addition to the special case that quantization error is 0 and LSB / 2 in turn, RMS error decreases with the spread of beam on aperture at other beam angles. In fact, for the angle equation (equation 3) where n is even, the quantization error is 0. If we plot the relative levels of the highest quantized sidelobe at different phase shifter resolutions, some interesting patterns will appear. Figure 9 shows the worst qsll for a 100 element linear array, which uses Hamming cones to distinguish the quantized sidelobes from the classical windowed sidelobes discussed earlier in this section.

Note that at 30 ° all quantization errors tend to zero, which can be shown as the result of sin (30 °) = 0.5. Note that for any particular n-shifter, the beam angle at the worst level will show zero quantization error at higher resolution n. It can be seen here that the beam angle at the worst sidelobe level described and the qsll improved by 6dB at each resolution.

Figure 10. Worst quantized sidelobe versus beam angle at 2-bit to 6-bit phase shifter resolutions.

Figure 11. Worst quantized sidelobe level versus phase shifter resolution.

The maximum quantization sidelobe level qsll of 2-bit to 8-bit phase shifter resolution is shown in FIG. 11, which follows the similar quantization noise rule of data converter,

Or each resolution is about 6dB. At 2 bits, the qsll level is about – 7.5db, which is higher than the classic + 12dB of random signal sampling by data converter. This difference can be considered as the result of periodic sawtooth error during aperture sampling, in which spatial harmonics increase the phase. Note that qsll is not a function of pore size.

**summary**

We can now summarize some of the challenges faced by antenna engineers in relation to beam width and sidelobes:

·Angular resolution requires a narrow beam. A narrow beam requires a large aperture, which in turn requires many elements. In addition, the beam becomes wider when it deviates from the line of sight, so additional elements are needed to keep the beam width constant as the scanning angle increases.

·It seems possible to expand the entire antenna area by increasing the element spacing without adding additional elements. This can narrow the beam, but unfortunately, if the components are not evenly distributed, it will lead to grating lobes. An attempt can be made to use the increased antenna area and minimize the grating lobe problem by reducing the scanning angle and using an aperiodic array with intentionally random component patterns.

·Sidelobe is another problem, which we know can be solved by gradually reducing the array gain toward the edge. However, this kind of taper needs more components at the cost of beam broadening. The resolution of phase shifter will cause quantization sidelobe, which must be considered in antenna design. For the antenna with phase shifter, the phenomenon of beam squint will lead to the interaction between angular displacement and frequency, thus limiting the available bandwidth at high angle resolution.

The above is about all three parts of phased array antenna pattern. In the first part, we introduce beam pointing, array factor and antenna gain. In the second part, we discuss the disadvantages of grating lobe and beam squint. In the third part, we discuss taper and quantization error. This paper is not aimed at the antenna design engineers who are proficient in electromagnetic and radiation element design, but for a large number of engineers working in the field of phased array. These intuitive explanations will help them understand the various factors that affect the performance of the whole antenna pattern.

**Reference circuit**

Theory, analysis and design of antenna. 3rd Edition, Wiley, 2005.

Mailloux, Robert J. phased array antenna manual. Second edition. Artech House, 2005.

O’Donnell, Robert M. “radar systems engineering: an introduction.” IEEE, June 2012.

Skolnik, Merrill. Radar manual. 3rd Edition, McGraw Hill, 2008.

**author**

PeterDelos

Peter Delos is the technical director of ADI’s aerospace and Defense Department and works in Greensboro, North Carolina. He received a bachelor’s degree in electrical engineering from Virginia Tech University in 1990 and a master’s degree in electrical engineering from New Jersey Institute of technology in 2004. Peter has more than 25 years of industry experience. Most of his career has been devoted to the architecture, PWB and IC Design of advanced RF / analog systems. He is currently focusing on the miniaturization of high-performance receivers, waveform generators and frequency synthesizers for phased array applications.

BobBroughton

Bob Broughton began to work for ADI in 1993. He has successively held the positions of product engineer and IC Design Engineer, and currently serves as the engineering director of the Department of aerospace and defense. Prior to joining ADI, Bob worked as an RF Design Engineer at Raytheon and a RFIC designer at Peregrine simicon uctor. Bob graduated from West Virginia University with a bachelor’s degree in electrical engineering in 1984.

JonKraft

Jon Kraft is a senior field application engineer based in Colorado and has been with ADI for 13 years. He focuses on Software Defined Radio and aerospace phased array radar applications. He holds a bachelor’s degree in electrical engineering from roshouman Institute of technology and a master’s degree in electrical engineering from Arizona State University. He has nine patents, six related to ADI, and one is under application.