“Regarding the phased array antenna pattern, we will introduce it in three parts. This is the second article. In the first part, we introduced the concept of phased array steering and looked at the factors that affect the gain of the array. In the second part, we will discuss grating lobes and beam squint. Grating lobes are difficult to visualize, so we use their similarity to signal aliasing in a digitizer and imagine the grating lobes as spatial aliasing.

“

**Introduction**

Regarding the phased array antenna pattern, we will introduce it in three parts. This is the second article. In the first part, we introduced the concept of phased array steering and looked at the factors that affect the gain of the array. In the second part, we will discuss grating lobes and beam squint. Grating lobes are difficult to visualize, so we use their similarity to signal aliasing in a digitizer and imagine the grating lobes as spatial aliasing. Next, we discuss the problem of beam squint. Beam squint is a phenomenon in which the antenna is not focused in the frequency range when we use phase shift instead of real time delay to steer the beam. We will also discuss the trade-offs between these two steering methods and understand the effect of beam squint on typical systems.

**Introduction to grating lobe**

So far, we have only seen the case where the component spacing is d = λ/2. Figure 1 begins to explain why λ/2 element spacing is so common in phased arrays. There are two situations shown in the figure. First, it is the blue line, repeating the 30° graph in Figure 11 in Part 1. Next, the d/λ interval is increased to 0.7 to show how the antenna direction changes. Note that as the interval increases, the beam width decreases, which is a positive phenomenon. The reduction of the zero interval makes their distance closer, which is also acceptable. But now there is a second angle, in this case C70°, at which the full array gain appears. This is the most unfavorable situation. This antenna gain duplication is defined as a grating lobe, which can be considered as spatial aliasing.

Figure 1. Normalized array factor of a 32-element linear array at two different d/λ intervals.

**Sampling system analogy**

In order to realize the visualization of grating lobes, it can be compared to the aliasing phenomenon in the sampling system. In an analog-to-digital converter (ADC), the receiver structure usually undersamples the frequency. Undersampling involves deliberately reducing the sampling rate (fS), which converts frequencies higher than fS/2 (higher Nyquist zone) into aliasing in the first Nyquist zone through the sampling process. This makes these higher frequencies appear to be lower frequencies at the ADC output.

A similar analogy method can be considered in the phased array, in which the wavefront is spatially sampled by the elements. If we propose to sample each wavelength twice (ie, components) in order to avoid aliasing, then the Nyquist criterion can be extended to the spatial region. Therefore, if the element spacing is greater than λ/2, we can consider this spatial aliasing.

**Calculate where the grating lobes appear**

But where will these spatial aliasing (grating lobes) appear? In Part 1, we showed the functional relationship between the phase shift of the elements in the entire array and the beam angle.

Conversely, we can calculate the beam angle as a function of the phase shift.

The arcsin function only produces real solutions between -1 and +1. Outside these ranges, real number solutions cannot be obtained, and “#NUM!” will appear in the spreadsheet software. Also note that the phase in Equation 2 is periodic, repeating every 2π. Therefore, we can use (m × 2π + ∆Φ) to replace ∆Φ in the beam steering formula, and then get formula 3.

Where m = 0, ±1, ±2…

In order to avoid grating lobes, our goal is to obtain a single real number solution.Mathematically, this is achieved by making the following formula true

If we do this, all spatial images (ie m = ±1, ±2, etc.) will produce non-real arcsin results, and we can ignore them. But if we cannot do this, then some values of m> 0 will produce real arcsin results, then we will have multiple solutions: grating lobes.

Figure 2. The application of arcsin function in grating lobes.

d> λ and λ = 0° grating lobe

Let’s try some examples to better illustrate this point. First, consider the example of mechanical axis calibration, where θ = 0, so ∆Φ = 0. Then, formula 3 is simplified to formula 5.

Through this simplification, it can be clearly seen that if λ/d> 1, then only when m = 0, the parameter between C1 and +1 can be obtained. This parameter is 0, and arcsin(0) = 0°, which is the calibration angle of the mechanical axis. This is the result we expect. In addition, when m ≥ 1, the arcsin parameter will be very large (>1), and no real number result will be obtained. We can see that θ = 0 and d

However, if d> λ (making λ/d

Figure 3. Array factor of axis calibration when d/λ = 1.5 and N = 8.

λ/2

When simplifying the grating lobe equation (Equation 5), we choose to only look at the mechanical axis calibration (∆Φ = 0). We also see that when the mechanical axis is calibrated, d

First, review how the phase changes with the steering angle in Figure 4 in Part 1. We see that when the main lobe is calibrated away from the mechanical axis, the range of ∆Φ is 0 to ±π. therefore,

The range is

When |m|≥1, its value exceeds this range

If we want to keep the entire arcsin parameter> 1 under all |m| ≥ 1, we will limit the minimum allowable λ/d. Consider two situations:

If λ/d ≥ 2 (that is, d ≤ λ/2), no matter what the value of m is, there will not be multiple solutions. All solutions with m> 0 will result in the arcsin parameter> 1. This is the only way to avoid grating lobes in the horizontal direction.

However, if we intentionally limit ∆Φ to be less than ±π, then we can accept a smaller λ/d without grating lobes. Reducing the range of ∆Φ means reducing the maximum steering angle of the array. This is an interesting trade-off, which will be explored in the next section.

**Component spacing consideration**

Should the component spacing always be less than λ/2? it’s not true! This is the consideration and trade-off that antenna designers need to make. If the beam is completely turned to the horizontal direction, and θ = ±90°, the element spacing is required to be λ/2 (if grating lobes are not allowed in the visible semi-circle). But in actual operation, the maximum achievable steering angle is always less than 90°. This is due to the element factor and other reductions at large steering angles.

From the arcsin diagram shown in Figure 2, we can see that if the y-axis θ is limited to a reduced limit, the grating lobes will only appear at scan angles that will not be used. For a given element spacing (dmax), what is the limit (θmax) of this reduction?As we said before, our goal is to make the following

We can use it to calculate where the first grating lobe (m = ±1) appears. Now using Equation 1 for ∆Φ in Part 1, we get:

Can be simplified to

Then get dmax

The dmax is the condition that there is no grating lobe at a reduced scanning angle (θmax), where θmax is less than π/2 (90°). For example, if the signal frequency is 10 GHz and we need to turn ±50° without grating lobes, the maximum element spacing is:

Figure 4. When θ = 50°, N = 32, d = 17 mm, and Φ = 10 GHz, the grating lobes begin to appear in the horizontal direction.

By limiting the maximum scanning angle, you can freely expand the component spacing, increase the physical size of each channel, and expand the aperture of a given number of components. For example, this phenomenon can be used to assign a fairly narrow predefined direction to the antenna. The element gain can be increased to provide directivity in a predefined direction, and the element spacing can also be increased to achieve a larger aperture. Both of these methods can obtain a larger overall antenna gain under a narrower beam angle.

Note that Equation 3 indicates that the maximum interval is one wavelength, even at zero steering angle. In some cases, it is sufficient if the grating lobes do not appear in the visible semicircle. Taking a geostationary satellite as an example, it will center on the mechanical axis calibration and cover the entire earth at a steering angle of 9°. In this case, as long as the grating lobe does not fall on the surface of the earth. Therefore, the element spacing can reach several wavelengths, making the beam width narrower.

There are also some noteworthy antenna structures that try to overcome the grating lobe problem by forming inconsistent element spacing. These are classified as aperiodic arrays, taking spiral arrays as an example. Due to the mechanical antenna structure, we may wish to have a general building module that can be expanded to a larger array, but this will form a consistent array, which will be affected by the grating lobe conditions described.

**Beam squint**

In Part 1, we described at the beginning how the time delay between the elements based on the crest angle θ calibrated relative to the axis occurs when the wave crest approaches the element array. For a single frequency, phase shift can be used instead of time delay to achieve beam steering. This method is suitable for narrowband waveforms, but for broadband waveforms that produce beam steering by phase shifting, the beam may shift direction (as a function of frequency). If we remember that time delay is the relationship between linear phase shift and frequency, it can be explained intuitively. Therefore, for a given beam direction, the phase shift is required to vary with frequency. Or conversely, for a given phase shift, the beam direction changes with frequency. The situation where the beam angle changes with frequency is called beam squint.

It is also considered that when the axis calibration position θ = 0, there is no phase shift across the elements, so no beam squint will occur. Therefore, the amount of beam squint must be a function of the angle θ and the frequency change. Figure 5 shows an example of X-band. In this example, the center frequency is 10 GHz, the modulation bandwidth is 2 GHz, and it is obvious that the beam changes direction with changes in frequency and initial beam angle.

Figure 5.32 An example of beam squinting in the X-band when the element spacing is λ/2 for a linear array of elements.

The beam squint can be calculated directly.Using formula 1 and formula 2, the beam direction deviation and beam squint can be calculated

This formula is shown in Figure 6. In Figure 6, the f/f0 ratio shown is intentional. The reciprocal of the previous equation (f0/f) provides an easier way to more intuitively express the change relative to the center frequency.

Figure 6. Beam squint and beam angle under several frequency deviations.

Some observations about beam squint:

The deviation of the beam angle from the frequency increases as the beam angle deviates from the axis calibration angle.

Frequencies below the center frequency produce greater deviations than frequencies above the center frequency.

Frequencies below the center frequency will cause the beam to be aligned further away from the axis.

**Beam squint considerations**

The beam squint, that is, the deviation of the steering angle and the frequency, is caused by the phase shift to achieve the time delay. This problem does not occur when the real time delay unit is used to perform beam steering.

Since the beam squint problem is so obvious, why does anyone use a phase shifter instead of a time delay unit? Generally speaking, this is due to the simplicity of the design and the availability of phase shifters and time delay units of the IC. The time delay is implemented in the form of some transmission lines, and the total delay time required is a function of the aperture size. So far, most of the available analog beamforming ICs are based on phase shifting, but there are also real time delay IC series that are more common in phased arrays.

In digital beamforming, the real time delay can be implemented using DSP logic and digital beamforming algorithms. Therefore, for the phased array architecture in which every element is digitized, it can solve the beam squint problem by itself and provide the highest programming flexibility. However, the function, size, and cost of this solution can cause problems.

In hybrid beamforming, the sub-array adopts analog beamforming, and the entire array adopts digital beamforming. This can provide some beam squint reductions worth considering. The beam squint is only affected by the sub-array, and the beam width of the sub-array is wider, so the tolerance of the beam angle deviation is greater. Therefore, as long as the beam squint of the sub-array is tolerable, a hybrid beam-forming structure with a phase shifter can be used in the sub-array followed by a real time delay (digital beamforming).

**Summarize**

The above is the second part of the three parts about the phased array antenna pattern. In Part 1, we introduced the beam pointing and array factor. In Part 2, we discuss the disadvantages of grating lobes and beam squint. In Part 3, we will discuss how to narrow the side lobes by narrowing the antenna, and give you an in-depth understanding of the phase shifter quantization error.

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