In the last tweet about array antenna shaped beam synthesis -- Array antenna shaped beam synthesis (1), the Fourier series method was simply described and the CST example was verified. This tweet offers an example of woodward-Lawson sampling.
The software used in this paper is CST 2018 and MATLAB 2019A
Woodward-lawson sampling involves sampling different discrete angles of a given directional graph. A lot of people look at this and think, how does this feel like that thing in the signal and system, and sample it? Last time the Fourier series method for finding the amplitude of excitation for a given unit to achieve the desired direction is also very much like the time domain to frequency domain transformation. Professional class finally or rely on mathematics as the final solution tool, mathematics - GOAT!
In fact, the above matrix factor superposition formula can deduce the first sidelobe value and Angle position of uniform linear array, the Angle position of each zero point, the relationship between the phase difference of linear array elements and beam pointing, and the main lobe 3dB beam.
When the simple linear array with equal amplitude and equal moment is changed to the linear array with equal spacing and variable amplitude and phase, the degree of freedom of the far field pattern that can be synthesized is improved.
Under this constraint, Woodward described the principle of Woodward-Lawson sampling in the following two papers, which has become a very popular method of antenna pattern synthesis for arbitrary beam formation.
1. P. m. Woodward, "A Method for developing the Field over A Plane Aperture Required to Produce A GivenPolar Diagram," J. IEE, Vol. 93, Pt. IIIA, pp. 1554 -- 1558, 1946.
2. P. M. Woodward and J. D. Lawson, "The Theoretical Precision with Which an Arbitrary radiation-pattern May be Obtained from a Source of a Finite Size," J. IEE, Vol. 95, PT. III, No. 37, pp. 363 -- 370, September 1948.
To understand this approach, the simplest idea is that for a straight array, a different set of amplitude and phase configurations will produce different beam directions. The 3dB beamwidth of the direction diagram shown below is used to fill the desired target direction diagram, and the amplitudephase excitation corresponding to all the group direction diagrams is superimposed to obtain the amplitude and phase that should be allocated to each unit in the end.
For example, for the target direction graph like the square wave shown below, the direction graph pointing to different beams is superimposed.
Run the above Matlab code (slide left and right to see the complete code) to get the following superposition diagram of different beams. If its envelope can be taken, the more sampling points, the smaller the ripple in the band.
But we're actually adding these directions together (swipe left and right to see the full code) :
This results in the following final direction:
It looks fine, but woodward-Lawson sampling is not as easy to take for granted, and all complete theories need to be mathematically described.
Here's straight to the heart of the tweet:
The first step is to determine the sampling Angle and excitation coefficient. Taking the expected direction diagram of the above square wave within one square as an example, it is assumed that the element spacing of the linear array is D =0.5, the number of elements is N=10, and the length of the linear array is L= Nd = 5. In order to make the integrated direction diagram correspond to the visible region 0E (0, open), the given direction diagram can be reconstructed accurately, and the sampling interval A can be determined as:
Sampling Angle can be expressed according to the parity of sampling points as follows:
Therefore, the sampling Angle can be calculated by the above formula, and the excitation coefficient corresponding to this Angle is taken as the value at the sampling Angle corresponding to the expected direction graph. If the sampling point is exactly on the boundary of the pre-given expected direction graph, the sampling value is half of it.
The following table is obtained after calculation:
The second step is to determine the feed coefficient of each unit. For discrete linear array, the feed coefficients of each unit can be determined by the following formula:
Feed coefficients of each unit are as follows:
The model of antenna array and array arrangement is slightly modified to make its working frequency near 15GHz, the half-wavelength of working frequency is 10mm, and 10 units are arranged at half-wavelength intervals.
The CST CombineResults are calculated with the feed coefficients calculated in the previous section, and the far field direction diagram of 15GHz can be obtained as follows:
Since the 10-element linear array is used to realize the desired direction diagram, the gain flatness of the middle part is slightly less than that of the array antenna with half the number of elements compared to the shaped beam synthesis (I). Because the arrays calculated in the previous part are arranged along the axis, while the arrays above are arranged along the axis, it is ok to add a value when looking at the direction diagram, or directly regard the abscissa in the following figure as the range from 0 to 180°.
The following figure is the expected direction diagram synthesized by Fourier series method under 20-element linear array:
Next, the square-like direction diagram and pen shape direction diagram with wider beam are integrated respectively:
Square - like wave direction diagram - feed coefficient
As can be seen from the above simulation results, the 3dB beam width reaches 107 degrees, and the in-band gain flatness is good.
As can be seen from the above simulation results, the 3dB beam width reaches 30 degrees, and the 2D image sub-warhead.
As can be seen from the following intuitive comparison of the three directions, the wider the beam width, the lower its gain, which is also consistent with theoretical experience.
Woodward-lawson sampling is simple and elegant. However, the lack of local control of sidelobe level is an inherent defect. See here, do you want to use GA, PSO and other algorithms to optimize the excitation amplitude and phase partners? If you are interested, you can try it.
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