The analytical methods of array antenna pattern synthesis are mainly for uniform linear arrays and uniform planar arrays, while numerical methods are generally used for non-uniform arrays. For more than half a century, many analytical methods for array antenna pattern synthesis have been studied. The two most basic methods, namely the Dolph–Chebyshev pattern synthesis method and the Taylor single parameter pattern synthesis method, are introduced here.
2.2.3.1Dolph–Chebyshev pattern synthesis method
For a uniform linear array, the first sidelobe is approximately 13.5 dB lower than the main lobe when the same excitation is used for each array element. For many practical applications, lower sidelobe levels are often required. In 1946, C.L. Dolphy proposed a method for obtaining lower sidelobe patterns in a classic paper. This method considers the properties of the Chebyshev polynomial and establishes the relationship from polynomial to array sidelobe level.
The Chebyshev polynomial T2N(u) has the characteristics of an undamped oscillation function when –1 ≤ u ≤ 1, and the monotonic increase is characteristic of the absolute value outside this oscillation interval. The undamped oscillation characteristics correspond to equal sidelobe levels, while the monotonic characteristics correspond to the main lobe. The Chebyshev polynomial whose order is 2N and number of elements is 2N + 1 can be expressed as
The relationship between the Chebyshev polynomial and the array antenna parameters is
where θ denotes the angle between the spatial orientation and the array.
The sidelobe level of the array antenna is expressed as 201gη in dB, where η = T2N(u0).
The above polynomial can also be expressed in the product form of the polynomial root.
where c is a constant and the root is given by
When the excitation current is symmetrically distributed, the root of the polynomial is a pair of complex conjugates. And through a series of mathematical derivations, the pattern can be expressed as
This is the Chebyshev pattern with 2N + 1 array elements.
The above Chebyshev pattern gives an array antenna pattern synthesis method which can control the sidelobe level to minimize the maximum sidelobe level. However, the method has the following problems as well. The excitation current between the antenna intermediate unit and the external unit varies greatly, which is difficult to implement. The far-sidelobe level is too high. These problems make the Chebyshev pattern encounter some difficulties in practical applications, which means that its physical achievability is poor.
2.2.3.2Taylor single-parameter pattern synthesis method
In 1953, T.T. Taylor presented a pattern synthesis method derived from the uniform excitation array pattern sin(πu)/πu. The zero interval of the pattern is an integer, and the descent velocity of sidelobe envelope is 1/u. Therefore, it is necessary to control the height of the first sidelobe level, which is realized by adjusting the zero point of the pattern function. The zero point of the array pattern is given by
B is an undetermined parameter and then the description of the antenna pattern becomes
when u = B, the pattern changes from a hyperbolic function to a sinc function.
SLR is the ratio of peak to sinc, and it is expressed in dB by
The method determines all parameters of the pattern by a single parameter B, including sidelobe level, beamwidth, and beam efficiency. The aperture distribution of the array is the inverse of the pattern, i.e.
where p is the distance from the center of the aperture to one end and I0 is the modified Bessel function. The excitation efficiency is
where
When using this method, B is calculated from Eq. (2.86) according to the SLR of the designed pattern, and the excitation value of the array is obtained from the aperture distribution equation. The characteristic parameters of the Taylor single-parameter pattern synthesis are shown in Table 2.1.
In addition to the two methods described above, the analytical methods for antenna pattern synthesis also include Taylor n, Villenenve n, and so on. The analytical methods for planar array pattern synthesis include Hansen single-parameter circle distribution, Taylor n circle distribution, and so on.
2.3Overview of the MIMO System
Table 2.1. Characteristic parameters of Taylor single-parameter pattern synthesis.
Note: u3 denotes half-power beamwidth and ηb denotes beam efficiency.
In a conventional wireless communication system, the transmitting end and the receiving end usually use one antenna each. This single-antenna system is also called a single-input single-output (SISO) system. For such a system, Shannon1 proposed the channel capacity formula in 1948 as follows: C = B lb(1 + S/N), where B represents the channel bandwidth and S/N represents the signal-to-noise ratio at the receiving end. It determines the upper limit rate for reliable communication in noisy channels. No matter what channel coding method and modulation method is used, it can only be accessed little by little but cannot be surpassed. This seems to be a recognized and insurmountable boundary and becomes a bottleneck in the development of wireless communications. According to Shannon’s channel capacity formula, increasing the SNR can improve the efficiency of the spectrum. For every 3-dB increase in SNR, the channel capacity increases by 1 bit/Hz/s. However, in the actual communication system, it is not recommended to increase the transmission power of the transmitting end in consideration of the actual conditions such as electromagnetic pollution, performance of radio frequency circuit,
