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, and the interferences among users. Diversity technique is another way to increase the spectrum usage efficiency. If a single antenna is used at the transmitting end and multiple antennas are used at the receiving end, this diversity is often called diversity reception which is also known as the single-input multiple-output (SIMO) system. The use of optimal combined diversity reception techniques generally improves the SNR at the receiving end, thereby increasing the channel capacity and the spectrum usage efficiency. If multiple antennas are used at the transmitting end and a single antenna is used at the receiving end, this diversity is often called the transmit diversity, which is also known as the multiple-input single-output (MISO) system. However, if the state information of the channel is not known at the transmitting end, beamforming technology and adaptive allocation cannot be used in the multi-transmitting antenna for transmitting power, and thus, the channel capacity cannot be improved much. The development and integration of SIMO and MISO technologies have evolved into MIMO technology, which is an effective method to break through the SISO channel capacity bottleneck. The core idea of the system is to synthesize the signals at both ends of the spatial sampling by way of generating effective multi-parallel spatial data channels (increasing the data traffic), so as to greatly improve the channel capacity, or by way of increasing the diversity to improve communication (reduce bit error rate).
2.3.1Diversity technique
The particularity of a wireless link is that it is affected by random fluctuations in signal levels across time, space, and frequency. This characteristic is fading and affects the system performance (symbol or bit error rate). Take the SISO Rayleigh fading channel transmitted by binary phase shift keying (BPSK) as an example.
When there is no fading (h = 1), in the additive white Gaussian noise (AWGN) channel, the bit error rate (SER) is
When considering the fading, the level of the received signal fluctuates with
where ps(s) is the distribution function of the fading. For Rayleigh fading, the integration of the above equation yields
When the SNR is large, the bit error rate in Eq. (2.91) becomes