Spatial Multidimensional Cooperative Transmission Theories And Key Technologies. Lin Bai
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Название: Spatial Multidimensional Cooperative Transmission Theories And Key Technologies

Автор: Lin Bai

Издательство: Ingram

Жанр: Зарубежная компьютерная литература

Серия:

isbn: 9789811202476

isbn:

СКАЧАТЬ the random variable X. Since the noise N(t) is assumed to follow the Gaussian random process, it can be seen that X is also a Gaussian random variable. Note that when Sm is true, R(t) = sm(t) + N(t), and thus, the statistical properties of X depend on Sm. In order to obtain the statistical properties of X, the mean and variance of the Gaussian random variable X need to be determined, respectively.

figure

      The average energy of the signal is defined as figure. Assume sm(t) is transmitted by equal probability with m = 0, 1. Define figure and figure, then

figure

      Therefore, in the case where S0 and S1 are, respectively, established, the probability density of X is

figure

      And then the error probability in signal detection is

figure

      For the fixed signal energy Es, the error probability can be minimized when τ = –1, namely

figure

      It is easy to prove that the signal that minimizes the error probability has the opposite polarity, namely s0(t) = –s1(t). For the orthogonal signal set, τ = 0 and the error probability is

figure

      It can be seen from Eqs. (2.42) and (2.43) that the SNR between the orthogonal signal set and the signal set with the opposite polarity is 3 dB.

      2.1.2.2M-ary signal detection

      The detection problem of the binary waveform signal when M = 2 has been introduced above, and then we will analyze the detection problem of the M-ary signal.

      It is assumed that in the M-ary communication, there is a set {s1(t), s2(t), . . . , sM(t)}, 0 ≤ t < T consisting of M signal waveforms. At this time, the data transmission rate is figurebit/s. It can be seen that the data transmission rate increases as M increases, which means that the larger the M value is, the better. However, in general, the detection performance of the signal will be worse as the value of M increases.

      In the case of M hypotheses, assuming that the received signal is R(t) = sm(t) + N(t), 0 ≤ t < T. The likelihood function and log-likelihood function of the L samples are expressed as

figure

      If the common terms in all hypotheses are ignored, the log-likelihood function becomes

figure figure

       Fig. 2.4. Maximum likelihood detector for M-ary signal detection based on a set of correlation detectors.

      Using r(t) to represent the observed value of R(t), when L tends to infinity, the following equation can be obtained:

figure

      where figure is the energy of the mth signal sm(t) and figure is the correlation function of r(t) and sm(t).

      According to the expression of the log-likelihood function, the maximum likelihood judging criteria for accepting Sm should be log fm(r(t)) ≥ logfm′(r(t)) or figure. The symbol “\” is defined as A\B = {x|x ∈ A, x ∉ B}.

      Based on the preceding conclusions, the process of connecting a series of correlation detectors to achieve the maximum likelihood judging criteria is shown in Fig. 2.4.

      2.1.2.3Signal detection in vector space

      In the process of wireless communication, in addition to receiving multiple target signals, the receiving end is also affected by the noise generated by the wireless channel and other devices. For these interferences, all of them are regarded as noise in the traditional signal combining method, which usually leads to a large performance loss. Considering this problem, the technology of signal detection in vector space is extensively studied to distinguish and detect target signals in mixed signals. Before introducing the signal detection in vector space, the expansion of space vector signal, namely Karhunen–Loeve Expansion, is first introduced.

      The Karhunen–Loeve expansion can represent a function with the sum of multiple basis functions with different weights. Assume that there is a set {ϕl(t)} (l = 1, 2, . . . , L, 0 ≤ t < T) consisting of L orthogonal basis functions, and if the received signal can be decomposed into

figure

      then the coefficient sm,l in Eq. (2.47) is called the Karhunen–Loeve expansion coefficient. According to the characteristics of the orthogonal basis function,

figure

      Therefore, we can obtain

figure

      As shown in Eq. (2.47), if sm,l is known, then we can re-solve sm(t).

      Define a vector sm = [s1,m s2,m · ·· sL,m]T. Obviously, sm and sm(t) are equivalent, where sm(t) represents the mth signal in the function space (or waveform space) and sm represents the mth signal in the vector space. Therefore, the energy of the two signals and the distance between СКАЧАТЬ