Читать книгу Spatial Multidimensional Cooperative Transmission Theories And Key Technologies - Lin Bai - Страница 31

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 {ϕ_l(t)} (l = 1, 2, . . . , L, 0 ≤ t < T) consisting of L orthogonal basis functions, and if the received signal can be decomposed into

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

Therefore, we can obtain

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

Define a vector s_m = [s_1,m s_2,m · ·· s_L,m]^T. Obviously, s_m and s_m(t) are equivalent, where s_m(t) represents the mth signal in the function space (or waveform space) and s_m represents the mth signal in the vector space. Therefore, the energy of the two signals and the distance between the signals are both equal.

where d_m,k represents the distance between the mth signal and the kth signal.

Using the Karhunen–Loeve expansion, we obtain

Define

where n =[n₁n₂ … n_L]^T. The noise signal can be expressed as

where and represents the noise that cannot be expanded by the Karhunen–Loeve expansion, . It is worth noting that the noise does not appear in vector y as shown in Eq. (2.52).

Since N(t) is additive white Gaussian noise, n_l is a white Gaussian random vector whose mean and variance are given as

where , as the base function is assumed to be orthogonal.

According to the Karhunen–Loeve expansion, it can be assumed that the received signal can be represented by a space vector, i.e.

Generally, it can be assumed that s_m and n are both complex vectors. In addition, n is assumed to be a circularly symmetric complex Gaussian (CSCG) random vector with E(n) = 0 and E(nn^H) = R_n. For convenience, is used to represent the probability density of a CSCG random variable whose mean vector is m and covariance matrix is R_x.

For a given vector y, the maximum likelihood judging criteria for receiving s_m(t) are f_m(y) ≥ f_m′(y), where m′ ≠ m, f_m(y) is the mth hypothesis or the likelihood function of s_m. Since the noise is assumed to be a CSCG random vector, the likelihood function of s_m can be written as

Then the log-likelihood function is

Therefore, the maximum likelihood judging criteria for receiving s_m(t) can be simplified as , where m′ ≠ m.

When M = 2 which is a binary signal, the L-likelihood ratio is

As a special case, if it is assumed that n_l are independent of each other, then the variances between n_l are the same, which means R_n = N₀I (N₀ > 0). In this case, the L-likelihood ratio is