Spatial Multidimensional Cooperative Transmission Theories And Key Technologies. Lin Bai

with m = 0, 1. Define and , then

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

      And then the error probability in signal detection is

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

      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

      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), . . . , s_M(t)}, 0 ≤ t < T consisting of M signal waveforms. At this time, the data transmission rate is bit/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) = s_m(t) + N(t), 0 ≤ t < T. The likelihood function and log-likelihood function of the L samples are expressed as

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

       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:

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

      According to the expression of the log-likelihood function, the maximum likelihood judging criteria for accepting S_m should be log f_m(r(t)) ≥ logf_m′(r(t)) or . 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 {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,