Читать книгу Spatial Multidimensional Cooperative Transmission Theories And Key Technologies - Lin Bai - Страница 29

In a wireless communication system, the channel transmits an analog signal rather than a discrete signal. For binary waveform signals, the received signal can be written as

where T represents the duration of the signal, N(t) is Gaussian white noise and with δ(t) representing a Dirac function, otherwise called a unit pulse function. The wireless channel of Eq. (2.27) is called an Additive White Gaussian Noise (AWGN) channel. Meanwhile, X(t) is a binary waveform signal, i.e.

whose transmission rate is 1/T bit/s.

(1) Waveform signal detection

First, we will discuss a heuristic algorithm for waveform signal detection and then extend the waveform to generalize a general waveform detection method.

If the signal is judged according to R(t)(0 ≤ t < T) at the receiving end, the observation values of R(t) and N(t) are represented by r(t) and n(t), respectively, L times sampling on r(t) has been performed, and define , we can obtain

Since N(t) is white noise and n_l is independent of each other, the mean of n_l is zero and the variance is

Defining r = [r₁ r₂ · ·· r_L]^T, the likelihood ratio of L is obtained as

where s_m = [s_m,1 s_m,2 · · · s_m,L]^T.

According to the likelihood ratio of L, the MAP criterion can be obtained.

If we consider the criterion based on the likelihood ratio and use the threshold ρ instead of ; then we have

(2) Correlation detector and its performance

In the sampling process of r(t), if it has a low sampling frequency within the duration T_s of the signal, the information may be missing due to the sampling process. To avoid this distortion, the value of L is assumed to be large enough to be close to . On this basis, the judging criteria based on the likelihood ratio can be rewritten as

Define , then the judging criteria in Eq. (2.34) become

The judging criteria can be implemented by the model shown in Fig. 2.3, which is called a correlation detector.

In order to make an analysis of the detector’s performance, the maximum likelihood judging criteria are first considered, namely setting ρ = 1 in the judging criteria based on the likelihood ratio. In this case, Define , and it is easy to know

Fig. 2.3. Correlation detector for binary waveform signals.

Second, the bit error rate is calculated based on the statistical properties of 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 S_m is true, R(t) = s_m(t) + N(t), and thus, the statistical properties of X depend on S_m. In order to obtain the statistical properties of X, the mean and variance of the Gaussian random variable X need to be determined, respectively.

The average energy of the signal is defined as . Assume s_m(t) is transmitted by equal probability 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.