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      In Section 2.1.1, we briefly introduced how to use the different criteria to obtain the optimal linear combination vector to design different signal combinations. Under ideal conditions, we can assume that the receiver can only receive the target signals. However, the receiver actually often receives interference signals from other devices while receiving the target signals. For these interference signals, traditional signal combining methods generally classify them into noise for processing, but such solutions have obvious drawbacks and sometimes lead to large performance losses.

      As a result, the method of extracting the target signal in the mixed signal, namely the signal detection method, is beginning to be studied. Multi-signal detection method has always been an important problem to be solved in wireless communication. As shown in Fig. 2.2, in a cellular system, if users in two cells use the same transmission channel to transmit signals, the base station can receive the user signal in the cell of the base station and can also receive the signal in another cell. And in this case, the base station needs to detect different signals. In addition, when a transmitter transmits signals through multiple transmitting antennas, the receiving end also needs to detect multiple received signals. And when multiple antennas are used by both the transmitter and the receiver, point-to-point signal detection can also be regarded as a virtual multi-signal detection technique due to interference between the antennas. In addition, when multiple antennas are used by both the transmitter and the receiver, point-to-point signal detection can also be regarded as a virtual multi-signal detection technology due to the interference between the antennas.

       Fig. 2.2. Multi-user monitoring model of the cellular system.

      The above examples illustrate several common application scenarios for multi-signal detection, and multi-signal detection technology will play a significant role for a long time.

      2.1.2.1Binary waveform signal detection^a

      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)

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