1) Maximum likelihood (ML) detection:

For ML detection, we calculate the Euclidean distance between the received signal and the possible transmitted symbols, and choose the symbol that minimizes this distance.

Given that the symbol sequences are s[0] = 1 and s[1] = -1, and the received signals are r[0] = 0.3 + 0.1j and r[1] = -0.1 - 0.4j, we calculate the distances as follows:

For r[0]:

d0 = |r[0] - √(Ex)s[0]|^2 = |0.3 + 0.1j - 2*1|^2 = 2.9
d1 = |r[0] - √(Ex)s[1]|^2 = |0.3 + 0.1j - 2*(-1)|^2 = 6.9

Since d0

Question

1) Maximum likelihood (ML) detection:

For ML detection, we calculate the Euclidean distance between the received signal and the possible transmitted symbols, and choose the symbol that minimizes this distance.

Given that the symbol sequences are s[0] = 1 and s[1] = -1, and the received signals are r[0] = 0.3 + 0.1j and r[1] = -0.1 - 0.4j, we calculate the distances as follows:

For r[0]:

d0 = |r[0] - √(Ex)s[0]|^2 = |0.3 + 0.1j - 2*1|^2 = 2.9
d1 = |r[0] - √(Ex)s[1]|^2 = |0.3 + 0.1j - 2*(-1)|^2 = 6.9

Since d0 < d1, we decide that s[0] = 1 was transmitted.

For r[1]:

d0 = |r[1] - √(Ex)s[0]|^2 = |-0.1 - 0.4j - 2*1|^2 = 5.17
d1 = |r[1] - √(Ex)s[1]|^2 = |-0.1 - 0.4j - 2*(-1)|^2 = 1.17

Since d1 < d0, we decide that s[1] = -1 was transmitted.

Therefore, the detected sequence at the receiver is ˆs[n] = [1, -1].

2) Plot of the received points:

The received points r[0] = 0.3 + 0.1j and r[1] = -0.1 - 0.4j can be plotted on the complex plane with the real part on the x-axis and the imaginary part on the y-axis. The points will be located at (0.3, 0.1) and (-0.1, -0.4) respectively.

3) Optional question:

If the transmission is corrupted by AWGN with a standard deviation σ = 0.1, the noisy received signals rnoisy[0] and rnoisy[1] can be calculated by adding the noise to the original received signals. The noise can be generated using the given Matlab command.

The ML detector can still make correct decisions given the noisy received signals, as long as the noise is not too large. If the noise is large, it may cause the received signal to be closer to the wrong symbol, leading to a detection error.

Knowee AI · Accepted Answer