In this assignment we implement adaptive equalizers in MATLAB simulation.

In this project we implement adaptive equalizers in MATLAB simulation. Simulation setting and parameters  Every component is simulated in the baseband at the symbol sampling rate (one sample per symbol);  Use Channel B that is a symbol-spaced baseband channel, assuming that the given channel taps are for both the real part and the imaginary part, i.e., c = [0.407+j0.407, 0.815+j0.815, 0.407+j0.407]. Also, use a test channel of c = [0, 1+j, 0] first to test your program. From the MSE or the signal constellation of the equalizer output of the test channel, you should know if your program is working or not, and thus troubleshooting accordingly.  You need to generate 4000 random and complex QPSK symbols for each case in your simulation, you can use, e.g., MATLAB function randsrc or pskmod.  Pass the symbols through the baseband channel. The way to handle complex baseband processing was covered at the end of Lecture Notes Chapter 2, but MATLAB computes complex quantities implicitly;  Add pseudo-random white Gaussian noise at the channel output in both the real part and the imaginary part, perform your simulation for per bit SNR of 20 dB; You need to compute the relative scaling between the signal and the noise that these SNRs correspond to, and include such calculation in your report. For simplicity, the SNR is only for the generated signal and noise before the channel.  Carrier synchronization and symbol synchronization are assumed perfect; The transmitting filter and receiving matched filter are designed so they together equal to () rc Xf that corresponds to (in the time domain) a delta function at symbol rate samples. Thus they can be ignored in this symbol rate only simulation;  Implement symbol-spaced adaptive LMS equalizers: (a) Linear MMSE; (b) MMSE-DFE; The equalizers should be 32 tap FIR, complex-valued (for both precursor and post-cursor equalizers in the DFE case); The two equalizers in (a) and (b) should be completely independent of each other, i.e., the feedforward part of MMSE-DFE should be implemented independently from the Linear MMSE.  To simplify, implement the training part only, use all 4000 symbols for training. Choose your step size to compromise between fast convergence speed vs. low residual MSE for each case separately; Chose the step size properly so you should see convergence within the first 2000 samples. Do “partial training” only for DFE, i.e., use the training symbols only for training the feedforward part and in computing the MSE, but not in the input of the feedback filter, which should be from your own decision output.  You also need to consider different delays in the channel and in the equalizers. For the channel and the linear equalizer, the delays are due to FIR filtering. For the DFE, there is an additional delay due to the feedback. Therefore the training symbols used in all adaptive algorithms need to be delayed appropriately. Such delays also appear in calculating BER. You need to figure out the best delay in your work by either experimental or analytical + experimental means.  In the adaptive algorithms, the complex conjugate operation may or may not need to be explicit, depending on what MATLAB commands are used. E.g. ‘ (prime) in MATLAB includes both transposition and conjugation. Do not use any equalization object in MATLAB Communications Toolbox.