QAM Modulation, Gaussian Noise, Demodulation w/matched filter, Detection, and Performance Analysis (Unless otherwise said, all the requirements below should be part of the report (including all numbers reported/required and organized in a table). Further, the code should be self-contained in a separate .m file and give all the results and figures in a single run). At the Transmitter Use 4-QAM baseband modulation to send between 10 and 100000 randomly generated bits at an overall transmission rate of 2 bits per millisecond using three pulse shapes given below. i. Square wave ii. SQUARE-ROOT Raised Cosine with r=.5 iii. SQUARE-ROOT Raised Cosine with r=1 Choose pulse amplitude to ensure that the pulse energy is 1, and list this value in your report. Use a carrier whose frequency is 10kHz and the amplitude A=1 as measured at the receiver. At the Receiver 1. Demodulate the carrier (for both I and Q) 2. Add iid (independent and identically distributed sequence where each element is distributed as a zero mean Gaussian Noise with variance 2. You will choose 2 in a range of values that will result in a range of detection SNRs, in a manner that will explained below. 3. Match filter the received noisy signal. The filtered output will be denoted with r_i(t) and r_q(t), for I and Q channels, respectively. The filter to be used (for both I and Q channels) should have an impulse response equal to the time-flipped pulse shape used at the transmitter. List the 95% bandwidth of the filter corresponding to the square pulse, as well as the 100% bandwidth of square-root raised cosine pulses (organize these results in a table in your report). 4. Give the detailed equation for the matched filtered signals r_i(t) and r_q(t) including the pulse shape of the output bit modulated signal. Give the mean, variance 2, and pdf of each noise sample at the output of the filter. What is the energy in each filtered pulse (assuming no noise, 2=0)? What is the energy per transmitted bit in the carrier modulated component and in its second harmonic at 20kHz, after filtering? (some of the above can be computed numerically and for others compute analytically, using equations.) (organize these results in a table in your report). 5. Plot the eye-diagram of the filtered signal output, for the noiseless case ( 2=0) (using matlab) for each selected pulse shape and a random sequence of 10 bits: 10011 10100 transmitted over I. Determine the best place to sample for each bit (at bit rate) in order to detect the bit with minimum probability of error. Below, I will call the best sampling instant for the first bit “D”. Hence, the decisions for the first two bits will be based on the samples r_i(D) and r_q(D) (assuming that I and Q are synchronized). 6. Call SNR_i = (r_i(D))2/ 2 and SNR_q = (r_q(D))2/ 2 linear SNRs for I and Q, respectively. They should be the same unless you made an error. Hence, I will call them the (linear) SNR. Next, We define SNR in dB as SNRdB = 10 log10(SNR). Find the range of the noise variance 2 that will give you the range of SNRdB as [0,10]. Can you compute SNR analytically for each pulse shape? How? 7. Now plot the Eye Diagram for r_i(t) for the noisy case for three cases of different SNR: SNRdB= 0, 4 and 9 dB for bits 10011 10100. 8. Plot the theoretical Bit error rate as Q(const*sqrt(SNR)) vs SNR in dB for the range [0,10] of SNRdB and using semiology() matlab function to get a log scale in y-axis. (choose the correct const by studying the textbook and notes in the class, and let me know what it is,) 9. Simulate a full system where you generate a random sequence of bits L (e.g., between 10 and 100000 depending on the required SNR and the corresponding error rate) for noise variances 2 in the range computed above in 6. For each noise variance run as many bits as needed to get at least between 10 and 20 errors (as a benchmark, you can use the theoretical BER from stage 8 above; as an example, if you expect to get the error rate of .0001 at a given SNR in dB, you need to run at least 100000 bits to get 10 errors). Record the ratio of number of bits in error and the total transmitted number L for each value of SNR in dB. Plot this curve for each pulse shape together with the theoretical BER curve from 8. Do they match? 10. Draw a block diagram of the system in your report, include all the numerical parameters you have used next to the corresponding blocks. For example, next to the noise generation block, add the range of variances that you had to choose to get your range of SNRs. At the output of the filtered and sampled signal, tell me what was the per sample amplitude squared and the corresponding filtered noise variance range, as well as the range of SNRs generated; etc.
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