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TO SIMULATE AN AWGN CHANNEL WITH BPSK MODULATION We need to send an uncoded BPSK modulated binary signal over a noisy channel. The noise is an additive white Gaussian noise. The simulation process can be divided into a number of smaller tasks: 1. Generating AWGN Noise: In generating a Gaussian noise, we take the help of central limit theorem. It states: If the variable X has a non-normal distribution with mean µ and standard deviation σ, then the limiting distribution of _ z = ( x - µ )/(σ/√n) as n ∞ is the standard normal distribution (i.e. with mean 0 and SD 1). X must have finite mean and variance. n >= 25 is regarded as large. Thus in order to generate Gaussian noise we generate a number of uniformly distributed samples varying uniformly between a = -0.5 to b = +0.5. Its mean is given by µ = (a + b)/2 = 0 and variance σ2 = (b-a) 2 /12 = 1/12 _ Now from the samples we can determine x and using the central limit theorem we can generate a number of values z corresponding to each sample vector. The values of z thus obtained are normally distributed with mean 0 and variance 1. We can obtain the normal distribution of desired variance simply by multiplying the values in Z vector by the corresponding standard deviation. 2. Generating the Message Signal: In order to generate a message signal where the probability of occurrence of 0 and 1 is same, first we generate a sample uniformly distributed within a range. Then we set the threshold exactly at the mid point such that if the sample value is greater than the threshold, then the message value is taken as 1 otherwise 0. 3. BPSK Modulation: BPSK modulation of the binary message signal can be performed simply by converting the unipolar signal (0,1) into the bipolar signal (-1,1). 4. Noise Addition: Now the Gaussian noise is added to the message signal. This addition must be performed sample wise i.e. a noise sample must be added to the corresponding message signal sample. 5. Hard Decision and decoding: At the receiver side, we need to demodulate and decode the received signal. This can be done by comparing the received signal with a threshold value. If the received signal amplitude is greater than the threshold, 1 is received otherwise 0 is received. Generally the threshold value is set to be zero if the probabilities of occurrence of 1 and 0 in a message bit stream are equal. Otherwise the optimum threshold value (Dopt) is decided by the relation: Dopt where = (σ2/A)ln(P0/P1) P0 = probability of occurrence of 0 P1 = probability of occurrence of 1 σ2 = noise variance A = signal amplitude 6. Calculating Theoretical and Experimental Bit Error Rate: After simulating the AWGN channel following the steps mentioned above, we need to verify whether our experimental model is close to the theoretical one or not. Thus we need to calculate the theoretical and experimental bit error rate and compare them. Theoretical bit error rate is given by the relation: BERtheo = 0.5*erfc(√(Eb/No)) Now if the amplitude of the message signal 1 is A, its power is given by A2. Power of the message signal 0 is 0. If probability of occurrence of 1 and 0 is same then average signal power is given by A2/2. And noise power is given by σ2. Hence signal to noise ratio is given by A2/2σ2. Thus experimental BER is given by: BERexp = 0.5*erfc(A/√2 σ) 7. Plotting the Results: Finally, to compare the experimental model with that of theoretical model, we need to plot the corresponding BER on a single canvas. BER is plotted against the signal to noise (SNR) ratio. SNR value is taken in dB. For a properly designed model, error between the two plots must be very small. .