CALIFORNIA INSTITUTE OF TECHNOLOGYDEPARTMENT OF ELECTRICAL ENGINEERINGEE 163ACommunication Theory IWinter 2005http://ee163.caltech.eduCommunication System Analysis ProjectPart 2: Binary Phase Shift KeyingWe are going to investigate the performance of BPSK, in terms of probability of bit error, P, as a function of signal-to-noise ratio (SNR).Consider the communication system shown below, and go through the step by step instructions on simulating it:SymbolMapperPulseShapingFilterReceiverFilterDecisionDeviceav(t)TransmitterSample att = (n+1)Tw(t)+r(t)ChannelReceiverrnvaˆInput Data Sequence• We will first generate a block of Na= 1, 000 random bits, ~a = [a0, a1, a2, . . . , aNa−1], with an∈ {0, 1}.Hint: In C/C++ lrand48()&1 generates one random bit.Symbol Mapper• Next, we will generate a block of symbols for transmission. For n = 0, 1, 2, . . . , Na− 1, letvn= (1 − 2an)pEb=(+√Eb, if an= 0−√Eb, if an= 1,where√Ebis the transmitted bit energy.Hint: Use a look-up table to implement this function.AWGN channel• The combined effect of pulse shaping at the transmitter, the AWGN channel, the receiver filter, and the signalsampler can be modeled by an equivalent discrete-time channel that produces:rn= vn+ wn,where ~w = [w0, w1, w2, . . . , wNa−1] are independent zero mean Gaussian random variables, with variance N0/2.Hints:– If X1and X2are independent random variables uniformly distributed over [0, 1), thenY1=p−2 ln X1cos(2πX2)Y2=p−2 ln X1sin(2πX2)are independent Gaussian random variables, each with zero mean and unit variance.– In C/C++ drand48() generates a random value in [0, 1).– To investigate system performance at different SNRs (Eb/N0), you may want to keep Ebfixed (at for example,Eb= 1), and adjust N0to achieve the desired signal-to-noise ratio.Decision Device• Next, we will generate the received data bits. For n = 0, 1, 2, . . . , Na− 1, selectˆan=(0, if rn≥ 01, if rn< 0Data Sink• We now compare each received bit, ˆan, with the corresponding transmitted bit, an, and count the number of errors,N.For a single transmitted block, the estimated probability of bit error is then given byˆP=NNaNotes• To obtain an accurate measurement of the probability of bit error when simulating communication systems, oneoften needs to transmit several million bits. For this assignment, it is recommended to break the number of bits youneed to transmit into blocks, with a block size of Na= 1, 000 bits, and then transmit several thousand blocks.If only a single bit at a time (Na= 1) is transmitted, your program will either run slowly, or require a lot ofrevisions in the future if you use the code for more elaborate communication systems. If all the bits in a singleblock (e.g., Na= several million) are transmitted, your simulation will use a lot of memory, and may run reallyslowly, if at all.• If Nfis the number of transmitted blocks, and N,iis the number of bit errors in the ithblock, then the estimatedprobability of bit error isˆP=1NfNaNfXi=1N,i.• As confirmation of the validity of your code, you should generate a plot of the probability of a bit error vs. SNR,for SNR ∈ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} dB. This plot should show three curves, corresponding to1. The theoretical probability of a bit error2. The estimated probability of a bit error for the case of 100 simulated blocks of data (transmitted), and3. The estimated probability of a bit error when sufficient simulated blocks of data are transmitted, so that 50block errors are counted for each SNR point.• This is not a group project - you must work on this by yourself alone.• Each individual should hand in a 1-2 page report that presents and discusses the results found. Please hand in(email) your source code as well, in one compressed file (zip or