EE 567 ProjectDue Tuesday, December 3, 2019 at 6:40 p.m.Work all 3 Parts.Instructions.Your project should be typed on one side of the paper only and stapledin the upper left hand corner. You should include a cover page and anappendix where you include your Matlab code. Do not place your projectinside any kind of binder. This is to be an individual effort. You may consultany written material (hard or soft copy) but you may not solicit input fromany person except that you may ask the professor or TA questions regard￾ing your project. Your project report should be self-contained, that is, thereader should be able to understand the problems and your solutions withoutconsulting the actual project assignment.Part 1.Assume we have downconverted a received signal via a mixing operationand now we wish to apply a LPF. One way to implement a LPF is simply tocompute an average. In continuous time we would just integrate the signalsince dividing by the integration time T to compute the average would notaffect the performance since the noise would be scaled by the same amountas the signal component. In discrete time we would implement a sum insteadof an integral. For this part we will assume we have a discrete time signalbut we will compute an average instead of just a sum.So let us assume the input to the LPF is a signal of the forms(k) = √E + double frequency terms + n(k), k = 0, 1, . . .where we have assumed scaling so that n(k) is a standard normal randomvariable for each k and niis independent of nj for i = j, i, j = 0, 1, . . ..We may ignore the double frequency terms and assume they are suppressed,either completely or at least sufficiently, by the LPF. The output of the LPF1isy(n) = 1N Xn k=n N+1s(k), n = 0, 1 . . .where we take y(n) = 0 for n < 0.Even though we are computing an average we will refer to this type of filteras an integrate and dump or I&D filter.Now for implementation purposes we can also construct a LPF using anIIR filter. Lety˜(n) = (1 α)s(n) + αy˜(n 1), n = 0, 1 . . .where we take ˜y( 1) = 0.a. Determine (analytically) the value of α = α(N) so that the mean andvariance of the IIR filter output matches the mean and variance of theI&D filter output as n → ∞. For this task you may assume withoutloss of generality that you only have noise present.b. Compute (analytically) the impulse response of the I&D filter.c. Compute (analytically) the step response of the I&D filter.d. Compute (analytically) the impulse response of the IIR filter using thevalue of α found in part (a).e. Compute (analytically) the step response of the IIR filter using thevalue of α found in part (a).f. Plot the impulse response for each filter on the same graph using N = 8.g. Plot the step response for each filter on the same graph using N = 8.Part 2.Assume we have downconverted a BPSK signal such that the input to aLPF is of the forms(k) = A + double frequency terms + n(k) 2where A is a constant and n(k), k = 0, 1, . . . are independent zero-meannormal random variables with variance σ2. We may ignore the double fre￾quency terms and assume they are suppressed, either completely or at leastsufficiently, by the LPF. Suppose the LPF is an integrator of the formy = 1N NX1 k=0s(k).After applying this LPF the output y is a test statistic with bit-energy-to￾noise ratio Eb/N0. We know that for BPSK the probability that we make awrong decision using this test statistic isPb = Q s2Eb N0 ! .We can approximate this LPF operation using an IIR filter of the formy˜(n) = (1 α)s(n) + αy˜(n 1), n = 0, 1 . . . N 1where we take ˜y( 1) = 0.a. Using Matlab simulate the signal s(k) and the LPF operation producingy above for N = 8 and plot your simulated bit error rate vs. Eb/N0 onthe same graph as Pb vs. Eb/N0 to compare. Your Eb/N0 range shouldbe large enough so that the BER ranges from 0.5 to 10 6.b. Using Matlab simulate the signal s(k) and the IIR filter operation pro￾ducing ˜y(n) above for N = 8 and plot your simulated bit error ratevs. Eb/N0 on the same graph as the simulated bit error rate for y tocompare. You should use the value of α that you found in Part 1 of thisproject. Your Eb/N0 range should be large enough so that the BERranges from 0.5 to 10 6.c. The value of α that you just used does not necessarily minimize theprobability of bit error for the IIR filter approach. Using Matlab andsome trial and error find the value of α that does minimize the proba￾bility of bit error for Eb/N0 = 7 dB. Using this new α (if it is differentthan that found in Part 1), plot your simulated bit error rate vs. Eb/N0on the same graph as the simulated bit error rate for y and the simu￾lated bit error for the α found in Part 1 to compare. Your Eb/N0 rangeshould be large enough so that the BER ranges from 0.5 to 10 6. 3Part 3.In this part we are going to investigate the detection of signals in noise.We will consider both integration detection and M of N logic detection.Suppose we receive a signal of the formr(t) = A cos(2πfct + φ) + n(t), 0 ≤ t ≤ Twhere A is a constant over T seconds taking on the value of A = 1 or A = 0where in the latter case we have only noise present, fc = 1 MHz, T = 1 msecand n(t) is a Gaussian random variable at time t with mean 0 and varianceσ2.a. Simulate the direct integration approach to signal detection using thesquare law detector as covered in class (this detector yielded a chi￾square random variable with 2 degrees of freedom). Assume there isno post detection integration. In your sims use a threshold requiredfor a probability of false alarm of 10 4. You should find this thresholdanalytically. You can assume that just prior to squaring and addingthe downconverted and filtered (integrated over T seconds) receivedwaveform has the form (after scaling)r1(T ) = A cos(φ) + n1in the upper path of the circuit andr2(T ) = A sin(φ) + n2in the lower path of the circuit, where, n1 and n2 are independentGaussian random variables with mean 0 and variance σ2. The outputof the detector is thenz(T ) = r21 + r22.Plot probability of detection results for SNR ranging from 0 to 15 dB.The y-axis of your plot should use a log scale and your x-axis shouldbe in dB.b. Repeat part (a) but now assume we also include a post detector inte￾grator of length N = 16. Now our overall probability of false alarmafter post detection integration is 10 4so a new threshold will need tobe found. You may find the new threshold to control false alarms viasimulation or numerically evaluating the appropriate integrals.c. Repeat part (b) but now assume the post detecting integration is re￾placed with an M of N logic detector where N = 16 and M = 8. Ouroverall probability of false alarm after the M of N logic is still 10 4.You will need to find a new threshold to use here right before the M ofN logic.