Due November 7, 2001October 28, 2001Communication TheoryECEN 4242Homework #7Due November 7, 20011) Problem 3.21 from Textbook2) Problem 3.32 from Textbook3) Computer Experiment: Linear Delta Modulation (LDM) and Adaptive Delta Modulation(ADM)a) Linear Delta Modulation (LDM): A LDM is described by the following set of equationsError: )1()()(  nmnmneq (1) Sign of Error: ))(()( nesignnd  (2) Quantized signal:  )()1()( ndnmnmqq (3)where m(n) is the nth sample of the message signal m(t), mq(n) is the quantized value of m(n),and  is a fixed step size.The transmitter and the receiver of LDM are given in Figure 3.23a and 3.23b of the Text-book.b) Adaptive Delta Modulation (ADM): An ADM system represents an adaptive version ofa LDM system, where the step size  is no longer constant; instead it is adjusted itera-tively using the following update equation minminmin)1( if ,)1( if ,)1(5.0)()()1()(nnndndndnn (4) where d(n) is a one-bit quantizer output given by a sequence of 1, as shown by Equation (2).c) Computer ExperimentGiven the input message signal m(t)=Asin(2fmt), where A=10, fm/fs=100 (fs is the sam-pling frequency), =1, for LDM, and min=0.125 for ADM.i) Write a MATLAB script for LDM case using Equations (1)-(3). Sample m(t)given above over one period, i.e., use 100 or 101 samples. Initialize the algorithmgiven by Equations (1)-(3) as follows: mq(1)=m(1), eq(1)=0. Plot the signal m(n)(sampled version of m(t)) using "plot" function of MATLAB. Plot on the samefigure the signal mq(n) (quantized version of m(n)) using "stairs" function ofMATLAB. On the same figure once again, plot the quantized error signaleq(n)=d(n)-15 using "stairs" function of MATLAB. Notice that the constant 15 is1substracted from d(n) just to separate the plots of m(n) and mq(n) from the plotof eq(n).ii) In a LDM system The quantized error eq(n)=d(n) is binary-encoded and trans-mitted (i.e. transmitted as binary pulses). At the receiver, a decoding process isperformed, followed by integration (accumulation) and low-pass filtering opera-tions (see Figure 3.23b). Assume that the transmission channel is noiseless and as-sume that the quantized error eq(n) is recovered at the accumulator input as it wastransmitted. Compute the output of the accumulator mrq(n) and compute the outputof the low-pass filter mrf(n) using the function "filter" of MATLAB. For the filter-ing process use the following finite impulse response (FIR) filter 40)(iizzH.On the same figure, plot mrq(n) using the function "stairs" of MATLAB and mrf(n)using the function "plot" of MATLAB. Notice here that the filter H(z) will intro-duce a delay of 2 units to the signal mrf(n) with respect to mrq(n) and therefore fora fair comparison you should adjust the signal mrf(n) before plotting it. Again onthe same figure, plot the error signal err(n)=m(n)-mrf(n)-15. Conclusion?iii) Repeat exactly the same experiment given in i) and ii) for ADM case by writinganother MATLAB script or by modifying the previous one. The only differencebetween LDM and ADM systems is that for ADM, the step size (n) is no longerconstant; it has to be updated iteratively using Equation (4). The block diagram ofADM transmitter and receiver are given in Figure 3.31 of the Textbook. Noticehere that at the ADM receiver, the step size (n) has to be computed as in thetransmitter and multiplied by the received binary signal. The result is integratedvia an accumulator then low-pass filtered (see Figure 3.31b).