EE 121: Introduction to Digital CommunicationSystemsFinal Exam Spring 2003• Please put your name on the front page.• Answer all the questions on the blank pages provided.• Explain clearly the steps in your answers. Yes/no answers with no explanations getno marks.• Waterfall error probability curves are provided at the back for your use throughoutthe exam. Please reference the appropriate plots when you use them. (You may wantto take a look at them now.)[10] 1.[6] a). Explain briefly what these concepts mean:i) entropyii) uniquely decodable codesiii) Nyquist sampling theoremiv) Nyquist criterionv) bi-orthogonal modulationvi) ISI[4] b) Is there any relationship between the delay spread and coherence bandwidth in awireless channel? How about b etween delay spread and coherence time? For each case, ifthere is a relationship, explain. If not, give an example of a wireless channel in which thetwo parameters can be specified separately.2[10] 2. Suppose you have calculated the transmit power link budget for a wireless system,assuming• uncoded binary pulse amplitude modulation (binary PAM, also known as BPSK)• an error probability of 10−4• transmission over an AWGN channel.[5] a) Suppose now there is flat Rayleigh fading and coherent detection is performed.How much extra transmit power is needed? (This is called the fade margin.) State clearlyhow you come up with your answer.[5] b) Suppose now each symbol is repeated twice and interleaved over different coherencetime periods. Compared to (a), how much is the fade margin reduced by?3[15] 3.A passband communication system uses ideal sinc pulses and signal at Nyquist rate.[3] a) What is the difference, in terms of symbols transmitted in the in-phase andquadrature-phase, between BPSK (binary phase-shift keying, also known as binary PAM)and QPSK in such a system? Draw a system diagram for each.[4] b) Compare the spectral efficiency (in bits/s/Hz) and Eb/N0performance of BPSKand QPSK over an AWGN channel, for the same symbol error probability pe. (You can as-sume that peis small in making any approximations. Also, you may find the approximationQ(x) ∼ exp(−x2/2) useful.)[3] c) Suppose the in-phase and quadrature-phase components of each QPSK symbolcarry two independent information bits . Repeat the comparison in part (a) between BPSKand QPSK but now for the same bit error probability.[5] d) Consider the discrete-time baseband equivalent flat Rayleigh fading channel:yn= hnxn+ wn,where wn∼ CN (0, N0) and hn∼ CN (0, 1). In class, we have stated that for coherentBPSK, the symbol error probability is:pe=121 −vuutEb/N01 + Eb/N0.Using part (c) or otherwise, compute the probability of bit error for coherent QPSK overthe Rayleigh flat fading channel. assuming again that the in-phase and quadrature phasescarry independent information bits. Again, what is the difference in Eb/N0performance forthe same bit error probability?4[10] 4. Consider a wireless channel with only one line-of-sight path which arrives after adelay of τ. Communication is over a passband at carrier frequency fc= 1900 MHz andbandwidth W = 1 MHz. Assume ideal sinc functions are used as transmit pulses.[4] a) The optimal demodulator matches filter to the transmit waveform and sample atthe correct times. Since τ is unknown in practice, it has to be estimated so that the sampletimes can be determined, a process known as timing recovery or symbol synchronization. Interms of the given system parameters, give an order-of-magnitude approximation of howaccurate τ have to be estimated for proper timing recovery.[5] b) Since we are communicating on a passband, the delay τ will also cause a phaserotation of the baseband transmitted signal. To do coherent detection, this phase has tobe estimated as well, a process known as carrier recovery. In terms of the given systemparamters, give an order-of-magnitude approximation of how accurate τ have to be esti-mated for proper carrier recovery.[1] c) In practice τ varies as the mobile moves, so τ have to be continuous tracked forongoing timing and carrier recovery. From your answers in part (a) and (b), which do youthink it’s a more difficult problem, timing or carrier recovery? Explain.5[27] 5.I. A sampled complex baseband-equivalent channel:yn= hxn+ wn,where h ∼ CN (0, 1) and {wn} is a sequence of independent CN (0, N0) noise.[2] a) What are the assumptions on the passband physical channel for which this would areasonable baseband model? Why is it reasonable to model the channel gain h as CN (0, 1)?[1] b) Suppose h is unknown to the receiver. Can you transmit information using BPSK?How about QPSK? Why?[4] c) Design a scheme such that the receiver can demodulate without knowing h. Givethe receiver structure. Give an expression for the error probability.II. Consider now another sampled baseband equivalent channel, given by:yn= ejθxn+ wn,where wn∼ CN (0, N0), independent over time, and θ is a random variable uniformlydistributed in [0, 2π] independent of the additive noise.[3] a) Give an example of a passband physical channel for which this is a reasonablebaseband model. What is the interpretation of θ? Why is it reasonable to model θ asuniformly distributed in [0, 2π]?[1]b) Suppose θ is unknown to the receiver. Can you transmit information using BPSK?How about QPSK? Why?[4] c) Design a scheme such that the receiver can demodulate without knowing θ. Givethe receiver structure. Give an expression for the error probability. It’s ok if it’s not inclosed form.[4] d) Qualitatively compare the performance in part I c) and part II c). Which channelis harder to communicate over? Explain.III. Let us now consider yet another baseband channel:yAn= hAxn+ wAnyBn= hBxn+ wBnwhere hA, hB∼ CN (0, 1) are independent, and wAn, wBnare independent additive CN (0, N0)noise, independent of hAand hB.[2] a) Give an example of a physical scenario for which this is a reasonable channelmodel. In your scenario, under what condition is it reasonable to model the channel gainshAhBas independent?[5] b) Design a scheme that can exploit both the received signals {yAn} and {yBn} indetecting the transmitted symbols, but without needing to know the gains hAand hB.[3] c) Qualitatively compare the performance of this scheme with that in part I c).Explain the nature of the performance improvement, if any.6[28] 6. Communication is done over the baseband [−W, W ] through an M-ary orthogonalcode in conjunction with binary PAM