Problem 1(Sample Communication System)A discrete memoryless source produces bits at a rate of 106 bits/s. The bits, which are uniformlydistributed and iid, are grouped into pairs and each pair is mapped into a distinct waveformand sent over an AWGN channel of noise power spectral density N0=2. Specically, the rsttwo bits are mapped into one of the four waveforms shown in the gure with T = 2106, thenext two bits are mapped into the same set of waveforms delayed by T, etc.(a) Describe an orthonormal basis for the inner product spaceW spanned by si(t), i = 0; : : : ; 3and plot the signal constellation in Rn where n is the dimensionality of W. (Hint: Thereare many possible bases for this problem. Design a basis that simplies the subsequentparts of this problem, not necessarily the one that Gram-Schmidt gives you.)(b) How would you choose the assignment between two bits and a waveform in order tominimize the bit error probability Pb (ie. what assignment of bit pairs to the signals inyour constellation gives the best error performance and why)? What is the resulting Pb?(c) Could you implement the receiver front end with a single matched lter? Explain.(d) What is the energy per bit Eb of the transmitted signal? What is the power?https://eeweb.ee.ucla.edu/images/upload/20121ee132A_1_EE132A_Problems_HW6.pdf #1
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