ASSIGNMENT - Analog and Digital Communications
1. Consider the signal set in Figure 1 for binary data transmission over a channel disturbed by AWGN. The noise is zero-mean and has two-sided PSD N0/2. As usual, s1(t) is used for the transmission of bit "0" and s2(t) is for the transmission of bit "1." Furthermore, the two bits are equiprobable.
(a) Find and draw an orthonormal basis {φ1(t), φ2(t)} for the signal set.
(b) Draw the signal space diagram and the optimum decision regions. Write the expression for the optimum decision rule.
(c) Draw the block diagram of the receiver that implements the optimum decision rule in (b).
(d) Let A = 1 volt and assume that N0 = 10-8 watts/hertz. What is the maximum bit rate that can be sent with a probability of error P [error] ≤ 10-6?
(e) Draw the block diagram of an optimum receiver that uses only one correlator or one matched filter.
(f) Assume that signal s1(t) is fixed. However, you can change s2(t). Modify it so that the average energy is maintained at A2T/2 but the probability of error is as small as possible. Explain.
2. The so-called Bi-phase modulation has the following mapping rule(s). There is always a transition at the beginning of a bit interval. If data bit bk = 1, there is a transition in the middle of the bit interval, while if bk = 0 there is no transition. As always the transitions are between ±V.
(a) Draw the signal space diagram for this modulation. Based on this signal set, how would you demodulate the received signal set with minimum error probability? Assume a symbol-by- symbol demodulator and equally probable bits.
(b) What is the error probability of this modulation?
3. Consider BPSK, where P [1T] = p. The "optimum" demodulator is designed on the assumption that p = 1/2.
(a) Fix the demodulator with the p = 1/2 assumption. What happens to the error performance of this emodulator if p = 1/2.
(b) Design the optimum demodulator for general p and derive the expression for its error performance.
4. To explore the influence of nearest neighbors consider the simple 4-ASK modulation with Gray mapping shown in Figure 2. The probability of symbol error is:
P[symbol error] = (2(M1)/M)Q(?/√(2N0)) (1)
Figure 2: Signal constellation of 4-ASK with Gray mapping, considered in Problem 4
(a) Determine the ratio ?/√(2N0) (which is related to the signal-to-noise ratio SNR = Eb/N0) so that P [symbol error] = 10-1, 10-2, 103, 104.
(b) For each P[symbol error] determine the following:
Q(?/√(2N0)), Q(3?/√(2N0)), Q(5?/√(2N0)). (2)
(c) Consider that symbol 00 was transmitted. Determine the following error probabilities:
P[{01}D|{00}T], P[{11}D|00T], P[{10}D|{00}T]. (3)
(d) How would the answer in (c) change if one of the other symbols was chosen to be the transmitted one? Therefore, in terms of neighbors the most important ones are which ones?
5. Your friend has been asked to design a binary digital communication system for a baseband channel whose bandwidth is W = 3, 000 hertz. The block diagram of the system is shown in Figure 3. Having learnt that intersymbol interference (ISI) generally occurs for a bandlimited channel, which can severely degrade the system performance, she proposes two possible choices for the overall system response SR(f ) = HT(f ) · HC(f ) · HR(f ) in Figures 4.
(a) For each choice of the overall response, at what rate should your friend transmit if she desires the ISI terms to be zero at the sampling instants? Explain.
(b) From your answers in (a), what is the excess bandwidth of the system for each choice of the overall response?
(c) Being aware of the raised-cosine spectrum, you suggest it to your friend as an alternative design for the spectrum in Figure 4(a). Neatly sketch the raised-cosine spectrum on top of the spectrum in Figure 4(a).
(d) Your friend considers your suggestion by examining the eye diagrams in Figure 5. What should be your friend's choice for the overall spectrum? Explain.
