110年 - 110台灣聯合大學系統_碩士班招生考試_電機類:通訊系統(通訊原理)#105574

is proportional to sA(r). The signals s_A(X), s_B(t), s_c(t), and s_D(t) can be expressed as a linear combination of these basis functions as

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where the basis functions are

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(b) (4%) The TWO candidate signals that would result in minimum average transmitted energy, given that the two messages are equiprobable, are_______________.

(c) (4%) The TWO candidate signals that would provide the lowest average probability of a bit error are____________________________

(d) (6%) To increase data rate, suppose now you want to send THREE messages instead of two, which again occur with equal a priori probability. Suppose S_A(t), S_B(t), and s_c(t) are chosen to represent these messages, and the power spectral density of the noise is . The union bound of the average symbol error probability is_____________________

(a) (3%) Let t₀ = 3 and T=4, i.e. h(t) = s(4- t). Determine the mean and variance of the filtered and sampled noise term W.

(b) (6%) Again, let t₀ = 3 and T=4. Please find the corresponding bit error rate (BER) of the ML receiver. (Express your answer in Q-function.)

(c) (3%) Can the BER in part (b) with h(t)=s(4-t) be improved by choosing a different sampling time t₀? Justify your answers.

(d) (3%) Suppose now T=5, i.e. h(t)=s(5-t). How to determine the sampling instant to in this case? Please provide a general rule t₀ determine the relation between the sampling instant t₀ and the delay T such that the sampled output yields the best BER.

(a) (4%) Find the entropy H(Y) and the conditional entropy H(Y|X).

(b) (4%) Find a that can maximize the channel capacity.

(c) (5%) For the case of p ＜ a, please give the maximum a posteriori probability (MAP) decision rule to decide which X is sent based on the output Y. (Hint: In this problem, the output Y of the channel can take just two values "0" or "1". Hence, it is sufficient to find the decision result for these two output possibilities.)

(d) (5%) Suppose the prior probability is not known in advance. We manage to produce an estimated prior probability which may or may not be equal to the true prior probability P(X = 0) = p. In this case, the cross- entropy between the true and estimated prior distributions P and is defined by , which can be considered as an approximated entropy of X. Please show that the cross- entropy is always no less than the true entropy H(X) of X, i.e.
(Hint: You may use the Jensen's inequality: plog₂ a + (1 -p)log₂≤ b ＜ log₂(pa +(1 -p)b) for 0 ≤p ≤ 1, a ＞ 0, and b ＞0.)

(a) (5%) Sketch the signal space diagram (constellation diagram) of this QAM signal.

(b) (10%) Sketch the block diagram of the coherent receiver of a typical QAM system. Furthermore, derive what happens if the local oscillation used by the "coherent" receiver is incorrect (say, the frequency is f₀ + △f).

(c) (5%) Let the symbol error rate of QAM be approximated aswhere Q represents the Q function and γ is the SNR. To improve the symbol error rate from , roughly how much shall the SNR increase?

(d) (7%) Assume that the channel gain and propagation delay from two base stations to the car are the same. Derive the mathematical expression of the received signal, ignoring the speed of the car.

(e) (7%) Now consider the car's speed v and the Doppler effect (the carrier frequency is modified with, where c is the speed of light, and choosing the sign according to the moving direction). Derive the mathematical expression of the received signal.

110年 - 110台灣聯合大學系統_碩士班招生考試_電機類:通訊系統(通訊原理)#105574

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