milk26364325>試卷(2021/09/15)

# 110 年 - 110 國立臺灣大學_碩士班招生考試_電信工程研究所乙組:通信原理(含信號與系統)#101261

【非選題】
1.(1) (12 %) Illustrate the following four terms: (a) aliasing efiect; (b) LTI; (c) wide-sense stationary random process; (d) additive white Gaussian noise

【非選題】
2.

(2) (6 %) Suppose that the continuous-time Fourier transform of x(t) is X(jω). What is the Fourier transform of Re(x(-2t+3) I where Tef ; means taking the real part?

【非選題】
3.

(3) Determine the following convolutions

【題組】(a)(6%) (* means the discrete convolution and 4[n] is the unit step function)

【非選題】
4.【題組】

(b) (6%) [cos(πt) +sin(4πt)+ cos(10πt)]* sinc(21) (* means the continuous convolution)

【非選題】
5.

(4) Suppose that x(t) is real and has the bandwidth of W and where *means convolution.

【題組】(a) (5%) What is the bandwidth of y(t)?

【非選題】
6.【題組】(b) (3 %) How do we sample y(t) properly to avoid the aliasing effiect?

【非選題】
7.

(5) Find the inverse Z transform of the following functions (suppose that x[r] is causal).

【題組】 (a) (6%) 【非選題】
8.【題組】

(b)(6%) 【非選題】
9.

(6) (25%) Consider a discrete memoryless source (DMS) with the associated source alphabet X := {a, b, c, d, e,f, g}. Each source output is independenty selocted from C with the probability distribution Px given in Table 1. The goal of source coding for the DMS is to construct a code C that assigns cach symbol bit string N. Its average length is defined as Moreover, the code is usually required to be uniquely decodable. Please answer the following questions.

【題組】

(a) (5%) What does the unique decodability mean? Assume that a code has the property that for each . Is the code uniquely decodable?

【非選題】
10.【題組】

(b) (5%) A class of codes called the prefix-free codes can be decoded with no delay (hence sometimes also called instantareous codes). What is the definition for the prefix-free codes?
Every prefix-free code for the alphabet 9X with its codeword lengths must satisfy the Kraftinequality. What is the Kraft inequality?
Assume that a code with its codeword lengths satisfies the Kraft inequality. Is it uniquely decodable? Does a uniquely decodable code have to satisfy the Kraft inequality?

【非選題】
11.【題組】

(c) (10%) Use the Hufiman algorithm to construct a prefix-free code for the above mentioned DMS. What is the associated average length ? Clearly state your coding strategy.

【非選題】
12.【題組】(d) (5%) Explain why the Huffman algorithm gives an optimal source code, i.e. no other source coding strategy has smaller .

【非選題】
13.

(7) (25%) Consider a standard M-ary digital modulation scheme where the signal space diagram is spanned by two orthonormal waveforms, i.e. each symbol on the constellation has the in-phase and quadrature-phase components. Assume that the underlying channel model is additive white Gaussian noise (AWGN) with the power spectral density N0 / 2 , and that no inter-symbol interference is happening. Further, assume that all symbols on the constellation are equally probable. Please answer the following questions. 【題組】 (a) (5%) Using the signal space diagrams given in Figures la and 1b, respectively. Which one has the better performance (i.e. the smallest symbol error probability)? Explain your reason.

【非選題】
14.【題組】

(b) (10%) Derive the optimal symbol error probability when using the signal space diagram given in Figure I b in terms of the average symbol energy, No, and the Q-function, 【非選題】
15.【題組】

(c) (5%) Given an arbitrary M-ary modulation scheme with the minimum distance , derive a union upper bound for the symbol error probability in terms of and the Q-function.

【非選題】
16.【題組】

(d) (5%) Since the minimun distance of a constellation dominates the performance of a modulation scheme, one would desire a signal diagram with the same minimum distance constraint that has smaller average symbol energy. Please design a 7-ary signal diagram on the I/Q-constellation with = 2 and its average symbol energy no larger than 3.43. red

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