課程名稱︰通信原理
課程性質︰選修
課程教師︰陳光禎
開課學院：電資學院
開課系所︰電機工程學系
考試日期（年月日）︰103.11.20
考試時限（分鐘）：120分鐘
是否需發放獎勵金：是
試題 :
1.[Gaussian Process,20%] Let X(t) be a zero-mean, stationary, Guassian process
with autocorrelation function R_x(τ).
(a)What is the power spectral density S_x(f)?
(b)By observing X(t) at some time t_k, what is the probabilty density func-
tion in this observation? This process is applied to a square-law device,
which is defined by the input-outout relation Y(t)=X^2(t), where Y(t) is
the output.
(c)Show that the mean of Y(t) is R_x(0).
(d)Show that the autocorrelation function of Y(t) is 2R_x(τ).
2.[PM,20%] Consider a phase modulation system, with the modulated waveform de-
fined by s(t)=A_c*cos[2πf_c*t+k_p*m(t)] where k_p is a constant and m(t) is
the message signal. The additive noise n(t) at the phase detector input can
be represented by the canonical form (i.e. I-channel and Q-channel).
Assuming the carrier-to-noise ratio at the detector input is high compared
with unity, please determine
(a)the ouput SNR
(b)the figure of merit of the system.
(c)Please derive the figure of merit for the FM system with sinusoidal modu-
lation, and compare the results.
3.[Waveform Coding,20%] A speech signal occupying bandwidth from 0-4kHz. We
wish to conduct digital communication for such signals. Please answer the
following series of questions.
(a)What is the Nyquist sampling rate in this case? What is the problem if
under-sampling?
(b)If we use 8 bits to represent a sample value, what is the data rate for
PCM?
(c)If we use delta modulation to encode such a signal, what is the minimum
rate if we use integer number of digits to represent a difference sample?
(d)In case we are using DPCM and only need 4 bits to represent each of the
resulting sampling, what is the data rate for this DPCM?
4.[Rayleigh,20%] Consider a random process Z(t)=X(t)+jY(t)=R(t)exp(jΦ(t)),
where random process X(t) and Y(t) are statistically independent Gaussian
distributed with zero mean and variance σ^2 for any t. Please show that
R(t) is Rayleigh distributed and Φ(t) is uniformly distributed.
5.[Envelope Detection,20%] Consider the output of an envelope detector, which
is reproduced here for convenience
y(t)={[A_c+A_c*k_a*m(t)+n_I(t)]^2+[n_Q(t)]^2}^(1/2)
(a)Assume that the probability of the event |n_Q(t)|>ε*A_c*|1+k_a*m(t)|
is equal to or less than δ_1, where ε<<1. What is the probability that
the effect of the quadrature component n_Q(t) is negligible?
(b)Suppose that k_a is adjusted related to the message signal m(t) such that
the probability of the event A_c*[1+k_a*m(t)]+n_I(t)<0 is equal to δ_2.
What is the probability that the estimation y(t)≒A_c*[1+k_a*m(t)]+n_I(t)
is valid?
(c)Comment on the significance of the result in part (b) for the case when
δ_1 and δ_2 are both small compared with unity.