108年 - 108 國立臺北大學_碩士班一般入學考試試題_通訊工程學系：通訊原理#147264

A. Signals that are neither energy nor power signals cannot be found.

B. The vestigial-sideband modulation requires a bandwidth that is smaller than the bandwidth of the double-sideband modulation but larger than the bandwidth of the single-sideband modulation.

C. Stereophonic FM radio uses orthogonal frequency for multiplexing.

D. For stationary processes, means and variances are independent of time, and the correlation coefficient depends only on the time difference.

E. An AM signal cannot be demodulated without coherent demodulation techniques.

F. A periodic signal has infinite power.

2. (10 pt) Find the time-average autocorrelation function and power spectral density of \( x(t) = 3 + 3\cos (10\pi t) + 3\sin (10\pi t) \).

3. (10 pt) An FM modulator with \( f_{d} = 10 \) Hz/V. Plot the frequency deviation in hertz and the phase deviation in radians for the message signal shown in the following figure.

i) (10pt) Use coherent demodulation to demodulate the received signal and formulate the detection problem as hypothesis testing. You must derive the probability density function for each hypothesis.

ii) (10pt) Derive the detection threshold \( \eta{\mathrm{ML}} \) of maximum likelihood detection for the hypothesis testing problem derived in part i). Also, derive the detection threshold \( \eta{\mathrm{MAP}} \) of maximum a posterior detection.

iii) (10pt) Derive the average probability of error based on \( \eta_{\mathrm{MAP}} \) in part ii).

Hint: $Q(x) \triangleq \intx^\infty \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} \, \mathrm{d}y$ and $Q(-x) = \int{-\infty}^x \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} \, \mathrm{d}y$.

i) (7pt) Derive the condition on \( h(t) \), in time domain, such that there will be no inter-symbol interference?

ii) (7pt) Now, derive from part i) the condition on \( H(f) \), in frequency domain, which guarantees no inter-symbol interference.

iii) (6pt) Provide an example of \( H(f) \) satisfying the condition in part ii). Show in time domain that it indeed will not result in any inter-symbol interference.