Communication Systems
2010 Spring – Midterm
Thursday, April 22, 2010.
1. Closed book.
2. 13:10~16:00 (170 minutes).
1. [13] The Fourier transform of a signal g(t) is denote by G( f ). Prove the following properties:
i. [3] If a real signal g(t) is an even function of time t, the Fourier transform G( f ) is purely real. If a real signal g(t) is an odd function of time t, the Fourier transform G( f ) is purely imaginary.
ii. [3] , where is the nth derivative of G( f ) with respect to f.
iii. [3]
iv. [2]
v. [2]
2. [7] Please show that the Fourier transform of a periodic signal of period T0 is given by: where G( f ) is the Fourier transform of .
Hint: , where .
3. [5] Properties of Filter
i. [1] For low-pass and band-pass filters, what’s the relationship between the number of zeros and the number of poles?
ii. [1] In the s-plane, where are the poles of the transfer function for causal systems?
iii. [1] Where are the zeros of the transfer function for the Butterworth filter?
iv. [1] What kind of filter is said to have a maximally flat passband response?
v. [1] Is the ideal low-pass filter causal or noncausal? Please explain your answer.
4. [10] A tapped delay-line filter consists of N weights, where N is odd. It is symmetric with respective to the center tap, that is, the weights satisfy the condition
i. [6] Find the amplitude response of the filter. Hint:
ii. [4] Show that this filter has a linear phase response.
5. [5] The analysis of a band-pass system can be replaced by an equivalent but much simpler low-pass analysis that completely retains the essence of the filtering process. Please illustrate the procedure.
6. [15] Please show that the generation of an AM wave may be accomplished by using the following switching modulator:
7. [10] Consider the models shown in the following figure for generating the VSB signal and for coherently detecting it. Please derive the condition of the filter H( f ) to permit accurate recovery of the message signal.
8. [10] Consider a square-law detector, using a nonlinear device whose transfer characteristic is defined by , where a1 and a2 are constants, is the input, and is the output. The input consists of the AM wave .
i. [5] Evaluate the output .
ii. [5] Find the conditions for which the message signal m(t) may be recovered from .
9. [10] Consider a message signal m(t) containing frequency components at 100, 200, and 400 Hz. This signal is applied to an SSB modulator together with a carrier at 100 kHz. In the coherent detector used to recover m(t), the local oscillator supplies a sine wave of frequency 100.02 kHz. The signal is transmitted by SSB modulator as follow:
i. [5] Assuming that only the upper sideband is transmitted. Determine the frequency components of the detector output.
ii. [5] Repeat your analysis, assuming that only the lower sideband is transmitted.
10. [10] Consider the following figure where (a) is the carrier wave and (b) is the sinusoidal modulating signal. Please answer the following questions and justify your answers.
i. [4] Which is the phase modulated signal?
ii. [3] Which is the frequency modulated signal?
iii. [3] Which is the amplitude modulated signal?
11. [5] In the case of sinusoidal modulation, the FM signal is given by
where is the modulation index, fc is the carrier frequency, and fm is the frequency of the sinusoidal modulating signal. Please show that the expression of a narrow-band FM signal is similar to that of an AM signal given by:
.