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.
n
 j  (n)
(n)
  G  f  , where G  f  is the nth derivative of G( f ) with respect to f.
 2π 
n
ii. [3] t g (t )

iii. [3] g1 (t ) g2* (t )   G1 ( )G2* (  f ) d 

n
 j 
iv. [2]  t g (t ) dt    G ( n )  0 

 2π 

v. [2]

n



g1 (t ) g2* (t ) dt   G1 ( )G2* ( ) d 

2. [7] Please show that the Fourier transform of a periodic signal gT0  t  of period T0 is given by:
gT0 (t ) 


m 
g (t  mT0 )

 g (t ),
g (t )   T0
0,
Hint: gT0 (t ) 
1
T0

 G(nf ) ( f  nf )
n 
0
0
where G( f ) is the Fourier transform of
T0
T
t  0
2
2.
elsewhere


c
n 
n
exp  j 2 nf 0t  , where cn 
1
T0

T0 2
T0 2
gT0  t  exp   j 2 nf 0t  dt .
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
wn  wN 1n
0  n  N 1
i. [6] Find the amplitude response of the filter. Hint: H ( f ) 

h
k 
k
exp   j 2 fk 
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 2  t   a1v1  t   a2v12 t  , where a1 and a2 are constants, 1  t  is the input, and
2  t  is the output. The input consists of the AM wave 1  t   Ac 1  ka m  t   cos  2πf ct  .
i. [5] Evaluate the output 2  t  .
ii. [5] Find the conditions for which the message signal m(t) may be recovered from 2  t  .
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:
s t  
Ac
 m  t  cos  2πf ct   mˆ  t  sin  2πf ct  
2 
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
s  t   Ac cos  2 f ct   sin  2 f mt  
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:


1
s AM  t   Ac cos  2 f ct    Ac cos  2  f c  f m  t   cos  2  f c  f m  t  .
2
2sin  cos =sin(   )+sin(   )
2cos  sin =sin(   )-sin(   )
 
 
sin  sin  2sin
cos
2
2
 
 
sin  sin  2cos
sin
2
2
sin  A  B   sinAcosB  cosAsinB
2sin  sin =cos(   )-cos(   )
2cos  cos =cos(   )+cos(   )
 
 
cos  cos  2cos
cos
2
2
 
 
cos  cos  2sin
sin
2
2
cos  A  B   cosAcosB  sinAsinB
sin  A  B   sinAcosB  cosAsinB
cos  A  B   cosAcosB  sinAsinB
2
2
  A   1  cos A
cos  2   
2
  
  A   1  cos A
sin  2   
2
  
 


n 
n 
Fourier Series: gT0  t   a0    an cos  2π t   bn sin  2π t  
n 1 
 T0 
 T0  
a0 
1
T0

T0 2
T0 2
gT0  t  dt ; an 
2
T0


n 
2
gT0  t  cos  2 t  dt ; bn 
2
T0
 T0 
T0 2
T0


n 
gT0  t  sin  2 t  dt
2
 T0 
T0 2
T0