[ ] sin [ ] - Oregon State University

4.4 The continuous-time signal xc  t   sin  20 t   cos  40 t  is sampled with a sampling period T to obtain
n 
 2 n 
the discrete-time signal x  n  sin 
  cos 

 5 
 5 
(a) Determine a choice for T consistent with this information.
(b) Is your choice for T in part (a) unique? If so, explain why. If not, specify another choice of T consistent
with the information given.
4.11 The following continuous-time input signals xc  t  and corresponding discrete-time output signals x  n  are
those of an ideal C/D as shown in figure 4.1:
Specify a choice for the sampling period T that is consistent with each pair of xc  t  and x  n  . In addition,
indicate whether your choice of T is unique. If not, specify a second possible choice of T consistent with
the information given.
n 
(a) xc  t   sin 10 t  , x  n  sin 

 4 
sin 10 t 
sin  n 2 
, x  n 
(b) xc  t  
10 t 
 n 2 
4.21 Consider a continuous-time signal xc  t  with Fourier transform X c  j  shown in Figure P4.21-1.
(a) A continuous-time signal xr  t  is obtained through the process shown in Figure P4.21-2:
Frist, xc  t  is multiplied by an impulse train of period T1 to produce the waveform xs  t  , i.e.,
xs  t  

 x  n  t  nT  . Next, x  t  is passed through a low pass filter with frequency response
n 
1
s
H r  j  . H r  j  is shown in Figure P4.21-3:
Determine the range of values for T1 for which xr  t   xc  t  .
(b) Consider the system in Figure P4.21-4:
The system in this case is the same as the one in part (a), except that the sampling period is now T2 . The
system H s  j  is some continuous-time ideal LTI filter. We want xo  t  to be equal to xc  t  for all t ,
i.e., xo  t   xc  t  for some choice of H s  j  . Find all values of T2 for which xo  t   xc  t  is
possible. For the largest T2 you determined that would still allow recovery of xc  t  , choose H s  j 
so that xo  t   xc  t  . Sketch H s  j  .
4.24 Consider the system shown in Figure P4.24-1:
The anti-aliasing filter is a continuous-time filter with the frequency response L  j  shown in Figure
P4.24-2:
The frequency response of the LTI discrete-time system between the converters is given by:
Hd e
j
e
j

3
,  
(a) What is the effective continuous-time frequency response of the overall system, H  j  ?
(b) Choose the most accurate statement:
d
d
 T
 1
(i) yc  t   xc  3t  (ii) yc  t   xc  t   (iii) yc  t   xc  t  3T  (iv) yc  t   xc  t  
dt
dt
 3
 3
(c) Express yd  n  in terms of yc  t  .
(d) Determine the impulse response h  n  of the discrete-time LTI system.
4.25 Two bandlimited signals, x1  t  and x2  t  , are multiplied, producing the product signal w  t   x1  t  x2  t  .
This
signal
is
sampled
by
a
periodic
impulse
train
yielding
the
signal

wp  t   w  t     t  nT  
n 

 w  nT   t  nT  . Assume that x  t  is bandlimited to  , and x  t  is
n 
1
1
2
bandlimited to  2 ; that is X 1  j   0,   1 X 2  j   0,    2 . Determine the maximum sampling
interval T such that w  t  is recoverable from wp  t  through the use of an ideal lowpass filter.
4.32 Consider the discrete-time system shown in Figure P4.32-1:
where
(i)
L and M are positive integers.
 x  n L  n  kL
,k Z
xe  n   
(ii)
, otherwise
 0
(iii)
y  n  ye  nM 
(iv)

 M
j
H e   
0

 

4

4
  
(a) Assume that L  2 and M  4 , and that X  e j  , the DTFT of x  n  , is real and is as shown in Figure
P4.32-2:
Make an appropriately labeled sketch of X e  e j  , Ye  e j  , and Y  e j  , the DTFTs of xe  n  , ye  n ,
and y  n  , respectively. Be sure to clearly label salient amplitudes and frequencies.
(b) Now assume L  2 and M  8 . Determine y  n  in this case.
Hint: see which diagrams in your answer to part (a) change.
4.33 For the system shown in Figure P4.33:
z1  n 
z2  n 
z3  n 
, find an expression for y  n  in terms of x  n  . Simplify the expression as much as possible.
4.38 Consider the two system of Figure P4.38:
(a) For M  2 , L  3 , and any arbitrary x  n  , will y A  n   yB  n  ? If your answer is yes, justify your
answer. If your answer is no, clearly explain or give a counterexample.
(b) How much M and L be related to guarantee y A  n   yB  n  for arbitrary x  n  ?