Practice Problems - 1. The response of a linear system with...

1. The response of a linear system with input
 
x n
is
   
2
3
n
k n
y n x k

 


Find and sketch
 
h n
2. Find
     
y n x n h n  
if
   
   
1
k
n
x n n kN
h n a U n a



 
 

Does the output have an analytic z-transform? (Analytic means it has no poles or other
singularities in some region where the z-transform converges)
3. To simulate a continuous time differentiator
 
a
H j   
the following discrete systems are
sometimes used with
 
 
a
f n f nT 
,
 
     
a a
t nT
d d
g n g nT f t f t
dt dt

 
:
 
   
       
 
1
2 2
1
2
1 1
f n f n
g n
T
g n g n f n f n
T
 

    
Find the corre4sponding discrete system functions
 
1
H z
and
 
2
H z
and sketch their
frequency responses.
4. Find all inverse transforms of the function
a
 
    2
1 2
z
F z
z z

 
5. Show that, if
   
f n F z 
Then
   
d
nf n z F z
dz
 
6. Use Problem 5 to find the transform of
   
n
f n na U n 
7. Let
1 a 
and
 
n
f n a
. Find
 
F z

8. Suppose we are given M+1 real numbers
 
0..
n
n M
a

Let
2 1 N M  
and
0
0
2
2 cos
M
m n
n
A a a nm
N


 
 
 
 

then
0
0
2 2
cos
M
n m
m
A
a A nm
N N N


 
 
 
 

9. Let
3
n m
N
a A 
be a DFT pair
Define auxiliary sequences