R1
T1) Give the analytical expression and plot the continuous-time Dirac (unit) impulse. Give the
filtering property of the Dirac impulse.
T2) What is BIBO stability; enunciate the condition for a discrete-time system to be BIBO stable.
T3) Signal’s modulation property for the continuous-time Fourier series (enunciation and proof)
 
P1. Consider the continuous-time periodic signal: x  t   sin  t   cos  t 
2 
a) Determine and sketch the magnitude, phase and power spectra for it;
b) Compute the power of the signal using Parseval’s relation.
P2. Consider the signal: x  n    n  2  n  1  3  n  2 .
a) Sketch the waveform of the signal.
b) Find and sketch: the even and odd components of the signal
c) Sketch x  2n .
R2
T1) Give the analytical expression and plot the continuous-time unit step. What is the connection
between the unit impulse and the unit step?
T2) Give the expression of the output signal y  n  when the input signal x  n is causal, and the
system is also causal. Is the output also causal?
T3) What is the continuous-time system’s response to the complex exponential with magnitude
one as a function of the Fourier coefficients. Proof.
 
 2 
t
P1. Consider the continuous-time periodic signal: x  t   cos  t   cos 
2 
 5 
a) Determine and sketch the magnitude, phase and power spectra for it;
b) Compute the power of the signal using Parseval’s relation.
P2. Consider the signal x  n  4  n   2  n  2 .
a) Sketch the waveform of the signal.
b) Find and sketch: the even and odd components of the signal
n
c) Sketch x   .
2
Solution-R1
T1) Give the analytical expression and plot the continuous-time Dirac (unit) impulse. Give the
filtering property of the Dirac impulse.
 , t  0 
 t   
,    t  dt  1
 0, t  0 

   t    t  dt   0 

T2) What is BIBO stability; enunciate the condition for a discrete-time system to be BIBO stable.
A discrete–time LTI system is stable if and only if its impulse response is absolutely
summable

 h  k    ,h  n  l1
k 
T3) Signal’s modulation property for the continuous-time Fourier series (enunciation and proof)

x  t  e jk00t  ckxk0
x t  

ce
jk0t
k
k 
x  t  e jk00t 


 ckxe jk0t e jk00t 
k 

1
2
x  t  e  jk0t dt , 0 

T0 T0
T
 ck 

c
x
m  k0
m 
e jm0t 

 ckxe
j  k  k0 0t
 k  k0   m

k 

c
k 
x
k  k0
e jk0t , qed
R1-Problems
 
P1. Consider the discrete-time periodic signal: x  t   sin  t   cos  t 
2 
a) Determine and sketch the magnitude, phase and power spectra for it;
b) Verify Parseval’s relation.
Solution.
a) 1 

2
2   ; T2  2 ; T0  lcm T1 , T2   4  0 
; T1  4

j t
2

j t
2

j t

j t

2
e j t  e j t e 2 e 2 e j t e  j t
x t  





2
2j
2j
2
2
c
c2
4
c4
c2
1 j
1

1
j
1

c2 

 c2  ;arg c2   ; c2  
  c2  ;arg c2 
2j 2
2
2
2j 2
2
2
1
1
c4  c4   c4  c4  ;arg c4  arg c4  0
2
2


2
1
1
2
2
b) Parseval’s relation: P   ck   x  t  dt ;  ck  4  1
4
T0 T0
k 
k 
e
e
2j
Solution-R2
R2-Theory
T1) Give the analytical expression and plot the continuous-time unit step. What is the
connection between the unit impulse and the unit step?
1, t  0
 t   
0 , t  0
Connection: '  t     t  , or
t
    d     t 

T2) Give the expression of the output signal y  n  when the input signal x  n is causal, and
the system is also causal. Is the output also causal?
x  n  0 , n  0
n
AND
h  n   0 , n  0  y  n    x  k  h  n  k  . yes it is causal
k 0
T3) What is the continuous-time system’s response to the complex exponential with
magnitude one as a function of the Fourier coefficients. Proof.
For x  t   e j0t the output is y  t   e j0t  H 0 
Proof: y  t  




h  e j0 t   d  e j0t   h   e  j0 d

 0
H 
R2-Problems
 
 2 
t
P1. Consider the continuous-time periodic signal: x  t   cos  t   cos 
2 
 5 
a) Determine and sketch the magnitude, phase and power spectra for it;
b) Verify Parseval’s relation.

2

2 
; T2  5  T0  lmc T1 , T2   20  0 
Solution: a) 1  ; T1  4
2
5
10
2
2
2
2


j t
j t
j t
j t
jk t
1 4 1
1 5 1
5
4
x  n    ck e 10  ee  ee
 ee  ee
2
2
2
2
k 
5
5
4
4
1 j 10 t
1 e j 10 t 1 e j 10 t
1 e j 10 t
 ee 
e
 e

e
2
2
2
2
c
c
c
c
5
5
4
4

2
1
2
b) Parseval’s relation: P   ck   x  t  dt ;
T0 T0
k 


k 
ck  4
2
1
1
4