UNIVERSITI MALAYSIA PERLIS (UniMAP)
EKT 232 (Signals and Systems)
Test 1 (1 ½ Hours)
Network Engineering
Instruction: Please write all your work to get full credit. Sketch pictures when appropriate.
Question 1:
a) Consider the continuous-time signal x(t ) as shown in figure 1. Identify and sketch each of the
following signal:
x(2t  1)
i.
2
1
-1.5
-0.5
-1
0
t
0.5
-1
ii.
t

x4  
2

2
1
12
4
6
-1
8
10
iii.
3
3
x(t )[ (t  )   (t  )]
2
2
0.5
0.5
3/2
0
-3/2
t
[9 Marks]
b) Determine the value of P and E for each of the following signals:
x3 (t )  cos(t )
i.


Therefore, E   x3 (t ) dt   cos 2 dt  
2
P  lim 21T T cos 2 (t )dt  lim 21T T
T 
T 
T
ii.
T

1 cos(2 t )
2
dt 
1
2
 
x3 [n]  cos  n 
4 


2
 
Therefore, E   x3[n]   cos 2  n   
n 
n 
4 
N
N
1
1
 1  cos( 2 n)  1

2 
P  lim
cos
n

lim




 2
T  2 N  1 n  N
2
 4  N  2 N  1 n  N 

[6 Marks]
Question 2
a) Express with a brief explanation of system properties for signals below:
i.
y (t )   x( )d 
t
Linear, causal, time-invariant.
ii.
y (t )  x(t )
Linear, non-causal, time varying
[2 Marks]
b) Consider a discrete-time signal x[ n ] , fed as input into a system. The system produces the discretetime output y[n] such that
 x[n], n even
y[n]  
n odd
 0,
i.
Is the system described above memoryless? Explain.
It is memoryless since the output at time instant n depends on the input only at time instant
n and not past or future time instants.
ii.
Is the system described above causal? Explain.
It is causal since the output at time instant n depends on the input only at time instant n and
not future time instants.
iii.
Are causal systems in general memoryless? Explain.
No. If the output at time instant n depends on the input at time instant n and past time
instants the system is causal but not memoryless.
[3 Marks]
c) Show that the power of a signal
n
n
k m
k m
f (t )   Dk e jk t is Pf   Dk
2
Assuming all frequencies to be distinct, that is, i  k for all i  k
Solution
T
Pf  lim T1 2T f (t ) f * (t )
T 
2
n
n
 lim T1    Dk D*r e j (k r )t dt
T 
T
2
 T2
k m r m
The integrals of the cross-product terms (when k  r ) are finite because the integrands are
periodic signals (made up of sinusoids). These terms, when divided by T   , yield zero. The
remaining terms (k  r ) yield
T
n
n
Pf  lim T1 2T  Dk dt   Dk
T 
2
k m
2
2
k m
d) Consider a linear time-invariant system denoted by the operator H, as indicated in Figure 2. The
input signal x(t ) applied to the system is periodic with period T. Show that the corresponding
response of the system, y (t ) , is also periodic with the same period T.
x(t)
H
y(t)
Solution
The output y (t ) is related to the input x(t ) as
y (t )  H {x(t )}
(1)
Let T 0 denote the fundamental period of x(t ) , assumed to be periodic. Then, by definition,
x(t )  x(t  T0 )
Substituting t  T0 for t into Eq. (1) and then using Eq.(2), we may write
y (t  T0 )  H {x(t  T0 )}
(2)
 H {x(t )}
 y (t )
Hence, the output y (t ) is also periodic with the same period T0
[10 Marks]