Signals and Systems (EKT230)
Tutorial III
1. Consider a continuous-time system with input x(t) and output y(t) related by
y (t )  x(sin( t ))
(a) Is the system causal ?
(b) Is the system linear ?
2. Determine whether the corresponding system is linear, time invariant or both.
(a) y[n] = x2[n-2]
(b) y[n] = Od{x(t)}
3.
(a) A continuous signal x(t) is shown in Figure 1. Sketch and label each of the following
signals.
(i) x(t) u(1-t)
(ii) x(t) [u (t) - u (t -1)]
(iii) x(t) ( t - 3/2)
(b) From the given signal y[n],
e  n  3  n  0
y[n]   n
 e
0n3
Draw and label the waveform for each of the following signals:
y[n  3]
(i)
(ii) y2n  2
(iii) 2 y[1  n]
4. A linear time invariant system has an impulse response, h(t) and input signal, x(t). Use
convolution to find the response, y(t) for following signals:
x(t )  cos(t )u (t  1)  u (t  3) 
h(t )  u (t )
5. A continuous-time periodic signal x(t) is real valued and has a fundamental period T = 8.
The non-zero Fourier series coefficient for x(t) are
a1  a1  2, a3  a *3  4 j
Express x(t) in the form
-2-
Tutorial III

x(t )   Ak cos( k t   k )
k 0
6.
A discrete-time periodic signal x[n] is real valued and has a fundamental period N=5.
The non-zero Fourier series coefficients for x[n] are
a 0  1, a 2  a 2 *  e j / 4 , a 4  a * 4  2e j / 3
Express x(t) in the form

x[n]  A0   Ak sin(  k n   k )
k 0
7. Consider a discrete-time LTI system with impulse response
h[n] =
1,
-1
0
0≤n≤2
-2 ≤ n ≤ -1
otherwise
Given that the input to this system is
x[n] 

  [ n  4k ]
k  
determine the Fourier series coefficients of the output y[n].
8. Consider a causal discrete-time LTI system whose input x[n] and output y[n] are related by
the following difference equation:
y[n] – ay[n-1] = x[n]
(a) Find the Fourier series representation of the output y[n] for the input:
3
x[n]  sin(
n)
4
(b) Find the impulse response h[n]
Signals and Systems (EKT230)
Tutorial III
Appendix 1
Lampiran 1
FOURIER TRANSFORM
Signal
Transform
LAPLACE TRANSFORM
Signal
Transform
1
u(t)
s
1
tu(t)
s2
Z-TRANSFORM
Signal
Transform
 [n]
(t)
1
1
2()
u(t)
1
  ( )
j
(t - )
e-s
 n u[n]
e-atu(t)
1
a  j
e-atu(t)
1
sa
n n u[n]
1
( s  a) 2
[cos(1n)]u[n]
1  z 1 cos 1
1  z 1 2 cos 1  z 2
1
u[n]
1
1
1  z 1
1
1  z  1
z 1
(1  z 1 ) 2
te-atu(t)
a  j 2
te-atu(t)
e-a|t|
2a
2
a 2
[cos(1t )]u(t )
s
2
s  12
[sin( 1n)]u[n]
z 1 sin 1
1  z 1 2 cos 1  z 2
[sin( 1t )]u(t )
1
2
s  12
[r n cos(1n)]u[n]
1  z 1 r cos 1
1  z 1 2r cos 1  r 2 z 2
sa
( s  a ) 2  12
[r sin( 1n)]u[n]
z 1 r sin 1
1  z 1 2r cos 1  r 2 z  2
1
2
e t
2
/2
e
 2 / 2
[e
 at
cos(1t )]u(t )
[e  at sin( 1t )]u (t )
1
( s  a) 2  12
n