EE2010/IM2004 Signals and Systems
Academic Year: Semester 1
Part I: Weeks 1-5
Lecturer: Prof. Ma Kai-Kuang Office: S2-B2C-83
Tel: 6790-6366 Email: [email protected]
• Notes and the attached tutorial set are from Assoc/Prof Teh Kah Chan. • Refer to NTUlearn regularly for important announcement, including rules
and assessment criteria.
Subject Outline
• Signals and Systems
– Classification of Signals
– Elementary and Singularity Signals
– Operations on Signals
– Properties of Systems
• Linear Time-Invariant (LTI) Systems
– Continuous-Time and Discrete-Time LTI Systems
– Convolution
– LTI System Properties
– Correlation Functions
2
Textbook1. M. J. Roberts, Fundamentals of Signals and Systems,McGraw-Hill, International Edition, 2008. (TK5102.9.R646F)
References1. M. J. Roberts, Signals and Systems, McGraw-Hill, InternationalEdition, 2003. (TK5102.9.R63)2. A. V. Oppenheim and A. S. Willsky, Signals and Systems,Prentice-Hall, 2nd Edition, 1997. (QA402.P62)3. S. Haykin and B. V. Veen, Signals and Systems, Wiley, 2ndEdition, 2003. (TK5102.5.H419)4. S. S. Soliman and M. D. Srinath, Continuous and Discrete Signalsand Systems, Prentice-Hall, 2nd Edition, 1998. (TK5102.9.S686)5. B. P. Lathi, Linear Systems and Signals, Oxford University Press,1st Edition, 2002. (TK5102.5.L352)
3
Overviews of Signals and Systems
TransducerInput Signal
Input
Transmitter
Transmitted Signal
Channel and NoiseDistortion
Received Signal
TransducerOutput
OutputSignal
Receiver
Input Message
MessageOutput
Figure �� A typical signal and system example
�
Classi�cation of Signals
� Continuous�Time vs Discrete�Time Signal
� Continuous�Value vs Discrete�Value Signal
� Deterministic vs Random Signal
� Even vs Odd Signal
� Periodic vs Aperiodic Signal
� Energy�Type vs Power�Type Signal
�
Continuous�Time vs Discrete�Time Signal
� Continuous�Time �CT� Signal� A signal x�t� that is speci�ed for
all value of time t
� Discrete�Time �DT� Signal� A signal y�n� that is speci�ed only
for integer value of n
0 -1 75
t
20 14 63
t n
( )x y [n]
Figure �� Continuous�time vs discrete�time signal
�
Example �� Sketch the waveforms of the CT signal x�t� � t and DT
signal x�n� � n� respectively
(t ) x[n]x
-3
.50
5
-5-5
-1-2-323
0 1 2 3t n-2-1
1
Figure �� Examples of CT and DT signals
�
Continuous�Value vs Discrete�Value Signal
� Continuous�Value Signal� A signal x�t� whose amplitude can take
on any value
� Discrete�Value Signal� A signal y�t� whose amplitude can take on
only a �nite number of values0 0 Tt t
x( ) y ( )
T2
tt
Figure �� Continuous�value vs discrete�value signal
�
Example �� Sketch the waveforms of the continuous�value signal
x�t� � A sin���f�t� and discrete�value signal y�n� � ����n�
respectively
12
3-1
t
-2-3
1
-1
00
A
t n
( ) y[n]
-A
x
Figure �� Examples of continuous�value vs discrete�value signals
Deterministic vs Random Signal
� Deterministic Signal� A signal x�t� that can be mathematically
modeled explicitly as a function of time� i e � x�t� � A sin���f�t�
� Random Signal� A signal y�t� that is known only in terms of
probabilistic description� i e � noise
0 0
A
t-A
t
x( t) y ( )t
Figure �� Deterministic vs random signal
��
Even vs Odd Signal
� Even Signal� A signal xe�t� that satis�es the condition
xe�t� � xe��t�
� Odd Signal� A signal xo�t� that satis�es the condition
xo�t� � �xo��t�
00 t t
ex ( )t ( )toxFigure �� Even vs odd signal
��
� Any deterministic signal x�t� can be decomposed into sum of an
even and an odd signalx�t� � xe�t� � xo�t�
where
xe�t� �
���x�t� � x��t��
and
xo�t� �
���x�t�� x��t��
��
� The product of two even signals is an even signal
� The product of two odd signals is an even signal
� The product of an even signal and an odd signal is an odd signal
� Note that
Z T�
�T�xe�t�dt � �
Z T�
�
xe�t�dt
and
Z T�
�T�xo�t�dt � �
��
Example �� Show that the signal x�t� � A sin���f�t� is an odd signal
Since
x��t� � A sin���f���t��
� �A sin���f�t�
� �x�t�
hence� x�t� is an odd signal
0 t
)x (A
-A
t
Figure �� An odd signal example
��
Example 4: Find the even and odd components of the signalx(t) = cos(t) + sin(t) cos(t).
The even component of x(t) is
xe(t) =12
[x(t) + x(−t)]
=12
[cos(t) + sin(t) cos(t) + cos(−t) + sin(−t) cos(−t)]
= cos(t)
The odd component of x(t) is
xo(t) =12
[x(t) − x(−t)]
=12
[cos(t) + sin(t) cos(t) − cos(−t) − sin(−t) cos(−t)]
= sin(t) cos(t)
15
Example 5: Evaluate∫ T0
−T0x(t)dt where x(t) = t3 cos3(10t).
Since
x(−t) = (−t)3 cos3[10(−t)]
= −t3 cos3(10t)
= −x(t)
hence, x(t) is an odd signal. Thus,∫ T0
−T0
x(t)dt = 0
16
Periodic vs Aperiodic Signal
� Periodic Signal� A signal x�t� with a constant period � � T� ��
that
x�t� � x�t� T��� �� � t ��
For a discrete�time signal� the constant period is an integer
� � K� �� thatx�n� � x�n�K��� �� � n ��
� Aperiodic Signal� A signal y�t� or y�n� that does not satisfy the
above equation
��
6... ...
......0
1 32 4 5
n
6 7 0 1 2
0 2
0
0
0 0
254 7
3
t t
nnK
T
T T
x( )
x[ ] y [ ]
y( )t t
n
=0
Figure � Periodic vs aperiodic signal
��
Energy�Type vs Power�Type Signal
� Energy�Type Signal
� A signal x�t� or x�n� that has �nite energy� i e � � � Ex ���
where
CT signal� Ex �
Z�
��
jx�t�j�dt
DT signal� Ex �
�Xn��
jx�n�j�
� Power�Type Signal
� A signal x�t� or x�n� that has �nite power� i e � � � Px ���
where
�
CT signal� Px � limT��
�T
Z T��
�T��jx�t�j�dt
DT signal� Px � limK��
�
�K � �
KXn�K
jx�n�j�
� Note that if x�t� or x�n� is a periodic signal with period T� or
K�� respectively� then
CT signal� Px �
�T�
Z t��T�
t�
jx�t�j� dt
DT signal� Px �
�K�
k�K���Xnk
jx�n�j�
with any real value of t� and any integer value of k
��
Example �� Determine the energy and power of the periodic signal
x�t� � A cos���f�t��
Ex �
Z�
��
jx�t�j�dt
�
Z�
��
jA cos���f�t�j�dt � �
Px � limT��
�T
Z T��
�T��jx�t�j�dt �
�T�
Z T���
�T���jx�t�j�dt
�
�T�
Z T���
�T���jA cos���f�t�j�dt �
A��
Hence� x�t� is a power�type signal In general� power�type signals are
periodic signals
��
Example �� Determine the energy and power of the signal
y�t� � exp��jtj��Ey �
Z�
��
jy�t�j�dt
�
Z�
��
jexp��jtj�j�dt
� ��Z�
�
exp���t�dt � �
Py � limT��
�T
Z T��
�T��jy�t�j�dt
� limT��
�T�Ey � �
Hence� y�t� is an energy�type signal In general� energy�type signals
are aperiodic signals
��
Example �� Determine the energy and power of the discrete�time
periodic signal x�n� � A sin���n����
Ex �
�Xn��
jx�n�j�
�
�Xn��
jA sin���n���j�
� �
Px �
�K�
k�K���Xnk
jx�n�j� �
��
�Xn�jA sin���n���j�
�
A����
�� � �� � �� � ������
�
A��
Hence� x�n� is a power�type signal ��
Example � A simpli�ed transmitter model of a digital
communication system is shown in Figure ��� determine the
classi�cations of each signal
cos (2π f0 t n] ]nn]
n]
)AQuantizationSampling
Ideal
Discrete Noise
x (t) = x[ x[[y
[w
Figure ��� Transmitter model of a digital communication system
��
The waveforms of various signals are shown in Figure ��
� x�t� is a continuous�time� continuous�value� deterministic� even�
periodic� and power�type signal
� x�n� � x�nTs� is a discrete�time� continuous�value� deterministic�
even� periodic� and power�type signal
� �x�n� is a discrete�time� discrete�value� deterministic� even�
periodic� and power�type signal
� w�n� is a discrete�time� continuous�value� random� and aperiodic
signal
� y�n� is a discrete�time� continuous�value� random� and aperiodic
signal
��
[ ]nxt
0 0.
0.
0
t
n
n
n
[ ]ny
x( )
x [ ]n
Figure ��� Waveforms of various signals for Example
��
Elementary and Singularity Signals
� Exponential signal
x�t� � A exp �at�
� x�t� is growing if a � �
� x�t� is decaying if a � �
00
1
1
tt
a 0
exp( exp(
> a<
at)at)
0
Figure ��� Exponential signal
��
� Sinusoidal signal
x�t� � A cos ���f�t� �� or A sin ���f�t� ��
where A is the amplitude� f� is the frequency in Hertz� and � is
the phase angle in radians
� A sinusoidal signal is periodic with period T� � ��f�
T0
... ...t0
cosA
A (2π )f0 t
-A
Figure ��� CT sinusoidal signal
��
� The discrete time version of the sinusoidal signal is
x�n� � A cos�
��nK�
� ��
or A sin�
��nK�
� ��
where A is the amplitude� K� is a positive integer de�ned as the
fundamental period� and � is the phase angle in radians
[n]=x 8(cos π2 )
0 8. . . ..... ...
A
n
nA
-A
Figure ��� DT sinusoidal signal
�
� Complex exponential signal
A exp�j��f�t� � A cos ���f�t� � jA sin ���f�t�
� The magnitude of complex exponential signal is given by
jA exp�j��f�t�j � A
� The sinusoidal signal can be expressed as
A cos ���f�t� �� � �fA exp�j��f�t� exp�j��g
and
A sin ���f�t� �� � �fA exp�j��f�t� exp�j��g
��
Example ��� Sketch the function x�t� � � exp��at�� cos �����t�
for t � � Assume that a � �
5)
0 0.1 0.2 0.3 t
-5
(x tFigure ��� An exponentially damped sinusoidal signal
��
� The DT unit impulse �or Dirac Delta� function ��n� is de�ned as
��n� �
���
�� n � ��
�� n �� ��
δ n]
3 4 1 2-1. . . .
[
.
3
. .1
. . .421-1 0 0 3 nn
A
[ ]nδAFigure ��� DT impulse functions
��
� The CT unit impulse �or Dirac Delta� function ��t� is de�ned as
��t� �
���
�� t � ��
�� t �� ��
0 0
1
0T
A
) (t
t t
δ( )0TδAtFigure ��� CT impulse functions
��
� Properties of the CT impulse function
� Property �
Z�
��
��t�dt � �
� Property �
x�t�� � �t� T�� � x�T��� � �t� T��
� Property �
Z�
��
x�t�� � �t� T�� dt � x�T��
��
� The CT unit step function u�t� is de�ned as
u�t� �
���
�� t � ��
�� t � ��
0
1
t
t( )u
Figure ��� A CT unit step function
��
� The DT unit step function u�n� is de�ned as
u�n� �
���
�� n � ��
�� n � ��
n]u
.[
.n
. ......2 3
. .0
1
6541
Figure �� A DT unit step function
��
� The CT signum function sgn�t� is de�ned as
sgn�t� �
�������
�� t � ��
�� t � ��
��� t � ��
-10
1)
t
sgn(t
Figure ��� A CT signum function
��
� The DT signum function sgn�t� is de�ned as
sgn�n� �
�������
�� n � ��
�� n � ��
��� n � ��
......
-40
]1
-1-2-3-5-6
-1
.1 2 3 4 65 n
sgn[n
Figure ��� A DT signum function
��
� The CT unit rectangular function rect�t
T
is de�ned as
rect�
tT
��
���
�� jtj � T���
�� otherwise
( )rect T
0
1
22T T t
t
Figure ��� A CT unit rectangular function
�
� The DT unit rectangular function rect�n
K�
�assume that K is
even� is de�ned asrecth n
Ki
�
���
�� jnj � K���
�� otherwise
rect[ ]
..n
. . . .......0-1
1
12 2K K
nK
Figure ��� A DT unit rectangular function
��
� The sinc function sinc�t� is de�ned as
sinc�t� �
sin��t�
�t
-1-2
)
-3-4 0 1 2 3 4
1
sinc
t
t(
Figure ��� A sinc function
��
Example ��� The function x�t� � �� sinc�t� is sampled at every
Ts � ��� second interval to produce the sampled signal xs�t�� sketch
the waveforms for x�t� and xs�t�� respectively
xs�t� �
�Xn��
x�t�� ��t� nTs�
�
�Xn��
x�nTs�� ��t� nTs�
�
�Xn��
�� sinc�nTs�� ��t� nTs�
��
1 2 3 4
1 2 3 4
0
0
t
. . . . . . . .
5
5
-1-2-3-4
t
t
-4 -3 -2 -1
xs
) =t(x 5sinc(
)(
t)
Figure ��� Waveforms for x�t� and xs�t�
��
Operations on Signals
� Amplitude scaling� The operation Ax�t� �or Ax�n�� is to multiply
the amplitude of x�t� �or x�n�� by an amount A
3
(t) (t) (t)x
0 210 21 0 12
t t t
2
4
x x2
-2
-4-3
1.5Figure ��� Amplitude scaling of signals
��
� Time shifting� The operation x�t� T � �or x�n�K�� is to shift
x�t� �or x�n�� by an amount T �or K�
2
3 3 3
.10-1
0-1 10 0.5 2.5
0.5
0 1 2 30 1 2
0 2
2 2
t t t
n n n
t
x n[ ] x[n
t t+1)
x[n+1]
)(x (x ) (x
]1
Figure ��� Time shifting of signals
��
Example ��� Show that rect�t
T
� u�
t� T�
� u�
t� T�
=( )rect
0
0
0
1
1
1
22
2
T
2t
t
tT T
T
t
T
u( t + )2T
u( t
u( t + )2T u(t 2
T )
2T )
Figure ��� Example on time shifting operation
��
� CT time scaling� The operation x�t�a� is to scale x�t� by a
� It expands the function horizontally by a factor jaj
� If a � �� the function will be also time inverted
)2
0 2 0 4
t
-1
)
0
A A A
t t t
x (t /2)t (x(xFigure �� CT time scaling of signals
��
� DT time scaling� x�Kn� or x�n�K� where K is an integer
� x�Kn� � Time compression or decimation
� x�n�K� � Time expansion
[n]
0 0-1
3 4 5 6
24
.6
x
.
]
...2
65431 212
-2-2 4 6 83 421-1 0n n n
x 2n [n/2]x[Figure ��� DT time scaling of signals
��
Example ��� If x�t� � ���� rect�t
�
as shown in Figure ��� sketch
the waveform y�t� � ��x�t��
�
0-2 2
0.5
x
t
t)(Figure ��� Example of operations on signals
�
t( )x2
2t 2(x )2)=t(y
/2
)
-4 4
-2 6
0.5
0
-1
0-2 2
0
-1
t
t
t
x t(
Figure ��� Example of operations on signals
��
Continuous�Time and Discrete�Time Systems
� A system refers to any physical device �i e � communication
channels� �lters� that produces an output signal y�t� in response
to an input signal x�t�
H
H
x[
y
[y ]
x t)( t)(
n] n
Figure ��� Block diagram representation of a system
��
Properties of Systems
� Stability
� A system is said to be bounded�input bounded�output �BIBO�
stable if and only if every bounded input �i e � jx�t�j �� for
all t� or jx�n�j �� for all n� results in bounded output
� An example of a BIBO stable system
y�n� � rnx�n�u�n�� jrj � �
� An example of a BIBO unstable system
y�n� � rnx�n�u�n�� jrj � �
��
� Memory
� A system is said to possess memory if its output signal depends
on past or future values of the input signal
� An example of a system with memory
y�n� � x�n� � x�n� �� � x�n� ��
� A system is memoryless if its output signal depends only on the
present value of the input signal
� An example of a memoryless system
y�t� � x��t�
��
� Causality
� A system is causal if the present value of the output signal
depends only on the present or past values of the input signal
� An example of a causal system
y�n� �
���x�n� � x�n� �� � x�n� ���
� A system is noncausal if the present value of the output signal
depends on the future values of the input signal
� A noncausal system is not physically realizable in real time
� An example of a noncausal system
y�n� �
���x�n� �� � x�n� � x�n� ���
��
� Linearity
� A system is linear if the principle of superposition holds� i e � if
input signal is x��t� � a�x��t� � a�x��t�� then the output signal
is y��t� � a�y��t� � a�y��t� for any constants a� and a�
= =H1 2+a x a1( )t x2( )t 1a +y a( )t 2 ( )
)
ty2( )ty3x )t3( 1
11 H Hx y x y( ( ( (2 2t) t) t) tFigure ��� A linear system
��
Example ��� For the system as shown in Figure ��� determine
whether it is a linear system
H ( )y = x (t )2 t)(x t
Figure ��� A linear system example
��
In this case� the principle of superposition holds� hence it is a
linear system x )(t x ( )t2 H 2( )t2x)(ty =21 )(ty =
H )(ty =3 +a1x1( )t2 a2
1
2
y
x ( )t2= 1 +xa1 ( )t 2a x ( )t2
1x (2t)H
x )(t3
= 1 +a1 ( )t 2a ( )t2y
Figure ��� A linear system example
��
� Time Invariant
� A system is time invariant if for any delayed input x�t� T ��
the output is delayed by the same amount y�t� T �
( )x1 =
t)(y
t)(y1 = (y tt T
x
)H
H
x ( t T )
t( )Figure ��� A time invariant system
��
Example ��� For the system as shown in Figure �� with
y�t� � x�t� � c� where c is an arbitrary constant� determine
whether it is a time invariant system
H = +t t t) )(x )(y (x c
Figure ��� A time invariant system example
�
In this case� the system is time invariant
t)(1yx( t T )=t)(x1
t)(x
=
c
H +
H =t)(y )t(x +
( t T )y=x ( t T ) c
Figure �� A time invariant system example
��
� Linear Time Invariant �LTI�
� A system is linear time invariant if it satis�es both conditions
of linear and time invariance
� A LTI system can be analyzed in both time domain and
frequency domainH H1( )tx 1y ( )t 2( )ty
( )tx3 = 1 1 1) + 2a x T a( t 2
t
2)x T(t H ( )ty3 = 1 1 )1 + 2a y T a(t 2 2y T(t )
2( )xFigure ��� A LTI system
��
Example ��� Determine whether the system given by
y�t� � x��t� in Example �� is a LTI system
From Example ��� the system is linear However� the system is
not time invariant� hence it is not a LTI system
H( t T )x( t)x1 = (y T )t(2 t
y
T )x( t)1y =
( t)x H =t) x (2t)(
Figure ��� A non�LTI system example
��
Analysis of DT and CT LTI Systems
� Any LTI system can be uniquely de�ned by its impulse response
H
Hδ( h(
[]δ [ h
t) t)
n]n
Figure ��� Impulse response of a LTI system
��
� The output of any LTI system is the convolution of the input
signal and its impulse response
(t)
h[n]
x ( ) = *( )y x ( )h (
h
)ht t t t
x[n] = *[n]y x[n] [n]
Figure ��� System response of a LTI system
��
� The discrete time convolution �convolution sum� is de�ned as
y�n� � x�n� h�n� �
�Xm��
x�m�h�n�m�
� The continuous time convolution �convolution integral� is de�ned
as
y�t� � x�t� h�t� �
Z�
��
x���h�t� ��d�
��
Example ��� Sketch the waveform of y�n� � x�n� h�n� using the
graphical approach for convolution sum
0 0
1 1
]
2
1 1-1 22-1 n n
[x n] [h n
Figure ��� Example on convolution sum
��
y�n� � x�n� h�n� �
�Xm��
x�m�h�n�m�
m
m
m
n 1n 2
.
.1
-1
2
21
0 1
0
1
0
10-2 -1
2
1
-1
-1
-1
11
1
1
12
1
2
1
2
-2 0-1
-3 1
-3 -2 -1 0-1 21
0
0
m m
nmm
nmm
h[ h[n]
h[x [ ]m(i)n=-1,
(ii)x [ ]m h[ [y n]
[y n]
]
]
]n=0,
1 m
n2
��
[n]
y [n]
y [n]
x[m]
x[m]
x[m](iii)n=1,
(iv)n=2,
(v)n=3,
m
m
m
y
]
.
.
1 ..
1
1
1
0
-1
0
0 0-2 0
100
1
0 2
2
1
1
1
2
1
-1
-1
-1
-1 2 21
1
321
32
321
2
2
2
1
2 21 1
0 1 2 3 2
-1-2
-1
-1
-1 3
m m n
m m n
m m n
h[
h[
h[
1 ]
2 ]
3
Figure ��� Solution for example on convolution sum
��
Example ��� Show that ��n� �� x�n� � x�n� �� where x�n� is shown
in Figure ��
[n]x
n
21
1-1 2-2 0
Figure ��� Example on convolution with Delta function
�
Firstly� using the graphical approach Denote
y�n� � ��n� �� x�n� �
�Xm��
��m� ��x�n�m�
m
m
m
n+1n 1n 2
121
21
121
21
2 2
-1
x
0
-1-2
1
-1
-1 1 -1 1 2
.. .
...3 0
-2 -1
2
0
2
11
1 1
1 2
2
-20
0 -1 0 1 03 2-2
-1-2 032
m
nm
nmm
m
m
x [ [x
y [nx [δ [m(i)n=0,
(ii) =1,n δ [m y [n
] n ]m
2]
2] 1 ]
] ]
]
n
[��
2
2
m
m
m
2
2
22
21
2
2 2
. . .
..
. . .
1 2 3 2
0 2 3 4 -1 1
]
2 3
2 0 1 2 3 4 0-1 1
2
5 4
-1 1 -1 -2 1
11
-1 1 1
11
1
1
1
-1 1
1
-13 020 0
00 3
30 32
m n
m n
m nm
m
m
δ[m x [ y [n]
y [n]
= x[y [n] n
x [3
[4xδ[m(v)n=4,
δ[m(iv)n=3,
(iii) =2,n] 2 ]
]
]
]
]
Figure ��� Solution for example on convolution with Delta function
��
Alternatively� based on the de�nition of convolution sum� we have
y�n� � ��n� �� x�n� �
�Xm��
��m� ��x�n�m�
Since
��m� �� �
���
�� m � ��
�� m �� ��
Hence
y�n� �
�Xm��
��m� ��x�n�m�
� x�n� ����
Example �� Sketch the waveform of y�t� � x�t� x�t� using the
graphical approach for convolution integral
0-2 2
3)
t
x (t
Figure ��� Example on convolution integral
��
y�t� � x�t� x�t� �
Z�
��
x���x�t� ��d�
x( −τ)
-2 2 τ
τ
τ
τ
τ
3
3
3 3
3
0
0-2 2
0
2-2 t
-2
0
0
2 -4 0 t
y
x (−τ)
(i) t<-4,
+2-2
( )x
+2
x ( −τ)( )x
( )x
-2
τ
τ t
tt τ
)(
t t
t
��
τ
(
( −τ)
( −τ)
τ
τ
3
3
3
0
0-2
-2
2
36
36
36
-2 0-4
-4 02
2
<4,
-4 0
4
4
0 t
t
t
x
x
x
(ii) -4<
-2 +2
x( )
(iv)tx( )
-2 +2
y ( )t
y ( )t
y ( )tτt−τ)
x( )
+2-2
τ t
tτ>4,
(iii) 0<
t t
tt
tt
t<0,
t
Figure �� Solution for example on convolution integral
��
Example ��� Sketch the waveform of y�t� � x��t� x��t� using the
graphical approach for convolution integral
50
1
5 0
)A
ttA
( 1
1
x 2x( t) t
Figure ��� Example on convolution integral
��
y�t� � x��t� x��t� �
Z�
��
x����x��t� ��d�
-5
-5
-5
τ
τ
τ τ
τ
0
0
0 0
0
0
5
5
5 t
A
A
)A
A
x
x
x
A
A
t
t
(i) t
x ( −τ)
x ( −τ)
x
y
1
1
1
1
1
1
2
2
2
(τ)
(τ)
(−τ)
t
1
(τ)<0,
t
t
t
1
1 (t��
)ty
( )ty
( )ty
x 2(t τ)
x 2(t τ)A1
A1
x 2(t τ)
(
<10,
τ
τ
τ50
0
00
0
0
10
105
5
5
5
5
-5
-5
-5
1
1
1
2
2
2
1
1
5
5
5
t
t
t
A
A
A
t t
A
tt
t t
(ii) 0<
(iii) 5<
(iv) t >10,
1
1
1
x (τ)
x (τ)
x (τ)
A
A
1
1At <5,
t
Figure ��� Solution for example on convolution integral
��
Properties of Convolution
� Commutative
x��n� x��n� � x��n� x��n�
x��t� x��t� � x��t� x��t�
� Distributive
x��n� fx��n� � x��n�g � x��n� x��n� � x��n� x��n�
x��t� �x��t� � x��t�� � x��t� x��t� � x��t� x��t�
�
� Associativex��n� fx��n� x��n�g � fx��n� x��n�g x��n�
x��t� �x��t� x��t�� � �x��t� x��t�� x��t�
� Convolution with Delta function
x�n� ��n�K�� � x�n�K��
x�t� ��t� T�� � x�t� T��
��
Example ��� Show that x�t� ��t� T�� � x�t� T�� using the
de�nition of convolution integral
Based on the de�nition of convolution integral� we have
y�t� � x�t� ��t� T��
�
Z�
��
��� � T��x�t� ��d�
�
Z�
��
��� � T��x�t� T��d�
� x�t� T��Z�
��
��� � T��d�
� x�t� T����
Step Response of LTI Systems
� The step response is de�ned as the output of the system with the
unit step function as input signal
� Step response of a DT system
s�n� � u�n� h�n� �
�Xm��
h�m�u�n�m�
�
nXm��
h�m�
� Step response of a CT system
s�t� � u�t� h�t� �
Z�
��
h���u�t� ��d�
�
Z t��
h���d���
Example ��� Find the step response of the one�stage RC �lter as
shown in Figure ��� where the impulse response is given by
h�t� � �RC � exp�
� tRC
u�t� Ct
R( )u ( )ts
Figure ��� A simple one�stage RC �lter
��
In this case� the step response is given by
s�t� � u�t� h�t�
�
Z t��
h���d�
�
Z t��
�RC� exp
�
�RC
�u���d�
�
�RC
Z t�
exp
�
�RC
�d�
�
���
�� exp�
� tRC
� t � ��
�� t � ��
��
Properties of LTI Systems
� Memoryless LTI Systems
� A LTI system is memoryless if and only if its impulse
response is given by
DT system� h�n� � c��n�
CT system� h�t� � c��t�
where c is an arbitrary constant
� All memoryless LTI systems simply perform scalar
multiplication on the input��
� Causal LTI Systems
� A LTI system is causal if and only if its impulse response
satis�es the following condition
DT system� h�n� � �� for n � �
CT system� h�t� � �� for t � �
� A causal LTI system cannot generate an output before the
input is applied
��
� Stable LTI Systems
� A LTI system is BIBO stable if and only if its impulse response
satis�es the following condition
DT system�
�Xn��
jh�n�j � �
CT system�
Z�
��
jh�t�jdt � �
� An example of a stable LTI system
h�n� � nu�n�� jj � �
��
Example ��� Determine whether the system with impulse response
h�t� � exp��at�u�t� where a � � is �i� memoryless� �ii� causal� and
�iii� BIBO stable
�i� The system is not memoryless since h�t� �� c��t�
�ii� The system is causal since h�t� � � for t � �
�iii� The system is BIBO stable since
Z�
��
jh�t�jdt �
Z�
�
exp��at�dt
� ��a � �
��
System Interconnections
� Parallel Connection
+
+
h1
h2
+ h2h1
(
(x
(
(
(t) (t)
y(
t y
[n]y+
+h2[n]
h1[n]x[n]
x[n] +h1[n] h2[n] [n]y
t)
t)
t)
(t)
t)
)
x
Figure ��� Parallel connection of systems
�
� Cascade Connection
h1( h2( (y
y* h2(h1( )
(x
( t t) (
tt ) ))h2[n] [n]yh1[n]x [n]
x [n] *h1[n] h2[n] [n]y
t)
x t) t)
tFigure ��� Cascade connection of systems
�
Example ��� Determine the equivalent impulse response h�n� of the
overall system as shown in Figure ��� where h��n� � u�n��
h��n� � u�n� ��� u�n�� h��n� � ��n� ��� and h��n� � nu�n�
[ ]nx [ ]ny
[ ]nh1
h [ ]n2
[ ]n
+
h3
[ ]nh4
+
+
Figure ��� Example on interconnections of systems
�
The resultant overall system impulse response is
h�n� � fh��n� � h��n�g h��n�� h��n�
� fu�n� � u�n� ��� u�n�g ��n� ��� nu�n�
� u�n� �� ��n� ��� nu�n�
� u�n�� nu�n�
� f�� ngu�n��
Di�erential and Di�erence Equations
� Block diagram representation of di�erence equation
y�n� � x�n�� �y�n� �� � �y�n� ��
]
+
+
+3
2
D
D
x yn n[ ] [
Figure ��� Block diagram representation of di�erence equation
�
� Block diagram representation of di�erential equation
y�t� � x�t���
��
ddty�t���
��
d�dt�y�t�
)+
+
45
12
ddt
ddt
) y(x t t(
Figure ��� Block diagram representation of di�erential equation
�
Example ��� Find the block diagram representation of di�erential
equation for the simple one�stage RC low�pass �lter as shown in
Figure ��
R
)C( yt)x (t
Figure ��� A simple one�stage RC �lter
�
In this case� the input and output relation of the �lter is given by
x�t� � RC �
ddty�t� � y�t�
y�t� � x�t��RC �
ddty�t�
+ y
RC ddt
t( )x ( )t
Figure �� Block diagram representation of di�erential equation for
the RC �lter
�
Correlation Function
� The correlation function is a mathematical expression of how
correlated two signals are as a function of how much one of them
is shifted
� The correlation function between two functions is a function of
the amount of shift
� Two types of correlation functions
� Autocorrelation function
� Cross correlation function�
� Autocorrelation function
� The autocorrelation is the correlation of a function with itself
� For an energy�type signal x�n� or x�t�
DT signal� Rxx�m� �
�Xn��
x�n�x��n�m�
CT signal� Rxx��� �
Z�
��
x�t�x��t� ��dt
where x��t� denotes the complex conjugation of x�t�
� For a power�type signal x�n� or x�t�
DT signal� Rxx�m� � limK��
�
�K � �
KXn�K
x�n�x��n�m�
CT signal� Rxx��� � limT��
�T
Z T��
�T��x�t�x��t� ��dt
�
� For an energy�type signal x�n� or x�t�
DT signal� Ex � Rxx���
CT signal� Ex � Rxx���
� For a power�type signal x�n� or x�t�
DT signal� Px � Rxx���
CT signal� Px � Rxx���
� Properties of autocorrelation function
� The peak of autocorrelation function occurs at the zero shift
DT signal� Rxx��� � Rxx�m�
CT signal� Rxx��� � Rxx���
� Autocorrelation functions are even functions
DT signal� Rxx�m� � Rxx��m�
CT signal� Rxx��� � Rxx����
� A time shift in the signal does not change its autocorrelation
function� i e � the autocorrelation functions of x�t� and x�t� T �
are the same
���
Example ��� Find the autocorrelation function and power of the
sinusoidal signal x�t� � A sin���f�t�
Since x�t� is a power�type signal� the autocorrelation function is
given by
Rxx��� � limT��
�T
Z T��
�T��x�t�x��t� ��dt
� limT��
�T
Z T��
�T��A� � sin���f�t�� sin���f��t� ���dt
� limT��
�T
Z T��
�T��A�
��cos���f���� cos���f���t� ���� dt
�
A��� cos���f���
���
The power of signal x�t� is given by
Px � Rxx���
�
A��
Alternatively� based on the de�nition of power� we have
Px � limT��
�T
Z T��
�T��jx�t�j�dt
�
�T�
Z T���
�T���jx�t�j�dt
�
�T�
Z T���
�T���jA sin���f�t�j� dt
�
A��
���
Example ��� Find the autocorrelation function and power of the
sinusoidal signal y�t� � A sin ���f��t� T ��� where T is an
arbitrary constant delay
Denote � � ��f�T � we have
y�t� � A sin���f�t� ��f�T �
� A sin���f�t� ��
Since y�t� is a power�type signal� the autocorrelation function is
given by
Ryy��� � limT��
�T
Z T��
�T��y�t�y��t� ��dt
���
� limT��
�T
Z T��
�T��A� � sin���f�t� ��� sin���f��t� ��� ��dt
� limT��
�T
Z T��
�T��A�
��cos���f���� cos���f���t� ��� ���� dt
�
A��� cos���f���
Comparing with the results with Example ��� we conclude that
the autocorrelation functions of x�t� and x�t� T � are the same
���
Example ��� Find the autocorrelation function of the signal x�n�
as shown in Figure �� using the graphical approach
[n]x
n
1
2
0 1
Figure ��� Example on autocorrelation function of a DT signal
���
Since x�n� is an energy�type signal�
Rxx�m� �
�Xn��
x�n�x��n�m� �
�Xn��
x�n�x�n�m�
m 1 m+1m
.
.
1
0 1
0
-2
-1-2
0 2
0
21
-1
2
xx
2
21
12
21
21
21
642
642
-2
-2 -1 1
11 0
-1
-1
2
1 2
2-1 0
0 22 -11
1
n
n
n
n
n
n
m
m
(ii)
(i)m=2,
x[n]
x[n]
x[n] n+1
x n+2
x[
R
R=1,m
]n+m
0
[ ]x
[ ][m]
[m]
xx
���
]m
[ ]m
[ ]m
.
..
.
0
2
2
-2
-2
12
12
12
[
1
xx
6
2
2
6
64
1 2 32
0
-2
24
42
21
12
1-1
-1
-1
-1 1
1 12-1 -2 -1 21
-2 -1 0 1 2 -1 0 21
0 0 210-1
00 m
m
m
n
n
n
n
n
n
(iii)
(iv)
(v)
R
R
R
x [=-1,m
x [m=0,
x [=-2,m
x
x
x [n
n
n]
]
] n]
n
n
1][
[ ]2
xx
xx
Figure ��� Solution for example on autocorrelation function
���
� Cross correlation function
� The cross correlation is the correlation of two di�erent functions
� For energy�type signals x�n� and y�n� �or x�t� and y�t��
DT signal� Rxy�m� �
�Xn��
x�n�y��n�m�
CT signal� Rxy��� �
Z�
��
x�t�y��t� ��dt
� For power�type signals x�n� and y�n� �or x�t� and y�t��
DT signal� Rxy�m� � limK��
�
�K � �
KXn�K
x�n�y��n�m�
CT signal� Rxy��� � limT��
�T
Z T��
�T��x�t�y��t� ��dt
���
Example �� Find the cross correlation function between the two
signals x�t� � exp�j��f�t� and y�t� � exp�j���f�t�
Since x�t� and y�t� are power�type signals�
Rxy��� � limT��
�T
Z T��
�T��x�t�y��t� ��dt
� limT��
�T
Z T��
�T��exp�j��f�t�� exp��j���f��t� ���dt
� limT��
�T
Z T��
�T��exp��j��f�t�� exp��j��f���dt
� exp��j��f��� limT��
�T
Z T��
�T���cos���f�t�� j sin���f�t�� dt
� �
��
Example ��� Find the cross correlation function between the two
signals x�t� and y�t� as shown in Figure �� using the graphical
approach
2
2
20
2
-2
)
-2 -2 0t t
t ( )x ty(
Figure ��� Example on cross correlation function of CT signals
���
Since x�t� and y�t� are energy�type signals�
Rxy��� �
Z�
��
x�t�y��t� ��dt �
Z�
��
x�t�y�t� ��dt
2
-2 2
τ
τ
2
0
2
-20
2
-22
0 2-2-2
0
-8
0
(ii) 2<x
<4,
+2
2+2
τ
ττ
(τ)
(τ)
4-2
42
0
2τ τ
(i) >4,
2τ τ
τ
tt
t
t
Rxy
Rxy
x
+τ)
t+τ)
+2τ
+τ)(t) (y t
ty (
t(y
τ
τ
)t(x
)(���
)(x
t)(x
t)(x
t)(x
τ
τ
τ-2
τ-2
20-2
-2
t
0
τ
2
0-2 2
0
-8
-8
-8
-8
2
2
2
-2
-2
(iii)
(iv)
(v)
0< <2,
-2< <0,
-4< <-2,
(vi) <-4,
2
2+2
+2
τ+2
2
2τ
τ
τ
ττ
τ
τ
(τ)2
-2
0 2
0
2
2
2
4
4
4
42
0-2
-2-4 0
-2-4
(τ)
8(τ)
(τ)8
τ
t
t
t
t
Rxy
Rxy
Rxy
Rxy
( ty +τ)
( ty +τ)
( ty +τ)
( ty +τ)τ
τ
8
+2τ
τ
τ
τ
Figure ��� Solution for example on cross correlation function
���
1
NANYANG TECHNOLOGICAL UNIVERSITY SCHOOL OF ELECTRICAL & ELECTRONIC ENGINEERING
EE2010/IM2004 SIGNALS AND SYSTEMS
TUTORIAL 1
Q1.1: Determine the even and odd components of the DT function Sketch the waveforms of xe[n], xo[n], and x[n].
Q1.2: Determine whether the following signals are energy-type or power-type signals:
(a)
(b)
Q1.3: A sinusoidal signal is passed through a half-wave rectifier circuit to produce:
(a) Sketch the waveforms of x(t) and y(t), respectively. (b) Determine the energy and power levels of x(t) and y(t), respectively.
≤≤−
= otherwise. ,0
,22for ,)(
|| tetx
t
.)1(][ nny −=
.)1(][ nnnx −+=
)20sin(2)( ttx π=
>
= otherwise. ,0
,0)( if ),()(
txtxty
.1, ,2, (b) :3.1.1, (b) .0,1 (a) :2.1
.][,)1(][ :1.1:Answers
4
=∞==∞=
=∞==−=
=−=
yyxx
yyxx
on
e
PEPEQPEPeEQ
nnxnxQ
2
NANYANG TECHNOLOGICAL UNIVERSITY SCHOOL OF ELECTRICAL & ELECTRONIC ENGINEERING
EE2010/IM2004 SIGNALS AND SYSTEMS
TUTORIAL 2
Q2.1: Assume that sketch the following waveforms and evaluate
(a)
(b)
(c)
Q2.2: Sketch the waveforms of
and
Q2.3: Assuming that the signal v(t) is an energy-type signal and its energy is denoted as Ev, determine the energy levels of the following signals as a function of Ev.
(a)
(b)
)]()([)( where,2
4)( and )()( 0
1
10 TtututtvtxtynTtvtx
n−−=
+
−=−= ∑−=.20 =T
),2sin()( 0tftv π= ∫∞
∞−.)( dtty
./1 where,2
rect)()( 000
fTTttvtw =
×=
( ).)(sgn)( twtx =
∑∞
−∞=
−×=n
nTttxty ).4/()()( 0δ
).(3)( tvtx −=
).3()( −= tvty
. (b) .9 (a) :3.2
.0)( :1.2
:Answers
-
vyvx EEEEQ
dttyQ
==
=∫∞
∞
3
NANYANG TECHNOLOGICAL UNIVERSITY SCHOOL OF ELECTRICAL & ELECTRONIC ENGINEERING
EE2010/IM2004 SIGNALS AND SYSTEMS
TUTORIAL 3
Q3.1: Evaluate the convolution sum
Q3.2: For the system as shown in Figure Q3.2, evaluate the system output y(t) where
Figure Q3.2.
Q3.3: Determine the properties of the system shown in Figure Q3.3 in terms of linearity and time invariance.
Figure Q3.3.
−=
=∗= otherwise. ,0
.1,0,1for ,1][ where][][][
nnxnxnxny
.4
2rect )( and 2
3rect 2
1rect )( 21
−
=
−
−
−
=tAtxtAtAtx
invariant. not timebut linear is system The :3.3:Answers
Q
4
NANYANG TECHNOLOGICAL UNIVERSITY SCHOOL OF ELECTRICAL & ELECTRONIC ENGINEERING
EE2010/IM2004 SIGNALS AND SYSTEMS
TUTORIAL 4
Q4.1: Radio signals can travel through a wireless channel by more than one path, with different time delays and attenuations (known as channel fading). Consider a three-path case with a system impulse response given by
Assume that the input signal is given by
(a) Determine whether the system is memoryless, causal, and stable. (b) Determine the system output y(t) and sketch its waveform. (c) Express the energy of y(t) as a function of the energy of x(t).
Q4.2: Evaluate the step response of the system with impulse response given by
Q4.3: (a) Find the equivalent impulse response h[n] of the overall system as shown in Figure Q4.3, where
(b) Determine whether the overall system is memoryless, causal, and stable. (c) Determine the step response of the overall system.
Figure Q4.3.
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NANYANG TECHNOLOGICAL UNIVERSITY SCHOOL OF ELECTRICAL & ELECTRONIC ENGINEERING
EE2010/IM2004 SIGNALS AND SYSTEMS
TUTORIAL 5
Q5.1: Show that the cross-correlation function between any energy-type signal x[n] and the delta function is equal to x[-m].
Q5.2: Find the cross-correlation function between the two signals
Figure Q5.2.
Q5.3: Consider two complex-valued signals given by
(a) Sketch the amplitude plots of x(t) and y(t), respectively. (b) Determine the power levels of x(t) and y(t), respectively. (c) Find the cross-correlation function of x(t) and y(t). (d) Comment on the result obtained in part (c).
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