02-Fundamental Concepts of Signals and Systems-II.ppt

signal

1. Chapter One
Classifications of Signals
Addis Ababa Science and Technology University
College of Electrical and Mechanical Engineering
Department of Electromechanical Engineering
Signals and Systems Analysis ( EEEg-2121)

2. Fundamental Concepts of Signals and Systems
Outline
Introduction
Continuous-time and Discrete-time Signals
Basic Operations on Signals
Basic Continuous and Discrete-time Signals
Classification of Signals
Representation of Systems
Classification of Systems
6/17/2024 2

3. Classification of Signals
i.Deterministic and Random Signals
Deterministic signals are those signals whose values are
completely specified for any given time.
Deterministic signals can be described by some mathematical
formula.
Thus, a deterministic signal can be modeled by a known
function of time t.
Random signals are those signals that take random values at
any given time and must be characterized statistically.
36/17/2024

4. Example: Deterministic and Random Signals
6/17/2024 4

5. Classification of Signals……
ii.Even and Odd Signals
A continuous-time signal x(t)is said to be an even signal if it
satisfies the condition:
A continuous-time signal x(t)is said to be an odd signal if it
satisfies the condition:
Even signals are symmetric about the vertical axis whereas
odd signals are asymmetric.
56/17/2024ttxtx allfor , )()( ttxtx allfor , )()( 

6. Classification of Signals……
Similarly, a discrete-time signal x(n)is said to be an even
signal if it satisfies the condition:
A discrete-time signal x(n)is said to be an odd signal if it
satisfies the condition:
A general continuous-time signal x(t)can be decomposed into
its even and odd components as follows.
66/17/2024nnxnx allfor , )()( nnxnx allfor , )()( 

7. Classification of Signals……
Let the signal x(t)be expressed as the sum of two components
x
e(t)and x
o(t)as follows:
Define x
e(t)to be even and x
o(t)to be odd, i.e.,
Putting in the expression for x(t), we will get:
76/17/2024)()()( txtxtx
oe )()( and )()( txtxtxtx
ooee  tt )()(
)()()(
txtx
txtxtx
oe
oe



8. Classification of Signals……
Solving for x
e(t)and x
o(t), we obtain:
Similarly, a general discrete-time signal x(n)can be
decomposed into its even and odd components as follows.
86/17/2024 
 )()(
2
1
)(

)()(
2
1
)(
txtxtx
and
txtxtx
o
e



9. Classification of Signals……
Let the signal x(n)be expressed as the sum of two components
x
e(n)and x
o(n)as follows:
Define x
e(n)to be even and x
o(n)to be odd, i.e.,
Putting in the expression for x(n), we will get:
96/17/2024)()()( nxnxnx
oe )()( and )()( nxnxnxnx
ooee  nn )()(
)()()(
nxnx
nxnxnx
oe
oe



10. Classification of Signals……
Solving for x
e(n)and x
o(n), we obtain:
106/17/2024 
 )()(
2
1
)(

)()(
2
1
)(
nxnxnx
and
nxnxnx
o
e



11. 6/17/2024 11
Examples of even signals (a and b) and odd signals (c and d)

12. Classification of Signals……
Exercise:
Determine whether the following signals are even, oddor neither.
If the signals are neither even nor odd, evaluate the even and
odd components.
126/17/2024 )cos()sin()sin()cos()( .
)25.0sin()( . )sin()cos(1 )( .
4
3
cos)( .
, 0
10 ,
)( .
2
2
tttttxc
nnnxftttttxb
n
nnxe
otherwise
tt
txa













 




13. Classification of Signals……
iii.Periodic and Non-periodic Signals
A continuous-time signal x(t)is said to be periodicwith
period Tif there is a positive non-zero value of Tfor which:
If x(t)is periodic with period T, then it is also periodic for all
integer multiples of T. That is:
The smallest positive value Tis known as the fundamental
period.
136/17/2024tTtxtx allfor , )()(  integeran is , )()( kkTtxtx 

14. Classification of Signals……
Any continuous-time signal which is not periodic is called a
non-periodicor aperiodicsignal.
All continuous-time sinusoidal signals are periodic.
Consider a continuous-time sinusoidal signal x(t)given by:
This sinusoidal signal is periodic for all values of tand its
fundamental period is given by:
146/17/2024)sin()(
0 tAtx 0
2


T

15. Classification of Signals……
All continuous-time complex exponential signals are also
periodic.
Consider a continuous-time complex exponential signal x(t)
given by:
This complex exponential signal is periodic for all values of t
and its fundamental period is given by:
156/17/2024)(
0
)(


tj
etx 0
2


T

16. Classification of Signals……
Exercise:
Determine whether the following continuous-time signals are
periodicor non-periodic. If the signals are periodic, find their
fundamental period.
166/17/2024)8()72(
2
)( . )( .
)( . )10tan()( .
52)( . )4sin()( .
tjtj
t
etxfetxc
etxettxb
ttxdttxa








17. Classification of Signals……
A discrete-time signal x(n)is said to be periodic with period N
if there is a positive integer value of Nfor which:
If x(n)is periodic with period N, then it is also periodic for all
integer multiples of N. That is:
The smallest positive value Nis known as the fundamental
period.
176/17/2024nNnxnx allfor , )()(  integeran is , )()( kkNnxnx 

18. Classification of Signals……
Any discrete-time signal which is not periodic is called a non-
periodicor aperiodicdiscrete-time signal.
All discrete-time sinusoidal signals are not periodic.
Consider a discrete-time sinusoidal signal x(n)given by:
This sinusoidal signal is periodic if and only if:
186/17/2024)sin()(
0 nAnx numberrational
2
0




19. Classification of Signals……
The term rational number is defined as a fraction of two
integers.
Given that the above discrete-time sinusoidal signal is periodic,
its fundamental period is evaluated from the relationship:
The fundamental period is calculated from the above equation
with mset to the smallest integer that results in an integer value
for N.
196/17/2024mN
0
2




20. Classification of Signals……
All discrete-time complex exponential signals are not also
periodic.
Consider a discrete-time complex exponential signal x(n)
given by:
This complex exponential signal is periodic iff is a
rational number and its fundamental period is given by:
206/17/2024)(
0
)(


nj
enx 0
2

 mN
0
2




21. Classification of Signals……
Exercise:
Determine whether the following discrete-time signals are
periodicor non-periodic. If the signals are periodic, find their
fundamental period.
216/17/2024)3(
28
7
)( . )5.0cos()( .
)(2)( .
10
3
cos)( .
)( .
412
sin)( .
nj
n
n
j
etxfnnxc
nunxe
n
nxb
enxd
n
nxa


























22. Classification of Signals……
The sum of two periodic continuous-time signals may not be
always periodic.
Consider a continuous-time signal x(t)given by:
If x
1(t) is periodic with fundamental period T
1and x
2(t)is
periodic with fundamental period T
2, then x(t)is periodic if
and only if:
226/17/2024)()()(
21 txtxtx  numberrational
n
m
T
T

2
1


23. Classification of Signals……
The fundamental period of x(t)given by:
In the above equation, the values of mand nmust be chosen
such that the greatest common divisor (gcd) between mand n
is 1.
236/17/202421
2
1


mTnTT
n
m
T
T



24. Classification of Signals……
The sum of two periodic discrete-time signals is always
periodic.
Consider a discrete-time signal x(n)given by:
If x
1(n) is periodic with fundamental period N
1and x
2(n)is
periodic with fundamental period N
2, then x(n)is always
periodic with fundamental period:
246/17/2024)()()(
21 nxnxnx  21
2
1
mNnNN
n
m
N
N


25. Classification of Signals……
Exercise:
Determine whether the following signals are periodicor non-
periodic. If the signals are periodic, find their fundamental
period.
256/17/2024 
 
 


18
cos
12
cos)( .
10cos )( .
10cos4sin)( .
3cos4sin)( .
42

















nn
nxd
tetxc
tttxb
tttxa
tj




26. Classification of Signals……
iv.Energy and Power Signals
The normalized energy Eof a continuous-time signal x(t)is
defined as:
The normalized average power Pof x(t)is defined as:
266/17/2024dttxE
2
)(


 dttx
T
P
T
TT
2
2/
2/
)(
1
lim



27. Classification of Signals……
Similarly, for a discrete-time signal x(n), the normalized
energy Eis defined as:
The normalized average power Pof x(n)is defined as:
276/17/20242
)(



n
nxE 2
)(
12
1
lim 

 

N
Nn
N
nx
N
P

28. Classification of Signals……
A signal is said to be an energy signal if the normalized total
energy Ehas a non-zero finite value, i.e., 0 < E < ∞and so P=0
On the other hand, a signal is said to be a power signal if it has
non-zero finite normalized average power, i.e., 0 < P < ∞, thus
implying that E = ∞.
A signal cannot be both an energy and a power signal
simultaneously.
The energy signals have zero average power whereas the power
signals have infinite total energy.
Some signals, however, can be classified as neither power
signals nor as energy signals.
286/17/2024

29. Classification of Signals……
Exercise:
Categorize each of the following signals as an energy orpower
signal or neither.
296/17/2024

, 0
11 , cos5
)( .
, 0
44 ,
2
sin
)( .
, 0
21 , 2
10 ,
)( .




 

























otherwise
tt
txb
otherwise
n
n
nxc
otherwise
tt
tt
txa



30. Representation of Systems
A system is a mathematical model of a physical process that
relates the input signal to the output signal.
In other words, a system is a mathematical operator or mapping
that transforms an input signal into an output signal by means of
a fixed set of rules or operations.
The notation T[ .] is used to represent a general system in which
an input signal is transformed into an output signal.
Mathematically, the input and output signals can be related as:
306/17/2024)]([)(or )]([)( nxTnytxTty 

31. Representation of Systems……
The relationship between the input and output may be expressed
in terms of a concise mathematical rule or function.
It is also possible to describe a system in terms of an algorithm
that provides a sequence of instructions or operations that is to
be applied to the input signal.
316/17/2024
Fig. Representation of continuous-time and discrete-time systems

32. Classification of Systems
In the analysis or design of a system, it is desirable to classify
the system according to some generic properties that the
system satisfies.
For a system to possess a given property, the property must
hold true for all possible input signals that can be applied to
the system.
If a property holds for some input signals but not for others,
the system does not satisfy that property.
326/17/2024

33. Classification of Systems……
We can classify systems into the following six basic categories.
i.Linear Vs non-linear systems
ii.Time-invariant Vs time-varying systems
iii.Memoryless Vs memory systems
iv.Causal Vs non-causal systems
v.Stable Vs unstable systems
vi.Invertible Vs non-invertible systems
336/17/2024

34. Classification of Systems……
i.Linear Vs non-linear systems
A system is linear if it satisfies the principle of superposition.
A continuous-time system with input x(t)and output y(t)is said
to be linear iff:
Similarly, a discrete-time system with input x(n)and output
y(n)is said to be linear iff:
346/17/2024)]([)]([)]()([
22112211 txTatxTatxatxaT  )]([)]([)]()([
22112211 nxTanxTanxanxaT 

35. Classification of Systems……
Exercise:
Determine whether the systems with the following input-output
relationships are linearor non-linear.
356/17/20245)(3)( .
)(
)( .
)](sin[)( . )( .
)2()()( . )(2)( .
)(




txtyd
dt
tdx
tyc
nxnyfetyb
nxnxnyetxtya
tx

36. Classification of Systems……
ii.Time-invariant Vs time-varying systems
A system is said to be time-invariantif a time delay or time
advancein the input signal leads to an identical time-shift in
the output signal.
A continuous-time system with an input x(t)and output y(t)is
said to be time-invariantiff:
Similarly, a discrete-time system with an input x(n)and output
y(n)is said to be time-invariant iff:
366/17/2024)]([)(
00 ttxTtty  )]([)(
00 nnxTnny 

37. Classification of Systems……
Exercise:
Determine whether the systems with the following input-output
relationships are time-invariantor time-varying.
376/17/2024)()( .
)()( .
)()( . )](sin[)( .
)2()()( . )](sin[)( .
2
nxnyd
txtyc
nnxnyftxttyb
nxnxnyetxtya





38. Classification of Systems……
iii.Memoryless Vs memory systems
A continuous-time system is said to be memorylessor
instantaneousif its output y(t)at time t = t
0depends only on the
values of the input x(t) at the same time t = t
0.
On the other hand, if the response of a system at t = t
0depends
on the values of the input x(t)in the past or in the future time, it
is called a dynamic system or a system with memory.
Similarly, a discrete-time system is said to be memoryless if its
output y(n)at time instant n = n
0depends only on the value of
its input x(n)at the same time instant n = n
0. Otherwise, the
discrete-time system is said to have memory.
386/17/2024

39. Classification of Systems……
Exercise:
Determine whether the systems with the following input-output
relationships are memorylessor memory.
396/17/20245)(3)( .
)( . )()( .
2
)( . )5()( .
)2()( . )2()( .
)(2










txtyd
enygtxtyc
n
xnyftxtyb
nxnyetxtya
nx

40. Classification of Systems……
iv.Causal Vs non-causal systems
A continuous-time system is said to be causal if the output at
time depends only on the input x(t)for .
Similarly, a discrete-time system is causal if the output at time
instant depends only on the input x(n)for .
That is, the output of a causal system at the present time
depends on only the present and/or past values of the input but
not on its future values.
A system that violates the causality condition is called a non-
causal or anti-causal system.
406/17/20240tt 0tt 0nn 0nn

41. Classification of Systems……
Exercise:
Determine whether the systems with the following input-output
relationships are causalor non-causal.
416/17/2024)2()( .
)( . )2()( .
2
)( . )5()2()( .
4)3()( . )2()2()( .
)2(2
txtyd
enygtxtyc
n
xnyftxtxtyb
nxnyetxtxtya
nx












42. Classification of Systems……
v.Stable Vs unstable systems
A system is referred to as bounded-input, bounded-output
(BIBO)stable if an arbitrary bounded-input signal always
produces a bounded-output signal.
A continuous-time system with input x(t)and output y(t)is
said to be BIBO stable iff:
Similarly, a discrete-time system with input x(n)and output
y(n)is said to be BIBO stable iff:
426/17/2024 )( )( 
yx
BtyBtx 
yx
BnyBnx )( )(

43. Classification of Systems……
Exercise:
Determine whether the systems with the following input-output
relationships are BIBO stableor unstable.
436/17/2024







n
k
n
nk
nx
kxnyftxtyc
kxnyetxtyb
enydtxtya
)()( . )()( .
)()( . )5()( .
)( . 5)(2)( .
2
2
2
)(

44. Classification of Systems……
vi.Invertible Vs non-invertible systems
A continuous-time system is said to be invertible if the input
signal x(t)can be uniquely determined from the output y(t)for
all time t ∈(−∞,∞).
Similarly, a discrete-time system is said to be invertible if the
input signal x(n)can be uniquely determined from the output
y(n)for all time n ∈(−∞,∞).
To be invertible, two different inputs cannot produce the same
output since, in such cases, the input signal cannot be uniquely
determined from the output signal.
446/17/2024

45. Classification of Systems……
Exercise:
Determine whether the systems with the following input-output
relationships are invertibleor non-invertible.
456/17/2024)2()()( .
)( . )()( .
)2()( . )](cos[)( .
7)(2)( . 5 )(3)( .
)(2




txtxtyd
enygtxtyc
nxnyftxtyb
nxnyetxtya
nx

46. Exercise
1.Determine whether the following signals are even, oddor
neither. If the signals are neither even nor odd, evaluate the
even and odd components.
466/17/2024


n
n
nxf
t
ttxc
nn
nxe
n
n
nxb
ttutxdtttxa
n



2sin
)( .
2
sin)( .

otherwise , 0
40 ,
)( .
0 , 0
0 , 1
)( .
)()( . 2cos)( .
2











 











47. Exercise……
2.Determine whether the following signals are periodicor non-
periodic. Calculate the fundamental period for the periodic
signals.
476/17/2024
n
n
j
n
jtj
nxf
tt
txc
n
nxe
tt
txb
eenxdetxa
1 )( .
64
63
cos
8
3
sin )( .
5
2
cos)( .
5
3
cos2
7
6
sin )( .
)( . )( .
4
3
4
7
4
5
























































48. Exercise……
3.Determine if the following signals are energyor powersignals
or neither. Calculate the energy and power of the signals in
each case.
486/17/2024




















 










8
3
sin
4
cos)( . )( )( .
1)( .
otherwise , 0
33 , 3cos
)( .
)( . sincos)( .
2
82
nn
nxftuetxc
nxe
nt
txb
enxdtttxa
t
n
n
j





49. Exercise……
4.Determine whether the systems described by the following
input-output relationships are:
i.Memoryless or memory
ii.Linear or non-linear
iii.Time-invariant or time-varying
iv.Causal or non-causal
v.BIBO Stable or unstable
vi.Invertible or non-invertible
496/17/2024

50. Exercise……
506/17/2024 )4(
2)( .
)()( . 2
5)2()( . )23(5
)52()( . )2()(
)1()1( cos
1
)(
2
2










ttxg. y(t)
nymx(t)f. y(t)
kxnylt)x( e. y(t)
nxnyktxd. y(t)
nxny j txtc. y
nxnxi. y(n) [x(t)] b. y(t)
)x(nx(n)h. y(n) t) x(a. y(t)
nx
n
nk