Continuous and Discrete
Signals and Systems
전북대학교 전자공학부
송민호 교수
전자공학부 백흥기 교수님의 강의자료를 바탕으로 구성하였음을 알립니다.
Ch. 1. Representing Signals
1.1 Introduction
1.2 Continuous-Time vs. Discrete-Time Signals
1.3 Periodic vs. Aperiodic Signals
1.4 Energy and Power Signals
1.5 Transformations of the Independent Variable
1.5.1 The Shifting Operation
1.5.2 The Reflection Operation
1.5.3 The Time-Scaling Operation
1.6 Elementary Signals
1.6.1 The Unit Step Function
1.6.2 The Ramp Function
1.6.3 The Sampling Function
1.6.4 The Unit Impulse Function
1.6.5 Derivatives of the Impulse Function
1.7 Other Types of Signals
1.8 Summary
2
Signals & Systems
Prof. M. Song
1.1 Introduction
§ Signal : detectable physical quantities or variables by which
messages or information can be transmitted.
• Examples of signals :
−
−
−
−
human voice,
television pictures,
teletype data,
and atmosphere temperature.
Electrical signals are the most easily measured and the most simply represented
type of signals. Therefore, many engineers prefer to transform physical variables
to electrical signals.
3
Signals & Systems
Prof. M. Song
§ Mathematically signals are represented as functions of one or more
independent variables.
• Examples of signals :
− AC current or voltage (time) ex.: oscilloscope
− the vibration of a rectangular membrane, image (x, y coordinates)
− electric field intensity (time and space)
In this S&S course, we focus attention on signals involving
one independent variable, time t.
4
Signals & Systems
Prof. M. Song
1.2 Continuous-Time vs. Discrete-Time Signals
§ The signal is a continuous-time signal, if the independent variable is
continuous.
x(t )
연속(시간) 신호
t
§ The signal is a discrete-time signal, if the independent variable is
discrete.
이산(시간) 신호
x[n]
n
5
Signals & Systems
Prof. M. Song
A continuous-time signal which is continuous or discontinuous
§ x(t) is continuous at t = t1 , if x(t1- ) = x(t1+ ) = x(t1 )
§ x(t) is discontinuous at t = t1 , otherwise
§ x(t) is continuous, if x(t) is continuous at all points t.
t1
t1
6
Signals & Systems
Prof. M. Song
A piecewise continuous signal
: continuous-time signal that is not continuous
§ x(t) is piecewise continuous, if the jump in amplitude at
each discontinuity is finite.
• Rectangular pulse function
æt
rect ç
èt
ö ì1, | t |< t / 2
÷=í
ø î0, | t |> t / 2
• Pulse train
x(t)
1
-4 -3 -2 -1
0
1
2
3
4
5
t
7
Signals & Systems
Prof. M. Song
A discrete-time signal
§ If the independent variable takes on only discrete values t=kTs,
the corresponding signal x(kTs) is called a discrete-time signal.
§ More details, in Ch. 6
x[n]
n
8
Signals & Systems
Prof. M. Song
1.3 Periodic vs. Aperiodic Signals
§ x(t) is a periodic signal, if
주기신호
x(t ) = x(t + nT ), n = 1, 2,3, L
where T is the fundamental period.
§ Periodic
Aperiodic (not periodic)
기본주기
§ Real-valued sinusoidal signal
x(t ) = A sin (w0t + f )
• x(t) is also periodic with period 2T, 3T, 4T,…
• The fundamental radian frequency is
w0 =
2p
T
K차 고조파
• The sinusoidal signal with wk = kw0 is called as k-th harmonic.
9
Signals & Systems
Prof. M. Song
Ex. 1.3.1
fk (t ) = exp [ jkw0t ]
§ fk (t ) is periodic with period T = 2p / w0
fk (t + T ) = exp [ jkw0 (t + T )]
= exp [ jkw0t ] exp [ jkw0T ]
= exp [ jkw0t ]
= fk (t )
é 2p ù
exp [ jkw0T ] = exp ê jk
T = exp [ jk 2p ] = 1
ë T úû
• A periodic signal is also periodic with period lT for any positive integer l.
10
Signals & Systems
Prof. M. Song
The sum of two periodic signals may or may not be periodic.
• Let x(t) and y(t) be periodic signals with periods T1 and T2
x(t ) = x ( t + kT1 ) , y (t ) = y ( t + lT2 )
z (t ) = ax(t ) + by (t ) = ax ( t + kT1 ) + by ( t + lT2 )
z (t + T ) = ax(t + T ) + by (t + T )
To be periodic,
T = kT1 = lT2
z (t ) = z (t + T )
Þ
T1 l
=
T2 k
유리수
• z(t) is periodic if T1/T2 is a rational number.
• z(t) is aperiodic if T1/T2 is a irrational number.
11
Signals & Systems
Prof. M. Song
Ex. 1.3.2
Which of the following signals are periodic?
2p
2p
2p
t Þ T=
=
=3
3
w0 2p / 3
(a)
x1 (t ) = sin
(b)
2p ö
æ 2p ö
æ 4p ö 1 æ 14p
x2 (t ) = sin ç
t ÷ cos ç
t ÷ = ç sin
- sin
÷
5
3
2
15
15
è
ø
è
ø
è
ø
2p
15
2p
T1 =
= , T2 =
= 15
14p /15 7
2p /15
T1 1
l 1
=
Þ
=
Þ l = 1, k = 7 Þ T = kT1 = lT2 = 15
T2 7
k 7
12
Signals & Systems
Prof. M. Song
Ex. 1.3.2
(c)
(d)
x3 (t ) = sin 3t Þ T =
2p
w0
=
æ 2p
x4 (t ) = x1 (t ) - 2 x3 (t ) = sin ç
è 3
T1
2p
9
T1 = 3, T2 =
Þ
=
3
T2 2p
2p
3
ö
t ÷ - 2sin 3t
ø
(irrational number) Þ aperiodic signal
13
Signals & Systems
Prof. M. Song
Ex. 1.3.3
In general, nonlinear operations on periodic signals can produce higher order
harmonics?
x(t ) = cos w1t , y (t ) = cos w2t
z (t ) = x(t ) y (t ) = cos w1t cos w2t =
1
éëcos (w1 - w2 ) t + cos (w1 + w2 ) t ùû
2
§ If w1 = w2 = w
x(t ) y (t ) = cos 2 wt =
1
[1 + cos 2wt ]
2
§ Nonlinear operations on periodic signals can produce higher order
harmonics.
14
Signals & Systems
Prof. M. Song
§ A periodic signal is a signal of infinite duration. Therefore, all
practical signals are aperiodic.
Nevertheless, the study of the system response to periodic inputs
is essential in the process of developing the system response to all
practical inputs.
15
Signals & Systems
Prof. M. Song
1.4 Energy and Power Signals
§ If x(t) is a real-valued signal,
• the energy over a time interval of length 2L :
L
E2 L = ò | x(t ) |2 dt
-L
• The total energy over t (-¥, ¥) :
L
E = lim ò | x(t ) |2 dt
L ®¥ - L
• The average power :
é 1 L
ù
P = lim ê ò | x(t ) |2 dt ú
L ®¥ 2 L - L
ë
û
16
Signals & Systems
Prof. M. Song
L
2
E = lim ò | x(t ) | dt
L ®¥ - L
§
§
§
§
§
é 1 L
ù
P = lim ê ò | x(t ) |2 dt ú
L ®¥ 2 L - L
ë
û
Signal is said to be an energy signal, if E exists and 0 < E < ¥
Signal is said to be an power signal, if P exists and
0< P<¥
Energy signals have zero power.
Power signals have infinite energy.
Periodic signals have infinite energy and finite average power, and
they are power signals.
§ In contrast, bounded finite duration signals are energy signals.
17
Signals & Systems
Prof. M. Song
Ex. 1.4.1
§ The average power of the periodic signal with period T
1 T
2
P = ò x(t ) dt
T 0
If x(t) is periodic with period T, then the integral is the same over any interval of length T.
1 T 2
P = ò sin wt dt
T 0
1 T 1 - 2cos wt
= ò
dt
0
T
2
1 T1
1 T cos wt
= ò dt + ò
dt
0
0
T 2
T
2
1
=
2
18
Signals & Systems
Prof. M. Song
Ex. 1.4.2
Determine whether the signals are energy or power signals.
energy signal
not power signal
x1(t)
x2(t)
A
A
exp[-t]
0
t
not energy signal
power signal
exp[-t]
0
t
A2
E1 = ò A exp [ -2t ] dt =
0
2
A2
é 1 L 2
ù
P1 = lim ê ò A exp [ -2t ] dt ú = lim
=0
L ®¥ 2 L 0
L ®¥ 4 L
ë
û
¥
2
0
L
1
é
ù
2
é
E2 = lim ò A dt + ò A2 exp [ -2t ] dt ù = lim A2 ê L + (1 - exp [ -2 L ]) ú = ¥
úû L ®¥ ë
0
L ®¥ ê
ë -L
2
û
2
L
1 é 0 2
A
2
P2 = lim
A dt + ò A exp [ -2t ] dt ù =
ò
úû 2
0
L ®¥ 2 L ê
ë -L
19
Signals & Systems
Prof. M. Song
Ex. 1.4.3
x(t ) = A sin (w0t + f )
§ The average power of x(t)
1 T 2 2
P = ò A sin (w0t + f ) dt
T 0
A2w0 2p / w0 é 1 1
ù
=
cos
2
w
t
+
2
f
dt
(
)
0
ò
ê
ú
0
2p
ë2 2
û
A2
=
2
20
Signals & Systems
Prof. M. Song
Ex. 1.4.4
L
2
1
L ®¥ - L
E1 = lim ò x (t )dt = ò
t /2
-t / 2
A2 dt = A2t
L
E2 = lim ò A2 exp [ -2a | t |] dt
L ®¥ - L
A2
A2
= lim
(1 - exp [ -2aL]) = a
L ®¥ a
§ x1(t) and x2(t) are the energy signals
§ Almost all time-limited signals are energy signals.
21
Signals & Systems
Prof. M. Song
1.5 Transformation of the Independent Variable
1.5.1 The Shifting Operation
전이연산
§ Signal x(t - t0 ) represents a time-shifted version of x(t).
x(t)
x(t-t0)
A
A
t0 t1
t0
t0 t2
0
x(t-t0)
t
t1
0
A
t2
t
0
t0 t1
t0
t0 t2
delay
advance
t0 < 0
t0 > 0
t
22
Signals & Systems
Prof. M. Song
Ex. 1.5.1
x(t )
Þ
x(t - 2), x(t + 3)
23
Signals & Systems
Prof. M. Song
Ex. 1.5.2
§ Vibration sensors are mounted on the front and rear axle of a moving
vehicle. Distance between two sensors : 6 ft
§ Sensor signals
§ Speed of vehicle is
d = vt
Þ
v=
d
t
=
6 ft
= 50 ft/sec
0.12 sec
24
Signals & Systems
Prof. M. Song
Ex. 1.5.3
§ Transmitted and received radar signals
§ Round-trip delay is
t=
2R
2 ´ 45
=
= 0.556 ms
C 161875
25
Signals & Systems
Prof. M. Song
1.5.2 The Reflection Operation
반전연산
§ Signal x(-t) is obtained from the signal x(t) by a reflection about t = 0.
x(t)
x(-t)
3
-1 0 1 2
3
t
-2 -1 0 1
t
• If x(t) represents a video signal, then x(-t) is the signal when the rewind switch
is pushed on.
26
Signals & Systems
Prof. M. Song
Ex. 1.5.4
§ The operations of shifting and reflecting are not commutative.
reflect
x(t )
Þ
shift
x(-t ) Þ
shift
x(t ) Þ
x(-(t - 3)) = x(-t + 3)
reflect
x(t - 3)
Þ
x(-t - 3)
27
Signals & Systems
Prof. M. Song
우대칭
§ Signal x(t) is even symmetric, if
기대칭
x(-t ) = x(t )
§ Signal x(t) is odd symmetric, if
x(-t ) = - x(t )
§ An arbitrary signal x(t) can be expressed as sum of even and odd
우대칭, 기대칭 신호
signals.
x(t ) = xe (t ) + xo (t )
1
[ x(t ) + x(-t )]
2
1
xo (t ) = [ x(t ) - x(-t ) ]
2
xe (t ) =
28
Signals & Systems
Prof. M. Song
Ex. 1.5.5
ì1, t > 0
x(t ) = í
î0, t < 0
1
xe (t ) = , t ¹ 0
2
x(0) =
1
Þ
2
ì-1/ 2, t < 0
xo (t ) = í
t >0
î1/ 2,
1
xe (0) = , xo (0) = 0
2
29
Signals & Systems
Prof. M. Song
Ex. 1.5.6
ì A exp [ -a t ] , t > 0
x(t ) = í
t<0
î0,
ì1
ïï 2 A exp [ -a t ] , t > 0
xe (t ) = í
ï 1 A exp [a t ] , t < 0
ïî 2
1
= A exp [ -a | t |]
2
ì1
ïï 2 A exp [ -a t ] , t > 0
xo (t ) = í
ï- 1 A exp [a t ] , t < 0
ïî 2
30
Signals & Systems
Prof. M. Song
1.5.3 The Time-Scaling Operation
시간척도조절 연산
§ x(at) is a compressed version of x(t) if | a |> 1
§ x(at) is an expanded version of x(t) if | a |< 1
-1
x(t)
x(3t)
x(t/2)
2
2
2
1
1
1
0
t
1
-1/3 1/3
t
-2
0
2
t
Recorded video
Played back 3 times faster
Played back @ ½ speed
31
Signals & Systems
Prof. M. Song
Ex. 1.5.7
x(3t - 6) = x [3(t - 2) ]
scaling
shifting
Scaling first, shifting next!!
32
Signals & Systems
Prof. M. Song
Ex. 1.5.8
x(t ) = 1 - A exp [ -a t ] cos (w0t + f )
§ Settling time: the time after which the signal stays within ±5%
x(t ) = 1 - A exp [ -a t ]
é 0.05 ù
ln ê
a ë A úû
( ts = 4.6052 for A = 0.5, a = 0.5)
ts = -
1
x(t ) = 1 - 2.3exp [ -10.356t ] cos5t
A = 2.3, a = 10.356, ts = 0.3697 s
x(t / 2) = 1 - 2.3exp [ -5.178t ] cos 2.5t Þ ts = 0.7394
x(2t ) = 1 - 2.3exp [ -20.712t ] cos10t Þ ts = 0.1849
33
Signals & Systems
Prof. M. Song
Order of Performing Operations
x ( a t + b ) = x (a ( t + b / a ) )
1.
2.
3.
4.
Scale by a.
If a is negative, reflect about the vertical axis.
Shift to the right by b/a if a, b have different signs.
Shift to the left by b/a if a, b have different signs.
§ Operation of reflecting and time scaling is commutative.
§ Operation of shifting and reflecting or shifting and time scaling is not
commutative.
34
Signals & Systems
Prof. M. Song
1.6 Elementary Signals
1.6.1 Unit Step Function
§ Continuous-time unit step function
단위계단 함수
ì1, t > 0
u (t ) = í
î0, t < 0
35
Signals & Systems
Prof. M. Song
Ex. 1.6.1
Rectangular pulse signal
æt
rect ç
èt
ö ì1, | t |< t / 2
÷=í
ø î0, | t |> t / 2
æ t ö
A rect ç ÷ = A [u (t + a ) - u (t - a ) ]
è 2a ø
ætö
A rect ç ÷ = A [u (t + 1) - u (t - 1) ]
è2ø
36
Signals & Systems
Prof. M. Song
Ex. 1.6.2
Signum function is one of the most often used signals in
communication and in control theory.
sgn(t)
t >0
ì1,
ï
sgn t = í0, t = 0
ï-1, t < 0
î
1
sgn t = -1 + 2u (t )
0
t
-1
37
Signals & Systems
Prof. M. Song
1.6.2 Ramp Function
램프 함수
ì t, t ³ 0
r (t ) = í
î0, t < 0
The ramp function is obtained by integrating the unit step function.
ò
t
-¥
u (t )dt = r (t )
dr (t )
= u (t )
dt
38
Signals & Systems
Prof. M. Song
Ex. 1.6.3
Expression of arbitrary signals
x(t ) = u (t + 2) - 2u (t + 1) + 2u (t ) - u (t - 2) - 2u (t - 3) + 2u (t - 4)
t
y (t ) = ò x(t )dt
-¥
\ y (t ) = r (t + 2) - 2r (t + 1) + 2r (t ) - r (t - 2) - 2r (t - 3) + 2r (t - 4)
39
Signals & Systems
Prof. M. Song
1.6.3 Sampling Function
§ Sampling function
§ Sinc function
샘플링 함수
Sa( x) =
sinc x =
sin x
x
sin p x
= Sa(p x)
px
§ Sampling function is a damped sine wave.
§ Sinc function is a compression version of Sa(x).
40
Signals & Systems
Prof. M. Song
1.6.4 Unit Impulse Function
단위 임펄스 함수
§ Unit impulse signal, Dirac delta function, delta function
ò
t2
t1
x(t )d (t )dt =x(0), t1 < 0 < t2
§ Many physical phenomena such as point sources, point charges,
concentrated loads on structures, and voltage or current sources
acting for very short time can be modeled as delta functions.
§ Properties of unit impulse function
1. d (0) ® ¥
2. d (t ) = 0, t ¹ 0
¥
d (t ) dt = 1
3.
ò
4.
d (t ) is an even function; i.e., d (t ) = d (-t )
-¥
41
Signals & Systems
Prof. M. Song
Examples of Delta Function
p1(t)
p2(t)
p3(t)
1/
1/
1/
0
t
0
d (t ) = lim pi (t )
e ®0
t
2
0
pt ö
æ 1
p3 (t ) = e ç sin ÷
e ø
è pt
2
t
2
§ All function have the following properties for "small" ε
1. The value at t=0 is very large and becomes infinity as ε approaches zero.
2. The duration is very short and becomes zero as ε becomes zero.
3. The total area under the function is constant and equal to 1.
4. The functions are all even.
42
Signals & Systems
Prof. M. Song
Ex. 1.6.4
2
2
pt ö
1 æ sin p t / e ö
1
æ 1
2æ t ö
p (t ) = lim+ e ç sin ÷ = lim+ ç
sinc ç ÷
÷ = elim
+
e ®0
e
®
0
®
0
e ø
e è pt / e ø
e
è pt
èe ø
• Property 1, 2, 4
• Property 2
ò
¥
-¥
p3 (t )dt = 1
>> syms x t;
>> e = 0.01;
>> p3 = e*(1/(pi*t)*sin(pi*t/e))^2;
>> Sp3 = int(p3,t,-inf,inf)
Sp3 =
1
43
Signals & Systems
Prof. M. Song
Important Properties – 1. Sifting Property
ò
t2
t1
ò
t2
t1
x(t )d ( t - t0 ) dt = ò
t 2 - t0
t1 -t0
x (t + t0 ) d (t )dt = x ( t0 ) , t1 < t0 < t2
ì x ( t0 ) , t1 < t0 < t2
x(t )d ( t - t0 ) dt = í
otherwise
î0,
• Signal x(t) can be expressed as a continuous sum of weighted delta
function.
¥
¥
-¥
-¥
x(t ) = ò x(t )d (t - t )dt = ò x(t )d (t - t )dt
¥
= ò x(t )d (t - t )dt
-¥
44
Signals & Systems
Prof. M. Song
Graphical Interpretation of Sifting Property
xˆ (t )
x(t)
0
x(k ) rect((t-k )/ )
k
t
¥
æ t - kD ö
x
(
k
D
)
rect
å
ç
÷
D
è
ø
k =-¥
1
é (t - k D) ù
rect ê
® d (t - t )
ú
D
ë D û
¥
x(k D) ® x(t )
é1
æ t - k D öù
= å x(k D) ê rect ç
÷ ú [ k D - (k - 1)D ]
k D - (k - 1)D ® dt
è D øû
k =-¥
ëD
xˆ (t ) =
¥
ß
¥
x(t ) = lim xˆ (t ) = ò x(t )d (t - t )dt
D® 0
¥
å ·®ò ·
k =-¥
-¥
-¥
45
Signals & Systems
Prof. M. Song
2. Sampling Property
• If x(t) is continuous at t0, then,
x(t0) rect((t-t0)/ )
x(t )d ( t - t0 ) = x ( t0 ) d ( t - t0 )
x(t)
t0
t
lim x(t )
D® 0
1
æ t - t0 ö
rect ç
÷ = x(t0 )d (t - t0 )
D
è D ø
• In the sifting property, the output is the value of the function evaluated at
some point.
• In the sampling property, the output is still a delta function with strength
equal to x(t0).
46
Signals & Systems
Prof. M. Song
3. Scaling Property
• Property 1
d (at ) =
1
d (t )
|a|
• Property 2
d (at + b) =
1 æ bö
d çt + ÷
|a| è aø
The pulse p(at) is a compressed (expanded) version of p(t) if a> 1
(a<1), and its area is 1/|a|.
47
Signals & Systems
Prof. M. Song
Ex. 1.6.5
§ Gaussian pulse
é -t 2 ù
p (t ) =
exp ê 2 ú
2
2pe
ë 2e û
1
ò
¥
-¥
p (t )dt = ò
¥
-¥
é -t 2 ù
exp ê 2 ú dt = 1
2
2pe
ë 2e û
1
lim p (t ) = d (t )
e ®0
48
Signals & Systems
Prof. M. Song
Ex. 1.6.6
(a)
1
ò ( t + t )d (t - 3)dt = 0
2
-2
(b)
4
ò ( t + t )d (t - 3)dt = ( t + t )
2
2
-2
t =3
= 3 + 32 = 12
(c)
1
1
1
exp
t
2
d
(2
t
4)
dt
=
exp
t
2
d
(
t
2)
dt
=
exp
0
=
[
]
[
]
[
]
ò0
ò0
2
2
2
(d)
ì1, t > 0
ò-¥ d (t )dt = íî0, t < 0
3
3
t
ò
t
-¥
d (t )dt = u (t ),
d
u (t ) = d (t )
dt
49
Signals & Systems
Prof. M. Song
1.6.5 Derivatives of Impulse Function
§ The derivative of the impulse function, or unit doublet, d ¢(t )
ò
d ¢(t )
t2
t1
x(t )d ¢ ( t - t0 ) dt = - x¢ ( t0 ) , t1 < t0 < t2
§ The unit doublet d '(t ) samples the derivative of the signal at time t=0.
§ High-order derivatives of d (t )
ò
t2
t1
x(t )d ( n ) ( t - t0 ) dt = (-1) n x ( n ) ( t0 ) , t1 < t0 < t2
50
Signals & Systems
Prof. M. Song
Ex. 1.6.7
i(t) [A]
§ The current through an inductor of 1-mH
10
i (t ) = 10 exp [ -2t ] u (t ) - d (t ) A
t
0
-1
§ Voltage drop across the inductor
d
v(t ) = 10
10 exp [ -2t ] u (t ) - d (t )}
{
dt
= -2 ´ 10-2 exp [ -2t ] u (t ) + 10-2 exp [ -2t ] d (t ) - 10-3 d ¢(t )
-3
v(t) [mV]
10-3
10-2
= -2 ´ 10-2 exp [ -2t ] u (t ) + 10-2 d (t ) - 10-3 d ¢(t ) V
0
t
-10-2
2x10-2
51
Signals & Systems
Prof. M. Song
Ex. 1.6.8
[Error in Textbook]
(a)
æ 1 1ö
2
¢
(
t
2)
d
ç - t + ÷ dt
ò-4
è 3 2ø
4
4
1 1ö
æ
æ 3ö
2
¢
¢
(
t
2)
d
t
+
dt
=
3(
t
2)
d
ç
÷
ç t - ÷ dt = -6(t - 2) t =3/ 2 = 3
ò-4
ò
4
è 3 2ø
è 2ø
4
2
(b)
ò
1
-4
ò
1
-4
t exp [ -2t ] d ¢¢(t - 1)dt
t exp [ -2t ] d ¢¢(t - 1)dt = ( t exp [ -2t ])¢¢
t =1
= (4t - 4) exp [ -2t ] t =1 = 0
52
Signals & Systems
Prof. M. Song