Introduction
CHAPTER
1.1 What is a signal?
A signal is formally defined as a function of one or more variables that
conveys information on the nature of a physical phenomenon.
1.2 What is a system?
A system is formally defined as an entity that manipulates one or more
signals to accomplish a function, thereby yielding new signals.
Figure 1.1 (p. 2)
Block diagram representation of a system.
1.3 Overview of Specific Systems
★ 1.3.1 Communication systems
Elements of a communication system
Fig. 1.2
1. Analog communication system: modulator + channel + demodulator
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Introduction
Figure 1.2 (p. 3)
Elements of a communication system. The transmitter changes the message
signal into a form suitable for transmission over the channel. The receiver
processes the channel output (i.e., the received signal) to produce an estimate
of the message signal.
◆ Modulation:
2. Digital communication system:
sampling + quantization + coding  transmitter  channel  receiver
◆ Two basic modes of communication:
1. Broadcasting
Radio, television
2. Point-to-point communication
Fig. 1.3
Telephone, deep-space
communication
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Introduction
Figure 1.3 (p. 5)
(a) Snapshot of Pathfinder
exploring the surface of Mars.
(b) The 70-meter (230-foot)
diameter antenna located at
Canberra, Australia. The
surface of the 70-meter
reflector must remain accurate
within a fraction of the signal’s
wavelength. (Courtesy of Jet
Propulsion Laboratory.)
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Introduction
★ 1.3.2 Control systems
Figure 1.4 (p. 7)
Block diagram of a feedback control system. The controller drives the plant,
whose disturbed output drives the sensor(s). The resulting feedback signal
is subtracted from the reference input to produce an error signal e(t), which,
in turn, drives the controller. The feedback loop is thereby closed.
◆ Reasons for using control system: 1. Response, 2. Robustness
◆ Closed-loop control system: Fig. 1.4.
1. Single-input, single-output (SISO) system
2. Multiple-input, multiple-output (MIMO) system
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Controller: digital
computer
(Fig. 1.5.)
4
CHAPTER
Introduction
Figure 1.5 (p. 8)
NASA space shuttle launch.
(Courtesy of NASA.)
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CHAPTER
Introduction
★ 1.3.3
Microelectromechanical
Systems (MEMS)
Structure of lateral capacitive
accelerometers: Fig. 1-6 (a).
Figure 1.6a (p. 8)
Structure of lateral
capacitive accelerometers.
(Taken from Yazdi et al.,
Proc. IEEE, 1998)
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Introduction
SEM view of
Analog Device’s
ADXLO5 surfacemicromachined
polysilicon
accelerometer:
Fig. 1-6 (b).
Figure 1.6b (p. 9)
SEM view of Analog
Device’s ADXLO5
surfacemicromachined
polysilicon
accelerometer.
(Taken from Yazdi et
al., Proc. IEEE, 1998)
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Introduction
★ 1.3.4 Remote Sensing
Remote sensing is defined as the process of acquiring information about an
object of interest without being in physical contact with it
1. Acquisition of information = detecting and measuring the changes that the
object imposes on the field surrounding it.
2. Types of remote sensor:
Radar sensor
Infrared sensor
Visible and near-infrared sensor
X-ray sensor
※ Synthetic-aperture radar (SAR)
Satisfactory operation
High resolution
See Fig. 1.7
Ex. A stereo pair of SAR acquired from earth orbit with Shuttle Imaging Radar
(SIR-B)
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Introduction
Figure 1.7 (p. 11)
Perspectival view of
Mount Shasta
(California), derived
from a pair of stereo
radar images acquired
from orbit with the
shuttle Imaging Radar
(SIR-B). (Courtesy of
Jet Propulsion
Laboratory.)
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Introduction
★ 1.3.5 Biomedical Signal Processing
Morphological types of nerve cells: Fig. 1-8.
Figure 1.8 (p. 12)
Morphological types of nerve cells (neurons) identifiable in monkey cerebral
cortex, based on studies of primary somatic sensory and motor cortices.
(Reproduced from E. R. Kande, J. H. Schwartz, and T. M. Jessel, Principles of
Neural Science, 3d ed., 1991; courtesy of Appleton and Lange.)
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Introduction
◆ Important examples of biological signal:
1. Electrocardiogram (ECG)
2. Electroencephalogram (EEG)
Fig. 1-9
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Figure 1.9
(p. 13)
The traces shown
in (a), (b), and (c)
are three
examples of EEG
signals recorded
from the
hippocampus of a
rat.
Neurobiological
studies suggest
that the
hippocampus
plays a key role in
certain aspects of
learning and
memory.
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Introduction
★ Measurement artifacts:
1. Instrumental artifacts
2. Biological artifacts
3. Analysis artifacts
★ 1.3.6 Auditory System
Figure 1.10 (p. 14)
(a) In this diagram, the basilar
membrane in the cochlea is depicted
as if it were uncoiled and stretched
out flat; the “base” and “apex” refer
to the cochlea, but the remarks “stiff
region” and “flexible region” refer to
the basilar membrane. (b) This
diagram illustrates the traveling
waves along the basilar membrane,
showing their envelopes induced by
incoming sound at three different
frequencies.
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Introduction
★ The ear has three main parts:
1. Outer ear: collection of sound
2. Middle ear: acoustic impedance match between the air and cochlear fluid
Conveying the variations of the tympanic membrane (eardrum)
3. Inner ear: mechanical variations → electrochemical or neural signal
★ Basilar membrane: Traveling wave
Fig. 1-10.
★ 1.3.7 Analog Versus Digital Signal Processing
Digital approach has two advantages over analog approach:
1. Flexibility
2. Repeatability
1.4 Classification of Signals
1. Continuous-time and discrete-time signals
Continuous-time signals: x(t)
Parentheses (‧)
Fig. 1-11.
Discrete-time signals: x n  x(nTs ), n  0,  1,  2, .......
Fig. 1-12.
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(1.1) where t = nTs
Brackets [‧]
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Introduction
Figure 1.11 (p. 17)
Continuous-time signal.
Figure 1.12 (p. 17)
(a) Continuous-time signal x(t). (b) Representation of x(t) as a
discrete-time signal x[n].
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Introduction
CHAPTER
Symmetric about vertical axis
2. Even and odd signals
Even signals: x( t )  x(t )
for all t
(1.2)
Odd signals: x( t )   x(t )
for all t
(1.3)
Example 1.1
Antisymmetric about origin
Consider the signal
 t 
sin   ,  T  t  T
x (t )    T 
 0 , otherwise

Is the signal x(t) an even or an odd function of time?
<Sol.>
  t 
sin    ,  T  t  T
x(t )    T 
 0
, otherwise


t 

sin

 ,  T  t  T
=
T 
 0
, otherwise

=  x (t )
odd function
for all t
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x(t )  e 2t cos t
Introduction
CHAPTER
◆ Even-odd decomposition of x(t):
x ( t )  xe ( t )  xo ( t )
where
xe (t )  xe (t )
Example 1.2
Find the even and odd components
of the signal
x(t )  e 2t cos t
xo (  t )   x o ( t )
x(t )  xe (t )  xo (t )
<Sol.>
x(t )  e2t cos(t )
 xe (t )  xo (t )
1
 x(t )  x(t )
2
1
xo   x ( t )  x (  t ) 
2
xe 
(1.4)
(1.5)
=e2t cos(t )
Even component:
1
xe (t )  ( e2t cos t  e 2t cos t )
2
 cosh(2t ) cos t
Odd component:
1
xo (t )  (e2t cos t  e2t cos t )  sinh(2t ) cos t
2
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Introduction
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◆ Conjugate symmetric:
A complex-valued signal x(t) is said to be conjugate symmetric if
x(t )  x (t )
Let
(1.6)
x(t )  a(t )  jb(t )
a( t )  a(t )
b( t )  b(t )
x* (t )  a (t )  jb(t )
a(t )  jb(t )  a(t )  jb(t )
Refer to
Fig. 1-13
Problem 1-2
3. Periodic and nonperiodic signals (Continuous-Time Case)
Periodic signals: x(t )  x(t  T ) for all t
(1.7)
T  T0 , 2T0 , 3T0 , ...... and T  T0  Fundamental period
Fundamental frequency:
1
f 
(1.8)
T
Angular frequency:
2
  2 f 
(1.9)
T
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Figure 1.13
(p. 20)
(a) One example
of continuoustime signal.
(b) Another
example of a
continuous-time
signal.
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CHAPTER
Introduction
◆ Example of periodic and nonperiodic signals: Fig. 1-14.
Figure 1.14 (p. 21)
(a) Square wave with amplitude A = 1 and period T = 0.2s.
(b) Rectangular pulse of amplitude A and duration T1.
◆ Periodic and nonperiodic signals (Discrete-Time Case)
x  n   x  n  N  for integer n (1.10)
Fundamental frequency of x[n]:
2

(1.11)
N
N = positive integer
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CHAPTER
Introduction
Figure 1.15 (p. 21)
Triangular wave alternative between –1 and +1 for Problem 1.3.
◆ Example of periodic and nonperiodic signals:
Fig. 1-16 and Fig. 1-17.
Figure 1.16 (p. 22)
Discrete-time square
wave alternative
between –1 and +1.
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Introduction
CHAPTER
Figure 1.17 (p. 22)
Aperiodic discrete-time signal
consisting of three nonzero samples.
4. Deterministic signals and random signals
A deterministic signal is a signal about which there is no uncertainty with
respect to its value at any time.
Figure 1.13 ~ Figure 1.17
A random signal is a signal about which there is uncertainty before it occurs.
Figure 1.9
5. Energy signals and power signals
Instantaneous power:
v 2 (t )
p(t ) 
R
(1.12)
p(t )  Ri (t )
(1.13)
2
If R = 1  and x(t) represents a current or a voltage,
then the instantaneous power is
p(t )  x 2 (t )
(1.14)
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Introduction
CHAPTER
The total energy of the continuous-time signal x(t) is
T
2
T
T 
2

E  lim  x (t )dt   x 2 (t )dt
2

(1.15)
Time-averaged, or average, power is
1 T2 2
P  lim T x (t )dt
T  T
2
(1.16)
For periodic signal, the time-averaged power is
1 T2 2
P  T x (t )dt
T 2
◆ Discrete-time case:
Total energy of x[n]:
E


x 2 [n]
n 
Average power of x[n]:
1
P  lim
n  2 N
N
 x [n]
2
(1.19)
n  N
1
P
N
(1.17)
(1.18)
N 1
 x [n]
2
n 0
★ Energy signal:
If and only if the total energy of the signal satisfies the condition
(1.20)
0 E 
★ Power signal:
If and only if the average power of the signal satisfies the condition
0 P
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Introduction
CHAPTER
1.5 Basic Operations on Signals
★ 1.5.1 Operations Performed on dependent Variables
y (t )  cx(t )
Amplitude scaling: x(t)
(1.21)
y[n]  cx[n]
Discrete-time case: x[n]
Addition:
y(t )  x1 (t )  x2 (t )
c = scaling factor
Performed by amplifier
(1.22)
Discrete-time case: y[n]  x1[n]  x2 [n]
Multiplication:
Ex. AM modulation
y(t )  x1 (t ) x2 (t ) (1.23)
y[n]  x1[n]x2 [n]
Differentiation:
y (t ) 
d
x(t )
dt
(1.24)
Integration:
t
y(t )   x( )d

Figure 1.18 (p. 26)
Inductor with current
d
v
(
t
)

L
i
(
t
)
Inductor:
(1.25) i(t), inducing voltage
dt
v(t) across its
terminals.
(1.26)
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CHAPTER
Introduction
1 t
i ( )d (1.27)


C
★ 1.5.2 Operations Performed on
independent Variables
Time scaling:
a >1  compressed
y (t )  x(at )
0 < a < 1  expanded
Capacitor: v(t ) 
Figure 1.19 (p. 27)
Capacitor with
voltage v(t) across
its terminals,
inducing current i(t).
Fig. 1-20.
Figure 1.20 (p. 27)
Time-scaling operation; (a) continuous-time signal x(t), (b) version of x(t) compressed
by a factor of 2, and (c) version of x(t) expanded by a factor of 2.
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Discrete-time case:
y[n]  x[kn],
k  0 k = integer
Some values lost!
Figure 1.21 (p. 28)
Effect of time scaling on a discrete-time signal: (a) discrete-time signal x[n] and (b)
version of x[n] compressed by a factor of 2, with some values of the original x[n] lost
as a result of the compression.
Reflection:
y(t )  x(t )
The signal y(t) represents a reflected version of x(t) about t = 0.
Ex. 1-3
Consider the triangular pulse x(t) shown in Fig. 1-22(a). Find the reflected
version of x(t) about the amplitude axis (i.e., the origin).
<Sol.> Fig.1-22(b).
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Introduction
CHAPTER
Figure 1.22 (p. 28)
Operation of reflection: (a) continuous-time signal x(t) and (b) reflected version of x(t)
about the origin.
x(t )  0
for
y (t )  0
for t  T1 and t  T2
Time shifting:
t  T1 and t  T2
t0 > 0  shift toward right
t0 < 0  shift toward left
y (t )  x(t  t0 )
Ex. 1-4 Time Shifting: Fig. 1-23.
Figure 1.23 (p. 29)
Time-shifting operation: (a) continuoustime signal in the form of a rectangular
pulse of amplitude 1.0 and duration 1.0,
symmetric about the origin; and (b) timeshifted version of x(t) by 2 time shifts.
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Introduction
CHAPTER
Discrete-time case: y[n]  x[n  m]
where m is a positive or negative integer
★ 1.5.3 Precedence Rule for Time Shifting and Time Scaling
1. Combination of time shifting and time scaling:
y (t )  x(at  b)
(1.28)
y (0)  x(b) (1.29)
b
y ( )  x(0)
a
(1.30)
2. Operation order: To achieve Eq. (1.28),
1st step: time shifting v(t )  x(t  b)
2nd step: time scaling y(t )  v(at )  x(at  b)
Ex. 1-5 Precedence Rule for Continuous-Time Signal
Consider the rectangular pulse x(t) depicted in Fig. 1-24(a). Find y(t)=x(2t + 3).
<Sol.> Case 1: Fig. 1-24.  Shifting first, then scaling
Case 2: Fig. 1-25.  Scaling first, then shifting
y(t )  v(t  3)  x(2(t  3))  x(2t  3)
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CHAPTER
Introduction
Figure 1.24 (p. 31)
The proper order in which the operations of time scaling and time shifting
should be applied in the case of the continuous-time signal of Example 1.5.
(a) Rectangular pulse x(t) of amplitude 1.0 and duration 2.0, symmetric
about the origin. (b) Intermediate pulse v(t), representing a time-shifted
version of x(t). (c) Desired signal y(t), resulting from the compression of v(t)
by a factor of 2.
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Introduction
Figure 1.25 (p. 31)
The incorrect way of applying the precedence rule. (a) Signal x(t).
(b) Time-scaled signal v(t) = x(2t). (c) Signal y(t) obtained by shifting
v(t) = x(2t) by 3 time units, which yields y(t) = x(2(t + 3)).
Ex. 1-6 Precedence Rule for Discrete-Time Signal
A discrete-time signal is defined by
 1, n  1,2

x[n]   1, n  1, 2
 0, n  0 and | n | 2

Find y[n] = x[2x + 3].
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Introduction
<Sol.> See Fig. 1-27.
Figure 1.27 (p. 33)
The proper order of applying the operations of time scaling and time shifting for the
case of a discrete-time signal. (a) Discrete-time signal x[n], antisymmetric about the
origin. (b) Intermediate signal v(n) obtained by shifting x[n] to the left by 3 samples.
(c) Discrete-time signal y[n] resulting from the compression of v[n] by a factor of 2,
as a result of which two samples of the original x[n], located at n = –2, +2, are lost.
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Introduction
1.6 Elementary Signals
★ 1.6.1 Exponential Signals x(t )  Beat
B and a are real parameters
(1.31)
1. Decaying exponential, for which a < 0
2. Growing exponential, for which a > 0
Figure 1.28 (p. 34)
(a) Decaying exponential form of continuous-time signal. (b) Growing exponential
form of continuous-time signal.
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Introduction
CHAPTER
Ex. Lossy capacitor: Fig. 1-29.
KVL Eq.:
RC
d
v(t )  v(t )  0
dt
v (t )  V0 e
 t /( RC )
(1.32)
RC = Time constant
(1.33)
Figure 1.29 (p. 35)
Lossy capacitor, with the
loss represented by shunt
resistance R.
Discrete-time case:
x[n]  Br n
(1.34)
where
r  e
Fig. 1.30
★ 1.6.2 Sinusoidal Signals
◆ Continuous-time case:
Fig. 1-31
x(t )  A cos(t   ) (1.35)
where
T
2

periodicity
x(t  T )  A cos( (t  T )   )
 A cos(t  T   )
 A cos(t  2   )
 A cos(t   )
 x (t )
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Introduction
Figure 1.30 (p. 35)
(a) Decaying exponential form of discrete-time signal. (b) Growing
exponential form of discrete-time signal.
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Introduction
Figure 1.31 (p. 36)
(a) Sinusoidal signal A cos( t + Φ) with phase Φ = +/6 radians.
(b) Sinusoidal signal A sin ( t + Φ) with phase Φ = +/6 radians.
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Introduction
CHAPTER
Ex. Generation of a sinusoidal signal  Fig. 1-32.
Circuit Eq.:
d2
LC 2 v(t )  v(t )  0
(1.36)
dt
v(t )  V0 cos(0t ), t  0
(1.37)
where
0 
1
LC
(1.38)
Natural angular frequency
of oscillation of the circuit
◆ Discrete-time case :
x[n]  A cos(n   )
N  2 m
(1.39)
x[n  N ]  A cos(n  N   )
Periodic condition:
or
Figure 1.32 (p. 37)
Parallel LC circuit,
assuming that the
inductor L and capacitor
C are both ideal.

(1.40)
2 m
radians/cycle, integer m, N
N
Ex. A discrete-time sinusoidal signal: A = 1,  = 0, and N = 12.
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(1.41)
Fig. 1-33.
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CHAPTER
Introduction
Figure 1.33 (p. 38)
Discrete-time sinusoidal signal.
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Example 1.7 Discrete-Time Sinusoidal Signal
A pair of sinusoidal signals with a common angular frequency is defined by
x1[n]  sin[5 n]
and
x2[n]  3 cos[5 n]
(a) Both x1[n] and x2[n] are periodic. Find their common fundamental period.
(b) Express the composite sinusoidal signal
y[n]  x1[n]  x2[n]
In the form y[n] = Acos(n + ), and evaluate the amplitude A and phase .
<Sol.>
(a) Angular frequency of both x1[n] and x2[n]:
  5 radians/cycle
N
2 m 2 m 2m



5
5
This can be only for m = 5, 10, 15, …, which results in N = 2, 4, 6, …
(b) Trigonometric identity:
A cos(n   )  A cos(n)cos( )  Asin(n)sin( )
Let  = 5, then compare x1[n] + x2[n] with the above equation to obtain that
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Introduction
A sin( )  1 and A cos( )  3
sin( ) amplitude of x1[n] 1


cos( ) amplitude of x2 [n]
3
A sin( )  1
tan( ) 
1
A
2
sin   / 6 
=/6
Accordingly, we may express y[n] as


y[n]  2cos  5 n  
6

★ 1.6.3 Relation Between Sinusoidal and Complex Exponential Signals
j t
1. Euler’s identity: e j  cos   j sin 
Complex exponential signal:
x(t )  A cos(t   ) (1.35)
Be
(1.41)
B  Ae j (1.42)  Ae j e j t
A cos(t   )  Re{Be jt }
(1.42)
 Ae j (  t )
 A cos( t   )  jA sin( t   )
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Introduction
◇ Continuous-time signal in terms of sine function:
x(t )  A sin(t   ) (1.44)
A sin(t   )  Im{Be jt }
(1.45)
2. Discrete-time case:
A cos(n   )  Re{Be jn } (1.46)
and
A sin(n   )  Im{Be jn } (1.47)
3. Two-dimensional representation of the complex
exponential e j  n for  = /4 and n = 0, 1, 2, …, 7.
: Fig. 1.34.
Projection on real axis: cos(n);
Projection on imaginary axis: sin(n)
 /4
 / 4
Figure 1.34 (p. 41)
Complex plane, showing eight points
uniformly distributed on the unit circle.
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Introduction
★ 1.6.4 Exponential Damped Sinusoidal Signals
x(t )  Ae t sin(t   ),   0
(1.48)
Example for A = 60,
 = 6, and  = 0: Fig.1.35.
Figure 1.35 (p. 41)
Exponentially damped
sinusoidal signal Ae at
sin(t), with A = 60 and 
= 6.
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Ex. Generation of an exponential damped sinusoidal signal
 Fig. 1-36.
d
1
1 t
Circuit Eq.: C v(t )  v(t )   v( )d  0
dt
R
L 
v(t )  V0 e  t /(2CR ) cos(0t ) t  0
where 0 
1
1
 2 2
LC 4C R
(1.51)
(1.49)
(1.50)
R  L /(4C )
Figure 1.36 (p. 42)
Parallel LRC, circuit, with
inductor L, capacitor C,
and resistor R all
assumed to be ideal.
Comparing Eq. (1.50) and (1.48), we have
A  V0 ,   1/(2CR),   0 , and    / 2
◆ Discrete-time case:
x[n]  Br n sin[n   ]
★ 1.6.5 Step Function
◆ Discrete-time case:
u[ n ]  1,0,
n 0
n 0
Fig. 1-37.
(1.53)
1 t
v ( )d


L
(1.52)
Figure 1.37 (p. 43)
Discrete-time version
of step function of unit
amplitude.
x[n]
1
n
3 2 1 0 1 2 3 4
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◆ Continuous-time case:
u( t ) 

1, t  0
0, t  0
Figure 1.38 (p. 44)
Continuous-time
version of the unit-step
function of unit
amplitude.
(1.54)
Example 1.8 Rectangular Pulse
Consider the rectangular pulse x(t) shown in Fig. 1.39 (a). This pulse has an
amplitude A and duration of 1 second. Express x(t) as a weighted sum of two
step functions.
<Sol.>
1. Rectangular pulse x(t):
 A, 0 t 0.5
x(t )  
 0, t 0.5
(1.55)
 1
 1
x (t )  Au  t    Au  t   (1.56)
 2
 2
Example 1.9 RC Circuit
Find the response v(t) of RC circuit shown in Fig. 1.40 (a).
<Sol.>
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Introduction
Figure 1.39 (p. 44)
(a) Rectangular pulse x(t) of amplitude A and duration of 1 s, symmetric about the
origin. (b) Representation of x(t) as the difference of two step functions of amplitude
A, with one step function shifted to the left by ½ and the other shifted to the right by
½; the two shifted signals are denoted by x1(t) and x2(t), respectively. Note that x(t)
= x1(t) – x2(t).
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Figure 1.40 (p. 45)
(a) Series RC circuit with a switch that is closed at time t = 0, thereby energizing
the voltage source. (b) Equivalent circuit, using a step function to replace the
action of the switch.
1. Initial value:
v(0)  0
2. Final value:
v()  V0
3. Complete solution:
v(t )  V0 1  e  t /( RC )  u(t )
(1.57)
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Figure 1.41 (p. 46)
Discrete-time form of
impulse.
★ 1.6.6 Impulse Function
◆ Discrete-time case:
 [ n] 

1, n  0
0, n  0
(1.58)
Fig. 1.41
(t)
a(t)
Figure 1.41 (p. 46)
Discrete-time form of impulse.
Figure 1.42 (p. 46)
(a) Evolution of a rectangular pulse of unit area into an impulse of unit strength (i.e.,
unit impulse). (b) Graphical symbol for unit impulse.
(c) Representation of an impulse of strength a that results from allowing the duration
Δ of a rectangular pulse of area a to approach zero.
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◆ Continuous-time case:
 (t )  0 for t  0



 (t )dt  1
Dirac delta function
(1.59)
(1.60)
1. As the duration decreases, the rectangular pulse approximates the impulse
more closely.
Fig. 1.42.
2. Mathematical relation between impulse and rectangular pulse function:
 (t )  lim x (t )
0
(1.61)
Fig. 1.42 (a).
3. (t) is the derivative of u(t):
(1.62)
1. x(t): even function of t,  = duration.
2. x(t): Unit area.
4. u(t) is the integral of (t):
t
u(t )    ( )d

(1.63)
Example 1.10 RC Circuit (Continued)
For the RC circuit shown in Fig. 1.43 (a), determine the current i (t) that flows
through the capacitor for t  0.
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Introduction
<Sol.>
Figure 1.43 (p. 47)
(a) Series circuit consisting of a capacitor, a dc voltage source, and a switch; the
switch is closed at time t = 0. (b) Equivalent circuit, replacing the action of the
switch with a step function u(t).
1. Voltage across the capacitor:
2. Current flowing through capacitor:
i(t )  C
dv(t )
dt
i (t )  CV0
du (t )
 CV0 (t )
dt
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◆ Properties of impulse function:
1. Even function:  (t )   (t )
(1.64)
2. Sifting property:



x(t ) (t  t0 )dt  x(t0 )
(1.65)
3. Time-scaling property:
1
 (at )   (t ), a  0 (1.66)
a
<p.f.> Fig. 1.44
1
lim x ( at )   (t )
 0
a
(1.68)
Ex. RLC circuit driven by impulsive
source: Fig. 1.45.
For Fig. 1.45 (a), the voltage across
the capacitor at time t = 0+ is
I
1 0
V0    I 0 (t )dt  0
C 0
C
(1.69)
1. Rectangular pulse approximation:
 (at )  lim x ( at )
 0
(1.67)
2. Unit area pulse: Fig. 1.44(a).
Time scaling: Fig. 1.44(b).
Area = 1/a
ax(at)
Restoring unit area
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Introduction
Figure 1.44 (p. 48)
Steps involved in proving the time-scaling property of the unit impulse. (a) Rectangular
pulse xΔ(t) of amplitude 1/Δ and duration Δ, symmetric about the origin. (b) Pulse xΔ(t)
compressed by factor a. (c) Amplitude scaling of the compressed pulse, restoring it to
unit area.
Figure 1.45 (p. 49)
(a) Parallel LRC circuit
driven by an impulsive
current signal. (b) Series
LRC circuit driven by an
impulsive voltage signal.
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★ 1.6.7 Derivatives of The Impulse
1. Doublet:
Problem 1.24
1
 (t   / 2)   (t   / 2) 
 0 
 (1) (t )  lim
(1.70)
2. Fundamental property of the doublet:






 (1) (t )dt  0




(1.71)
f (t ) (1) (t  t0 )dt 


d
f (t )
dt
t  t0
f (t )
(2)
f (t )
( n)
d2
(t  t0 )dt  2 f (t ) |t t0
dt
dn
(t  t0 )dt  n f (t ) |t t0
dt
(1.72)
3. Second derivative of impulse:
2
d (1)
 (1) (t   / 2)   (1) (t   / 2)
 (t )   (t )  lim
2
0
t
dt

(1.73)
★ 1.6.8 Ramp Function
1. Continuous-time case:
 t, t  0
r (t )  
0, t  0
(1.74)
or
r (t )  tu (t )
(1.75)
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Fig. 1.46
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2. Discrete-time case:
n, n  0
r[n]  
0, n  0
or
r[n]  nu[n]
(1.76)
Figure 1.46 (p. 51)
Ramp function of unit
slope.
(1.77)
x[n]
Fig. 1.47.
Figure 1.47 (p. 52)
Example 1.11 Parallel Circuit
Discrete-time version
4
Consider the parallel circuit of
of the ramp function.
Fig. 1-48 (a) involving a dc
current source I0 and an
initially uncharged capacitor C.
3 2 1 0 1 2 3 4
The switch across the capacitor is suddenly
opened at time t = 0. Determine the current i(t)
flowing through the capacitor and the
voltage v(t) across it for t  0.
<Sol.>
i (t )  I 0u (t )
1. Capacitor current:
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Introduction
2. Capacitor voltage:
1 t
v(t )   i ( )d
C 
1 t
v (t )   I 0u ( )d
C 
for t  0
0

  I0
 C t for t  1
I0
 tu(t )
C
I0
 r(t )
C
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Figure 1.48 (p. 52)
(a) Parallel circuit
consisting of a
current source,
switch, and
capacitor, the
capacitor is initially
assumed to be
uncharged, and the
switch is opened at
time t = 0. (b)
Equivalent circuit
replacing the
action of opening
the switch with the
step function u(t).
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Introduction
1.7 Systems Viewed as Interconnections of Operations
A system may be viewed as an interconnection of operations that transforms an
input signal into an output signal with properties different from those of the
input signal.
1. Continuous-time case:
y (t )  H {x(t )} (1.78)
2. Discrete-time case:
y[n]  H {x[n]} (1.79)
Fig. 1-49 (a) and (b).
Figure 1.49 (p. 53)
Block diagram representation of operator H for (a)
continuous time and (b) discrete time.
Example 1.12 Moving-average system
Consider a discrete-time system whose output signal y[n] is the average of the
three most recent values of the input signal x[n], that is
1
y[n]  ( x[n]  x[n  1]  x[n  2])
3
Formulate the operator H for this system; hence, develop a block diagram
representation for it.
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<Sol.> 1. Discrete-time-shift operator Sk: Fig. 1.50.
Shifts the input x[n] by k time units to
produce an output equal to x[n  k].
2. Overall operator H for the moving-average
system:
1
H  (1  S  S 2 )
Fig. 1-51.
3
Fig. 1-51 (a): cascade form; Fig. 1-51 (b): parallel
form.
Figure 1.50 (p. 54)
Discrete-time-shift operator
Sk, operating on the discretetime signal x[n] to produce
x[n – k].
1.8 Properties of Systems
★ 1.8.1 Stability
1. A system is said to be bounded-input, bounded-output (BIBO) stable if and
only if every bounded input results in a bounded output.
2. The operator H is BIBO stable if the output signal y(t) satisfies the condition
y(t )  M y   for all t
(1.80)
whenever the input signals x(t) satisfy the condition
x(t )  M x   for all t
(1.81)
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Both Mx and My
represent some finite
positive number
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Introduction
Figure 1.51 (p. 54)
Two different (but equivalent) implementations of the
moving-average system: (a) cascade form of
implementation and (b) parallel form of implementation.
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Introduction
One famous example of an unstable
system:
Figure 1.52a (p. 56)
Dramatic photographs showing the
collapse of the Tacoma Narrows
suspension bridge on November 7, 1940.
(a) Photograph showing the twisting
motion of the bridge’s center span just
before failure.
(b) A few minutes after the first piece of
concrete fell, this second photograph
shows a 600-ft section of the bridge
breaking out of the suspension span and
turning upside down as it crashed in Puget
Sound, Washington. Note the car in the
top right-hand corner of the photograph.
(Courtesy of the Smithsonian Institution.)
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Example 1.13 Moving-average system (continued)
Show that the moving-average system described in Example 1.12 is BIBO stable.
<p.f.>
1. Assume that:
x[n]  M x  
for all n
2. Input-output relation:
y[n] 
1
 x[n]  x[n  1]  x[n  2]
3
1
y[n]  x[n]  x[ n  1]  x[ n  2]
3
1
  x[n]  x[n  1]  x[ n  2] 
3
1
 M x  M x  M x 
3
 Mx
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The moving-average
system is stable.
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Introduction
Example 1.14 Unstable system
Consider a discrete-time system whose input-output relation is defined by
y[n]  r n x[n]
where r > 1. Show that this system is unstable.
<p.f.>
for all n
1. Assume that: x[n]  M x  
2. We find that
y[n]  r n x[n]  r n．x[n]
With r > 1, the multiplying factor rn diverges for increasing n.
The system is unstable.
★ 1.8.2 Memory
A system is said to possess memory if its output signal depends on past or
future values of the input signal.
A system is said to possess memoryless if its output signal depends only on
the present values of the input signal.
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Ex.: Resistor
Ex.: Inductor
1
v(t )
R
1 t
i(t )   v( )d
L 
i(t ) 
Memoryless !
Memory !
Ex.: Moving-average system
1
y[n]  ( x[n]  x[n  1]  x[n  2])
3
Memory !
Ex.: A system described by the input-output relation
y[n]  x 2[n]
Memoryless !
★ 1.8.3 Causality
A system is said to be causal if its present value of the output signal depends
only on the present or past values of the input signal.
A system is said to be noncausal if its output signal depends on one or more
future values of the input signal.
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Ex.: Moving-average system
1
y[n]  ( x[n]  x[n  1]  x[n  2])
3
Causal !
Ex.: Moving-average system
1
y[n]  ( x[n  1]  x[n]  x[n  1])
3
Noncausal !
 A causal system must be capable of operating in real time.
★ 1.8.4 Invertibility
A system is said to be invertible if the
input of the system can be recovered
from the output.
1. Continuous-time system: Fig. 1.54.
x(t) = input; y(t) = output
H = first system operator;
H inv = second system operator
Figure 1.54 (p. 59)
The notion of system invertibility. The
second operator H inv is the inverse of the
first operator H. Hence, the input x(t) is
passed through the cascade correction
of H and H inv completely unchanged.
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2. Output of the second system:
H inv  y(t )  H inv H x(t )  H inv H x(t )
H inv = inverse
operator
3. Condition for invertible system:
H inv H  I
I = identity operator
(1.82)
Example 1.15 Inverse of System
Consider the time-shift system described by the input-output relation
y (t )  x(t  t0 )  S t0 x (t )
where the operator S t0 represents a time shift of t0 seconds. Find the inverse of
this system.
<Sol.>
1. Inverse operator S t 0:
S t0 { y(t )}  S t0 {S t0 {x(t )}}  S t0 S t0 {x(t )}
2. Invertibility condition:
S t0 S t0  I
S  t0 
Time shift of t0
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Example 1.16 Non-Invertible System
Show that a square-law system described by the input-output relation
y(t )  x 2 (t )
is not invertible.
<p.f.> Since the distinct inputs x(t) and  x(t) produce the same output y(t).
Accordingly, the square-law system is not invertible.
★ 1.8.5 Time Invariance
A system is said to be time invariance if a time delay or time advance of the
input signal leads to an identical time shift in the output signal.
 A time-invariant system do not change with time.
Figure 1.55 (p.61)
The notion of time invariance. (a) Time-shift operator St0 preceding operator H. (b)
Time-shift operator St0 following operator H. These two situations are equivalent,
provided that H is time invariant.
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1. Continuous-time system:
y1 (t )  H {x1 (t )}
2. Input signal x1(t) is shifted in time by t0 seconds:
x2 (t )  x1 (t  t0 )  S t0 {x1 (t )}
S t 0 = operator of a time shift equal to t0
3. Output of system H:
y2 (t )  H {x1 (t  t0 )}
 H {S t0 {x1 (t )}}
(1.83)
 HS t0 {x1 (t )}
4. For Fig. 1-55 (b), the output of system H is y1(t  t0):
y1 (t  t0 )  S t0 { y1 (t )}
 S t0 {H {x1 (t )}}
(1.84)
 S t0 H {x1 (t )}
5. Condition for time-invariant system:
HS t0  S t0 H
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Example 1.17 Inductor
The inductor shown in figure is described
by the input-output relation:
1 t
y1 (t ) 
L 
x1 ( )d
y1(t) = i(t)
x1(t) = v(t)
where L is the inductance. Show that the inductor so described is time invariant.
<Sol.>
Response y2(t) of the inductor to x1(t  t0) is
1. Let x1(t)
x1(t  t0)
1 t
y2 (t )   x1 (  t0 )d
L 
(A)
2. Let y1(t  t0) = the original output of the inductor, shifted by t0 seconds:
1 t t0
(B)
y1 (t  t 0 )   x1 ( )d

L
3. Changing variables:  '    t0
(A)
1 t t0
y2 (t )   x1 ( ')d '
L 
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Inductor is time invariant.
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y1(t) = i(t)
Example 1.18 Thermistor
Let R(t) denote the resistance of the thermistor,
expressed as a function of time. We may express the
input-output relation of the device as
x1(t) = v(t)
y1 (t )  x1 (t ) / R(t )
Show that the thermistor so described is time variant.
<Sol.>
1. Let response y2(t) of the thermistor to x1(t  t0) is
x1 (t  t0 )
y2 (t ) 
R(t )
2. Let y1(t  t0) = the original output of the thermistor due to x1(t), shifted by t0
seconds:
x1 (t  t0 )
y1 (t  t0 ) 
R(t  t0 )
3. Since R(t)  R(t  t0)
y1 (t  t0 )  y2 (t ) for t0  0
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★ 1.8.6 Linearity
A system is said to be linear in terms of the system input (excitation) x(t) and
the system output (response) y(t) if it satisfies the following two properties of
superposition and homogeneity:
1. Superposition:
x (t )  x1 (t )
y (t )  y1 (t )
x (t )  x1 (t )  x2 (t )
x ( t )  x2 ( t )
y (t )  y2 (t )
y (t )  y1 (t )  y2 (t )
2. Homogeneity:
y (t )
x(t )
ay (t )
ax(t )
a = constant
factor
 Linearity of continuous-time system
1. Operator H represent the continuous-tome system.
2. Input:
x1(t), x2(t), …, xN(t)  input signal; a1, a2, …, aN 
N
Corresponding weighted factor
x(t )   ai xi (t ) (1.86)
i 1
3. Output:
N
y (t )  H {x(t )}  H { ai xi (t )}
(1.87)
i 1
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N
y (t )   ai yi (t )
(1.88)
i 1
Superposition and
homogeneity
where
yi (t )  H {xi (t )}, i  1, 2, ..., N .
(1.89)
4. Commutation and Linearity:
N
y (t )  H { ai xi (t )}
i 1
N
  ai H {xi (t )}
(1.90)
Fig. 1.56
i 1
N
  ai yi (t )
i 1
 Linearity of discrete-time system
Same results, see Example 1.19.
Example 1.19 Linear Discrete-Time system
Consider a discrete-time system described by the input-output relation
y[n]  nx[n]
Show that this system is linear.
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
Figure 1.56 (p. 64)
The linearity property of a system. (a) The combined operation of
amplitude scaling and summation precedes the operator H for multiple
inputs. (b) The operator H precedes amplitude scaling for each input; the
resulting outputs are summed to produce the overall output y(t). If these
two configurations produce the same output y(t), the operator H is linear.
<p.f.>
1. Input:
N
x[n]   ai xi [n]
i 1
2. Resulting output signal:
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N
N
N
y[n]  n ai xi [n]   ai nxi [n]   ai yi [n]
i 1
i 1
where
yi [n]  nxi [n]
i 1
Linear system!
Example 1.20 Nonlinear Continuous-Time System
Consider a continuous-time system described by the input-output relation
y(t )  x(t ) x(t  1)
Show that this system is nonlinear.
<p.f.>
N
1. Input: x(t ) 
ai xi (t )

i 1
2. Output:
N
N
N
N
i 1
j 1
i 1 j 1
y(t )   ai xi (t ) a j x j (t  1)  ai a j xi (t ) x j (t  1)
Here we cannot write
y (t )   i 1 ai yi (t )
N
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Example 1.21 Impulse Response of RC Circuit
For the RC circuit shown in Fig. 1.57,
determine the impulse response y(t).
<Sol.>
1. Recall: Unit step response
y(t )  (1  e t / RC )u(t ), x(t )  u(t )
(1.91)
2. Rectangular pulse input: Fig. 1.58.
x(t) = x(t)
1

u (t  )

2
1

x2 (t )  u (t  )

2
x1 (t ) 
1/
Figure 1.57 (p. 66)
RC circuit for Example 1.20, in
which we are given the capacitor
voltage y(t) in response to the
step input x(t) = y(t) and the
requirement is to find y(t) in
response to the unit-impulse
input x(t) = (t).
Figure 1.58 (p. 66)
Rectangular pulse of unit
area, which, in the limit,
approaches a unit impulse
as Δ0.
3. Response to the step
functions x1(t) and x2(t):
/2
/2
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 


 t   /( RC )  
1

y1  1  e  2 
u
t

 
,
 
2
 
x(t )  x1 (t )
 t   /( RC ) 
1
 
 2
y2  1  e
 u  t  ,
 
2
 
x ( t )  x2 ( t )


Next, recognizing that
x (t )  x1 (t )  x2 (t )
1
1
(1  e ( t  / 2) /( RC )  )u(t   / 2)  (1  e ( t  / 2) /( RC )  )u( t   / 2)


1
1
 (u(t   / 2)  u(t   / 2))  ( e ( t  / 2) /( RC ) u( t   / 2)  e ( t  / 2) /( RC )  )u( t   / 2))


y (t ) 
i) (t) = the limiting form of the pulse x(t):
(1.92)
 (t )  lim x (t )
0
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ii) The derivative of a continuous function of time, say, z(t):
d
1


z (t )  lim{ ( z (t  )  z (t  ))}
0 
dt
2
2
y (t )  lim y (t )
0
d  t /( RC )
  (t )  (e
u (t ))
dt
d t /( RC )
 t /( RC ) d
  (t )  e
u (t )  u (t ) (e
)
dt
dt
1 t /( RC )
 t /( RC )
  (t )  e
 (t ) 
e
u (t ), x(t )   (t )
RC
(1.92)
Cancel each other!
y (t ) 
1  t /( RC )
e
u(t ), x(t )   (t )
RC
(1.93)
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1.9 Noise
Noise  Unwanted signals
1. External sources of noise: atmospheric noise, galactic noise, and humanmade noise.
2. Internal sources of noise: spontaneous fluctuations of the current or voltage
signal in electrical circuit. (electrical noise)
Fig. 1.60.
★ 1.9.1 Thermal Noise
Thermal noise arises from the random motion of electrons in a conductor.
Two characteristics of thermal noise:
1. Time-averaged value:
2T = total observation interval of noise
1
v  lim
T  2T

T
T
v(t )dt
(1.94)
As T  , v  0
2. Time-average-squared value:
1
v  lim
T  2T
2
As T  ,

T
T
v 2 (t )dt
k = Boltzmann’s constant =
1.38  10 23 J/K
Tabs = absolute temperature
(1.95)
v 2  4kTabs Rf
Refer to Fig. 1.60.
volts2
(1.96)
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Figure 1.60
(p. 68)
Sample waveform
of electrical noise
generated by a
thermionic diode
with a heated
cathode. Note
that the timeaveraged value of
the noise voltage
displayed is
approximately
zero.
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 Thevenin’s equvalent circuit: Fig. 1.61(a), Norton’s equivalent circuit:
Fig. 1.61(b).
Noise voltage generator:
v(t )  v 2
Noise current generator:
1 T v(t ) 2
i  lim
(
) dt
T  2T T
R
(1.97)
 4kTabsGf amps2
2
where G = 1/R = conductance [S].
 Maximum power transfer
Figure 1.61 (p. 70)
theorem: the maximum
possible power is transferred (a) Thévenin equivalent circuit of a noisy resistor.
(b) Norton equivalent circuit of the same resistor.
from a source of internal
resistance R to a load of
resistance Rl when R = Rl.
Under matched condition, the available power is
kTabs f watts
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 Two operating factor that affect available noise power:
1. The temperature at which the resistor is maintained.
2. The width of the frequency band over which the noise voltage across the
resistor is measured.
★ 1.9.2 Other Sources of Electrical Noise
1. Shot noise: the discrete nature of current flow electronic devices
2. Ex. Photodetector:
1) Electrons are emitted at random times, k, where  < k < 
2) Total current flowing through photodetector:
x(t ) 

 h(t  
k 
k
)
(1.98)
where h(t   k ) is the current pulse generated at time k.
3. 1/f noise: The electrical noise whose time-averaged power at a given
frequency is inversely proportional to the frequency.
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1.10 Theme Example
★ 1.10.1 Differentiation and Integration: RC Circuits
1. Differentiator  Sharpening of a pulse
x(t)
d
y (t )  x(t )
dt
y(t)
differentiator
(1.99)
1) Simple RC circuit: Fig. 1.62.
2) Input-output relation:
d
1
d
v2 (t ) 
v2 (t )  v1 (t ) (1.100)
dt
RC
dt
If RC (time constant) is small enough
such that (1.100) is dominated by the
second term v2(t)/RC, then
1
d
v2 (t )  v1 (t )
RC
dt
v2 (t )  RC
Figure 1.62 (p. 71)
Simple RC circuit with small time
constant, used as an approximator
to a differentiator.
d
v1 (t ) for RC small
dt
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Input: x(t) = RCv1(t); output: y(t) = v2(t)
2. Integrator  smoothing of an input signal
x(t)
t
y(t ) 
x( )d
(1.102)
y(t)

integrator

1) Simple RC circuit: Fig. 1.63.
2) Input-output relation:
d
RC v2 (t )  v2 (t )  v1 (t )
dt
t
t


RCv2 (t )   v2 ( )d   v1 ( )d
(1.103)
If RC (time constant) is large enough
such that (1.103) is dominated by the
first term RCv2(t), then
t
RCv2 (t )   v1 ( )d
Figure 1.63 (p. 72)
Simple RC circuit with large time
constant used as an approximator
to an integrator.

1 t
v2 ( t ) 
v1 ( )d


RC
for large RC
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output: y(t) = v2(t)
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★ 1.10.2 MEMS Accelerometer
1. Model: second-order mass-damper-spring system
Fig. 1.64.
M = proof mass, K = effective spring constant, D =
damping factor, x(t) =external acceleration, y(t)
=displacement of proof mass, Md 2y(t)/dt 2 = inertial
force of proof mass, Ddy(t)/dt = damping force,
Ky(t) = spring force.
2. Force Eq.:
Figure 1.64 (p. 73)
Mechanical lumped model
of an accelerometer.
d 2 y (t )
dy (t )
Mx(t )  M
D
 Ky (t )
2
dt
dt
d 2 y (t ) D dy (t ) K


y (t )  x(t )
2
dt
M dt
M
1) Natural frequency:
n 
K
M
[rad/sec]
(1.105)
2) Quality factor:
(1.106)
Q
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(1.107)
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(1.105)
Introduction
d 2 y(t ) n dy(t )
2



n y (t )  x(t )
dt 2
Q dt
(1.108)
★ 1.10.3 Radar Range Measurement
1. A periodic sequence of radio frequency (RF) pulse: Fig. 1.65.
T0 = duration [sec], 1/T = repeated frequency, fc = RF frequency [MHz~GHz]
Figure 1.65 (p. 74)
Periodic train of rectangular FR pulses used for measuring molar ranges.
 The sinusoidal signal acts as a carrier.
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2. Round-trip time = the time taken by a radar pulse to reach the target and for
the echo from the target to come back to the radar.
2d

(1.109)
c
d = radar target range, c = light speed.
3. Two issues of concern in range measurement:
1) Range resolution: The duration T0 of the pulse places a lower limit on the
shortest round-trip delay time that the radar can measure.
Smallest target range: dmin = cT0/2 [m]
2) Range ambiguity: The interpulse period T places an upper limit on the
largest range that the radar can measure.
Largest target range: dmax = cT/2 [m]
★ 1.10.4 Moving-Average Systems
1. N-point moving-average system:
Fig. 1.66.
1
y[n] 
N
N 1
 x[n  k ]
(1.110)
x(t) = input signal
k 0
The value N determines the degree to which the system smooths the input data.
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Figure 1.66a
(p. 75)
(a) Fluctuations in
the closing stock
price of Intel over
a three-year
period.
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Figure 1.66b
(p. 76)
(b) Output of a
four-point
moving-average
system.
N = 4 case
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Figure 1.66c
(p. 76)
(c) Output of an
eight-point
moving-average
system.
N = 8 case
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2. For a general moving-average system, unequal weighting is applied to past
values of the input:
N 1
y[n]   ak x[n  k ]
(1.111)
k 0
★ 1.10.5 Multipath Communication Channels
1. Channel noise degrades the performance of a communication system.
2. Another source of degradation of channel: dispersive nature, i.e., the channel
has memory.
3. For wireless system, the dispersive characteristics result from multipath
propagation.
Fig. 1.67.
 For a digital communication, multipath propagation manifests itself in the
form of intersymbol interference (ISI).
4. Baseband model for multipath propagation: Tapped-delay line  Fig. 1.68.
p
y (t )   i x(t  iTdiff )
i 0
(1.112)
Tdiff = smallest time difference
between different path
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Figure 1.67 (p. 77)
Example of multiple propagation paths in a wireless communication environment.
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Figure 1.68 (p. 78)
Tapped-delay-line model of a linear communication channel, assumed to be timeinvariant.
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5. PTdiff = the longest time delay of any significant path relative to the arrival of
the signal.
The coefficients wi are used to approximate the gain of each path.
For P =1, then
y(t )  0 x(t )  1x(t  Tdiff )
0x(t) = direct path, 1x(t Tdiff) = single reflected path
6. Discrete-time case:
Linearly weighted
moving-average system
p
y[n]   k x[n  k ]
(1.113)
k 0
For P =1, then
y[n]  x[n]  ax[n  1]
(1.114)
★ 1.10.6 Recursive Discrete-Time Computations
The term recursive
signifies the dependence
of the output signal on its
own past values.
1. First-order recursive discrete-time filter: Fig. 1.69.
y[n]  x[n]   y[n  1]
(1.115)
x[n] = input, y[n] = output
where  is a constant.
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Figure 1.69 (p. 79)
Block diagram of first-order
recursive discrete-time filter. The
operator S shifts the output signal
y[n] by one sampling interval,
producing y[n – 1]. The feedback
coefficient  determines the
stability of the filter.
 Fig. 1.69: linear discrete-time feedback system.
2. Solution of Eq.(1.115):

y[n]    k x[n  k ]
r
(1.116)
k 0

y[n]  x[n]    k x[n  k ]
(1.117)
k 1
Setting k  1 = l, Eq.(1.117) becomes


y[n]  x[n]    [n  1  l ] x[n]     l [n  1  l ]
l 0
1 l
(1.118)
l 0
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y[n]  x[n]   y[n  1]
3. Three special cases (depending on ):
1)  = 1:
(1.116)
Accumulator

y[n]   x[n  k ]
(1.119)
k 0
2)   1: Leaky accumulator
Stable in BIBO sense
3)   1: Amplified accumulator
1.11 Exploring Concepts with MATLAB
 MATLAB Signal Processing Toolbox
Unsatble in
BIBO sense
1. Time vector: Sampling interval Ts of 1 ms on the interval from 0 to 1 s
t = 0:.001:1;
2. Vector n: n = 0:1000;
★ 1.11.1 Periodic Signals
1. Square wave: A =amplitude, w0 = fundamental frequency, rho = duty cycle
A*square(w0*t, rho);
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Ex. Obtain the square wave shown in Fig. 1.14 (a) by using MATLAB.
<Sol.> >> A = 1;
>> w0 =10*pi;
>> rho = 0.5;
>> t = 0:.001:1;
>> sq = A*square(w0*t, rho);
>> plot (t, sq)
>> axis([0 1 -1.1 1.1])
2. Triangular wave: A =amplitude, w0 = fundamental frequency, w = width
A*sawtooth(w0*t, w);
Ex. Obtain the triangular wave shown in Fig. 1.15 by using MATLAB.
<Sol.>
>> A = 1;
>> w0 =10*pi;
>> w = 0.5;
>> t = 0:0.001:1;
>> tri = A*sawtooth(w0*t, w);
>> plot (t, tri)
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3. Discrete-time signal: stem(n, x);
x = vector, n = discrete time vector
Ex. Obtain the discrete-time square wave shown in Fig. 1.16 by using MATLAB.
<Sol.>
>> A = 1;
>> omega =pi/4;
>> n = -10:10;
>> x = A*square(omega*n);
>> stem(n, x)
4. Decaying exponential:
B*exp(-a*t);
Growing exponential: B*exp(a*t);
Ex. Obtain the decaying exponential signal shown in Fig. 1.28 (a) by using
MATLAB.
<Sol.> >> B = 5;
>> a = 6;
>> t = 0:.001:1;
>> x = B*exp(-a*t); % decaying exponential
>> plot (t, x)
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Ex. Obtain the growing exponential signal shown in Fig. 1.28 (b) by using
MATLAB.
<Sol.>
>> B = 1;
>> a = 5;
>> t = 0:0.001:1;
>> x = B*exp(a*t); % growing exponential
>> plot (t, x)
Ex. Obtain the decaying exponential sequence shown in Fig. 1.30 (a) by using
MATLAB.
<Sol.>
>> B = 1;
>> r = 0.85;
>> n = -10:10;
>> x = B*r.^n; % decaying exponential sequence
>> stem (n, x)
★ 1.11.3 Sinusoidal Signals
1. Cosine signal:
A*cos(w0*t + phi);
2. Sine signal:
A*sin(w0*t + phi);
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frequency, phi =
phase angle
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Ex. Obtain the sinusoidal signal shown in Fig. 1.31 (a) by using MATLAB.
<Sol.> >> A = 4;
>> w0 =20*pi;
>> phi = pi/6;
>> t = 0:.001:1;
>> cosine = A*cos(w0*t + phi);
>> plot (t, cosine)
Ex. Obtain the discrete-time sinusoidal signal shown in Fig. 1.33 by using
MATLAB.
<Sol.>
>> A = 1;
>> omega =2*pi/12; % angular frequency
.*  element-by-element
>> n = -10:10;
multiplication
>> y = A*cos(omega*n);
>> stem (n, y)
★ 1.11.4 Exponential Damped Sinusoidal Signals
1. Exponentially damped sinusoidal signal:
x(t )  A sin(0t   )e  at
MATLAB Format: A*sin(w0*t + phi).*exp(-a*t);
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Ex. Obtain the waveform shown in Fig. 1.35 by using MATLAB.
<Sol.> >> A = 60;
>> w0 =20*pi;
>> phi = 0;
>> a = 6;
>> t = 0:.001:1;
>> expsin = A*sin(w0*t + phi).*exp(-a*t);
>> plot (t, expsin)
Ex. Obtain the exponentially damped sinusoidal sequence shown in Fig. 1.70 by
using MATLAB.
<Sol.>
>> A = 1;
>> omega =2*pi/12; % angular frequency
>> n = -10:10;
>> y = A*cos(omega*n);
>> r = 0.85;
>> x = A*r.^n; % decaying exponential sequence
>> z = x.*y; % elementwise multiplication
>> stem (n, z)
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Figure 1.70 (p. 84)
Exponentially damped sinusoidal sequence.
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★ 1.11.5 Step, Impulse, and Ramp Functions
◆ MATLAB command:
1. M-by-N matrix of ones: ones (M, N)
2. M-by-N matrix of zeros: zeros (M, N)
Ex. Generate a rectangular pulse
centered at origin on the interval
[-1, 1].
<Sol.>
 Unit amplitude step function:
u = [zeros(1, 50), ones(1, 50)];
 Discrete-time impulse:
delta = [zeros(1, 49), 1, zeros(1, 49)];
>> t = -1:1/500:1;
>> u1 = [zeros(1, 250), ones(1, 751)];
>> u2 = [zeros(1, 751), ones(1, 250)];
>> u = u1 – u2;
 Ramp sequence:
ramp = 0:.1:10
★ 1.11.6 User-Defined Functions
1. Two types M-files exist: scripts and functions.
Scripts, or script files automate long sequences of commands; functions, or
function files, provide extensibility to MATLAB by allowing us to add new functions.
2. Procedure for establishing a function M-file:
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1) It begins with a statement defining the function name, its input arguments, and
its output arguments.
2) It also includes additional statements that compute the values to be returned.
3) The inputs may be scalars, vectors, or matrices.
Ex. Obtain the rectangular pulse depicted in Fig. 1.39 (a) with the use of an
M-file.
1. The function size returns a two<Sol.>
element vector containing the row
>> function g = rect(x)
and column dimensions of a matrix.
>> g = zeros(size(x));
2. The function find returns the
>> set1 = find(abs(x)<= 0.5);
indices of a vector or matrix that
>> g(set1) = ones(size(set1));
satisfy a prescribed relation
3. The new function rect.m can be used
liked any other MATLAB function.
Ex. find(abs(x)<= T) returns the indices
Ex. To generate a rectangular pulse:
of the vector x, where the absolute
>> t = -1:1/500:1;
value of x is less than or equal to T.
>> plot(t, rect(t));
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