Signals_and_Systems_Simon Haykin & Barry Van Veen 1 Introduction CHAPTER 1.1 What is a signal? A signal is formally defined as a function of one or more.

Signals_and_Systems_Simon Haykin & Barry Van Veen

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IntroductionIntroduction CHAPTER

1.1 What is a signal?1.1 What is a signal?

A signal is formally defined as a function of one or more variables that A signal is formally defined as a function of one or more variables that conveys information on the nature of a physical phenomenon.conveys information on the nature of a physical phenomenon.

1.2 What is a system?1.2 What is a system?

Figure 1.1 (p. 2)Block diagram representation of a system.

A system is formally defined as an entity that manipulates one or more A system is formally defined as an entity that manipulates one or more signals to accomplish a function, thereby yielding new signals.signals to accomplish a function, thereby yielding new signals.

1.3 Overview of Specific Systems1.3 Overview of Specific Systems

★ ★ 1.3.1 Communication systems1.3.1 Communication systems

1. Analog communication system: modulator + channel + demodulator1. Analog communication system: modulator + channel + demodulator

Elements of a communication system Fig. 1.2Fig. 1.2


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Figure 1.2 (p. 3)Figure 1.2 (p. 3)Elements of a communication system. The transmitter changes the message Elements of a communication system. The transmitter changes the message

signal into a form suitable for transmission over the channel. The receiver 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 processes the channel output (i.e., the received signal) to produce an estimate

of the message signal.of the message signal.

◆ ◆ Modulation:Modulation:

2. Digital communication system: 2. Digital communication system:

sampling + quantization + codingsampling + quantization + coding transmitter transmitter channel channel receiver receiver

◆ ◆ Two basic modes of communication:Two basic modes of communication:

1.1. BroadcastingBroadcasting

2.2. Point-to-point communicationPoint-to-point communication

Radio, televisionRadio, television

Telephone, deep-space communication

Fig. 1.3Fig. 1.3


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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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★ ★ 1.3.2 Control systems1.3.2 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. RobustnessReasons for using control system: 1. Response, 2. Robustness

◆ ◆ Closed-loop control system: Closed-loop control system: Fig. 1.4.Fig. 1.4.

1. Single-input, single-output (SISO) system

2. Multiple-input, multiple-output (MIMO) system

Controller: digital computer(Fig. 1.5Fig. 1.5.)


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Figure 1.5 (p. 8)NASA space shuttle launch.

(Courtesy of NASA.)


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★ ★ 1.3.3 Microelectromechanica1.3.3 Microelectromechanical Systems (l Systems (MEMSMEMS))

Structure of lateral capacitive Structure of lateral capacitive accelerometers: accelerometers: Fig. 1-6 (a)Fig. 1-6 (a)..

Figure 1.6a (p. 8)Figure 1.6a (p. 8)Structure of lateral capacitivStructure of lateral capacitive accelerometers.e accelerometers.(Taken from Yazdi et al., (Taken from Yazdi et al., ProProc. IEEEc. IEEE, 1998), 1998)


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SEM view of AnalSEM view of Analog Device’s ADXLog Device’s ADXLO5 surface-microO5 surface-micromachined polysilimachined polysilicon accelerometer:con accelerometer: Fig. 1-6 (b).Fig. 1-6 (b).

Figure 1.6b (p. 9)Figure 1.6b (p. 9)SEM view of Analog DSEM view of Analog Device’s ADXLO5 surfaevice’s ADXLO5 surface-micromachined polce-micromachined polysilicon accelerometer.ysilicon accelerometer. (Taken from Yazdi et a(Taken from Yazdi et al., l., Proc. IEEEProc. IEEE, 1998), 1998)


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★ ★ 1.3.4 Remote Sensing1.3.4 Remote SensingRemote 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:2. Types of remote sensor:

Radar sensor

Infrared sensor

Visible and near-infrared sensor

X-ray sensor

※ ※ Synthetic-aperture radar (SAR)Synthetic-aperture radar (SAR)

Satisfactory operation

High resolution

See Fig. 1.7See Fig. 1.7

Ex. A stereo pair of SAR acquired from earth orbit with Shuttle Imaging RadarEx. A stereo pair of SAR acquired from earth orbit with Shuttle Imaging Radar

(SIR-B)(SIR-B)


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Figure 1.7 (p. 11)Figure 1.7 (p. 11)Perspectival view of MPerspectival view of Mount Shasta (Californiount Shasta (California), derived from a pair a), derived from a pair of stereo radar images of stereo radar images acquired from orbit witacquired from orbit with the shuttle Imaging h the shuttle Imaging Radar (SIR-B). (CourteRadar (SIR-B). (Courtesy of Jet Propulsion Lasy of Jet Propulsion Laboratory.)boratory.)


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★ ★ 1.3.5 Biomedical Signal Processing 1.3.5 Biomedical Signal Processing

Morphological types of nerve cells:Morphological types of nerve cells: Fig. 1-8.Fig. 1-8.

Figure 1.8 (p. 12)Figure 1.8 (p. 12)Morphological types of nerve cells (neurons) identifiable in monkey cerebral cMorphological types of nerve cells (neurons) identifiable in monkey cerebral cortex, based on studies of primary somatic sensory and motor cortices. (Reproortex, based on studies of primary somatic sensory and motor cortices. (Reproduced from E. R. Kande, J. H. Schwartz, and T. M. Jessel, duced from E. R. Kande, J. H. Schwartz, and T. M. Jessel, Principles of Neural Principles of Neural

Science,Science, 3d ed., 1991; courtesy of Appleton and Lange.) 3d ed., 1991; courtesy of Appleton and Lange.)


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◆ ◆ Important examples of biological signal:Important examples of biological signal:

1. Electrocardiogram (ECG)

2. Electroencephalogram (EEG)Fig. 1-9Fig. 1-9

Figure 1.9 Figure 1.9 (p. 13)(p. 13)The traces shown The traces shown in (a), (b), and (c) in (a), (b), and (c) are three are three examples of EEG examples of EEG signals recorded signals recorded from the from the hippocampus of a hippocampus of a rat. rat. Neurobiological Neurobiological studies suggest studies suggest that the that the hippocampus hippocampus plays a key role in plays a key role in certain aspects of certain aspects of learning and learning and memory. memory.


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★ ★ Measurement artifacts:Measurement artifacts:

1. Instrumental artifacts1. Instrumental artifacts

2. Biological artifacts2. Biological artifacts

3. Analysis artifacts3. Analysis artifacts

★ ★ 1.3.6 Auditory System1.3.6 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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★ ★ The ear has three main parts: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.Fig. 1-10.

★ ★ 1.3.7 Analog Versus Digital Signal Processing1.3.7 Analog Versus Digital Signal Processing

Digital approach has two advantages over analog approach:1. Flexibility2. Repeatability

1.4 Classification of Signals1.4 Classification of Signals

1. Continuous-time and discrete-time signals

Continuous-time signals: x(t)

Discrete-time signals: ( ), 0, 1, 2, .......sx n x nT n (1.1)

Fig. 1-11.Fig. 1-11.

Fig. 1-12.Fig. 1-12.

Parentheses ( )‧

Brackets [ ]‧

where t = nTs


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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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Symmetric about vertical axis


2. Even and odd signals

Even signals: ( ) ( ) for allx t x t t (1.2)

Odd signals: ( ) ( ) for allx t x t t (1.3)

Antisymmetric about originExample 1.1Example 1.1Consider the signal

sin ,( )

0 , otherwise

tT t T

x t T

Is the signal x(t) an even or an odd function of time?<Sol.><Sol.>

sin , ( )

0 , otherwise

sin , =

0 , otherwise

= ( ) for all t

tT t T

x t T

tT t T

T

x t

odd function


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◆ Even-odd decomposition of x(t):

(1.4)

( ) ( ) ( )e ox t x t x t ( ) ( )e ex t x t ( ) ( )o ox t x t

( ) ( ) ( )

( ) ( )e o

e o

x t x t x t

x t x t

1( ) ( )

2ex x t x t

1( ) ( )

2ox x t x t

where

(1.5)

Example 1.2Example 1.2Find the even and odd components of the signal

2( ) costx t e t

2( ) costx t e t

<Sol.><Sol.> 2

2

( ) cos( )

= cos( )

t

t

x t e t

e t

2 21( ) ( cos cos )

2 cosh(2 ) cos

t tex t e t e t

t t

2 21( ) ( cos cos ) sinh(2 ) cos

2t t

ox t e t e t t t

Even component:

Odd component:


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◆ Conjugate symmetric:A complex-valued signal x(t) is said to be conjugate symmetric if

( ) ( )x t x t (1.6)

( ) ( ) ( )x t a t jb t *( ) ( ) ( )x t a t jb t ( ) ( ) ( ) ( )a t jb t a t jb t

Let

( ) ( )

( ) ( )

a t a t

b t b t

3. Periodic and nonperiodic signals (Continuous-Time Case)

Periodic signals: ( ) ( ) for allx t x t T t (1.7)

0 0 0, 2 , 3 , ......T T T T 0 Fundamental periodT T and

1f

T

Fundamental frequency:

(1.8)

Angular frequency:2

2 fT

(1.9)

Refer to Fig. 1-13

Problem 1-2

Figure 1.13 (p. 20)(a) One example of continuous-time signal. (b) Another example of a continuous-time signal.


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◆ Example of periodic and nonperiodic signals: Fig. 1-14Fig. 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)

for integerx n x n N n (1.10)

N = positive integerFundamental frequency of x[n]:2

N

(1.11)


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Figure 1.15 (p. 21)Triangular wave alternative between –1 and +1 for Problem 1.3.

Figure 1.16 (p. 22)Discrete-time square wave alternative between –1 and +1.

◆ Example of periodic and nonperiodic signals: Fig. 1-16 and Fig. 1-17Fig. 1-16 and Fig. 1-17.


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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 1.17Figure 1.13 ~ Figure 1.17A random signal is a signal about which there is uncertainty before it occurs. Figure 1.9Figure 1.9

5. Energy signals and power signals

Instantaneous power:

2 ( )( )

v tp t

R

2( ) ( )p t Ri t

(1.12)

(1.13)

If R = 1 and x(t) represents a current or a voltage, then the instantaneous power is

2( ) ( )p t x t (1.14)


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The total energy of the continuous-time signal x(t) is

2 22

2

lim ( ) ( )T

TT

E x t dt x t dt

(1.15)

Time-averaged, or average, power is

22

2

1lim ( )

T

TT

P x t dtT

22

2

1( )

T

TP x t dtT

2[ ]n

E x n

21lim [ ]

2

N

nn N

P x nN

1

2

0

1[ ]

N

n

P x nN

(1.16)

For periodic signal, the time-averaged power is

(1.18)

◆ Discrete-time case:

Total energy of x[n]:

(1.17)

Average power of x[n]:

(1.19)

(1.20) ★ Energy signal:

If and only if the total energy of the signal satisfies the condition

0 E ★ Power signal:

If and only if the average power of the signal satisfies the condition0 P


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Figure 1.18 (p. 26)Inductor with current i(t), inducing voltage v(t) across its terminals.

1.5 Basic Operations on Signals1.5 Basic Operations on Signals

★ ★ 1.5.1 Operations Performed on dependent Variables1.5.1 Operations Performed on dependent Variables

Amplitude scaling: x(t) ( ) ( )y t cx t (1.21) c = scaling factor

Performed by amplifierDiscrete-time case: x[n] [ ] [ ]y n cx nAddition:

1 2( ) ( ) ( )y t x t x t (1.22)

Discrete-time case: 1 2[ ] [ ] [ ]y n x n x n Multiplication:

1 2( ) ( ) ( )y t x t x t

1 2[ ] [ ] [ ]y n x n x n

(1.23) Ex. AM modulation

Differentiation:

( ) ( )d

y t x tdt

(1.24) Inductor:Inductor: ( ) ( )d

v t L i tdt

(1.25)

Integration:

( ) ( )t

y t x d

(1.26)


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1( ) ( )

tv t i d

C

Capacitor

:(1.27) Figure 1.19 (p. 27)

Capacitor with voltage v(t) across its terminals, inducing current i(t).

★ ★ 1.5.2 Operations Performed on 1.5.2 Operations Performed on independent Variablesindependent VariablesTime scaling:

( ) ( )y t x at a >1 compressed0 < a < 1 expanded

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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[ ] [ ], 0y n x kn k Discrete-time case: 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)<Sol.> Fig.1-22(b).


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Figure 1.22 (p. 28)Operation of reflection: (a) continuous-time signal x(t) and (b) reflected version of x(t) about the origin.

1 2( ) 0 for and x t t T t T

1 2( ) 0 for and y t t T t T

Time shifting: 0( ) ( )y t x t t

t0 > 0 shift toward right

t0 < 0 shift toward left

Ex. 1-4 Time Shifting: Fig. 1-23.

Figure 1.23 (p. 29)Time-shifting operation: (a) continuous-time signal in the form of a rectangular pulse of amplitude 1.0 and duration 1.0, symmetric about the origin; and (b) time-shifted version of x(t) by 2 time shifts.


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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 Scaling1.5.3 Precedence Rule for Time Shifting and Time Scaling

1. Combination of time shifting and time scaling:

( ) ( )y t x at b

(0) ( )y x b

( ) (0)b

y xa

(1.28)

(1.29)

(1.30)

2. Operation order:2. Operation order: To achieve Eq. (1.28),

( ) ( )v t x t b ( ) ( ) ( )y t v at x at b

1st step: time shifting

2nd step: time scaling

Ex. 1-5 Precedence Rule for Continuous-Time Signal

Consider the rectangular pulse x(t) depicted in Fig. 1-24(a).Fig. 1-24(a). Find y(t)=x(2t + 3).<Sol.><Sol.> Case 1: Fig. 1-24. Shifting first, then scaling

Case 2: Fig. 1-25. Scaling first, then shifting

( ) ( 3) (2( 3)) (2 3)y t v t x t x t


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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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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 SignalA discrete-time signal is defined by

1, 1,2

[ ] 1, 1, 2

0, 0 and | | 2

n

x n n

n n

Find y[n] = x[2x + 3].


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<Sol.> See Fig. 1-27.<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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1.6 Elementary Signals1.6 Elementary Signals

★ ★ 1.6.1 Exponential Signals1.6.1 Exponential Signals ( ) atx t Be (1.31)

B and a are real parameters

1. Decaying exponential, for which a < 02. 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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Ex. Lossy capacitor: Fig. 1-29Fig. 1-29.

Figure 1.29 (p. 35)Lossy capacitor, with the loss represented by shunt resistance R.

( ) ( ) 0d

RC v t v tdt

KVL Eq.:

(1.32)

/( )0( ) t RCv t V e (1.33)

RC = Time constant

Discrete-time case:

[ ] nx n Br (1.34)

r e

where

Fig. 1.30Fig. 1.30

★ ★ 1.6.2 Sinusoidal Signals1.6.2 Sinusoidal Signals

( ) cos( )x t A t

2T

(1.35)

where

( ) cos( ( ) )

cos( )

cos( 2 )

cos( )

( )

x t T A t T

A t T

A t

A t

x t

periodicity

◆ Continuous-time case: Fig. 1-31Fig. 1-31


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Figure 1.30 (p. 35)(a) Decaying exponential form of discrete-time signal. (b) Growing

exponential form of discrete-time signal.


33


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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Ex. Generation of a sinusoidal signal Fig. 1-32Fig. 1-32.

Figure 1.32 (p. 37)Parallel LC circuit, assuming that the inductor L and capacitor C are both ideal.

Circuit Eq.:2

2( ) ( ) 0

dLC v t v t

dt

0 0( ) cos( ), 0v t V t t

(1.36)

(1.37)

where0

1

LC (1.38)

Natural angular frequency of oscillation of the circuit

◆ Discrete-time case :

[ ] cos( )x n A n

[ ] cos( )x n N A n N

2N m 2

radians/cycle, integer ,m

m NN

(1.40)

(1.39)

Periodic condition:

or

Ex. A discrete-time sinusoidal signal: A = 1, = 0, and N = 12. Fig. 1-33.Fig. 1-33.

(1.41)


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Figure 1.33 (p. 38)Discrete-time sinusoidal signal.


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Example 1.7 Discrete-Time Sinusoidal SignalA pair of sinusoidal signals with a common angular frequency is defined by

1[ ] sin[5 ]x n n 2[ ] 3cos[5 ]x n n

1 2[ ] [ ] [ ]y n x n x n

and

(a) Both x1[n] and x2[n] are periodic. Find their common fundamental period.

(b) Express the composite sinusoidal signal

In the form y[n] = Acos(n + ), and evaluate the amplitude A and phase .

<Sol.><Sol.>(a) Angular frequency of both x1[n] and x2[n]:

5 radians/cycle 2 2 2

5 5

m m mN

This can be only for m = 5, 10, 15, …, which results in N = 2, 4, 6, …(b) Trigonometric identity:

cos( ) cos( )cos( ) sin( )sin( )A n A n A n x1[n] + x2[n] with the above equation to obtain thatLet = 5, then compare


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sin( ) 1 and cos( ) 3A A

1

2

sin( ) amplitude of [ ] 1tan( )

cos( ) amplitude of [ ] 3

x n

x n

sin( ) 1A

1

2sin / 6

A

[ ] 2cos 56

y n n

= / 6

Accordingly, we may express y[n] as

★ ★ 1.6.3 Relation Between Sinusoidal and Complex Exponential Signals1.6.3 Relation Between Sinusoidal and Complex Exponential Signals

1. Euler’s identity: cos sinje j (1.41)

Complex exponential signal: jB Ae (1.42)

cos( ) Re{ }j tA t Be (1.42)

( )

cos( ) sin( )

j t

j j t

j t

Be

Ae e

Ae

A t jA t

( ) cos( )x t A t (1.35)


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/ 4/ 4


◇ ◇ Continuous-time signal in terms of sine function:Continuous-time signal in terms of sine function:

( ) sin( )x t A t (1.44)

sin( ) Im{ }j tA t Be (1.45)

2. Discrete-time case:

cos( ) Re{ }j nA n Be (1.46) (1.47) and

3. Two-dimensional representation of the complex exponential e j n for = /4 and n = 0, 1, 2, …, 7. : Fig. 1.34Fig. 1.34.

Projection on real axis: cos(n);Projection on imaginary axis: sin(n)

Figure 1.34 (p. 41)Complex plane, showing eight points uniformly distributed on the unit circle.

sin( ) Im{ }j nA n Be


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★ ★ 1.6.4 Exponential Damped Sinusoidal Signals1.6.4 Exponential Damped Sinusoidal Signals

( ) sin( ), 0tx t Ae t (1.48)

Example for A = 60, = 6, and = 0: Fig.1.35Fig.1.35.

Figure 1.35 (p. 41)Exponentially damped sinusoidal signal Ae at sin(t), with A = 60 and = 6.


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1( )

tv d

L


Ex. Generation of an exponential damped sinusoidal signal Fig. 1-36Fig. 1-36.

Figure 1.36 (p. 42)Parallel LRC, circuit, with inductor L, capacitor C, and resistor R all assumed to be ideal.

Circuit Eq.:1 1

( ) ( ) ( ) 0td

C v t v t v ddt R L

(1.49)

/(2 )0 0( ) cos( ) 0t CRv t V e t t

0 2 2

1 1

4LC C R

(1.50)

where (1.51) /(4 )R L C

Comparing Eq. (1.50) and (1.48), we have

0 0, 1/(2 ), , and / 2A V CR


[ ] sin[ ]nx n Br n (1.52)

★ ★ 1.6.5 Step Function1.6.5 Step Function ◆ Discrete-time case:

1, 00, 0[ ] n

nu n (1.53)

Figure 1.37 (p. 43)Discrete-time version of step function of unit amplitude.

Fig. 1-37.Fig. 1-37.

x[n]

n1 2 3 40123

1


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1, 0( )0, 0

tu tt

◆ Continuous-time case:

(1.54)

Figure 1.38 (p. 44)Continuous-time version of the unit-step function of unit amplitude.

Example 1.8 Rectangular PulseConsider the rectangular pulse x(t) shown in Fig. 1.39 (a).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.><Sol.> , 0 0.5( )

0, 0.5A t

x tt

1. Rectangular pulse x(t): (1.55)

1 1( )

2 2x t Au t Au t

(1.56)

Example 1.9 RC CircuitFind the response v(t) of RC circuit shown in Fig. 1.40 (a).Fig. 1.40 (a).<Sol.><Sol.>


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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: (0) 0v

0( )v V

/( )0( ) 1 ( )t RCv t V e u t (1.57)

2. Final value:

3. Complete solution:


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★ ★ 1.6.6 Impulse Function1.6.6 Impulse Function

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.

Figure 1.41 (p. 46)Discrete-time form of impulse.


1, 0[ ]0, 0

nnn

(1.58)

Fig. 1.41Fig. 1.41

Figure 1.41 (p. 46)Discrete-time form of impulse.

(t) a(t)


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◆ Continuous-time case:

( ) 0 for 0t t (1.59)

( ) 1t dt

(1.60)

Dirac delta function

1. As the duration decreases, the rectangular pulse approximates the impulse more closely.

Fig. 1.42.Fig. 1.42.2. Mathematical relation between impulse and rectangular pulse function:

0( ) lim ( )t x t (1.61) 1. x(t): even function of t, = duration.

2. x(t): Unit area.Fig. 1.42 (a).Fig. 1.42 (a).

3. (t) is the derivative of u(t):

(1.62)

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),Fig. 1.43 (a), determine the current i (t) that flows through the capacitor for t 0.


46


<Sol.><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:

( )( )

dv ti t C

dt 0 0

( )( ) ( )

du ti t CV CV t

dt


47


◆ Properties of impulse function:

1. Even function: ( ) ( )t t (1.64) 2. Sifting property:

0 0( ) ( ) ( )x t t t dt x t

(1.65)

3. Time-scaling property:

1( ) ( ), 0at t a

a (1.66)

<p.f.><p.f.>

0

1lim ( ) ( )x at t

a

(1.68)

00

0 00

1( )

IV I t dt

C C

(1.69)

Fig. 1.44Fig. 1.44

1. Rectangular pulse approximation:

0( ) lim ( )at x at

(1.67)

2. Unit area pulse: Fig. 1.44(a).Fig. 1.44(a).

Time scaling: Fig. 1.44(b).Fig. 1.44(b).Area = 1/aRestoring unit area ax(at)

Ex. RLC circuit driven by impulsive source: Fig. 1.45.Fig. 1.45.

For Fig. 1.45 (a)Fig. 1.45 (a), the voltage across the capacitor at time t = 0+ is


48


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.


49


★ ★ 1.6.7 Derivatives of The Impulse1.6.7 Derivatives of The Impulse

1. Doublet:

(1)

0

1( ) lim ( / 2) ( / 2)t t t

(1.70)

2. Fundamental property of the doublet:

(1) ( ) 0t dt

(1.71)

0

(1)0( ) ( ) ( ) t t

df t t t dt f t

dt

(1.72)

2 (1) (1)(1)

2 0

( / 2) ( / 2)( ) ( ) lim

d t tt t

t dt

(1.73)

3. Second derivative of impulse:

★ ★ 1.6.8 Ramp Function1.6.8 Ramp Function

1. Continuous-time case:

Problem 1.24Problem 1.24

0

2(2)

0 2( ) ( ) ( ) |t t

df t t t dt f t

dt

0

( )0( ) ( ) ( ) |

nn

t tn

df t t t dt f t

dt

, 0( )

0, 0

t tr t

t

(1.74) ( ) ( )r t tu tor (1.75) Fig. 1.46Fig. 1.46


50



, 0[ ]

0, 0

n nr n

n

(1.76)

Figure 1.46 (p. 51)Ramp function of unit slope.

or

[ ] [ ]r n nu n (1.77)

Figure 1.47 (p. 52)Discrete-time version of the ramp function.

x[n]

n1 2 3 40123

4

Fig. 1.47.Fig. 1.47.

Example 1.11 Parallel CircuitConsider the parallel circuit of Fig. 1-48 (a) involving a dc current source I0 and an initially uncharged capacitor C. 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.><Sol.>1. Capacitor current: 0( ) ( )i t I u t


51


2. Capacitor voltage:

1( ) ( )

tv t i d

C

0

0

0

0

1( ) ( )

0 for 0

for 1

( )

( )

tv t I u d

Ct

It t

CItu t

CIr t

C

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).


52


1.7 Systems Viewed as Interconnections of Operations1.7 Systems Viewed as Interconnections of Operations

A system may be viewed as an interconnection of operationsinterconnection 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)

[ ] { [ ]}y n H x n (1.79) Figure 1.49 (p. 53)Block diagram representation of operator H for (a) continuous time and (b) discrete time.Fig. 1-49 (a) and Fig. 1-49 (a) and

(b).(b).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[ ] ( [ ] [ 1] [ 2])

3y n x n x n x n

Formulate the operator H for this system; hence, develop a block diagram representation for it.


53


21(1 )

3H S S

<Sol.><Sol.> 1. Discrete-time-shift operator Sk: Fig. 1.50Fig. 1.50.

Figure 1.50 (p. 54)Discrete-time-shift operator Sk, operating on the discrete-time signal x[n] to produce x[n – k].

Shifts the input x[n] by k time units to produce an output equal to x[n k].

2. Overall operator HH for the moving-average system:

Fig. 1-51.Fig. 1-51.

Fig. 1-51 (a): cascade form; Fig. 1-51 (b): parallel Fig. 1-51 (a): cascade form; Fig. 1-51 (b): parallel form.form.

1.8 Properties of Systems1.8 Properties of Systems

★ ★ 1.8.1 Stability1.8.1 Stability1. A system is said to be bounded-input, bounded-output (BIBOBIBO) stable if and only if every bounded input results in a bounded output.2. The operator HH is BIBO stable if the output signal y(t) satisfies the condition

( ) for allyy t M t (1.80)

whenever the input signals x(t) satisfy the condition

( ) for allxx t M t (1.81)

Both Mx and My represent some finite positiv

e number


54


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.


55


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.)

One famous example of an unstable One famous example of an unstable system:system:


56


Example 1.13 Moving-average system (continued)Show that the moving-average system described in Example 1.12 is BIBO stable.<p.f.><p.f.>

1. Assume that: [ ] for all xx n M n

2. Input-output relation:

1[ ] [ ] [ 1] [ 2]

3y n x n x n x n

1[ ] [ ] [ 1] [ 2]

31

[ ] [ 1] [ 2]31

3

x x x

x

y n x n x n x n

x n x n x n

M M M

M

The moving-average The moving-average system is stable.system is stable.


57


Example 1.14 Unstable systemConsider a discrete-time system whose input-output relation is defined by

[ ] [ ]ny n r x nwhere r > 1. Show that this system is unstable.

<p.f.><p.f.>

1. Assume that: [ ] for all xx n M n 2. We find that

[ ] [ ] [ ]n ny n r x n r x n ．

With r > 1, the multiplying factor rn diverges for increasing n.

The system is unstable.The system is unstable.

★ ★ 1.8.2 Memory1.8.2 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.


58


Ex.: Resistor1

( ) ( )i t v tR

Memoryless !

Ex.: Inductor1

( ) ( )t

i t v dL

Memory !

Ex.: Moving-average system

1[ ] ( [ ] [ 1] [ 2])

3y n x n x n x n Memory !

Ex.: A system described by the input-output relation2[ ] [ ]y n x n Memoryless !

★ ★ 1.8.3 Causality1.8.3 CausalityA 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.


59



1[ ] ( [ ] [ 1] [ 2])

3y n x n x n x n Causal !


1[ ] ( [ 1] [ ] [ 1])

3y n x n x n x n Noncausal !

A causal system must be capable of operating in real timereal time.

★ ★ 1.8.4 Invertibility1.8.4 InvertibilityA system is said to be invertible if the input of the system can be recovered from the output. 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.

1. Continuous-time system: Fig. 1.54Fig. 1.54.

x(t) = input; y(t) = outputH = first system operator;H

inv = second system operator


60


2. Output of the second system:

( ) ( ) ( )inv inv invH y t H H x t H H x t 3. Condition for invertible system:

invH H I (1.82) I = identity operator

H inv = inverse operator

Example 1.15 Inverse of System

Consider the time-shift system described by the input-output relation

00( ) ( ) ( )ty t x t t S x t

where the operator S t0 represents a time shift of t0 seconds. Find the inverse of

this system.<Sol.><Sol.>1. Inverse operator S

t 0:

0 0 0 0 0{ ( )} { { ( )}} { ( )}t t t t tS y t S S x t S S x t 2. Invertibility condition:

0 0t tS S I 0tS Time shift of t0


61


Example 1.16 Non-Invertible SystemShow that a square-law system described by the input-output relation

2( ) ( )y t x tis not invertible.<p.f.><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 Invariance1.8.5 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.


62


1. Continuous-time system:

1 1( ) { ( )}y t H x t2. Input signal x1(t) is shifted in time by t0 seconds:

02 1 0 1( ) ( ) { ( )}tx t x t t S x t S

t 0 = operator of a time shift equal to t0

3. Output of system H:

0

0

2 1 0

1

1

( ) { ( )}

{ { ( )}}

{ ( )}

t

t

y t H x t t

H S x t

HS x t

(1.83)

4. For Fig. 1-55 (b),Fig. 1-55 (b), the output of system H is y1(t t0):

0

0

0

1 0 1

1

1

( ) { ( )}

{ { ( )}}

{ ( )}

t

t

t

y t t S y t

S H x t

S H x t

(1.84)

5. Condition for time-invariant system: 0 0t tHS S H (1.85)


63


Example 1.17 Inductor

x1(t) = v(t)

y1(t) = i(t)

The inductor shown in figure is described by the input-output relation:

1 1

1( ) ( )

ty t x d

L

where L is the inductance. Show that the inductor so described is time invariant.<Sol.><Sol.>1. Let x1(t) x1(t t0) Response y2(t) of the inductor to x1(t t0) is

0

1 0 1

1( ) ( )

t ty t t x d

L

2 1 0

1( ) ( )

ty t x t d

L

2. Let y1(t t0) = the original output of the inductor, shifted by t0 seconds:

3. Changing variables: 0' t

(A)

(B)

(A)0

2 1

1( ) ( ') '

t ty t x d

L

Inductor is time invariant.


64


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) = i(t)

1 1( ) ( ) / ( )y t x t R tShow that the thermistor so described is time variant.<Sol.><Sol.>1. Let response y2(t) of the thermistor to x1(t t0) is

1 02

( )( )

( )

x t ty t

R t

2. Let y1(t t0) = the original output of the thermistor due to x1(t), shifted by t0

seconds:

1 01 0

0

( )( )

( )

x t ty t t

R t t

3. Since R(t) R(t t0) 1 0 2 0( ) ( ) for 0y t t y t t Time variant!


65


★ ★ 1.8.6 Linearity1.8.6 LinearityA 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:

1( ) ( )x t x t 1( ) ( )y t y t

2( ) ( )x t x t 2( ) ( )y t y t1 2( ) ( ) ( )x t x t x t

1 2( ) ( ) ( )y t y t y t 2. Homogeneity:

( )x t ( )y t ( )ax t ( )ay ta = constant

factor

Linearity of continuous-time system1. Operator H represent the continuous-tome system.2. Input:

1

( ) ( )N

i ii

x t a x t

(1.86)

x1(t), x2(t), …, xN(t) input signal; a1, a2, …, aN

Corresponding weighted factor

3. Output:

1

( ) { ( )} { ( )}N

i ii

y t H x t H a x t

(1.87)


66


1

( ) ( )N

i ii

y t a y t

(1.88) Superposition and

homogeneity

where

( ) { ( )}, 1, 2, ..., .i iy t H x t i N (1.89)

4. Commutation and Linearity:

1

1

1

( ) { ( )}

{ ( )}

( )

N

i ii

N

i ii

N

i ii

y t H a x t

a H x t

a y t

(1.90) Fig. 1.56Fig. 1.56

Linearity of discrete-time system Same results, see Example 1.19Example 1.19.Example 1.19 Linear Discrete-Time systemConsider a discrete-time system described by the input-output relation

[ ] [ ]y n nx nShow that this system is linear.


67


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.><p.f.>1. Input:

1

[ ] [ ]N

i ii

x n a x n

2. Resulting output signal:


68


1 1 1

[ ] [ ] [ ] [ ]N N N

i i i i i ii i i

y n n a x n a nx n a y n

where [ ] [ ]i iy n nx n

Linear system!

Example 1.20 Nonlinear Continuous-Time SystemConsider a continuous-time system described by the input-output relation

Show that this system is nonlinear.<p.f.><p.f.>

( ) ( ) ( 1)y t x t x t

1. Input:1

( ) ( )N

i ii

x t a x t

2. Output:

1 1 1 1

( ) ( ) ( 1) ( ) ( 1)N N N N

i i j j i j i ji j i j

y t a x t a x t a a x t x t

Here we cannot write 1

( ) ( )N

i iiy t a y t

Nonlinear system!


69

1/

/2 /2


Example 1.21 Impulse Response of RC CircuitFor the RC circuit shown in Fig. 1.57Fig. 1.57, determine the impulse response y(t).

Figure 1.57 (p. 66)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).

<Sol.><Sol.>1. Recall: Unit step response

/( ) (1 ) ( ), ( ) ( )t RCy t e u t x t u t (1.91)

2. Rectangular pulse input: Fig. 1.58Fig. 1.58.

Figure 1.58 (p. 66)Rectangular pulse of unit area, which, in the limit, approaches a unit impulse as Δ0.

x(t) = x(t)

1

1( ) ( )

2x t u t

2

1( ) ( )

2x t u t

3. Response to the step

functions x1(t) and x2(t):


70


/( )2

1 1

11 , ( ) ( )

2

t RC

y e u t x t x t

/( )2

2 2

11 , ( ) ( )

2

t RC

y e u t x t x t

Next, recognizing that

1 2( ) ( ) ( )x t x t x t

( / 2) /( ) ( / 2) /( )

( / 2) /( ) ( / 2) /( )

1 1( ) (1 ) ( / 2) (1 ) ( / 2)

1 1( ( / 2) ( / 2)) ( ( / 2) ) ( / 2))

t RC t RC

t RC t RC

y t e u t e u t

u t u t e u t e u t

(1.92) i) (t) = the limiting form of the pulse x(t):

0( ) lim ( )t x t


71


ii) The derivative of a continuous function of time, say, z(t):

0

1( ) lim{ ( ( ) ( ))}

2 2

dz t z t z t

dt

(1.92)

0

/( )

/( ) /( )

/( ) /( )

( ) lim ( )

( ) ( ( ))

( ) ( ) ( ) ( )

1 ( ) ( ) ( ), ( ) ( )

t RC

t RC t RC

t RC t RC

y t y t

dt e u t

dtd d

t e u t u t edt dt

t e t e u t x t tRC

/( )1( ) ( ), ( ) ( )t RCy t e u t x t t

RC (1.93)

Cancel each other!


72


1.9 Noise1.9 NoiseNoise Unwanted signals1. External sources of noise: atmospheric noise, galactic noise, and human- made noise.2. Internal sources of noise: spontaneous fluctuations of the current or voltage signal in electrical circuit. (electrical noise)

Fig. 1.60.Fig. 1.60.

★ ★ 1.9.1 Thermal Noise1.9.1 Thermal NoiseThermal noise arises from the random motion of electrons in a conductor.

Two characteristics of thermal noise: 1. Time-averaged value:

1lim ( )

2

T

TTv v t dt

T (1.94)

2T = total observation interval of noise

As T , 0v Refer to Fig. 1.60.Refer to Fig. 1.60.

2. Time-average-squared value:

2 21lim ( )

2

T

TTv v t dt

T (1.95)

As T , 2 24 voltsabsv kT R f (1.96)

k = Boltzmann’s constant = 1.38 10 23 J/K

Tabs = absolute temperature


73


Figure 1.60 (p. 68)Sample waveform of electrical noise generated by a thermionic diode with a heated cathode. Note that the time-averaged value of the noise voltage displayed is approximately zero.


74


Thevenin’s equvalent circuit: Fig. 1.61(a),Fig. 1.61(a), Norton’s equivalent circuit: Fig. 1.61(b).Fig. 1.61(b).

Figure 1.61 (p. 70)(a) Thévenin equivalent circuit of a noisy resistor. (b) Norton equivalent circuit of the same resistor.

Noise voltage generator:2( )v t v

Noise current generator:

2 2

2

1 ( )lim ( )

2

4 amps

T

TT

abs

v ti dt

T R

kT G f

(1.97)

where G = 1/R = conductance [S]. Maximum power transfer theorem: the maximum possible power is transferred from a source of internal resistance R to a load of resistance Rl when R = Rl.

Under matched condition, the available power iswattsabskT f


75


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 Noise1.9.2 Other Sources of Electrical Noise1. Shot noise: the discrete nature of current flow electronic devices2. Ex. Photodetector:

1) Electrons are emitted at random times, k, where < k < 2) Total current flowing through photodetector:

( ) ( )kk

x t h t

(1.98)

( )kh t where 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.


76


1.10 Theme Example1.10 Theme Example

★ ★ 1.10.1 Differentiation and Integration: 1.10.1 Differentiation and Integration: RCRC Circuits Circuits 1. Differentiator Sharpening of a pulse

differentiatorx(t) y(t)

( ) ( )d

y t x tdt

1) Simple RC circuit: Fig. 1.62Fig. 1.62.

Figure 1.62 (p. 71)Simple RC circuit with small time constant, used as an approximator to a differentiator.

2) Input-output relation:

(1.99)

2 2 1

1( ) ( ) ( )

d dv t v t v t

dt RC dt (1.100)

If RC (time constant) is small enough such that (1.100) is dominated by the second term v2(t)/RC, then

2 1

1( ) ( )

dv t v t

RC dt 2 1( ) ( ) for small

dv t RC v t RC

dt (1.101)


77


Input: x(t) = RCv1(t); output: y(t) = v2(t)

2. Integrator smoothing of an input signal

1) Simple RC circuit: Fig. 1.63Fig. 1.63.

integratorx(t) y(t)

( ) ( )t

y t x d

(1.102)

2) Input-output relation:

2 2 1( ) ( ) ( )t t

RCv t v d v d

2 2 1( ) ( ) ( )d

RC v t v t v tdt

(1.103)

If RC (time constant) is large enough such that (1.103) is dominated by the first term RCv2(t), then

2 1( ) ( )t

RCv t v d

2 1

1( ) ( ) for large

tv t v d RC

RC

Figure 1.63 (p. 72)Simple RC circuit with large time constant used as an approximator to an integrator.

Input: x(t) = [1/(RC)v1(t)];

output: y(t) = v2(t)


78


★ ★ 1.10.2 MEMS Accelerometer1.10.2 MEMS Accelerometer1. Model: second-order mass-damper-spring system

Figure 1.64 (p. 73)Mechanical lumped model of an accelerometer.

Fig. 1.64.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 for

ce of proof mass, Ddy(t)/dt = damping force, Ky(t) = spring force.

2. Force Eq.:2

2

( ) ( )( ) ( )

d y t dy tMx t M D Ky t

dt dt

2

2

( ) ( )( ) ( )

d y t D dy t Ky t x t

dt M dt M (1.105)

1) Natural frequency:

n

K

M (1.106) [rad/sec]

2) Quality factor:

KMQ

D (1.107)


79


(1.105) 2

22

( ) ( )( ) ( )n

n

d y t dy ty t x t

dt Q dt

(1.108)

★ ★ 1.10.3 Radar Range Measurement1.10.3 Radar Range Measurement1. A periodic sequence of radio frequency (RF) pulse: Fig. 1.65Fig. 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.


80


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

c (1.109)

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: ddminmin = = cTcT00/2/2 [m]2) Range ambiguity: The interpulse period T places an upper limit on the largest range that the radar can measure.

Largest target range: ddmaxmax = = cTcT/2/2 [m]

★ ★ 1.10.4 Moving-Average Systems1.10.4 Moving-Average Systems1. N-point moving-average system:

1

0

1[ ] [ ]

N

k

y n x n kN

(1.110) x(t) = input signal

The value N determines the degree to which the system smooths the input data.

Fig. 1.66.Fig. 1.66.


81


Figure 1.66a (p. 75)(a) Fluctuations in the closing stock price of Intel over a three-year period.


82


Figure 1.66b (p. 76)(b) Output of a four-point moving-average system.N = 4 case


83


Figure 1.66c (p. 76)(c) Output of an eight-point moving-average system.

N = 8 case


84


2. For a general moving-average system, unequal weighting is applied to past values of the input:

1

0

[ ] [ ]N

kk

y n a x n k

(1.111)

★ ★ 1.10.5 Multipath Communication Channels1.10.5 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.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. 1.68Fig. 1.68.

0

( ) ( )p

i diffi

y t x t iT

(1.112) Tdiff = smallest time difference bet

ween different path


85


Figure 1.67 (p. 77)Example of multiple propagation paths in a wireless communication environment.


86


Figure 1.68 (p. 78)Tapped-delay-line model of a linear communication channel, assumed to be time-invariant.


87


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

0 1( ) ( ) ( )diffy t x t x t T

0x(t) = direct path, 1x(t Tdiff) = single reflected path


0

[ ] [ ]p

kk

y n x n k

(1.113)

For P =1, then

[ ] [ ] [ 1]y n x n ax n (1.114)

Linearly weighted moving-average system

★ ★ 1.10.6 Recursive Discrete-Time Computations1.10.6 Recursive Discrete-Time Computations

1. First-order recursive discrete-time filter: Fig. 1.69Fig. 1.69.

The term recursive signifies the dependence of the output signal on its

own past values.

x[n] = input, y[n] = output[ ] [ ] [ 1]y n x n y n (1.115)

where is a constant.


88


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):

1

[ ] [ ] [ ]k

k

y n x n x n k

(1.116) 0

[ ] [ ]k

k

y n x n k

(1.117)

Setting k 1 = l, Eq.(1.117) becomes

1

0 0

[ ] [ ] [ 1 ] [ ] [ 1 ]l l

l l

y n x n n l x n n l

(1.118)

r


89


[ ] [ ] [ 1]y n x n y n 3. Three special cases (depending on ):

1) = 1:

0

[ ] [ ]k

y n x n k

(1.116) (1.119)

Accumulator

2) 1: Leaky accumulator

3) 1: Amplified accumulator

1.11 Exploring Concepts with MATLAB1.11 Exploring Concepts with MATLAB

MATLAB Signal Processing Toolbox

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 1.11.1 Periodic Signals 1. Square wave: A =amplitude, w0 = fundamental frequency, rho = duty cycle

A*square(w0*t, rho);

Stable in BIBO senseStable in BIBO sense

Unsatble in BIUnsatble in BIBO senseBO sense


90


Ex. Obtain the square wave shown in Fig. 1.14 (a)Fig. 1.14 (a) by using MATLAB.

<Sol.><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. 1.15Fig. 1.15 by using MATLAB.

>> A = 1;>> w0 =10*pi;>> w = 0.5;>> t = 0:0.001:1;>> tri = A*sawtooth(w0*t, w);>> plot (t, tri)

<Sol.><Sol.>


91


3. Discrete-time signal: stem(n, x);x = vector, n = discrete time vector

Ex. Obtain the discrete-time square wave shown in Fig. 1.16Fig. 1.16 by using MATLAB.

>> A = 1;>> omega =pi/4;>> n = -10:10;>> x = A*square(omega*n);>> stem(n, x)

<Sol.><Sol.>

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)Fig. 1.28 (a) by using MATLAB.

>> B = 5;>> a = 6;>> t = 0:.001:1;>> x = B*exp(-a*t); % decaying exponential>> plot (t, x)

<Sol.><Sol.>


92


Ex. Obtain the growing exponential signal shown in Fig. 1.28 (b)Fig. 1.28 (b) by using MATLAB.

>> B = 1;>> a = 5;>> t = 0:0.001:1;>> x = B*exp(a*t); % growing exponential>> plot (t, x)

<Sol.><Sol.>

Ex. Obtain the decaying exponential sequence shown in Fig. 1.30 (a)Fig. 1.30 (a) by using MATLAB.<Sol.><Sol.>

>> B = 1;>> r = 0.85;>> n = -10:10;>> x = B*r.^n; % decaying exponential sequence>> stem (n, x)

★ ★ 1.11.3 Sinusoidal Signals 1.11.3 Sinusoidal Signals

1. Cosine signal:

A*cos(w0*t + phi);

2. Sine signal:

A*sin(w0*t + phi);

A = amplitude, w0 = frequency, phi =

phase angle


93


Ex. Obtain the sinusoidal signal shown in Fig. 1.31 (a)Fig. 1.31 (a) by using MATLAB.

>> A = 4;>> w0 =20*pi;>> phi = pi/6;>> t = 0:.001:1;>> cosine = A*cos(w0*t + phi);>> plot (t, cosine)

<Sol.><Sol.>

Ex. Obtain the discrete-time sinusoidal signal shown in Fig. 1.33Fig. 1.33 by using MATLAB.

>> A = 1;>> omega =2*pi/12; % angular frequency>> n = -10:10;>> y = A*cos(omega*n);>> stem (n, y)

<Sol.><Sol.>

★ ★ 1.11.4 Exponential Damped Sinusoidal Signals 1.11.4 Exponential Damped Sinusoidal Signals

1. Exponentially damped sinusoidal signal:

0( ) sin( ) atx t A t e MATLAB Format:MATLAB Format: A*sin(w0*t + phi).*exp(-a*t);

.*.* element-by-element multiplication


94


Ex. Obtain the waveform shown in Fig. 1.35Fig. 1.35 by using MATLAB.

<Sol.><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. 1.70Fig. 1.70 by using MATLAB.<Sol.><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)


95


Figure 1.70 (p. 84)Exponentially damped sinusoidal sequence.


96


★ ★ 1.11.5 Step, Impulse, and Ramp Functions 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)

Unit amplitude step function:

u = [zeros(1, 50), ones(1, 50)];

Discrete-time impulse:

delta = [zeros(1, 49), 1, zeros(1, 49)];

Ramp sequence:

ramp = 0:.1:10

Ex.Ex. Generate a rectangular pulse centered at origin on the interval [-1, 1].

<Sol.><Sol.>

>> t = -1:1/500:1;>> u1 = [zeros(1, 250), ones(1, 751)];>> u2 = [zeros(1, 751), ones(1, 250)];>> u = u1 – u2;

★ ★ 1.11.6 User-Defined Functions 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:


97


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)Fig. 1.39 (a) with the use of an M-file.<Sol.><Sol.>

>> function g = rect(x)>> g = zeros(size(x));>> set1 = find(abs(x)<= 0.5);>> g(set1) = ones(size(set1));

1. The function sizesize returns a two-element vector containing the row and column dimensions of a matrix.

2. The function findfind returns the indices of a vector or matrix that satisfy a prescribed relation

Ex. find(abs(x)<= T) returns the indices of the vector x, where the absolute value of x is less than or equal to T.

3. The new function rect.m can be used liked any other MATLAB function.

>> t = -1:1/500:1;>> plot(t, rect(t));

Ex. To generate a rectangular pulse:


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Signals_and_Systems_Simon Haykin & Barry Van Veen 1 Introduction CHAPTER 1.1 What is a signal? A signal is formally defined as a function of one or more.

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