Ch01 Introduction Compatibility Mode

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IntroductionIntroductionCHAPTER

11..11 What is a signal?What is a signal?

A signal is formally defined as a function of one or more variables thatA signal is formally defined as a function of one or more variables that

. .

11..22 What is a system?What is a system?

A s stem is formall defined as an entit that mani ulates one or moreA s stem is formall defined as an entit that mani ulates one or more

signals to accomplish a function, thereby yielding new signals.signals to accomplish a function, thereby yielding new signals.

Figure 1.1 (p. 2)

oc agram representat on o a system.

11..33 Overview of Specific SystemsOverview of Specific Systems

★ .. ..

11. Analog communication system: modulator + channel + demodulator . Analog communication system: modulator + channel + demodulator

Elements of a communication system Fig.Fig. 11..22

Signals_and_Systems_Simon

Haykin & Barry Van Veen

1




FigureFigure 11..22 (p.(p. 33))

Elements of a communication system. The transmitter changes the messageElements of a communication system. The transmitter changes the message

signal into a form suitable for transmission over the channel. The receiversignal into a form suitable for transmission over the channel. The receiver

processes the channel output (i.e., the received signal) to produce an estimateprocesses the channel output (i.e., the received signal) to produce an estimate

of the message signal.of the message signal.

Modulation:Modulation:

22. Di ital communication s stem:. Di ital communication s stem:

sampling + quantization + codingsampling + quantization + coding → transmitter transmitter → channelchannel → receiver receiver

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

1.1. BroadcastingBroadcasting

2.2. PointPoint--toto--point communicationpoint communication

Radio, televisionRadio, television

Telephone, deep-space

Fig.Fig. 11..33



2

communication




Figure 1.3 (p. 5)

(a) Snapshot of Pathfinder

ex lorin 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 JetPropulsion Laboratory.)



3




★ 11..33..22 Control systemsControl systems

. .

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 in ut to roduce an error si nal e t which

in turn, drives the controller. The feedback loop is thereby closed.

Reasons for using control system:Reasons for using control system: 11. Response,. Response, 22. Robustness. Robustness ClosedClosed--loop control system:loop control system: Fig.Fig. 11..44..

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

- -

Controller: digital

computer

(Fig.Fig. 11..55.)



4

. ,




Figure 1.5 (p. 8)

NASA s ace shuttle launch.

(Courtesy of NASA.)



5




★ 11..33..33MicroelectromechanicalMicroelectromechanical

Systems (Systems (MEMSMEMS))

Structure of lateral capacitiveStructure of lateral capacitive

accelerometers:accelerometers: Fig.Fig. 11--66 (a)(a)..

FigureFigure 11..66a (p.a (p. 88))

Structure of lateralStructure of lateralcapacitive accelerometers.capacitive accelerometers.

(Taken from Yazdi et al.,(Taken from Yazdi et al.,

Proc. IEEEProc. IEEE,, 19981998))



6




SEM view ofSEM view of

Analog Device’sAnalog Device’s

--

micromachinedmicromachined

polysiliconpolysilicon

Fig.Fig. 11--66 (b).(b).

FigureFigure 11..66b (p.b (p. 99))SEM view of AnalogSEM view of Analog

Device’s ADXLODevice’s ADXLO55

surfacesurface--

micromachinedmicromachined

polysiliconpolysiliconaccelerometer.accelerometer.

(Taken from Yazdi et(Taken from Yazdi et

al.,al., Proc. IEEEProc. IEEE,, 19981998))



7




★ 11..33..44 Remote SensingRemote 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.. ypes o remo e sensor:. ypes o remo e sensor:

Radar sensor

Infrared sensor

Visible and near-infrared sensor X-ray sensor

SyntheticSynthetic--aperture radar (SAR)aperture radar (SAR)

Satisfactory operation See Fig.See Fig. 11..77

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

SIRSIR--BB



8





Perspectival view ofPerspectival view of

(California), derived(California), derived

from a pair of stereofrom a pair of stereo

from orbit with thefrom orbit with the

shuttle Imaging Radarshuttle Imaging Radar

SIRSIR--B . Courtes ofB . Courtes of

Jet PropulsionJet PropulsionLaboratory.)Laboratory.)



9




★ 11..33..55 Biomedical Signal ProcessingBiomedical Signal ProcessingMorphological types of nerve cells:Morphological types of nerve cells: Fig.Fig. 11--88..


cortex, based on studies of primary somatic sensory and motor cortices.cortex, based on studies of primary somatic sensory and motor cortices.

(Reproduced from E. R. Kande, J. H. Schwartz, and T. M. Jessel,(Reproduced from E. R. Kande, J. H. Schwartz, and T. M. Jessel, Principles ofPrinciples of



10

,, .,., ..




Important examples of biological signal:Important examples of biological signal:1. Electrocardiogram (ECG)

Fig.Fig. 11--99FigureFigure 11..99

(p.(p. 1313))

e races s owne races s own

in (a), (b), and (c)in (a), (b), and (c)

are threeare three

signals recordedsignals recorded

from thefrom the

rat.rat.NeurobiologicalNeurobiological

that thethat the

hippocampushippocampus

la s a ke role inla s a ke role incertain aspects ofcertain aspects of

learning andlearning and

memory.memory.



11




★ Measurement artifacts:Measurement artifacts:11. Instrumental artifacts. Instrumental artifacts

22. Biolo ical artifacts. Biolo ical artifacts

33. Analysis artifacts. Analysis artifacts

★ 11..33..66 Auditory SystemAuditory 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

reg on an ex e reg on re er o

the basilar membrane. (b) This

diagram illustrates the traveling ,

showing their envelopes induced by

incoming sound at three different



12

.




★ The ear has three main parts:The ear has three main parts:1. Outer ear: collection of sound

.

Conveying the variations of the tympanic membrane (eardrum)

3. Inner ear: mechanical variations →→ electrochemical or neural signal

★ Basilar membrane: Traveling wave Fig.Fig. 11--1010..

★ 11..33..77 Analog Versus Digital Signal ProcessingAnalog Versus Digital Signal Processing

g a approac as wo a van ages over ana og approac :

1. Flexibility

2. Repeatability

11..44 Classification of SignalsClassification of Signals

1. Continuous-time and discrete-time signalsParentheses (‧)

Continuous-time signals: x(t)

Discrete-time signals: [ ] ( ), 0, 1, 2, .......s

x n x nT n= = ± ± (1.1)

Fig.Fig. 11--1111..

where t = nTs



13

g.g. -- .. Brackets [‧]




Figure 1.11 (p. 17)

Continuous-time signal.

gure . p.

(a) Continuous-time signal x(t). (b) Representation of x(t) as a

discrete-time signal x[n].



14




2. Even and odd signals

Even signals: ( ) ( ) for all x t x t t − = (1.2)

− = − .

Antisymmetric about originExampleExample 11..11

Consider the signal

sin ,( )

t T t T

x t T

π ⎧ ⎞⎛ − ≤ ≤⎪ ⎜ ⎟= ⎝ ⎠⎨⎪ ,

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

sint

T t T π ⎧ ⎞⎛ − − ≤ ≤

( )

0 , otherwise

x t T − =⎪⎩

sin ,=

0 , otherwise

T t T T

− − ≤ ≤⎪ ⎜ ⎟⎝ ⎠⎨

⎪⎩



15

= ( ) for all t x t −




2( ) cost x t e t

−=

Even-odd decomposition of x(t):( ) ( ) ( )e o x t x t x t = +

ExampleExample 11..22Find the even and odd components

of the signale e x x− =

( ) ( )o o x t x t − = −

w ere2( ) cost

x t e t −=

<Sol.><Sol.>

( ) ( )

e o

e o

x t x t x t

x t x t

− = − + −

= − 2

cos

= cos( )t

x t e t

e t

− = −

(1.4)[ ]1 ( ) ( )2

e x x t x t = + −2 21

( ) ( cos cos )t t e x t e t e t

−= +Even component:

[ ]( ) ( )2

o x x t x t = − − (1.5)

cosh(2 ) cost t =

2 21( ) ( cos cos ) sinh(2 ) cost t

o x t e t e t t t −= − = −

Odd component:



16




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 b t = −

Let

( ) ( )a t a t − =

Fig. 1-13

Problem 1-2

( ) ( ) ( ) ( )a t jb t a t jb t − + − = − ( ) ( )b t b t − = −

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

Periodic signals: ( ) ( ) for all x t x t T t = + (1.7)

0 0 0, 2 , 3 , ......T T T T = 0 Fundamental period T T = ≡andFigure 1.13

(p. 20)

1 f

T =

(1.8)

a ne examp e

of continuous-

time signal.Angular frequency:

22 f

T

π ω π = = (1.9)

example of a

continuous-time

Signals_and_Systems_SimonHaykin & Barry Van Veen

17

.




Example of periodic and nonperiodic signals: Fig.Fig. 11--1414.

Figure 1.14 (p. 21)(a) Square wave with amplitude A = 1 and period T = 0.2s.

b Rectan ular ulse of am litude A and duration T .

Periodic and nonperiodic signals (Discrete-Time Case)

[ ] [ ] for integer x n x n N n= + (1.10)N = positive integer Fundamental frequency of x[n]:

2π Ω = (1.11)


18




Figure 1.15 (p. 21)

Triangular wave alternative between –1 and +1 for Problem 1.3.

Fig.Fig. 11--1616 and Fig.and Fig. 11--1717.

Discrete-time square

wave alternative

between –1 and +1.


19




Figure 1.17 (p. 22)

Aperiodic discrete-time signal

.

4. Deterministic signals and random signalsA deterministic signal is a signal about which there is no uncertainty with

respect to its value at any time.

FigureFigure 11..1313 ~ Figure~ Figure 11..1717

A random signal is a signal about which there is uncertainty before it occurs.FigureFigure 11..99

.

Instantaneous power:

2v t If R = 1 Ω and x(t) represents a current or a voltage, p R

=

2( ) ( ) p t Ri t =

.

(1.13)

then the instantaneous power is2( ) ( ) p t x t = (1.14)


20




The total energy of the continuous-time signal x(t) is2 22lim ( ) ( )

T

T E x t dt x t dt ∞

− −∞∞= =∫ ∫ (1.15)

Discrete-time case:Total energy of x[n]:

2

Time-averaged, or average, power is

1 T

2[ ]n

E x n=−∞

= ∑ (1.18)

Avera e ower of x n :

2

m T T

x t t T

−→∞

=21

lim [ ]2

N

nn N

P x n N →∞

=−

= ∑(1.16)

For periodic signal, the time-averaged power is(1.19)

22

2

1 ( )T

T P x t dt T

−= ∫1

2

0

1 [ ] N

n

P x n N

−

== ∑(1.17)

.★ Energy signal:

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

★ Power signal:

If and only if the average power of the signal satisfies the condition


21

∞




11..55 Basic Operations on SignalsBasic Operations on Signals★ 11..55..11 Operations Performed on dependent VariablesOperations Performed on dependent Variables

Am litude scalin : x t t cx t = 1.21

c = scaling factor

Performed by amplifier Discrete-time case: x[n] [ ] [ ] y n cx n=

Addition:1 2 y x x= .

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)

x. mo u a on

Figure 1.18 (p. 26)

Inductor with current

i (t ), inducing voltage

Differentiation:

( ) ( )

d

y t x t = (1.24) Inductor:Inductor: ( ) ( )

d

v t L i t = (1.25)v (t ) across its

terminals.Integration:

t

=


22

−∞.




( ) ( )v t i d C τ τ −∞=Capacitor: (1.27) Figure 1.19 (p. 27)Capacitor with

voltage v (t ) across11..55..22 Operations Performed onOperations Performed on

its terminals,

inducing current i (t ).

independent Variablesindependent Variables

Time scaling:

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.


23




, y n x n= >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)

vers on o x n compresse y a ac or o , w some va ues o e or g na x n os

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

- version of x(t) about the amplitude axis (i.e., the origin).

<Sol.> Fig.<Sol.> Fig.11--2222(b)(b).


24




Figure 1.22 (p. 28) -

about the origin.

1 2( ) 0 for and x t t T t T = < − >t > 0 ⇒ shift toward ri ht

1 2( ) 0 for and y t t T t T = > < −Time shift ing: 0( ) ( ) y t x t t = −

⇒

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-


25

shifted version of x (t ) by 2 time shifts.






Figure 1.24 (p. 31)

The ro er order in which the o erations of time scalin and time shiftin

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-shiftedversion of x (t ). (c) Desired signal y (t ), resulting from the compression of v (t )

by a factor of 2.


27




Figure 1.25 (p. 31)

w y y u . .

(b) Time-scaled signal v (t ) = x (2t ). (c) Signal y (t ) obtained by shiftingv (t ) = x (2t ) by 3 time units, which yields y (t ) = x (2(t + 3)).

x. - rece ence u e or scre e- me gna

A discrete-time signal is defined by

1, 1,2n =⎧[ ] 1, 1, 2

0, 0 and | | 2

x n n

n n

= − = − −⎨⎪ = >⎩


28

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




<Sol.> See Fig.<Sol.> See Fig. 11--2727..

Figure 1.27 (p. 33)

The proper order of applying the operations of time scaling and time shifting for thecase 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,


29

as a result of which two samples of the original x [n], located at n = –2, +2, are lost.




11..66 Elementary SignalsElementary Signals★ 11..66..11 Exponential SignalsExponential Signals ( ) a t

x t Be= (1.31)

a

1. Deca in ex onential for which < 0

2. Growing exponential, for which a > 0

Figure 1.28 (p. 34)

(a) Decaying exponential form of continuous-time signal. (b) Growing exponential


30

orm o con nuous- me s gna .




Ex. Lossy capacitor: Fig.Fig. 11--2929.

d

KVL Eq.:

v t v t dt

+ = .

/( )0( )

t RC

v t V e−

= (1.33)

RC = Time constant

Figure 1.29 (p. 35)

Lossy capacitor, with the

loss represented by shunt

Discrete-time case:

[ ] n x n Br = (1.34)

resistance R .

r eα =where

Fig.Fig. 11..3030

★ 11..66..22 Sinusoidal SignalsSinusoidal Signals ( ) cos( ( ) )

cos( )

x t T A t T

A t T

ω φ

ω ω φ

+ = + +

= + +Continuous-time case:Fig.Fig. 11--3131

2T

π

ω =

.

where cos

cos( )

t

A t

ω π

ω φ

= + +

= +periodicity


31




Figure 1.30 (p. 35)

a Deca in ex onential form of discrete-time si nal. b Growin exponential form of discrete-time signal.


32

I d iI d i




Figure 1.31 (p. 36)

(a) Sinusoidal signal A cos(ω t + Φ) with phase Φ = +π/6 radians.

b Sinusoidal si nal A sin ω t + Φ with hase Φ = +π/6 radians.


33

I t d tiI t d ti




Ex. Generation of a sinusoidal signal ⇒ Fig.Fig. 11--3232.Circuit Eq.:2d 2

0 LC v t v t dt

+ =

0 0( ) cos( ), 0v t V t t ω = ≥

(1.36)

(1.37)

Figure 1.32 (p. 37)

Parallel LC circuit,

assumin that the

where0

1

LC ω = (1.38)

Natural an ular fre uenc

inductor L and capacitorC are both ideal.

of oscillation of the circuit

Discrete-time case :

cos x n n= +

[ ] cos( ) x n N A n N φ + = Ω + Ω + (1.40)

(1.39)

Periodic condition:

2 N mπ Ω = radians/cycle, integer ,m

m N N

π Ω =or

- = φ= = --

(1.41)


34

φ

I t d tiI t d ti




Figure 1.33 (p. 38)

-


35

.

I t d tiI t d ti




Example 1.7 Discrete-Time Sinusoidal Signal

A pair of sinusoidal signals with a common angular frequency is defined by

sin 5 x n nπ = =and

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

(b) Express the composite sinusoidal signal1 2 y n x n x n= +

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

..

(a) Angular frequency of both x1[n] and x2[n]:

5 radians/c cleπ Ω = 2 2 2m m mπ π = = =

5 5π ΩThis can be only for m = 5, 10, 15, …, which results in N = 2, 4, 6, …

cos( ) cos( )cos( ) sin( )sin( )n A n A nφ φ φ Ω + = Ω − Ω

x n + x n with the above e uation to obtain thatLet Ω = 5π then com are


36

Introd ctionIntrod ction




sin( ) 1 and cos( ) 3 A Aφ φ = − =1sin( ) amplitude of [ ] 1

tan x nφ −

= = = = π

2cos( ) amplitude of [ ] 3 x nφ

sin( ) 1 A φ = −

π

( )1

2sin / 6

Aπ

−= =

−Accordingly, we may express y[n] as

[ ] 2cos 56

y n n π π ⎛ ⎞= −⎜ ⎟⎝ ⎠★ 11..66..33 Relation Between Sinusoidal and Complex Exponential SignalsRelation Between Sinusoidal and Complex Exponential Signals

1. Euler’s identity: cos sin je j

θ θ θ = + (1.41) j t

Be ω

e=

j t ω =

( )

j t

e e

Ae φ ω +

=

=( ) cos( ) x t A t ω φ = + (1.35)


37

.cos( ) sin( )t jA t ω φ ω φ = + + +





on nuouson nuous-- me s gna n erms o s ne unc on:me s gna n erms o s ne unc on:

( ) sin( ) x t A t ω φ = + (1.44)

j t ω = .

2. Discrete-time case:

cos( ) Re{ } j n

A n Beφ Ω

Ω + = (1.46) (1.47)and sin( ) Im{ } j n

A n Beφ Ω

Ω + =3. Two-dimensional representation of the complex

exponential e j n for Ω = π /4 and n = 0, 1, 2, …, 7.

.. .. .

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

/ 4π

/ 4π −

Figure 1.34 (p. 41)

Complex plane, showing eight points

uniforml distributed on the unit circle.


38





★ 11..66..44 Exponential Damped Sinusoidal SignalsExponential Damped Sinusoidal Signals

( ) sin( ), 0t x t Ae t

α ω φ α −= + > (1.48)

xamp e or = ,

α = 6, and φ = 0: Fig.Fig.11..3535.

Figure 1.35 (p. 41)

Exponentially damped

sinusoidal signal Ae−at

sin(ω t ), with A = 60 and α

= 6.


39





( )v d L τ τ −∞Ex. Generation of an exponential damped sinusoidal signal

⇒ Fig.Fig. 11--3636.

1 1 t d rcu q.: v v v

dt R Lτ τ

−∞= .

/(2 )

0 0( ) cos( ) 0

t CR

v t V e t t ω

−

= ≥ (1.50)

Figure 1.36 (p. 42)0 2 2

1 1

4 LC C Rω = −where (1.51) /(4 ) R L C >

Parallel LRC, circuit, with

inductor L, capacitor C ,and resistor R all

ompar ng q. . an . , we ave

0 0, 1/(2 ), , and / 2 A V CRα ω ω φ π = = = = -

assume o e ea .

[ ] sin[ ]n x n Br n φ = Ω + (1.52)

★11..66..55 Step FunctionStep Function Figure 1.37 (p. 43)

x [n]

Discrete-time case:

{1, 00, 0[ ] n

nu n ≥

<= (1.53)

Discrete-time version

of step function of unit

amplitude. n

1


40

Fig.Fig. 11--3737.. −−−





1, 0( )

t u t

>=

Continuous-time case:

(1.54)

Figure 1.38 (p. 44)

Continuous-time

version of the unit-step, unc on o un

amplitude.

Example 1.8 Rectangular PulseConsider the rectangular pulse x(t) shown in Fig.Fig. 11..3939 (a).(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, 0.5 A t x t

t ⎧ ≤ <= ⎨ >⎩1. Rectangular pulse x(t): (1.55)

1 1( )

2 2 x t Au t Au t

⎞⎛ ⎛ = + − −⎜ ⎟ ⎜ ⎟⎝ ⎝ ⎠

(1.56)

Example 1.9 RC Circuit

Find the response v(t) of RC circuit shown in Fig.Fig. 11..4040 (a).(a).

<Sol.><Sol.>


41





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 x 1(t ) and x 2(t ), respectively. Note that x (t )

= x (t ) – x (t ).


42




IntroductionIntroduction

Figure 1.40 (p. 45)

,

the voltage source. (b) Equivalent circuit, using a step function to replace theaction of the switch.

1. Initial value: (0) 0v =

0( )v V ∞ =2. Final value:

( )/( )0( ) 1 ( )t RC

v t V e u t −= − (1.57)

. omp e e so u on:


43





★11..66..66 Impulse FunctionImpulse Function

Figure 1.41 (p. 46)

Discrete-time form of

impulse. Discrete-time case:

,[ ]

0, 0n

nδ =

≠ (1.58)

Fig.Fig. 11..4141δ δ

Figure 1.41 (p. 46)

Discrete-time form of impulse.

δ aδ

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


44

o a rec angu ar pu se o area a o approac zero.





Continuous-time case:

( ) 0 for 0t t δ = ≠ (1.59)Dirac delta function

( ) 1t dt δ −∞

= (1.60)

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

more c ose y.

Fig.Fig. 11..4242..

2. Mathematical relation between impulse and rectangular pulse function:

0

( ) lim ( )t x t δ Δ

Δ→

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

2. x (t): Unit area.Fig.Fig. 11..4242 (a).(a).

3. δ(t) is the derivative of u(t):

(1.62)

4. u(t) is the integral of δ(t):

( ) ( )t

u t d δ τ τ −

= (1.63)

Example 1.10 RC Circuit (Continued)

For the RC circuit shown in Fig.Fig. 11..4343 (a),(a), determine the current i (t) that flows

≥


45

roug e capac or or ≥ .





<Sol.><Sol.>

Fi ure 1.43 . 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 t i t C = 0 0

( )( ) ( )

du t i t CV CV t δ = =


46

t t





Properties of impulse function:

1. Even function: ( ) ( )t t δ δ − = (1.64)

2. Siftin ro ert :0

1lim ( ) ( ) x at t a

δ ΔΔ→= (1.68)

0 0( ) ( ) ( ) x t t t dt x t δ ∞

−∞− =∫ (1.65)

Ex. RLC circuit driven by impulsive

source: Fig.Fig. 11..4545..

. -

1( ) ( ), 0at t a

aδ δ = > (1.66)

001 I +

or g.g. .. aa , e vo age across

the capacitor at time t = 0+ is

<p.f.><p.f.> 0 00C C − .

Fig.Fig. 11..4444

1. Rectangular pulse approximation:

0ma x aΔΔ→

= (1.67)

2. Unit area pulse: Fig.Fig. 11..4444(a).(a).

me sca ng: g.g. .. ..

Area = 1/ a

Restoring unit area ax ( at)


47





Figure 1.44 (p. 48)

- .

pulse x ∆

(t ) of amplitude 1/ ∆

and duration ∆

, symmetric about the origin. (b) Pulse x ∆

(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

r ven y an mpu s ve

current signal. (b) Series

LRC circuit driven by an


48

u v v .




★ 11..66..77 Derivatives of The ImpulseDerivatives of The Impulse

1. Doublet:

1

ProblemProblem 11..24242

(2) d ∞− =

0( ) lim ( / 2) ( / 2)t t t δ δ δ

Δ→= + Δ − − Δ

Δ(1.70)

2. Fundamental property of the doublet:

02 t t dt

=−∞

( )0( ) ( ) ( ) |

nn

t t n

d

f t t t dt f t δ

∞

=− =(1)( ) 0t dt δ ∞

−∞=∫ (1.71)

(1) d ∞− =

−∞

00 t t dt

=−∞.

2 (1) (1)− −

3. Second derivative of impulse:

(1

2 0( ) ( ) limt t

t dt δ δ

Δ→= =

∂ Δ(1.73)

★ 11..66..88 Ramp FunctionRamp Function

1. Continuous-time case:

, 0( )

t t r t

≥⎧= ⎨ (1.74) ( ) ( )r t tu t =or (1.75) Fig.Fig. 11..4646


49

,





, 0[ ]

n nr 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)

-

x [n]Fig.Fig. 11..4747..

Example 1.11 Parallel Circuit

of the ramp function.ons er e para e c rcu o

Fig. 1-48 (a) involving a dccurrent source I0 and an

n1 2 3 40−1−2−3

n a y unc arge capac or .

The switch across the capacitor is suddenly

opened at time t = 0. Determine the current i(t)

ow ng roug e capac or an evoltage v(t) across it for t ≥ 0.

<Sol.><Sol.>


50

. apac or curren : 0u=




2. Capacitor voltage:

1( ) ( )

t

v t i d τ τ −∞

= ∫Figure 1.48 (p. 52)

(a) Parallel circuit

consisting of a

0

1( ) ( )

t

v t I u d τ τ −∞

=

∫

current source,

switch, and

capacitor, the

0 for 0

t

I

<⎧⎪

=

capacitor is initially

assumed to be

uncharged, and the

0

ort t

C I

>⎪⎩

sw c s opene a

time t = 0. (b)Equivalent circuit

0

C

I

r t =

action of opening

the switch with the

C .


51




11..77 Systems Viewed as Interconnections of OperationsSystems 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 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.Fig. 11--4949 (a) and (b).(a) and (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] [ 2])3

y n x n x n x n= + − + −

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


52

representation for it.




<Sol.><Sol.> 1. Discrete-time-shift operator Sk: Fig.Fig. 11..5050.

Shifts the input x[n] by k time units to

roduce an out ut e ual to x n k .

21=

Figure 1.50 (p. 54)

Discrete-time-shift operator

Sk o eratin on the discrete-

2. Overall operator HH for the moving-average

system:

3 time signal x [n] to produce

x [n – k ].

.. ..

Fig.Fig. 11--5151 (a): cascade form; Fig.(a): cascade form; Fig. 11--5151 (b): parallel(b): parallel

form.form.

11..88 Properties of SystemsProperties of Systems★ 11..88..11 StabilityStability

. - , -

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 all y y t M t ≤ < ∞ (1.80)

whenever the input signals x(t) satisfy the condition Both Mx and My

represent some finite


53

x . positive number






One famous example of an unstableOne famous example of an unstablesystem:system:

. .

Dramatic photographs showing the

collapse of the Tacoma Narrows

, .(a) Photograph showing the twisting

motion of the bridge’s center span just

.

(b) A few minutes after the first piece of

concrete fell, this second photograph

shows a 600-ft section of the brid e

breaking out of the suspension span and

turning upside down as it crashed in Puget

Sound, Washin ton. Note the car in thetop right-hand corner of the photograph.

(Courtesy of the Smithsonian Institution.)

Signals_and_Systems_SimonHaykin & Barry Van Veen 55




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 x x n M n≤ < ∞

2. Input-output relation:

( )1

[ ] [ ] [ 1] [ 2]3

y n x n x n x n= + − + −

[ ] [ ] [ 1] [ 2]3 y n x n x n x n= + − + −

[ ] [ 1] [ 2]3

1

x n x n x n≤ + − + − The movingThe moving--averageaverage

system is stable.system is stable.

3

x x x

x M =





Example 1.14 Unstable system

Consider a discrete-time system whose input-output relation is defined by

n

where r > 1. Show that this system is unstable.

<p.f.><p.f.>1. Assume that: [ ] for all x x n M n≤ < ∞

2. We find that

[ ] [ ] [ ] y n r x n r x n= = ．

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

The system is unstable.The system is unstable.

★ 11..88..22 MemoryMemory

sys em s sa o possess memory s ou pu s gna epen s on pas orfuture values of the input signal.

A system is said to possess memoryless if its output signal depends only on


e presen va ues o e npu s gna .




Ex.: Resistor 1

( ) ( )i t v t R= Memoryless !

Ex.: Inductor ( ) ( )t

i t v d L

τ τ −∞

= ∫ Memory !

Ex.: Movin -avera e s stem

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

3 y n x n x n x n= + − + − Memory !

Ex.: A system described by the input-output relation

2[ ] [ ] y n x n= Memoryless !

★ 11..88..33 CausalityCausality

A system is said to be causal if its present value of the output signal depends

on y on e presen or pas va ues o e npu s gna .A system is said to be noncausal if its output signal depends on one or more

future values of the input signal.





Ex.: Moving-average system

11 2n x n x n x n= + − + − Causal !

Ex.: Moving-average system

[ ] ( [ 1] [ ] [ 1])3

y n x n x n x n= + + + − Noncausal !

♣ A causal s stem must be ca able of o eratin in real timereal time.

★11..88..44 InvertibilityInvertibility

A system is said to be invertible if the

from the output.Figure 1.54 (p. 59)

The notion of system invertibility. The1. Continuous-time system: Fig.Fig. 11..5454.

second operator H inv is the inverse of the

first operator H . Hence, the input x (t ) is

passed through the cascade correction

x(t) = input; y(t) = output

H = first system operator;

H inv = second system operator


o an nv comp e e y unc ange .




2. Output of the second system:

{ } { }{ } { }( ) ( ) ( )inv inv inv H y t H H x t H H x t = = H

inv

= inverseoperator

3. Condition for invertible system:

inv H H I = (1.82)I = identity operator

.

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

0

0( ) ( ) ( )t y 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 t

S y t S S x t S S x t − − −= =

2. Invertibility condition:

0 0t t S S I

− = 0t S

− ≡ Time shift of t0





Example 1.16 Non-Invertible System

Show that a square-law system described by the input-output relation2( ) ( ) y t x t =

is 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.★ 11..88..55 Time InvarianceTime 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 St 0 preceding operator H . (b)

Time-shift operator St 0 following operator H . These two situations are equivalent,


provided that H is time invariant.




1. Continuous-time system:

1 1( ) { ( )} y t H x t =. 1 0

0

2 1 0 1( ) ( ) { ( )}t x t x t t S x t = − = S t 0 = operator of a time shift equal to t0

. u pu o sys em :

0

2 1 0( ) { ( )}t

y t H x t t = −

0

1

1{ ( )}t

HS x t =

.

4. For Fi .Fi . 11--5555 bb the out ut of s stem H is t t :

0

0

1 0 1( ) { ( )}t

t

y t t S y t

S H x t

− =

= 1.840

1{ ( )}t S H x t =

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



t = i t



t i tExample 1.17 Inductor

x (t) = v(t)

The inductor shown in figure is described

by the input-output relation: 1 t

1 1 y t x L

τ τ −∞

=

where L is the inductance. Show that the inductor so described is time invariant.

..

1. Let x1(t) x1(t t0) Response y2(t) of the inductor to x1(t t0) is

1 t

2 1 0 y x

L

τ τ −∞

= −

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

0

1 0 1

1( ) ( )

t t

y t t x d L

τ τ −

−∞− = ∫ (B)

3. Changing variables: 0t τ τ = −

(A)01

' 't t

t x d τ τ −

= Inductor is time invariant.


−∞


y (t) = i(t)



y ( ) ( )Example 1.18 Thermistor

Let R(t) denote the resistance of the thermistor,

expressed as a function of time. We may express the x1(t) = v(t)

input-output relation of the device as

1 1( ) ( ) / ( ) y t x t R t =

Show 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 t y 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

( )( )

y t t R t t

−− =−

≠ for 0t t t t − ≠ ≠


≠




★ .. .. near ynear y

A system is said to be linear in terms of the system input (excitation) x(t) andthe system output (response) y(t) if it satisfies the following two properties of

1. Superposition:

1( ) ( ) x t x t = 1( ) ( ) y t y t = 1 2( ) ( ) ( ) x t x t x t = +2( ) ( ) x t x t = 2( ) ( ) y t y t = 1 2( ) ( ) ( ) y t y t y t = +

2. Homogeneity: a = constant

( ) x t ( ) y t ( )ax t ( )ay t factor

♣ Linearity of continuous-time system

-. .

2. Input: N

x t a x t =

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

Corresponding weighted factor

1i=

3. Output:( ) { ( )} { ( )}

N

i i y t H x t H a x t = = ∑ (1.87)


1i=


N



1( ) ( )

N

i ii

y t a y t == ∑ (1.88)

Superposition and

homogeneity

where

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

4. Commutation and Linearity:

1

( ) { ( )}i i

i

N

y t H a x t =

= ∑

1

{ ( )}i i

i N

a H x t

=

=

=

(1.90) Fig.Fig. 11..5656

1i i

i=

♣ Linearity of discrete-time system Same results, see ExampleExample 11..1919.

-.Consider a discrete-time system described by the input-output relation

[ ] [ ] y n nx n=



66

Show that this system is linear.




∑

Figure 1.56 (p. 64)

.

amplitude scaling and summation precedes the operator H for multipleinputs. (b) The operator H precedes amplitude scaling for each input; the

resu ng ou pu s are summe o pro uce e overa ou pu y . ese

two configurations produce the same output y (t ), the operator H is linear.<p.f.><p.f.>

N

1. Input:

1

[ ] [ ]i i

i

x n a x n=

= ∑



67

.




Example 1 21 Impulse Response of RC Circuit



Example 1.21 Impulse Response of RC Circuit

For the RC circuit shown in Fig.Fig. 11..5757,determine the impulse response y(t).


RC circuit for Example 1.20, in

<Sol.><Sol.>

1. Recall: Unit step response

/t RC −which we are given the capacitor

voltage y (t ) in response to the

step input x (t ) = y (t ) and the

,− .

2. Rectangular pulse input: Fig.Fig. 11..5858.

x t = x t

1/Δ

requirement is to find y (t ) in

response to the unit-impulseinput x (t ) = δ (t ).

11( ) ( ) x t u t Δ= +

Figure 1.58 (p. 66)

Rectangular pulse of unit2

1( ) ( )

2 x t u t

Δ= −

Δarea, w c , n e m ,approaches a unit impulse

as ∆→0.

3. Response to the step

functions x1(t) and x2(t):



69

−


Δ ⎞⎛



/( )2

1 11 1 , ( ) ( )2

t RC

y e u t x t x t

Δ ⎞⎛ − +⎜ ⎟⎝ ⎠⎡ ⎤ Δ ⎞⎛ = − + =⎢ ⎥ ⎜ ⎟Δ ⎝ ⎠⎢ ⎥

/( )2

2 2

11 , ( ) ( )

t RC

y e u t x t x t

Δ ⎞⎛ − −⎜ ⎟⎝ ⎠

⎡ ⎤ Δ ⎞⎛ = − − =⎢ ⎥ ⎜ ⎟

Next, recognizing that

1 2 x x xΔ = −

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

t RC t RC y t e u t e u t

− +Δ − −ΔΔ = − + Δ − − − Δ

( ) ( )( / 2) /( ) ( / 2) /( )1 1( ( / 2) ( / 2)) ( ( / 2) ) ( / 2))t RC t RC u t u t e u t e u t

− +Δ − −Δ= + Δ − − Δ − + Δ − − ΔΔ Δ

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

( ) lim ( )t x t δ =



70

0Δ→


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



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

1lim

d z t z t z t

Δ Δ= + − −

0 2 2dt Δ→ Δ

0

( ) lim ( ) y t y t Δ

Δ→

=

/( ) ( ) ( ( ))t RC d t e u t

dt δ −= −

./( ) /( )

( ) ( ) ( ) ( )

t RC t RC d d

t e u t u t edt dt δ

− −

= − −/( ) /( ) ( ) ( ) ( ), ( ) ( )t RC t RC

t e t e u t x t t RC

δ δ δ − −= − + =

/( )1( ) ( ), ( ) ( )t RC y t e u t x t t

RC δ −= = (1.93)



71


11 99 N iN i



11..99 NoiseNoiseNoise ⇒ 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.Fig. 11..6060..

★ 11..99..11 Thermal NoiseThermal 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

lim ( )2 T T

v v t dt T −→∞

= ∫ (1.94) As T → ∞, 0v → Refer to Fig.Refer to Fig. 11..6060..

2. Time-average-squared value: k = Boltzmann’s constant =

2 21lim ( )

2

T

T T v v t dt

T −→∞= ∫ (1.95)

→ 2 2

1.38 × 10 23 J/K

Tabs = absolute temperature



72

→ ∞, vo sabs

v = .




Figure 1.60

(p. 68)

Sample waveform

of electrical noise

generated by a

thermionic diode

with a heated

cathode. Note

a e me-

averaged value ofthe noise voltage

approximately

zero.



73


Thevenin’s equvalent circuit: Fig.Fig. 11..6161(a),(a), Norton’s equivalent circuit:



♣ Thevenin s equvalent circuit: Fig.Fig. 11..6161(a),(a), Norton s equivalent circuit:

Fig.Fig. 11..6161(b).(b).Noise voltage generator:

2( )v t v=Noise current generator:

2 2

2

lim ( )2

4 amps

T T

abs

i dt T R

kT G f

−→∞=

= Δ(1.97)

Figure 1.61 (p. 70)

where G = 1/R = conductance [S].

♣ Maximum power transfertheorem: the maximum

(a) Thévenin equivalent circuit of a noisy resistor.

(b) Norton equivalent circuit of the same resistor.possible power is transferred

from a source of internal

resistance Rl when R = Rl.

Under matched condition, the available power is



74

abs


Two operating factor that affect available noise power:



♣ Two operating factor that affect available noise power:

1. The temperature at which the resistor is maintained.

2. The width of the fre uenc band over which the noise volta e across the

resistor is measured.

★ 11..99..22 Other Sources of Electrical NoiseOther Sources of Electrical Noise

.2. Ex. Photodetector:

1) Electrons are emitted at random times, τk, where ∞ < τk < ∞

o a curren ow ng roug p o o e ec or:

( ) ( )k

k

x t h t τ ∞

=−∞= −∑ (1.98)

( )k h t τ −where is the current pulse generated at time τk.

3. 1/f noise: The electrical noise whose time-averaged power at a given

.



75


11..1010 Theme ExampleTheme Example



pp

★ 11..1010..11 Differentiation and Integration:Differentiation and Integration: RCRC CircuitsCircuits

1. Differentiator ⇒ Sharpening of a pulse

differentiator ( ) ( )

d y t x t

dt = (1.99)

mp e c rcu : g.g. .. .

2) Input-output relation:

1d d v t v t v t + = 1.100

Figure 1.62 (p. 71)

Simple RC circuit with small time

constant, used as an approximator

dt RC dt

If RC (time constant) is small enough

such that (1.100) is dominated by the

to a differentiator.second term v2(t)/RC, then

2 1

1( ) ( )

d v t v t ≈ 2 1( ) ( ) for small

d v t RC v t RC ≈ (1.101)



76


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



p ( ) 1( ) p y( ) 2( )

2. Integrator ⇒ smoothing of an input signal

x t t

1) Simple RC circuit: Fig.Fig. 11..6363.

integrator ( ) ( ) y t x d τ τ −∞

= (1.102)

npu -ou pu re a on:

2 2 1( ) ( ) ( )d

RC v t v t v t dt

+ =

2 2 1( ) ( ) ( )

t t

RCv t v d v d τ τ τ τ −∞ −∞+ =∫ ∫ (1.103)If RC time constant is lar e enou h

such that (1.103) is dominated by the

first term RCv2(t), then

. .

Simple RC circuit with large time

constant used as an approximator

2 1( ) ( ) RCv t v d τ τ −∞≈

1 t

≈

.

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



77

RC −∞ output: y(t) = v2(t)


★ 11..1010..22 MEMS Accelerometer MEMS Accelerometer



1. Model: second-order mass-damper-spring system

Fig.Fig. 11..6464..

M = proof mass, K = effective spring constant, D =

damping factor, x(t) =external acceleration, y(t)

=dis lacement of roof mass Md2

t /dt2

= inertial force of proof mass, Ddy(t)/dt = damping force,

Ky(t) = spring force.

Figure 1.64 (p. 73)

Mechanical lumped model

. .2

2( ) ( )( ) ( )d y t dy t x t M D Ky t dt dt = + +

.2

2

( ) ( )( ) ( )

d y t D dy t K y t x t

dt M dt M + + = (1.105)

a ura requency:

n

K

M ω = (1.106)[rad/sec]

ua y ac or:KM

Q D

= (1.107)



78


2

(1.105)2

2( ) ( )n

n

y y y t x t

ω ω + + = (1.108)



(1.105) 2( ) ( )n y

dt Q dt

(1.108)

★ 11..1010..33 Radar Range MeasurementRadar Range Measurement

1. A periodic sequence of radio frequency (RF) pulse: Fig.Fig. 11..6565.

T0 = duration [μsec], 1/T = repeated frequency, f c = RF frequency [MHz~GHz]

Figure 1.65 (p. 74)

Periodic train of rectangular FR pulses used for measuring molar ranges.

♣



79

♣ .






Figure 1.66a(p. 75)

(a) Fluctuations in

the closing stock

price of Intel over

a three-year

period.



81




Figure 1.66b

(b) Output of a

four-point

movin -avera esystem.

N = 4 case



82




Figure 1.66c(p. 76)

(c) Output of an

eight-point

moving-average

sys em.

N = 8 case



83


2. For a general moving-average system, unequal weighting is applied to past



values of the input:1 N =

0k

k =

= − .

★ 11..1010..55 Multipath Communication ChannelsMultipath Communication Channels

. anne no se egra es e per ormance o a commun ca on sys em.

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.Fi .Fi . 11..6767..

♣ For a digital communication, multipath propagation manifests itself in the

form of intersymbol interference (ISI).

⇒. - ⇒ .. .. .

0

( ) ( ) p

i diff

i

y t x t iT ω =

= −∑ (1.112) Tdiff = smallest time difference



84




Figure 1.67 (p. 77)



85

.




Figure 1.68 (p. 78)

Tapped-delay-line model of a linear communication channel, assumed to be time-



86

invariant.


5. PTdiff = the longest time delay of any significant path relative to the arrival of

th i l



the signal.

For P =1, then

0 1( ) ( ) ( )diff

y t x t x t T ω ω = + −

ω0x(t) = direct path, ω1x(t Tdiff ) = single reflected path

6. Discrete-time case: Linearly weighted

0

[ ] [ ]k

k

y n x n k ω

=

= −∑ (1.113)

=

mov ng-average sys em

The term recursive,

[ ] [ ] [ 1] y n x n ax n= + − (1.114)

★ 11..1010..66 Recursive DiscreteRecursive Discrete--Time ComputationsTime Computations

signifies the dependence

of the output signal on its

own past values.

1. First-order recursive discrete-time filter: Fig.Fig. 11..6969.

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



87

where ρ is a constant.


Figure 1.69 (p. 79)

Bl k di f fi t d



Block diagram of first-orderrecursive discrete-time filter. The

opera or s s e ou pu s gna

y [n] by one sampling interval,

producing y [n – 1]. The feedback

ρ stability of the filter.

♣ Fig. 1.69: linear discrete-time feedback system.

2. Solution of Eq.(1.115):

(1.116)[ ] [ ]k y n x n k ρ ∞

= −∑ r ≡ ρ

1

[ ] [ ] [ ]k

k

y n x n x n k ρ ∞

=

= + −∑

=

(1.117)

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

1[ ] [ ] [ 1 ] [ ] [ 1 ]l l y n x n n l x n n l ρ ρ ρ

∞ ∞+= + − − = + − −∑ ∑ (1.118)



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0 0l l= =


[ ] [ ] [ 1] y n x n y n ρ = + −



3. Three special cases (depending on ρ):Accumulator

ρ = :

0

[ ] [ ]k

y n x n k ∞

=

= −∑(1.116) (1.119)

2) 1: ρ < Leaky accumulator

3) 1: ρ > Amplified accumulator

Stable in BIBO senseStable in BIBO sense

Unsatble inUnsatble in11..1111 Exploring Concepts with MATLABExploring Concepts with MATLAB

♣ MATLAB Signal Processing Toolbox1. Time vector: Sam lin interval T of 1 ms on the interval from 0 to 1 s

BIBO senseBIBO sense

t = 0:.001:1;

2. Vector n: n = 0:1000

★ 11..1111..11 Periodic SignalsPeriodic Signals

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



89

square w , r o ;


Ex. Obtain the square wave shown in Fig.Fig. 11..1414 (a)(a) by using MATLAB.

<Sol ><Sol > >> A = 1;



<Sol.><Sol.> >> A = 1;

>> w0 =10* i

>> rho = 0.5;

>> t = 0:.001:1;

>> s = A*s uare w0*t, rho ;

>> plot (t, sq)

>> axis([0 1 -1.1 1.1])

. , ,

A*sawtooth(w0*t, w);

Ex. Obtain the triangular wave shown in Fig.Fig. 11..1515 by using MATLAB.

>> A = 1;

>> w0 =10*pi;

=

<Sol.><Sol.>

.

>> t = 0:0.001:1;

>> tri = A*sawtooth(w0*t, w);

>>



90

,


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

x = vector n = discrete time vector



x = vector, n = discrete time vector Ex. Obtain the discrete-time s uare wave shown in Fi .Fi . 11..1616 b usin MATLAB.

>> A = 1;

>> omega =pi/4;

<Sol.><Sol.>

>> 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.Fig. 11..2828 (a)(a) by using

.

>> B = 5;

>> a = 6;

<Sol.><Sol.>

>> = :. : ;>> x = B*exp(-a*t); % decaying exponential

>> plot (t, x)



91


Ex. Obtain the growing exponential signal shown in Fig.Fig. 11..2828 (b)(b) by using

MATLAB



MATLAB.>> B = 1;

<Sol.><Sol.>

>> 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.Fig. 11..3030 (a)(a) by using

.<Sol.><Sol.>

>> B = 1;

>> r = 0.85;

>> n = - : ;

>> x = B*r.^n; % decaying exponential sequence

>> stem (n, x)

★ 11..1111..33 Sinusoidal SignalsSinusoidal Signals

1. Cosine signal: 2. Sine signal:

A = amplitude, w0 =frequency, phi =

phase angle



92

cos w p ; s n w p ;


Ex. Obtain the sinusoidal signal shown in Fig.Fig. 11..3131 (a)(a) by using MATLAB.

>> A = 4;<Sol ><Sol >



>> A = 4;

>> w0 =20* i

<Sol.><Sol.>

>> phi = pi/6;

>> t = 0:.001:1;

>> cosine = A*cos w0*t + hi ;

>> plot (t, cosine)

Ex. Obtain the discrete-time sinusoidal signal shown in Fig.Fig. 11..3333 by using

.

>> A = 1;

>> omega =2*pi/12; % angular frequency

<Sol.><Sol.>

.*.* ≡ element-by-element>> n = - : ;

>> y = A*cos(omega*n);

>> stem (n, y)

mu p ca on

★ 11..1111..44 Exponential Damped Sinusoidal SignalsExponential Damped Sinusoidal Signals1. Exponentially damped sinusoidal signal:

at −= * * * - *



93

0 .


Ex. Obtain the waveform shown in Fig.Fig. 11..3535 by using MATLAB.

<Sol ><Sol > >> A = 60;



<Sol.><Sol.> >> A 60;>> w0 =20* i

>> phi = 0;

>> a = 6;

>> t = 0:.001:1;

>> expsin = A*sin(w0*t + phi).*exp(-a*t);

>> plot (t, expsin)

. .. ..

using MATLAB.

<Sol.><Sol.> >> A = 1;

>> omega =2*pi/12; % angular frequency

>> n = -10:10;

>> y = A*cos(omega*n);

>> r = . ;>> x = A*r.^n; % decaying exponential sequence

>> z = x.*y; % elementwise multiplication



94

s em n, z




Figure 1.70 (p. 84)



95

.


★ 11..1111..55 Step, Impulse, and Ramp FunctionsStep, Impulse, and Ramp Functions

MATLAB command:

Ex.Ex. Generate a rectangular pulse

centered at origin on the interval[ 1 1]



MATLAB command:1. M-by-N matrix of ones: ones (M, N)

centered at origin on the interval[-1, 1].

2. M-by-N matrix of zeros: zeros (M, N)

♣ Unit amplitude step function:

<Sol.><Sol.>

= -u = zeros , , ones , ;

♣ Discrete-time impulse:

=

>> u1 = [zeros(1, 250), ones(1, 751)];

>> u2 = [zeros(1, 751), ones(1, 250)];

> > u = u 1 – u 2 , , , ,

♣ Ramp sequence:

ramp = 0:.1:10

★ 11..1111..66 User User--Defined FunctionsDefined Functions

1. Two types M-files exist: scripts and functions.

cr p s, or scr p es au oma e ong sequences o comman s; unc ons, orfunction files, provide extensibility to MATLAB by allowing us to add new functions.

2. Procedure for establishing a function M-file:



96


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



p g2) It also includes additional statements that compute the values to be returned.

e npu s may e sca ars, vec ors, or ma r ces.

Ex. Obtain the rectangular pulse depicted in Fig.Fig. 11..3939 (a)(a) with the use of an

M-file.

-< o .>< o .>>> function g = rect(x)

>> g = zeros(size(x));

>> = <=

. element vector containing the row

and column dimensions of a matrix.

.

>> g(set1) = ones(size(set1));

.

indices of a vector or matrix that

satisfy a prescribed relation3. The new function rect.m can be used

Ex. find(abs(x)<= T) returns the indices

of the vector x, where the absolute

.

>> = -

Ex. To generate a rectangular pulse:

.

>> plot(t, rect(t));



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