Pusan National University

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Intelligent Robot Lab

Pusan National University

Intelligent Robot Lab

Chapter 13 Signals and Sampling Theory

Pusan National University Intelligent Robot Laboratory

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Intelligent Robot Lab

v Introduction

v Examples of Digital Control Systems v Data Conversion and Quantization

v Mathematical Modeling of the Sampling Process v Sampling Theorem

v Ideal Sampling

v Data Reconstruction and Filtering of Sampled Signals v Zero-Order Hold

v First-Order Hold

v Introduction to signal and sampling theory

v This chapter introduces discrete-data control system and analysis of signal conversion and processing.

v The digital computer can perform two functions:

§ Supervisory external to the feedback loop

• Examples of supervisory functions consist of scheduling tasks, monitoring parameters and variables for out of compensation heretofore discussed.

§ Control internal to the feedback loop

• Examples of control functions are lead and lag compensations.

Introduction

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v Advantages of digital computers

§ Reduced cost

• In the steel industry, a single digital computer can replace numerous analog controllers with a subsequent reduction in cost.

§ Flexibility in response to design changes

• Any changes or modifications that are required in the future can be implemented with simple software changes rather than expensive hardware modifications.

§ Noise immunity

• Digital systems exhibit more noise immunity than analog systems by virtue of the methods of implementation.

Introduction

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v Closed-Loop Drug Delivery System

§ To design a closed-loop drug system, a sensor is utilized to measure the levels of the regulated drug or nutrient in the blood. This measurement is converted to digital form and fed to the control computer, which drives a pump that injects the drug into the patient’s blood as shown in figure 13.1.

Examples of Digital Control Systems

Figure 13.1 Drug delivery digital control system:

a. Schematic of drug delivery system; b. block diagram of a drug delivery system;

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v Computer Control of an Aircraft Turbojet Engine

§ To achieve the high performance required for today’s aircraft, turbojet engines employ sophisticated computer control strategies. The control requires feedback of the engine state(speed, temperature, and pressure), measurements of the aircraft state(speed and direction), and pilot command as shown in Figure 13.2.

Examples of Digital Control Systems

Figure 13.2 Turbojet engine control system:

a. F-22 military fighter aircraft; b. Block diagram of an engine control system;

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v Control of a Robotic Manipulator

§ To perform task of Robotic manipulators, manipulator hand(or end-effector) positions and velocities are controlled digitally. A simple robotic manipulator is shown in figure 13.3.a, and a block diagram of its digital control system is shown in figure 13.3.b.

Examples of Digital Control Systems

Figure 13.3 Robotic manipulator control system:

a. 3-D.O.F. robotic manipulator; b. Block diagram of a manipulator control system;

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v Digital computer

§ Typically, the computer replaces the cascade compensators and is thus positioned at the place shown in Figure 13.4 a.

Data Conversion and Quantization

Figure 13.4

a. Placement of the digital computer within the loop;

b. detailed block diagram showing placement of A/D and D/A converters

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v Digital to Analog Conversion is simple and effectively instantaneous.

v Properly weighted voltages are summed together to yield the analog output.

v Example

§ The three bit binary code is represented by the switches.

§ Thus, if the binary number is 110₂, the center and bottom switches are on, and the analog output is 6 volts.

§ In actual usage, the switches are electronic and are set by the input binary codes.

Data Conversion and Quantization

Switch ON

6 volt

Figure 13.5 Digital-to-analog converter

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v Analog to Digital Conversion

§ In an analog to digital converter, the analog signal is first

converted to a sampled signal and then converted to a sequence of binary numbers, i.e., the digital signal.

§ The sampling rate must be at least twice the bandwidth of the signal, or else there will be distortions.

§ This minimum sampling frequency is called the Nyquist Sampling Rate.

Data Conversion and Quantization

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v Steps in Analog to Digital Conversion

§ Figure 13.6(a) is the analog signal.

§ In Figure 13.6(b), we see the analog signal sampled at periodic intervals and held over the sampling interval by a device called a zero-order sample and hold (Z.O.H) that yields a staircase approximation to the analog signal.

Data Conversion and Quantization

Figure 13.6 Steps in analog-to-digital conversion :

a. analog signal; b. analog signal after sample-and-hold;

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v Steps in Analog to Digital Conversion

§ After sampling and holding, the analog-to-digital converter converters the sample to a digital number (as shown in Figure 13.6(c)), which is arrived at in the following manner.

§ The dynamic range of the analog signal’s voltage is divided into discrete levels, and each level is assigned a digital number.

Data Conversion and Quantization

Figure 13.6 Steps in analog-to-digital conversion : c. conversion of samples to digital numbers

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v Steps in Analog to Digital Conversion

§ Quantization level

• [V] : M is the maximum analog voltage

• [V] : n is the number of binary bits

§ Looking at Figure 13.3(b), we can see that there will be an associated error for each digitized analog value except the voltages at the

boundaries such as M/8 and 2M/8.

§ We call this the quantization error.

§ Maximum of quantization error :

Data Conversion and Quantization

/ 2ⁿ M

/ 8 M

2

1

/ )

2 / )(

2 / 1

( M

ⁿ

= M

ⁿ⁺

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v The analog number is rounded off or truncated into a digital number.

§ In the round-off operation

Data Conversion and Quantization

Figure 13.7

a. input-output relationship of an A/D 3 bit round-off quantizer;

• Figure 13.7(a) illustrates the round-off quantization relation between the analog and the digital binary integral codes for a 3-bit word for positive and negative

number.

• Example: the number 3.55 is rounded off to 3.6, and the number -3.55 is rounded off to -3.6

• the maximum quantization error is ±q 2

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v Sampling

§ The computer is working with a quantized amplitude representation of the analog signal formed from values of the analog signal at discrete intervals of time.

§ Ignoring the quantization error, the computer performs just as the

compensator does, except that signals pass through the computer only at the sampled intervals of time.

§ The sampling of data has an unusual effect upon the performance of a closed-loop feedback system, since stability and transient response are now dependent upon the sampling rate.

§ If the sampling rate is too slow, the system can be unstable since the values are not being updated rapidly enough.

Sample-And-Hold Devices

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v Sample and Hold Process

§ The fact that signals are sampled at specified intervals and held causes the system performance to change with changes in sampling rate.

§ Basically, then, the computers effect upon the signal comes from this sampling and holding.

§ In order to model digital control systems, we must come up with a mathematical representation of this sample and hold process

.

Sample-And-Hold Devices

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v Sample-and-hold devices are used extensively in digital and sampled- data control systems.

§ The S/H operation is conceptually illustrated by the circuit shown in Figure 13.8(a)

§ When the switch is closed, the S/H device samples and tracks the input signal

§ When the switch is opened, the output is held at the voltage that the capacitor is charged.

§ Figure 13.8(b) illustrates typical input and output signal of the sample S/H devices as the is zero.

Sample-And-Hold Devices

s( ) e t

Figure 13.8

a. A simple circuit illustrating the sample- and-hold principle;

Rs

Figure 13.8

b. Simplified sample-and-hold signals. =0 ;^R^s

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v Sample-and-hold devices are used extensively in digital and sampled- data control systems.

§ Figure 13.8(c) illustrates a typical input signal and the corresponding output of a practical S/H device.

Sample-And-Hold Devices

s( ) e t

Figure 13.8

c. Input and output S/H signals with finite time delays ;

• Acquisition Time ( ).

The time when The S/H output enters and remains within a specified error band around input signal.(As Sample command is given)

Ta

• Aperture Time ( ).

The time between the issuance of the hold command and the time the sampler is opened.

Tp

• Settling Time ( ).

The time required for the transient oscillation to settle to within a certain percent of FS.

Ts

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v Sampling operation in sampled data and digital control systems is used to model either the sample-and-hold operation or the fact that the

signal is digitally coded.

v If the sampler is used to present S/H and A/D operations, it may involve delays, finite sampling duration, and quantization effects.

v It is also common to find digital control systems that contain multiple samplers with different sampling rates.

v We shall now examine some of the various sampling operations and develop mathematical models for the sampler so that the operations can be used for analytical purposes.

Mathematical Modeling of the Sampling Process

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v Finite-pulse width Sampler

§ The most common type of modulation is the pulse-amplitude-modulation(PAM).

§ p: pulse or sampling duration; T: sampling period.

§ The finite sampling duration characteristic of a sampler should be considered if the value of p is not negligible when compared with the sampling period T.

§ Figure 13.9(a) shows the block diagram representation of a periodic sampler with finite sampling duration( :continuous-time function. : train of finite-width pulses.

§ Figure 13.9(b) shows an equivalent block diagram representation of the sampler as a pulse-amplitude modulator.

Mathematical Modeling of the Sampling Process

Figure 13.9

a. A uniform-rate sampler with finite sampling duration.

Figure 13.9

b. Pulse-amplitude modulator as a sampler ( )

f t f_p^*( )t

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v Finite-pulse width Sampler

§ Figure 13.9(c) illustrates typical waveforms of the input signal , the carrier , and the output .

§ Typical input and output signals of a pulse width modulator are illustrated in figure 13.9(d)

Mathematical Modeling of the Sampling Process

( )

f t p t( )

p( ) f^* t

Figure 13.9

c. Typical input and output waveforms of a uniform-rate sampler

Figure 13

d. Typical input and output signals of a pulse width sampler.

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v Finite-pulse width Sampler

§ The unit pulse train contains pulses each with a unit magnitude can be pressed as

Sampling operation begins at , and the leading edge of the pulse at

§ The output of the sampler is written as

Mathematical Modeling of the Sampling Process

( ) [ (_s ) _s( )] ( )

k

p t u t kT u t kT p p T

¥

=-¥

=

å

- - - - <

( ) p t

(13.1)

*( ) ( ) ( )

fp t = f t p t _(13.2)

Figure 13.10 The unit pulse train.

t = -¥ t =0

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v Finite-pulse width Sampler

§ Substituting Eq.(3.1) into Eq.(3.2) , we get

Pulse train generally contains much higher frequency components than . So, the sampler is regarded as harmonic generator.

§ is periodic function with period T, it can be represented by a Fourier series

§ denotes the complex Fourier series coefficients and is given by

Mathematical Modeling of the Sampling Process

(13.3)

*

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

p s s

k

f t f t u t kT u t kT p p T

¥

=-¥

= å - - - - <

(13.4)

*( )

fp t f t( )

( ) p t

( ) _n ^jn ^s^t

k

p t C e ^w

¥

=-¥

=

å

⁽^w_s ⁼ ²^p^T⁾

Cn

0

1 ^T ( ) ^jn ^s^t

Cn p t e dt

T

w

=

ò

- ^(13.5)

(24)

v Finite-pulse width Sampler

§ Since for , Eq. (13. 5) becomes

§ Using well-known trigonometric identities, is written

§ Substituting Eq.(13.7) into Eq.(13.4) , we get

Mathematical Modeling of the Sampling Process

(13.6)

(13.7)

(13.8)

( ) 1

p t = 0£ £t p

0

1 ^p _jn _s_t 1 ^jn ^s^p

n

s

C e dt e

T jn T

w w

w

-

- -

=

ò

=

Cn

sin( 2) 2

2

jn s p s

n

s

n p

C p e

T n p

w

= -

sin( 2) 2

( ) 2

s s

jn p jb t s

n s

n p

p t e e

n p

w w

¥

-

=-¥

=

å

(25)

v Finite-pulse width Sampler

§ Substitution of from Eq.(13.4) into Eq.(13.2) yields

§ The Fourier transform of is obtained as

§ Using the complex shifting theorem of the Fourier Transform

, the Fourier transform of Eq.(13.9) is written

Mathematical Modeling of the Sampling Process

(13.9)

(13.10)

(13.11) ( )

p t

( ) ( ) ^jn ^s^t

p n

n

f t C f t e ^w

¥

*

=-¥

=

å

p ( ) f ^* t

( ) ( )

^{j t}

p p p

F

^*

j w f

^*

t

^¥

f e

^{* -} ^w

dt

-¥

é ù

= ^F ë û = ò

( ) ( )

p n s

n

F j w C F j w jn w

¥

*

=-¥

= å -

( ) ( )

jn st

e ^w f t F jw jnws

é ù = -

ë û

F

(26)

v Finite-pulse width Sampler

§ Since n extends from , the last equation can also be written as

§ Taking the limit as of the Fourier coefficient in Eq.(13.7)

§ The term in Eq.(13.12) is written

Mathematical Modeling of the Sampling Process

(13.12)

(13.13)

(13.14) to

-¥ ¥

( ) ( )

p n s

n

F jw C F jw jnw

¥

*

=-¥

=

å

+

0 n ®

0 lim0 _n n

C C P

T

= ® =

0 n =

0 0

( ) | ( ) ( )

p n

F j C F j P F j

w w T w

*

= = =

(27)

v Finite-pulse width Sampler

§ The frequency components contained in the original continuous-time input, are still present in the sampler output , except that the amplitude is multiplied by the factor .

§ For , is a complex quantity, but the magnitude of may be written as

§ The magnitude of is written

Mathematical Modeling of the Sampling Process

(13.15) ( )

f t f_p^*( )t

/ p T

0

n ¹ ^Cⁿ ^Cⁿ

sin( 2)

| |

2

s n

s

n p

C P

T n p

w

= w

( ) Fp^* jw

| _p ( ) | _n ( _s)

n

F jw C F jw jnw

¥

*

=-¥

=

å

+ ^(13.16)

(28)

v Finite-Pulse width Sampler

§ General amplitude of spectrum is indicated as shown in figure

Mathematical Modeling of the Sampling Process

Figure 13.11 Amplitude spectra of input and output signals of a finite-pulsewidth sampler.

a. Amplitude spectrum of unit pulse train p(t);

b. product of time waveform and sampling waveform;

c. Amplitude spectrum of sampler output ; d. Amplitude spectrum of sampler output

(w_s >2w_c) (w <2w )

(29)

v The Folding Frequency, Nyquist Frequency, and Alias Frequency

§ The band around zero frequency still carries all the information contained in the continuous input signal, but the same information is also repeated along the frequency axis.

§ The phenomenon of the overlapping of the high-frequency components with fundamental component in the frequency spectrum of the sampled signal is referred to as folding.

§ The frequency is often known as folding frequency (rad/s).

§ The frequency is sometimes referred to as Nyquist frequency.

Mathematical Modeling of the Sampling Process

s 2 w

wc

(30)

v The Folding Frequency, Nyquist Frequency, and Alias Frequency

§ Effect due to the result of frequency folding is known as aliasing.

§ Figure 13.12(a) shows a periodic signal that is sampled at a rate greater than twice per cycle (sampling period )

§ The output signal will have a different frequency from the input(aliasing), and the output frequency is referred to as alias frequency. The period of the output is alias period. as shown in figure 13.12(b).

Mathematical Modeling of the Sampling Process

Figure 13.12

a. Sampling frequency is greater than two times the frequency of the input signal. No aliasing;

b. Aliasing caused by inadequate sampling rate. T is the sampling period, period of input signal;

c / 2 T <T

Tc

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v When sampling is used in a given system, what should be the proper sampling rate, and what are the limits on the rate of sampling?

v The lowest sampling frequency for possible signal reconstruction is , where is the highest frequency contained in

v The sampling theorem states that if a signal contains no frequency higher than rad/s, it is completely characterized by the values of the signal measured at instants of time separated by .

v We have to select a sampling rate that is much higher than the

theoretical minimum governed by the sampling theorem because a low sampling frequency normally has a detrimental effect on the stability of a closed-loop system.

Sampling Theorem

2 w

_c

w

c

f t ( ).

ws

T =

p w

c s

(32)

v Addendum to the Sampling Theorem

§ A signal can be defined completely by the sampled data at a rate less than rad/s provided that the derivatives of the signal.

§ If a signal contains no frequency higher than rad/s, it is completely characterized by the values and

where

§ Eq.(13.17) means when the values of the first derivative of at , for are known in addition to the values of , the minimum

sampling rate is , which is one-half of that required when alone is measured.

Sampling Theorem

2w_c

( )

f t

w

_c

( ) ( 1) (1)

( ), ( ),. . ., ( ),

n n

f kT f ^- kT f kT f kT k =( ), 0,1, 2,. . .,

( ) ( )

( )

t kT

n n

n

d f t f kT

dt ₌

= _(13.17)

( )

f t

t = kT

0,1, 2,. . .,

k = ^{f kT}⁽ ⁾

wc _{f kT}₍ ₎

(33)

v The ideal sampling

§ If s(t) is a sequence of pulses of width , constant amplitude, and uniform rate as shown, the sampled output, , will consist of sequence of section of f(t) at regular intervals.

Ideal Sampling

* ( )

TW

f t

Figure 13.13 Two views of uniform-rate sampling:

a. switch opening and closing;

b. product of time waveform and sampling waveform

Tw

(34)

where k is an integer between -∞ and +∞, T is the period of the pulse train, and is the pulse width.

v Laplace transform

§ The pulse width, T_w , is small in comparison to the period, T , such that f(t) can be considered constant during the sampling interval. Over the sampling interval, then f(t)=f(kT).

Ideal Sampling

* ( ) ( ) ( ) ( ) ( ) ( )

TW w

k

f t f t s t f t u t kT u t kT T

¥

=-¥

= =

å

- - - - ^(13.18)

[ ]

* ( ) ( ) ( ) ( )

TW W

k

f t f kT u t kT u t kT T

¥

=-¥

=

å

- - - - ^(13.19)

Tw

(35)

§ Taking the Laplace transform of Eq.(13.19)

§ Replacing with its series expansion

Ideal Sampling

* ( ) ( )

( ) 1

W

kTs T s kTs

T

k

T s

kTs k

e e

F s f kT

s s

f kT e e

s

- -

¥ -

=-¥

¥ -

-

=-¥

é ù

= ê - ú

ë û

é - ù

= ê ú

ë û

å å

T sW

e^-

2

*

( )

1 1

( ) ( ) 2!

W

W W

kTs T

k

T s T s

F s f kT e

s

¥

-

=-¥

é ì üù

-í - + - ý

ê ú

î þ

ê ú

= ê ú

ê ú

ë û

å

L

(13.20)

(13.21)

(36)

§ For small

§ Converting back to the time domain

where are Dirac delta functions.

Ideal Sampling

* ( ) ( ) ( )

W

kTs kTs

W

T W

k k

F s f kT T s e f kT T e

s

¥ ¥

- -

=-¥ =-¥

é ù

=

å

êë úû =

å

* ( ) ( ) ( )

TW W

k

f t T f kT d t kT

¥

=-¥

=

å

-

) ( t - kT d

Figure 13.14 Model of sampling with a uniform rectangular

(13.22)

(13.23)

Tw

(37)

v Data-reconstruction problem

§ Given a sequence of numbers, a continuous-data signal , , is to be reconstructed from information contained in sequence.

§ The continuous-data signal is to be constructed based on information available instants.

§ Original signal between two consecutive sampling instants, KT and (K+1)T

is to be estimated based on the values at all preceding sampling like as

Data Reconstruction and Filtering of Sampled signals

(13.24) (0), ( ), (2 ), , ( ),

f f T f T K f kT K

( )

f t

t ³ 0

( ) f t

( ), [( 1) ], [( 2) ], , (0).

f kT f k - T f k - T K f

(38)

v Data-reconstruction problem

§ The power-series expansion of in the interval between the sampling instants and . The approximation is written as

where,

Data Reconstruction and Filtering of Sampled signals

(2)

(1)

( )

2

( ) ( ) ( )( ) ( )

k

2!

f kT

f t = f kT + f kT t - kT + t - kT +L

( ) ( ) f tk = f t

( )

n n

n

t kT

d f t f kT

dt

=

(13.26)

(13.27) ( )

f t

kT (k +1)T

(13.25)

for kT £ < t ( k + 1) T

(39)

v Data-reconstruction problem

§ A simple expression involving only two data pluses gives an estimate of the first derivative of at :

Similarly, we can approximate the second derivative of at as

where

§ Substituting of Eq.(13.28) and (13.30) into Eq.(13.29), we get

Data Reconstruction and Filtering of Sampled signals

( )

f t t = kT

(1) 1

( ) { ( ) [( 1) ]}

f kT f kT f k T

= T - - (13.28)

( )

f t t =kT

(2) 1 (1) (1)

( ) { ( ) [( 1) ]}

f kT f kT f k T

= T - - (13.29)

(1) 1

[( 1) ] { [( 1) ] [( 2) ]}

f k T f k T f k T

- = T - - - _(13.30)

(2) 1

( ) { ( ) 2 [( 1) ] [( 2) ]}

f kT f kT f k T f k T

=T - - + -

(13.31)

(40)

v Data-reconstruction problem

§ We have expressed the second derivative of at in terms of and the sampled values at two preceding sampling instants.

§ We can express any higher order derivatives in terms of more of the past sampled values of .

§ For instance, would involve , , , and

§ The higher the order of the derivative to be approximated, the larger will be the number of delayed pulses required.

§ In general, The number of delayed pulse data required to approximate the value of is n+1.

Data Reconstruction and Filtering of Sampled signals

( )

f t t = kT f kT( )

( ) f kT

(3)( )

f kT f kT( ) f k[( -1) ]T f[(k -2) ]T f k[( -3) ]T

( )ⁿ ( ) f kT

(41)

v Modeling the Zero-Order Hold

§ The impulse response of the zero-order hold is the transfer function of the zero- order hold (ZOH).

§ The approximation of during the time interval à

§ Impulse response is expressed as ( is unit-step function)

Zero-Order Hold

Figure 13.15

a. Unit impulse input to zero-order hold;

b. Impulse response of zero-order hold;

( )

f t kT £ £t (k +1)T

( ) ( )

f tk = f kT

0( ) ( ) ( )

gh t =u t -u t T-

u t ( )

(42)

v Modeling the Zero-Order Hold

§ Using an impulse at zero time, the transform of the resulting step that starts at t=0 and ends at t=T is

Zero-Order Hold

Figure 13.16 Zero-order hold operation in the time domain a. Unit impulse input to zero-order hold;

b. Impulse response of zero-order hold;

0

( ) 1

Ts h

G s e

s - -

= _(13.31)

(43)

v Modeling the Zero-Order Hold

§ In a physical system, samples of the input time waveform, f(kT), are held over the sampling interval.

Zero-Order Hold

Figure 13.17 Ideal sampling and the zero-order hold

(44)

v Modeling the First-Order Hold

§ If the first two terms of the power series in Eq.(13.25) are used to extrapolate the time function over the time interval , the device is called a first-order hold (FOH).

§ The equation for the FOH is

where the first-order derivative of at is approximated as

§ Substituting Eq.(13.32) in Eq.(13.33), we get

First-Order Hold

( )

f t kT £ £t (k +1)T

( ) ( ) (1)( )( )

fk t = f kT + f kT t -kT (13.32) ( )

f t t = kT

(1) ( ) [( 1) ]

( ) f kT f k T

f kT

T

- -

= _(13.33)

( ) [( 1) ]

( ) ( ) ( )

k

f kT f k T

f t f kT t kT

T

- -

= + - ^(13.34)

(45)

v Modeling the First-Order Hold

§ The corresponding output, or the impulse response, is determined from Eq.(13.34) by setting k=0,1,2,…, for the various time intervals.

§ When k=0, the impulse response for is described by

§ For a unit impulse input, and . Thus, the impulse response of the FOH for is

First-Order Hold

0£ <t T

0

(0) ( )

( ) (0) f f T

f t f t

T

- -

= + ^(13.35)

(0) 1

f = f(-T) = 0

0 £ <t T

1

( ) 1 1

h

g t t

T

= + + _(13.36)

(46)

v Modeling the First-Order Hold

§ Setting k=1 in Eq.(13.34), we have

§ Since and , the impulse response of the FOH over the time interval is

§ The impulse response of the FOH is shown in Figure 13.18.

First-Order Hold

(13.37)

(13.38)

1

( ) (0)

( ) ( ) f T f ( )

f t f T t T

T

= + - -

(0) 1

f = f T =( ) 0 0£ <t T ^g_h₁^{( ) 1}^t ^t

= -T

Figure 13.18

a. Unit impulse input to first-order hold;

b. Impulse response of first-order hold;

(47)

v Modeling the First-Order Hold

§ Functionally, the impulse response in Figure 13.18(b) is written

§ The transfer function of the FOH is obtained by taking the Laplace transform of the Eq.(13.39), and simplifying

First-Order Hold

1

2( )

( ) ( ) ( ) 2 ( ) ( )

( 2 )

( 2 ) ( 2 ) ( 2 )

h

t t T

g t u t u t T u t T u t T

T T

t T

u t T u t T u t T T

= + - - - -

+ - - - + - ^(13.39)

2 1

1 1

( )

Ts h

Ts e

G s

T s

é - ù

+ -

= ê ú

ë û

(13.40)

[ ]

²

1 0

( ) 1

h h

G s Ts G

T

= +

^(13.41)

(48)

v Modeling the First-Order Hold

§ Figure 13.19 illustrate the reconstruction of a time function from the sampled signal using a FOH.

First-Order Hold

Figure 13.19 Reconstruction of a continuous time signal by mean of a first-order hold.

( ) f t

( ) f kT

(49)

Pusan National University

Intelligent Robot Lab