Tabla Transformadas Propiedades

8/10/2019 Tabla Transformadas Propiedades

1/8

CHAP. 51

FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS

Table

5 2.

Common Fourier Transforms airs

sin t

5 5

THE FREQUENCY RESPONSE OF CONTINUOUS TIME LTI SYSTEMS

A.

Frequency Response:

In Sec. 2.2 we showed tha t th e ou tput

y t )

of a continuous-time LTI system equals the

convolution of the input

x t )

with the impulse response

h t ;

that is

Applying th e convolution property

5.58),

we obtain


2/8


3/8

FOURIER ANALYSIS OF DISCRETE TIME SlGNALS

AND

SYSTEMS

[CHAP.

6

Table

6 2.

Common Fourier Transform Pairs

s n

n

o


4/8

CHAP.

6

FOUR IER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS

Table

6 1.

Properties of the ourier Transform

Property Sequence Fou rier transform

Periodicity

Linearity

Tim e shifting

Frequency shifting

Conjugat ion

Tim e reversal

Tim e scaling

Frequ ency differentiation

First difference

Accumulation

Convolution

Multiplication

Real sequence

Even component

Odd component

Parseval s r elations


5/8

CHAP. 3

LAPLACE TRANSFORM AN D CONTINUOU S-TIME LTI SYSTEMS

Table

3 1

Som e Laplace Transforms Pairs

All s

cos wotu t)

sin wotu t

s a

e- '

cos wotu t)

Re s ) Re a )

s + a 1 2 + w ;

A.

Linearity:

I f

The set notation

A

B

means that set

A

contains set

B,

while A

n

B

denotes the

intersection of sets A and B, that is the set containing all elements in both A and B.

Thus Eq

3 . 15 )

indicates that the

ROC

of the resultant Laplace transform is at least as

large as the region in common between

R ,

and

R 2 .

Usually we have simply

R R ,

n

R , .

This is illustrated in Fig. 3-4.


6/8

CHAP. 31

LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS

Table 3 2 Properties of the Laplace Transform

Property Signal Transform ROC

x ( t ) X ( s ) R

x , ( t ) x , w R

x 2 W x , w R2

Linearity

a , x , ( t )

+

a 2 x 2 ( l )

a ,X , ( s )

+

a , X 2 ( s) R ' I R , n R 2

Time shifting

x ( t o ) e - X ( s )

R' R

Shifting in

s

es 'x( X ( s o) R' R Re (s , )

Time scaling x( a t - X ( s ) R' aR

la

Time reversal R ' =

- R

Differentia tion in

Differentia tion in s x ( t )

d X( s )

R f = R

ds

Integration

Convolution

then

% ( t )

~ 2 0 ) X I ~ ) X ~ ~ ) R ' I R , n R 2 ( 3 . 2 3 )

Th is convolution p roperty plays a central role in th e analysis and design of continuous-time

LTI systems.

Table 3 2 summarizes the properties of the Laplace transform presented in this

section.

3 5 THE INVERSE LAPLACE TRANSFORM

Inversion of the Laplace transform to find the signal

x ( t )

from its Laplace transform

X s) is called the inverse Laplace transform, symbolically denoted as

A Inversion Formula:

There

is

a procedure that is applicable to all classes of transform functions that

involves the evaluation of a line integral in complex s-plane; that is,

In this integral, th e real c is to be selected such that if th e ROC of X s) is

a

Re s )

< a 2

then a < c


7/8

TH E Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS

[CHAP.

4

Thus,

[n] all

B.

Unit Step Sequence

d n l :

Setting a in Eqs. ( 4 . 8 ) to (4.101, we obtain

C.

z Transform Pairs:

Th e z-transforms of some common sequences are tabulated in Table 4-1.

Table 4 1. Some Common z Transform Pairs

All z

lzl

>

1

Iz l< 1

Z- '

All z except 0 if

m

>

0) r m if

m

0)

Z

1 - a z - ' ' 2 - a

Izl

>

lal

z 2 COS o ) z

(COS on)u[n l

z 2 2 co s R o )t

1

l z l >

(sin n o )

(sin R,n)u[ n]

z 2 ( 2cos R , ) z 1

Iz l>

z 2 r c o s R 0 ) z

( r n cos R,n)u[n]

z 2 2 r cos R o ) z r 2

Izl> r

( r sin R , )z

r n

in R,n)u[nI

z 2 2 r co s R , ) z r 2

Iz l>

r

O < n s N - 1 1 ~ ' z - ~

otherwise az- '

lz l> 0


8/8

CHAP. 41 T H E Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS 173

Table

4 2.

Some Properties of the z Transform

Property Sequence Transform RO C

Linearity

Tim e shifting

Multiplication by

z,

Multiplication by

einon

Tim e reversal

Multiplication

by n

Accumulation

Convolution

( z )

2

d ?

H. Summary of Some z transform Properties

For convenient reference, the properties of the z-transform presented above are

summarized in Table

4-2.

4 5

THE

INVERSE Z TRANSFORM

Inversion of the z-transform to find the sequence x n ] from its z-transform X ( z ) is

called th e inverse z-transform , symbolically den oted as

~ [ n ]s - ' { X ( z > } ( 4 . 2 7 )

A.

Inversion Formula:

As in the case of the Laplace transform, there is a formal expression for the inverse

z-transform in terms of an integration in the z-plane; that is,

where C is a counterclockwise contour of integration enclosing the origin. Formal

evaluation of

Eq

(4.28 ) requires a n un derstanding of com plex variable theory.

B.

Use of Tables of z Transform Pairs:

In th e second method for the inversion of X (z ), we attem pt to express X (z ) as a sum

X ( z ) = X , ( z )

.

+X,(z) (4 .29)

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